{"text":"\\section{Introduction}\nOne important problem in machine learning is to find the minimum of the expected loss, \n\\begin{align}\\label{origin}\n\\min_{\\theta} \\mathbb{E}_{\\Xb,Y\\sim \\cD}\\left[l(Y,\\langle \\Xb,\\theta \\rangle) \\right]. \n\\end{align}\nHere $l(\\cdot,\\cdot)$ is a loss function and $(\\Xb,Y) \\in \\cX\\times \\cY \\subseteq \\mathbb{R}^d \\times \\cY$ has a distribution $\\cD$. In practice, the minimizer $\\theta^*$ needs to be estimated by observing $N$ samples $\\{\\xb_i,y_i\\}$ drawn from distribution $\\cD$. In many applications $N$ or $d$ are very large, so distributed algorithms are necessary in such case. Without loss of generality, assume that $N = nm$ and that the observations of $j$-th machine are $\\{\\xb_{ji},y_{ji}\\}_{i =1}^n$. We consider the high-dimensional learning problem where the dimension $d$ can be very large, and the effective variables are supported on $S:= \\text{support}\\{\\theta^*\\} = \\{i\\in [d]:\\theta^*_i \\ne 0\\}$ and $s:=|S|\\ll d$. Extensive efforts have been made to develop batch algorithms \\cite{friedman2007pathwise,Beck:09,xiao2013proximal}, which provide good convergence guarantees in optimization. However, when $N$ is large, batch algorithms are inefficiency, which takes at least $\\cO(N)$ time per iteration. Therefore, an emerging recent interest is observed to address this problem using the distributed optimization frameworks \\cite{jordan2016communication,lee2015distributed,wang2016efficient}, which is more efficient than the stochastic algorithms. One important issue of existing distributed optimization for sparse learning is that they did not take advantage of the sparse structure, thus they have the same communication costs with general dense problems. In this paper, we propose a novel communication-efficient distributed algorithm to explicitly leverage the sparse structure for solving large scale sparse learning problems. This allows us to reduce the communication cost from $\\cO(d)$ in existing works to $\\cO(s)$, while we still maintaining nearly the same performance under mild assumptions.\n\n\\noindent \\textbf{Notations}\nFor a sequence of numbers $a_n$, we use $\\cO(a_n)$ to denote a sequence of numbers $b_n$ such that $b_n\\leq C \\cdot a_n$ for some positive constant $C$. Given two sequences of numbers $a_n$ and $b_n$, we say $a_n\\lesssim b_n$ if $a_n =\\cO(b_n) $ and $a_n\\gtrsim b_n$ if $b_n =\\cO(a_n) $. The notation $a_n \\asymp b_n$ denotes that $a_n =\\cO(b_n) $ and $b_n =\\cO(a_n) $. For a vector $\\vb\\in \\mathbb{R}^d$, the $l_p$-norm of $\\vb$ is defined as $\\|\\vb\\|_p = ( \\sum_{i =1}^d|\\vb_i|^p)^{1\/p}$, where $p>0$; the $l_0$-norm of $\\vb$ is defined as the number of its nonzero entries; the support of $\\vb$ is defined as $\\text{supp}(\\vb) = \\{i:\\vb_i\\ne 0 \\}$. For simplicity, we use $[d]$ to denote the set $\\{1,\\cdots,d\\}$. For a matrix $A = (a_{ij})\\in \\mathbb{R}^{n_1\\times n_2}$, we define the $l_\\infty$-norm of $A$ as $\\|A\\|_\\infty = \\max_{i\\in[n_1],j\\in[n_2]}|a_{ij}|$. Given a number $k\\leq d$, the hard thresholding $\\cH_k(\\vb)$ of a vector $\\vb\\in\\RR^d $ is defined by keeping the largest $k$ entries of $\\vb$ (in magnitude) and setting the rest to be zero. Given a subset $S$ of index set $\\{1,\\cdots,d\\}$, the projection $\\cP_S(\\vb)$ of a vector $\\vb$ on $S$ is defined by \n\\begin{align*} \n\\cP_S(\\vb)_j =0,\\hspace{0.1in} \\text{if} \\hspace{0.05in} j\\notin S \\hspace{0.1in} \\text{and}\\hspace{0.1in} \\cP_S(\\vb)_j =\\vb_j, \\hspace{0.1in}\\text{if}\\hspace{0.05in} j\\in S.\n\\end{align*} \n$\\cP_S(\\vb)$ is also denoted as $(\\vb)_S$ for short.\n\n\\vspace{-0.1in}\n\\subsection{{Related work}}\nThere is much previous work on distributed optimizations such as (Zinkevich et al. \\cite{zinkevich2010parallelized}; Dekel et al. \\cite{dekel2012optimal}; Zhang et al. \\cite{zhang2012communication}; Shamir and Srebro \\cite{shamir2014distributed}; Arjevani and Shamir \\cite{arjevani2015communication}; Lee et al. \\cite{lee2015distributed}; Zhang and Xiao \\cite{zhang2015disco}). Initially, most distributed algorithms used averaging estimators formed by local machines (Zinkevich et al. \\cite{zinkevich2010parallelized}; Zhang et al. \\cite{zhang2012communication}). Then Zhang and Xiao \\cite{zhang2015disco}, Shamir et al. \\cite{shamir2014communication} and Lee et al. \\cite{lee2015communication} proposed more communication-efficient distributed optimization algorithms. More recently, using ideas of the approximate Newton-type method, Jordan et al. \\cite{jordan2016communication} and Wang et al. \\cite{wang2016efficient} further improved the computational efficiency of this type of method.\n\nMany gradient hard thresholding approaches are proposed in recent years such as (Yuan et al. \\cite{yuan2014gradient}; Li et al. \\cite{li2016stochastic}; Jain et al. \\cite{jain2014iterative}). They showed that under suitable conditions, the hard thresholding type first-order algorithms attain linear convergence to a solution which has optimal estimation accuracy with high probability. However, to the best of our knowledge, hard thresholding techniques applied to approximate Newton-type distributed algorithms has not been considered yet.  So in this paper, we present some initial theoretical and experimental results on this topic. \n\\iffalse\nInitially, averaging estimators formed by locally machines is a intuitive approach to distributed estimation (Zinkevich et al. \\cite{zinkevich2010parallelized}; Zhang et al. \\cite{zhang2012communication}).\n\\fi\n\\vspace{-0.15in}\n\n\\section{Algorithm} \n\\vspace{-0.05in}\nIn this section, we explain our approach to estimating the $\\theta^*$ that minimizes the expected loss. The detailed steps are summarized in Algorithm \\ref{algor}. \n\nFirst the empirical loss at each machine is defined as\n\\begin{align*}\\textstyle\n\\cL_j(\\theta) = \\frac{1}{n} \\sum_{i =1}^n l(y_{ji},\\langle \\xb_{ji},\\theta \\rangle), ~~\\text{where}~~ j\\in [m]. \n\\end{align*}\nAt the beginning of algorithm, we solve a local Lasso subproblem to get an initial point. Specifically, at iteration $h =0$, the master machine solves the minimization problem\n\\begin{align} \\label{initial}\n\\textstyle \\gamma^0 = \\argmin \\cL_1(\\theta) + \\mu_0\\|\\theta\\|_1. \n\\end{align}\nThe initial point $\\theta^0$ is formed by keeping the largest $k$ elements of the resulting minimizer $\\gamma^0$ and setting the other elements to be zero, i.e., $\\theta^0 = \\cH_k(\\gamma^0)$. Then, $\\theta^0$ is broadcasted to the local machines, where it is used to compute a gradient of local empirical loss at $\\theta^0$, that is, $\\nabla\\cL_j(\\theta^0)$. The local machines project $\\nabla\\cL_j(\\theta^0)$ on the support $S^0$ of $\\theta^0$ and transmit the projection $\\cP_{S^0}\\left[\\nabla\\cL_j(\\theta^0)\\right]$ back to the master machine. Later at $(h+1)$-th iteration ($h\\geq 0$), the master solves a shifted $l_1$ regularized minimization subproblem:\n\\begin{align}\\label{subproblem}\n\\textstyle \\nonumber \\gamma^{h+1}& = \\argmin_{ \\theta} \\hspace{0.1in}\\cL_1(\\theta) + \\mu_{h+1} \\|\\theta\\|_1\\\\\n&+\\Big\\langle \\cP_{S^h}\\left[{\\textstyle\\frac{1}{m}\\sum_{j=1}^m \\nabla \\cL_j(\\theta^h)}\\right] -\\nabla \\cL_1(\\theta^h),\\theta \\Big \\rangle.  \n\\end{align} \nAgain the minimizer $\\gamma^{h+1}$ is truncated to form $\\theta^{h+1}$, and this quantity is communicated to the local machines, where it is used to compute the local gradient as before. \n\nSolving subproblem $\\eqref{subproblem}$ is inspired by the approach of Wang et al. \\cite{wang2016efficient} and Jordan et al.\\cite{jordan2016communication}. Note that the formulation takes advantage of both global first-order information and local higher-order information. Specially, assuming the $\\mu_{h+1} = 0$ and $\\cL_j$ has an invertible Hessian, the solution of $\\eqref{subproblem}$ has the following closed form\n\\begin{align*}\n\\gamma^{h+1} = \\theta^{h} - \\nabla^2\\cL_1(\\theta^h) ^{-1} \\left(\\cP_{S^h}\\left[{\\textstyle\\frac{1}{m}\\sum_{j=1}^m \\nabla \\cL_j(\\theta^h)}\\right]\\right),\n\\end{align*}\nwhich is similar to a Newton updating step. Note that here we add a projection procedure $\\cP_{S^h}\\left[\\frac{1}{m}\\sum_{j=1}^m\\nabla\\cL_j(\\theta^h)\\right]$ to reduce the number of nonzeros that need to be communicated to the master machine. This procedure is reasonable intuitively. First, when $\\theta^h$ is close to $\\theta^*$, the elements of $\\frac{1}{m}\\sum_{j=1}^m\\nabla \\cL_j(\\theta^h)$ outside the support $S^{h}$ should be very small, so nominally little error is incurred in the truncation step. Second, when $\\theta^{h+1}$ is also close to $\\theta^*$, the lost part has even more minimal effects on the inner product in subproblem $\\eqref{subproblem}$. Third, we leave $-\\nabla \\cL_1(\\theta^h)$ in $\\eqref{subproblem}$ out of the truncation to maintain the formulation as unbiased.\n\n\\begin{algorithm}[!htb]\n\t\\caption{Two-way Truncation Distributed Sparse Learning}\n\t\\label{algor}\n\n\t\\begin{algorithmic}\n\t\t\\REQUIRE Loss function $l(\\cdot,\\cdot)$, data $\\{\\xb_{ji}, y_{ji}\\}_{i\\in[n],j\\in[m]}$. \n\t\t\\STATE \\hspace{-1.25em} \\textbf{\\underline{Local machines:}}\n\t\t\\STATE \\hspace{-1.25em} \\textbf{Initializaiton:} The master solves the local $l_1$ regularized loss minimization problem \\eqref{initial} to get a solution $\\gamma^0$. Set $\\theta^{0} = \\cH_k(\\gamma^{0})$. \n\t\t\\FOR{$h=0,1, \\dots$}\n\t\t\\FOR{$j=2,3, \\dots, m$}\n\t\t\\STATE   \\textbf{if} Receive $\\theta^h$ from the master \\textbf{then}\n\t\t\\STATE Calculate gradient $\\nabla \\cL_j(\\theta^h)$ and get the projection $\\cP_{S^{h}}\\left[ \\nabla\\cL_j(\\theta^h)\\right]$ of the gradient on support $S^h$ and transmit it to the master.\n\t\t\\STATE \\textbf{end}\n\t\t\n\t\t\\ENDFOR\n\t\t\\STATE  \\textbf{\\underline{Master:}}\n\t\t\\STATE   \\textbf{if} Receive $\\{\\nabla\\cL_j(\\theta^h)\\}_{j=2}^m$ from local machines \\textbf{then}\n\t\t\\STATE \\hspace{1.25em}Solve the shifted $l_1$ regularized problem\n\t\t\\STATE \\hspace{1.25em}\\eqref{subproblem} to obtain $\\gamma^{h+1}$.\n\t\t\\STATE \\hspace{1.25em}Do hard thresholding $\\theta^{h+1} =         \\cH_k(\\gamma^{h+1})$. \n\t\t\\STATE \\hspace{1.25em}Let $S^{h+1} = \\text{supp}(\\theta^{h+1})$. \n\t\t\\STATE \\hspace{1.25em}Broadcast $\\theta^{h+1}$ to every local machine. \n\t\t\\STATE \\textbf{end}\t\n\t\t\\ENDFOR\t\n\t\\end{algorithmic}\n\\end{algorithm}\n\\vspace{-0.1in}\n\\section{Theoretical Analysis}\n\\vspace{-0.05in}\n\\subsection{Main Theorem}\n\nWe present some theoretical analysis of the proposed algorithm in this section.\n\\vspace{-0.05in}\n\\begin{assumption}\\label{assum_smooth}\n\tThe loss $l(\\cdot,\\cdot)$ is a $L$-smooth function of the second argument, i.e.,\n\t\\begin{align*} {\\textstyle\n\tl^\\prime(x,y) - l^\\prime(x,z) \\leq L|y - z|, \\hspace{0.2in} \\forall x,y,z\\in \\RR}\n\t\\end{align*}\n\tMoreover, the third derivative with respect to its second argument, $\\partial^3 l(x,y)\/\\partial y^3$, is bounded by a constant $M$, i.e.,\n\t{\\begin{align*} \\textstyle\n\t|\\partial^3 l(x,y)\/\\partial y^3| \\leq M, \\hspace{0.2in} \\forall x,y\\in \\RR\n\t\\end{align*}}\n\\end{assumption}\n\n\\begin{assumption}\\label{assum_restrict}\n\tThe empirical loss function computed on the first machine satisfies that: $\\forall \\Delta \\in \\cC(S,3)$, we have\n\t\\begin{align*}\n\t\\cL_1(\\theta^* + \\Delta) - \\cL_1(\\theta^*) -\\langle \\nabla \\cL_1(\\theta^*),\\Delta \\rangle \\geq \\kappa \\|\\Delta\\|_2^2,\n\t\\end{align*}\n\twhere $\\cC(S,3)$ is defined as\n\t\\begin{align*}\n\t\\cC(S,3) = \\{\\Delta \\in \\RR^d |~ \\|\\Delta_{S^c}\\|_1 \\leq 3\\|\\Delta_S\\|_1\\}. \n\t\\end{align*}\t\n\\end{assumption}\n\n\\begin{assumption}\\label{assum_supp}\n\tThe $\\gamma^{h+1}$, $S^{h+1}$ and $S^h$ defined in Algorithm \\ref{algor} satisfy the following condition:\n\tthere exists some positive constants $H$ and $\\tau_1$ and $\\tau_2$ such that for $h\\geq H$, \n\t\\begin{align*}\n\t\\left\\|\\left(\\gamma^{h} - \\theta^*\\right)_{(S^h)^c}\\right\\|_1 &\\leq \\tau_1 \\left\\|\\gamma^{h} - \\theta^*\\right\\|_1\\\\\n\t\\left\\|\\left(\\gamma^{h+1} - \\theta^*\\right)_{S^{h+1}\\backslash S^{h}}\\right\\|_1 &\\leq \\tau_2\\left\\|\\gamma^{h+1}-\\theta^*\\right\\|_1.   \n\t\\end{align*}\n\\end{assumption}\n\\begin{remark}\n\tIn practice, both $\\tau_1$ and $\\tau_2$ are very small even after only one round of communication and will decrease to $0$ fast in the later steps. \n\\end{remark}\n\\vspace{-0.1in}\nFor simplicity, we define the following notation:\n\\begin{align*}\n\\overbar{\\cL_1}(\\theta^*,\\theta^h) &:= \\cL_1(\\theta^*) +\\left\\langle {\\textstyle \\frac{1}{m}\\sum_{j = 1}^m  \\nabla\\cL_j(\\theta^h) - \\nabla\\cL_1(\\theta^h)},\\theta \\right \\rangle,\\\\\n\\tilde{\\cL_1}(\\theta^*,\\theta^h) &:= \\cL_1(\\theta^*)\\\\\n&\\hspace{0.2in}+\\left\\langle \\cP_{S^h}\\left[{\\textstyle \\frac{1}{m}\\sum_{j = 1}^m  \\nabla\\cL_j(\\theta^h) - \\nabla\\cL_1(\\theta^h)}\\right],\\theta \\right \\rangle.\n\\end{align*}\n\nNow we state our main theorem. \n\\begin{thmi} \\label{main_theorem}\n\tSuppose that Assumption~\\ref{assum_smooth}, \\ref{assum_restrict}, and \\ref{assum_supp} hold. Let $k = C_1\\cdot s$ with $C_1>1$ and \n\t\\begin{align}\n\t\\nonumber& \\textstyle\\mu_{h+1}  = \\hspace{0.1in} 4\\left\\|{ \\frac{1}{m}\\sum_{j = 1}^m  \\nabla \\cL_j(\\theta^*)}\\right\\|_\\infty \\\\\n\t\\nonumber&\\textstyle +  2L\\left(\\max_{j,i}\\|x_{j,i}\\|_\\infty^2\\right)\\cdot \\Big[{ 2\\sqrt{\\frac{\\log(2d\/\\delta)}{n}} + \\rho }\\Big]\\|\\theta^h - \\theta^*\\|_1\\\\\n\t&\\textstyle + 2M\\left(\\max_{j,i}\\|x_{j,i}\\|_\\infty^3 \\right)\\|\\theta^h - \\theta^*\\|_1^2, \\label{mu_def}\n\t\\end{align} \n\twhere $\\rho := \\tau_1+\\tau_2$.  \n\t\n\tThen with probability at least $1-\\delta$, we have that\n\t\n\t\\begin{align*} \n    &\\|\\theta^{h+1} - \\theta^*\\|_1  {\\textstyle \\leq \\frac{C_2 s}{\\kappa}\\left \\|\\frac{1}{m}\\sum_{j=1}^m \\nabla \\cL_j(\\theta^*)\\right \\|_\\infty}\\\\\n    &\\hspace{0.2in}{\\textstyle +\\frac{C_2 s}{2\\kappa} L\\cdot \\max_{j,i}\\|\\xb_{ji}\\|_\\infty^2\\cdot \\Big[2\\sqrt{\\frac{\\log(2d\/\\delta)}{n}} + \\rho \\Big]\\|\\theta^h - \\theta^*\\|_1}\\\\\n    &\\hspace{0.2in}{\\textstyle +\\frac{C_2 s}{2\\kappa}  M\\cdot  \\max_{j,i}\\|\\xb_{ji}\\|_\\infty^3\\cdot \\|\\theta^h - \\theta^*\\|_1^2}, ~\\text{and}\\\\\n    &\\|\\theta^{h+1} - \\theta^*\\|_2 {\\textstyle \\leq \\frac{C_3\\sqrt{s}}{\\kappa}\\left \\|\\frac{1}{m}\\sum_{j=1}^m \\nabla \\cL_j(\\theta^*)\\right \\|_\\infty }\\\\\n    &\\hspace{0.2in} {\\textstyle + \\frac{C_3\\sqrt{s}}{2\\kappa} L \\cdot \\max_{j,i}\\|\\xb_{ji}\\|_\\infty^2\\cdot\\Big[2\\sqrt{\\frac{\\log(2d\/\\delta)}{n}} + \\rho \\Big]\\cdot \\|\\theta^h - \\theta^*\\|_1} \\\\\n    &\\hspace{0.2in} \\textstyle+\\frac{C_3\\sqrt{s}}{2\\kappa}M\\cdot \\max_{j,i}\\|\\xb_{ji}\\|_\\infty^3\\cdot \\|\\theta^h - \\theta^*\\|_1^2, \n    \\end{align*}\n\twhere $\\textstyle C_2 = 24\\sqrt{1+2(C_1-1)^{-\\frac{1}{2}}}\\cdot \\sqrt{C_1+1}$ and $\\textstyle C_3= 24\\sqrt{1+2(C_1-1)^{-\\frac{1}{2}}}$ are positive constants independent of $m,n,s,d$.\n\\end{thmi}\nThe theorem immediately implies the following convergence result.\n\\begin{cori} \\label{first_cor}\n\tSuppose that for all $h$\n\t\\begin{align}\\label{cor_ass}\n\t\\textstyle \\nonumber&M\\cdot  \\Big(\\max_{j,i}\\|x_{ji}\\|_\\infty^3\\Big) \\|\\theta^h - \\theta^*\\|_1\\leq\\\\ &  L\\cdot \\max_{j,i}\\|x_{ji}\\|_\\infty^2\\left[2\\sqrt{\\frac{\\log(2d\/\\delta)}{n}} \n\t+ \\rho\\right],\n\t\\end{align}\n\twhere $\\rho := \\tau_1+\\tau_2$.\n\t\n\tThen under the assumption of Theorem~\\ref{main_theorem} we have\n\t\\begin{align*} \n\t\\|\\theta^{h+1} - \\theta^*\\|_1 &\\leq \\textstyle \\frac{1-a_n^{h+1}}{1 - a_n}\\cdot \\frac{C_2 s}{\\kappa}\\cdot \\textstyle \\left\\|\\frac{1}{m}\\sum_{j=1}^m \\nabla \\cL_j(\\theta^*)\\right\\|_\\infty \\\\\n\t& \\hspace{0.1in}+ a_n^{h +1} \\|\\theta^0 - \\theta^*\\|_1,\\\\\n\t|\\theta^{h+1} - \\theta^*\\|_2 &\\leq \\textstyle \\frac{1-a_n^{h+1}}{1 - a_n}\\cdot \\frac{C_3\\sqrt{s}}{\\kappa}\\cdot \\textstyle \\left\\|\\frac{1}{m}\\sum_{j=1}^m \\nabla \\cL_j(\\theta^*)\\right\\|_\\infty \\\\\n\t&\\hspace{0.1in}+ a_n^h b_n  \\|\\theta^0 - \\theta^*\\|_1,\n\t\\end{align*}\n\twhere \n   \\begin{align*}\n\t a_n =&\\textstyle \\frac{C_2 s}{\\kappa} L\\cdot \\max_{j,i}\\|x_{ji}\\|_\\infty^2 \\cdot \\Bigg[2\\sqrt{\\frac{\\log(2d\/\\delta)}{n}}+ \\rho\\Bigg]\n\t\\end{align*}\n\tand \n\t\\begin{align*}\n\t b_n =&\\textstyle \\frac{C_3\\sqrt{s}}{\\kappa} L\\cdot \\max_{j,i}\\|x_{ji}\\|_\\infty^2\\cdot\\Bigg[ 2\\sqrt{\\frac{\\log(2d\/\\delta)}{n}}+ \\rho \\Bigg],\n\t\\end{align*}\n\twhere $C_2 $ and $C_3$ are defined in Theorem~\\ref{main_theorem} and independent of $m,n,s,d$. \n\\end{cori}\n\\begin{remark}\n\tFrom the conclusion, we know that the hard thresholding parameter $k$ can be chosen as $C_1\\cdot s$, where $C_1$ can be a moderate constant larger than $1$. By contrast, previous work such as \\cite{li2016stochastic} solving a nonconvex minimization problem subject to $l_0$ constraint $\\|\\theta\\|_0 \\leq k$ requires that $k\\geq \\cO(\\kappa_s^2 s)$, where $\\kappa_s$ is the condition number of the object function.\n\tMoreover, instead of only hard thresholding on the solution of Lasso subproblems, we also do projection on the gradients in \\eqref{subproblem}. These help us reduce the communication cost from $\\cO(d)$ to $\\cO(s)$. \n\\end{remark}\n\n\\subsection{Sparse Linear Regression}\nIn the sparse linear regression, data $\\{\\xb_{ji}, y_{ji}\\}_{i\\in [n], j\\in[m]}$ are generated\naccording to the model\n\\begin{align} \\label{lin_model} \n\\textstyle y_{ji} = \\langle\\xb_{ji},\\theta^*\\rangle + \\epsilon_{ji},\n\\end{align}\nwhere the noise $\\epsilon_{ji}$ are i.i.d subgaussian random variables with zero mean. Usually the the loss function for this problem is the squared loss function $l(y_{ji},\\langle \\theta,\\xb_{ji}\\rangle) = \\frac{1}{2}(y_{ji}-\\langle \\theta,\\xb_{ji}\\rangle)^2$, which is $1$-smooth.\n\nCombining Corollary \\ref{first_cor} with some intermediate results obtained from \\cite{rudelson2011reconstruction, vershynin2010introduction} and \\cite{wainwright2009sharp}, we have the following bound for the estimation error. \n\\begin{cori}\n\tSuppose the design matrix and noise are subgaussian, Assumption \\ref{assum_supp} holds and $\\mu_{h+1}$ is defined as \\eqref{mu_def}. Then under the sparse linear model, we have the following estimation error bounds with probability at least $1-2\\delta$:\n\t\\begin{align*} \\textstyle\n\t\\|\\theta^{h+1} - \\theta^*\\|_1 \\lesssim \\frac{1-a_n^{h+1}}{1-a_n} \\cdot\\frac{C_2s\\sigma\\sigma_X}{\\kappa}\\sqrt{\\frac{\\log(d\/\\delta)}{mn}}\\\\\n\t\\textstyle + a_n^{h+1} \\frac{s \\sigma \\sigma_X}{\\kappa}\\sqrt{\\frac{\\log(nd\/\\delta)}{n}}\n\t\\end{align*}\n\tand\n\t\\begin{align*} \\textstyle\n\t\\|\\theta^{h+1} - \\theta^*\\|_2 \\lesssim\\frac{1-a_n^{h+1}}{1-a_n} \\cdot\\frac{C_3\\sqrt{s}\\sigma \\sigma_X}{\\kappa}\\sqrt{\\frac{\\log(d\/\\delta)}{mn}}\\\\\n\t\\textstyle + a_n^{h}b_n \\frac{s \\sigma \\sigma_X}{\\kappa}\\sqrt{\\frac{\\log(nd\/\\delta)}{n}},\n\t\\end{align*}\n\twhere $C_2 $ and $ C_3$ are defined in Theorem~\\ref{main_theorem}, and where\n\t\\begin{align*} \n\t\\textstyle a_n =\\frac{C_2s}{\\kappa}\\sigma_X^2\\log\\left(\\frac{mnd}{\\delta}\\right)\\left[2\\sqrt{\\frac{\\log(2d\/\\delta)}{n}} + \\rho \\right] \n\t\\end{align*}\n\tand\n\t\\begin{align*} \n\t\\textstyle b_n = \\frac{C_3\\sqrt{s}}{\\kappa}\\sigma_X^2\\log\\left(\\frac{mnd}{\\delta}\\right)\\left[2\\sqrt{\\frac{\\log(2d\/\\delta)}{n}} + \\rho \\right]. \n\t\\end{align*}\t\n\\end{cori}\n\n\\begin{remark}\nUnder certain conditions we can further simplify the bound and have an insight of the relation between $n,m,s,d$. When $n\\geq s^2\\log d$, it is easy to see by choosing\n\\begin{align*}\n\\mu_{h+1} \\asymp \\sqrt{\\frac{\\log d}{mn}} + \\sqrt{\\frac{\\log d}{n}} \\left[s\\left( \\sqrt{\\frac{\\log d}{n}} + \\rho \\right)\\right]^{h+1} \n\\end{align*}\nand $k = \\cO(s)$ there holds the following error bounds with high probabiltiy:\n\\begin{align*}\n&\\|\\theta^{h+1} - \\theta^*\\|_1 \\\\\n&\\hspace{0.2in} \\lesssim s\\sqrt{\\frac{\\log d}{mn}} + s\\sqrt{\\frac{\\log d}{n}} \\left[s\\left( \\sqrt{\\frac{\\log d}{n}} + \\rho \\right)\\right]^{h+1},\\\\\n&\\|\\theta^{h+1} - \\theta^*\\|_2 \\\\\n&\\hspace{0.2in}\\lesssim \\sqrt{\\frac{s\\log d}{mn}} +\\sqrt{\\frac{s\\log d}{n}} \\left[s\\left( \\sqrt{\\frac{\\log d}{n}} + \\rho \\right)\\right]^{h+1}.\n\\end{align*}\n\\end{remark}\n\n\\subsection{Sparse Logistic Regression}\nCombining Corollary \\ref{first_cor} with some intermediate results obtained from \\cite{wang2016efficient} and \\cite{regularizers2012m}, we now can give a similar result about the estimation error bound for sparse logistic regression. The explicit form is omitted due to the limitation of spaces.\n\\iffalse\n\\begin{cori}\n\tSuppose Assumption \\ref{assum_supp} holds and $\\mu_{h+1}$ is defined as \\eqref{mu_def}.\n\tIf the following condition holds for some $T\\geq 0$:\n\t\\begin{align*}\n\t\\|\\theta^T - \\theta^*\\|_1 \\leq 4\\sqrt{\\frac{\\log (2d\/\\delta)}{n}}. \n\t\\end{align*}\n\tThen under the sparse logistic model with random design, we have the following estimation error bound for all $h\\geq T$ with probability at least $1-2\\delta$:\t\n\t\\begin{align*} \\textstyle\n\t\\|\\theta^{h+1}- \\theta^*\\|_1 \\lesssim\\sqrt{1+2(C_1-1)^{-\\frac{1}{2}}}\\frac{1-a_n^{h-T+1}}{1-a_n}\\hspace{0.27in}\\\\\n\t\\cdot\\frac{24C_2s \\sigma_X}{\\kappa} \\sqrt{\\frac{\\log(d\/\\delta)}{mn}}+4a_n^{h-T+1}\\sqrt{\\frac{\\log(2d\/\\delta)}{n}}\n\t\\end{align*}\n\tand\n\t\\begin{align*} \\textstyle\n\t\\|\\theta^{h+1}- \\theta^*\\|_2 \\lesssim \\sqrt{1+2(C_1-1)^{-\\frac{1}{2}}}\\frac{1-a_n^{h-T+1}}{1-a_n}\\hspace{0.27in}\\\\\n\t\\cdot \\frac{24\\sqrt{s} \\sigma_X}{\\kappa}\\sqrt{\\frac{\\log(d\/\\delta)}{mn}} +4a_n^{h-T}\\sqrt{\\frac{\\log(2d\/\\delta)}{n}},\n\t\\end{align*}\n\twhere\n\t\\begin{align*} \\textstyle\n\ta_n = 6\\sqrt{1+2(C_1-1)^{-\\frac{1}{2}}}\\hspace{1.5in}\\\\\n\t\\cdot\\frac{ C_2 s}{\\kappa}\\sigma_X^2\\log(\\frac{mnd}{\\delta})\\left[\\sqrt{\\frac{\\log(2d\/\\delta)}{n}} + \\rho\\right]\n\t\\end{align*}\n\tand\n\t\\begin{align*} \\textstyle\n\tb_n = 6\\sqrt{1+2(C_1-1)^{-\\frac{1}{2}}}\\hspace{1.5in}\\\\\n\t\\cdot\\frac{\\sqrt{s}}{\\kappa}\\sigma_X^2\\log(\\frac{mnd}{\\delta})\\left[\\sqrt{\\frac{\\log(2d\/\\delta)}{n}} + \\rho\\right]. \n\t\\end{align*}\n\\end{cori}\n\\fi\n\n\\section{Experiments}\n\\vspace{-0.02in}\nNow we test our algorithm on both simulated data and real data. In both settings, we compare our algorithm with various advanced algorithms. These algorithms are:\n\n\\begin{itemize}\n\t\\item[1.] EDSL: the state-of-the-art approach proposed by Jialei Wang et al. \\cite{wang2016efficient}.\n\t\\vspace{-0.05in}\n\t\\item[2.] Centralize: using all data, one machine solves the centralized loss minimization problem with $l_1$ regularization. This procedure is communication expensive or requires much larger storage.\n\t\\vspace{-0.05in} \n\t\\item[3.] Local: the first machine solves the local $l_1$ regularized loss minimization problem with only the data stored on this machine, ignoring all the other data.\n\t\\vspace{-0.05in}\n\t\\item[4.] Two-way Truncation: the proposed sparse learning approach which further improves the communication efficiency. \t\n\t\\vspace{-0.1in}\n\\end{itemize}\n\\subsection{Simulated data}\n\\begin{figure}[!htb]\n\t\\centering \n\t\\vspace{-0.1in}\n\t\\begin{center}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t\\hspace{-0.096in}\\subfloat[$\\Sigma_{ij} = 0.5^{|i -j|}$]{\n\t\t\t\\includegraphics[width=0.46\\linewidth]{{figure1a}.pdf}\n\t\t}\n\t\t\\hspace{-0.08in}\\subfloat[$\\Sigma_{ij} = 0.5^{|i -j|\/5}$]{\n\t\t\t\\includegraphics[width=0.46\\linewidth]{{figure1b}.pdf}\n\t\t}\\\\\n\t\t\\vspace{0.1in} $m= 20, n =600, d = 20000, s=10, \\Xb \\sim \\cN(0,\\Sigma)$ \\vspace{0.1in}\\\\\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\end{center}\n\t\\vspace{-0.1in}\t\n\t \\caption{Comparison among four algorithms in sparse linear regression setting}\\label{figure1}\n\n\\end{figure}\n\n\\begin{figure}[!htb] \n\n\t\\centering\n\t\\begin{center}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t\\hspace{-0.096in}\\subfloat[$\\Sigma_{ij} = 0.5^{|i -j|}$]{\n\t\t\t\\includegraphics[width=0.46\\linewidth]{{figure2a}.pdf}\n\t\t}\n\t\t\\hspace{-0.08in}\\subfloat[$\\Sigma_{ij} = 0.5^{|i -j|\/5}$]{\n\t\t\t\\includegraphics[width=0.46\\linewidth]{{figure2b}.pdf}\n\t\t}\\\\\n\t\t\\vspace{0.1in} $m= 10, n =1000, d = 2000, s=20, \\Xb \\sim \\cN(0,\\Sigma)$ \\vspace{0.1in}\\\\\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\end{center}\n\t\\vspace{-0.1in}\n\t\\caption{\\hspace{-0.1in} Comparison among four algorithms in sparse logistic regression setting}\\label{figure2}\n\n\\end{figure}\nThe simulated data $\\{\\xb_{ji}\\}_{i\\in [n], j\\in[m]}$ is sampled from multivariate Gaussian distribution with zero mean and covariance matrix $\\Sigma$. We choose two different covariance matrices: $|\\Sigma_{ij}| = 0.5^{|i-j|}$ for a well-conditioned situation and $|\\Sigma_{ij}| = 0.5^{|i-j|\/5}$ for an ill-conditioned situation. The noise $\\epsilon_{ji}$ in sparse linear model ($y_{ji} = \\langle\\xb_{ji},\\theta^*\\rangle + \\epsilon_{ji}$) is set to be a standard Gaussian random variable. We set the true parameter $\\theta^*$ to be $s$-sparse where all the entries are zero except that the first $s$ entries are i.i.d random variables from a uniform distribution in [0,1]. Under both two models, we set the hard thresholding parameter $k$ greater than s but less than $3 s$. \n\\vspace{-0.02in}\n\nHere we compare the algorithms in different settings of $(n,d,m,s)$ and plot the estimation error $\\|\\theta^h - \\theta^*\\|_2$ over rounds of communications. The results of sparse linear regression and sparse logistic regression are showed in Figure~\\ref{figure1} and Figure~\\ref{figure2}. We can observe from these plots that: \n\n\\begin{itemize}\n\t\\item First, there is indeed a large gap between the local estimation error and the centralized estimation error. The estimation errors of EDSL and the Two-way Truncation decrease to the centralized one in the first several rounds of communications.\n\n\t\\item Second, the Two-way Truncation algorithm is competitive with EDSL in both statistical accuracy and convergence rate as the theory indicated. Since it can converge in at least the same speed as EDSL's and requires less communication and computation cost in each iteration, overall it's more communicationally and computationally efficient.   \n\n\\end{itemize}\nThe above results support the theory that the Two-way Truncation approach is indeed more efficient and competitive to the centralized approach and EDSL.\n\\vspace{-0.05in}\n\n\\vspace{-0.05in}\n\n\\subsection{Real data}\n\\begin{figure}[!htb]\n\t\\vspace{-0.1in}\n\t\\begin{center}\n\t\t\\hspace{-0.08in}\\subfloat[dna (linear regression)]{\n\t\t\t\\includegraphics[width=0.46\\linewidth]{{figure3a}.pdf}\n\t\t}\n\t\t\\hspace{-0.08in}\\subfloat[a9a (classification)]{\n\t\t\t\\includegraphics[width=0.46\\linewidth]{{figure3b}.pdf}\n\t\t}\n\t\n\t\n\t\n\t\\end{center}\n\t\\vspace{-0.1in}\n\t\\caption{Comparison among four algorithms on real datasets}\\label{fig:real_data1}   \n\t\\vspace{-0.5in}\n\\end{figure}\n\\vspace{0.4in}\nIn this section, we examine the above sparse learning algorithms on real-world datasets. The data comes from UCI Machine Learning Repository \\footnote{http:\/\/archive.ics.uci.edu\/ml\/} and the LIBSVM website \\footnote{https:\/\/www.csie.ntu.edu.tw\/$\\sim$cjlin\/libsvmtools\/datasets\/}. The high-dimensional data 'dna' and 'a9a' are used in the regression model and classification model respectively. We randomly partition the data in $[60\\%, 20\\%,20\\%]$ for training, validation and testing respectively. Here the data is divided randomly on $m = 10$ machines and processed by algorithms mentioned above. The results are summarized in Figure \\ref{fig:real_data1}. These results in real-world data experiments again validate the theoretical analysis that the proposed Two-way Truncation approach is a quite effective sparse learning method with very small communication and computation costs. \n\n\\section{Conclusions}\n\nIn this paper we propose a novel distributed sparse learning algorithm with Two-way Truncation.  Theoretically, we prove that the algorithm gives an estimation that converges to the minimizer of the expected loss exponentially and attain nearly the same statistical accuracy as EDSL and the centralized method. Due to the truncation procedure, this algorithm is more efficient in both communication and computation. Extensive experiments on both simulated data and real data verify this statement.\n\n\n\\section*{Acknowledgment}\n\nThe authors graciously acknowledge support from NSF Award CCF-1217751 and DARPA Young Faculty Award N66001-14-1-4047 and thank Jialei Wang for very useful suggestion. \n\n\\bibliographystyle{CADSLTT}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction} \\label{sec:intro}\n\nIn the local Universe, there is a noticeable dearth of baryons within massive ($M_{DM}>10^{12.5}$ M$_{\\odot}$) galactic halos \\citep{Benson03,Croton06}. Feedback from supermassive black holes (SMBH) is commonly invoked to explain the missing baryons within the massive halos. Similarly, the tight correlation between the mass of a galaxy's SMBH and the total mass of the bulge and galaxy \\citep{Magorrian98,Gebhardt00} suggests that galaxies and SMBHs have evolved together. How SMBHs and galaxies co-evolve and regulate their mutual growth is an outstanding problem in modern astrophysics. Theoretical work suggests that the strong correlation between the mass of the SMBH and the velocity dispersion of the bulge ($M_{\\bullet}-\\sigma~$) may be achieved through quenching of star formation by powerful outflows driven once the galaxies reside on the $M_{\\bullet}-\\sigma~$ relationship \\citep{Hopkins06,Zubovas12,Zubovas14}.\n\nThe feedback that regulates the overall growth of a galaxy is expected to be most important when both the galaxies and the SMBHs were experiencing the majority of their growth \\citep{Zubovas12,Choi12,Barai17,Costa18}. Given that both quasar activity and star formation peak in peak in normal galaxies at cosmic noon  ($1.5<z<3$), studying in detail the nature of outflows from quasars and starbursts at these redshifts will provide important clues about the nature of feedback during this important epoch.\n\nFeedback can manifest itself in the form of galaxy-scale winds that expel gas from the galaxy. Moreover, winds can drive shocks and turbulence and prolong the time necessary for gas to cool, collapse and form stars. To date, most of the outflows studied in the distant ($z>1$) Universe have focused on optical emission lines that trace the warm ionized phase of outflows with typical densities $n_e\\sim 100-1000$ cm$^{-3}$ and temperatures $T=10^4$ K. At the epoch of peak quasar activity $z\\sim 2-3$, the bright emission lines such as [O{\\sc III}]\\xspace\\ $\\lambda$5007\\AA, H$\\alpha$ and H$\\beta$ are redshifted into near-infrared bands where they can be resolved spatially and spectrally. Using near-infrared spectroscopy, several authors have studied quasar-driven ionized winds on galaxy-wide scales both in radio-loud quasars (with powerful jets) and in radio-quiet quasars \\citep{Nesvadba08,Cano-Diaz12,Carniani15,Vayner17,Vayner21,Kakkad20}.\n\nRecent theoretical and observational works have suggested that the energy and momentum in galaxy outflows are shared between multiple phases of the gas. These gas phases span a wide range of densities and temperatures, from the dense cold molecular and neutral gas ($T\\sim10-300$ K) to diffuse hot post-shock medium ($T>10^7$ K; \\citealt{Crichton16,Hall19}). There have been several studies of multi-phase outflows in distant quasars, focusing on individual systems \\citep{Vayner17,Brusa18,Herrera-Camus19} and at studying the cold molecular and ionized gas phase in galactic outflows. It is unknown which phase of the outflow is responsible for the bulk of the momentum and mass of these multi-phase outflows, and it may well be a function of the conditions in the host galaxies. Theoretical work suggests that the molecular gas clouds may be disrupted and entrained by the outflow \\citep{Scannapieco15} or, instead, molecules may form within the outflowing gas \\citep{Richings18}. Observations of nearby galaxies suggest that the cold molecular gas phase (50-100 K) may be the dynamically dominant phase of quasar winds \\citep{Sun14,Cicone14,Alatalo11,Aalto12,Feruglio13,Morganti13,Veilleux17,Fiore17}. The warm (T$\\sim$500 K) molecular gas phase may also be important and carry a significant fraction of the momentum in a galactic outflow \\citep{Richings18}. In nearby galaxies, Spitzer observations of the warm molecular H$_{2}$\\ gas (T$\\sim$500 K) have found a significant amount of gas mass in galactic outflows \\citep{Beirao15,Dasyra14,Rogemar20}. \n\nBecause cold molecular gas is the fuel for star formation, the fate of the molecular gas phase is the key link in understanding the impact of quasar feedback on star formation. Since outflows are invariably multi-phase, understanding the relationship between molecular gas and atomic gas in outflows is crucial for accurately estimating the energetics of feedback and the fate of the interstellar medium (ISM). Statistical studies are lacking, as the study of multi-gas phase outflows in the distant Universe is still in its infancy. A recent attempt at comparing molecular gas properties of galaxies with powerful (L$_{bol}=10^{45-47}$ erg s$^{-1}$ ) active galactic nuclei (AGN) to star forming galaxies with similar stellar mass have found the AGN to reside in galaxies with lower molecular gas masses hinting at potential effects of feedback from outflows or radiation \\citep{Circosta21,Bischetti21}.\n\nTo address both the molecular gas reservoir and the measurement of cold molecular gas in galactic outflows requires kpc-scale spatial resolution observations that have the right sensitivity to detect the molecular gas or place stringent limits. The lowest energy transition of the H$_{2}$\\ molecule is the rotational quadrupole transitions that require gas temperatures $>100$ K to excite; hence H$_{2}$\\ is invisible in the cold molecular gas phase. The next most abundant molecule in molecular clouds is carbon monoxide (CO) that has a weak permanent dipole moment, with the near-ground rotational transitions having small excitation energies, enabling to trace colder molecular gas (5.5-55 K). In this paper, we present Atacama Large Millimeter Array (ALMA) observations of the cold molecular gas traced through rotational transitions of CO in six radio-loud quasars at $z=1.439-2.323$ with known powerful ionized gas outflows. We present observations, data reduction, and emission line analysis in Section \\ref{sec:obs}. We describe how we search for molecular outflows and calculate their energetics in Section \\ref{sec:dynamics}. We discuss individual objects in Section \\ref{sec:indiv_obj}. We compare the entire sample, discuss potential sources that drive the multi-phase gas outflows and the dominant source of molecular gas depletion in Section \\ref{sec:discussion}. We summarize our conclusions in Section \\ref{sec:conc}. We use an $\\rm H_{0}=67.8$ \\kms\\ Mpc$^{-1}$, $\\Omega_{\\rm m}=0.308$, $\\Omega_{\\Lambda}=0.692$ cosmology throughout this paper. \n\n\\section{Observations, Data Reduction \\& Line Fitting} \\label{sec:obs}\n\n\\begin{deluxetable*}{ccccccccc}\n\\centering\n\\tablecaption{Summary of ALMA observations \\label{tab:VLA-archive}}\n\\tablehead{\n\\colhead{Object} & \n\\colhead{Date} & \n\\colhead{Central frequency} &\n\\colhead{Continuum beam \\tablenotemark{a}}&\n\\colhead{Continuum Sensitivity}&\n\\colhead{Line beam}&\n\\colhead{Line sensitivity}&\n\\colhead{Channel width} \n\\\\\n\\colhead{} & \n\\colhead{} & \n\\colhead{(GHz)}& \n\\colhead{}&\n\\colhead{mJy}&\n\\colhead{}&\n\\colhead{mJy}&\n\\colhead{\\kms}}\n\\startdata\n7C 1354 & 2018 Jan 8-24  & 153.364 & 0.48\\arcsec$\\times$0.36\\arcsec &0.0069&0.59\\arcsec$\\times$0.47\\arcsec&0.08 & 34.0\\\\\n4C 22.44 & 2017 Dec 17-30 & 135.55 & 0.39\\arcsec$\\times$0.35\\arcsec&0.0068&0.408\\arcsec$\\times$0.366\\arcsec&0.083 & 66.0\\\\\n4C 05.84 & 2018 Jan 5-16  & 138.87 & 0.41\\arcsec$\\times$0.30\\arcsec&0.0075&0.438\\arcsec$\\times$0.54\\arcsec&0.134 & 33.0\\\\\n4C 09.17 & 2017 Dec 25 & \t148.24 & 0.32\\arcsec$\\times$0.23\\arcsec&0.0139&0.39\\arcsec$\\times$0.34\\arcsec&0.130 & 15.8\\\\\n & 2018 Jan 8 & \t & &&&\\\\\n3C 318 & 2014 Jul 6  & 134.34 & 0.25\\arcsec$\\times$0.18\\arcsec&0.0095&0.454\\arcsec$\\times$0.320\\arcsec&0.107 & 66.66\\\\\n3C 298 & 2016 Sep 9  & 141.85 & 0.25\\arcsec$\\times$0.18\\arcsec&0.046&0.39\\arcsec$\\times$0.30\\arcsec&0.22 & 34.0\n\\enddata\n\\tablenotetext{a}{From continuum images integrated over all frequencies and cleaned with robust = 0.5 parameter.}\n\\end{deluxetable*}\n\nA leading goal of the ALMA observational program was to detect molecular gas outflows in quasars with powerful ionized outflows that were previously detected via optical emission line (e.g., [O{\\sc III}]\\xspace, H$\\alpha$\\xspace) kinematics using integral field spectroscopy observations with adaptive optics \\citep{Vayner19b,Vayner19a}. All sources selected within this study display ionized gas outflows on kpc scales, with outflow rates in the range of 50-1000 \\myr, velocities $>$ 500 \\kms\\ with momentum fluxes $>$ 10$^{35} $ dyne and coupling efficiencies between the kinetic luminosity of the outflow and the bolometric luminosity of the quasar $>$0.05$\\%$. Given the similarities between the field-of-view and angular resolution between the Keck\/OSIRIS and ALMA observations we are able to study molecular gas outflows on similar spatial and dynamical time scales.\n\nALMA band 4 observations were conducted in Cycle 5 in the C43-5 configuration with a typical angular resolution of 0.4\\arcsec\\ and a maximum recoverable scale of 4-5.5\\arcsec, corresponding to rough physical scales of 3 kpc and 34-43 kpc, respectively. One 1.875 GHz spectral window was tuned to the redshifted frequency of CO (3-2) or CO (4-3) emission line with an effective velocity bandwidth of 4,000 \\kms, while additional three bands were tuned to the nearby continuum.\\\\  \n\nData reduction was performed using CASA (Common Astronomy Software Applications; \\citealt{McMullin07}) version 5.1.2-4. The ALMA automated pipeline was used to create the measurement sets (MS) for each observing block, which were then combined into a single measurement set for each source. We performed phase self-calibration for 4C 05.84 and 7C 1354+2552, while for 3C 318 and 4C 09.17, we performed both phase and amplitude self-calibration. For each source, the band 4 quasar continuum, was used as a self-calibration model using the CASA task \\textit{clean}. Given the similarity between the archival VLA images of the jets in these systems, we believe that the majority of the continuum in our sources comes from the synchrotron emission of the quasar core\/jets. While there is extended continuum emission for each source, the majority of the flux is associated with the unresolved core emission from the quasar making the modeling of the continuum relatively simple for self-calibration. We image the continuum with Briggs weighting using a robust value of 0.5 and a pixel size set to 1\/4th of the beam's full-width half max (FWHM). In this work, we also include our pilot observations of 3C 298 conducted in cycles 2 and 3 \\citep{Vayner17} with ALMA band 4. We achieve an SNR of 600-25,000 for the peak continuum flux. The continuum SNR improved by a factor of 1.2-5 from phase-only self-calibration and in the case of 4C09.17 after amplitude self-calibration the rms further improved by a factor of 1.5, while for 3C318 we did not see a significant improvement in the rms after amplitude calibration.\\\\\n\nWe performed continuum subtraction using the CASA task \\textit{uvcontsub} by fitting a first-order polynomial to line-free emission channels of the spectral window with the CO emission. We then subtracted the best fit continuum model from the full spectral window.\\\\\n\nWe imaged the cube using \\textit{clean} with a robust value of 1.5 to help improve detection of fainter and more diffuse emission, resulting in a larger beam than the continuum imaging. We used a spectral pixel size of either 16 \\kms, 34 \\kms, or 66 \\kms, depending on the signal-to-noise ratio (SNR) of the CO emission. We used a value of 0.05\\arcsec\\ for the spatial pixel size. For all sources except 4C 09.17, we used a wide circular aperture centered on the quasar for the cleaning mask with a radius of 1\/4 the primary beam size. For 4C 09.17, the CO emission was detected in individual 16 \\kms\\ channels in the first cleaning cycle, and a tight mask was designed for each channel encompassing the CO (4-3) emission.\n\n\nTo search for extended emission, we construct an SNR map for each channel in the data cube. An SNR map is made by first computing a standard deviation in a large aperture away from the phase center, followed by dividing the flux per spaxel by the standard deviation. SNR maps are constructed from data cubes that were not corrected for the primary beam's response. For any spaxel containing emission with a peak SNR$\\geq$4 per beam, we fit the emission line in neighboring spaxels that lie in the beam with a Gaussian plus constant continuum model using the least-squares fitting routine \\textit{curvefit} within \\textit{SciPy}. We create a 0$^{th}$ moment (flux) map by integrating the emission line in each spaxel with a successful emission line fit over the fitted Gaussian model and a 1$^{st}$ moment map by computing the line centroid's Doppler shift relative to the redshift of the quasar host galaxy. The integrated intensity maps are all optimally extracted, with a varying window for integrating the CO emission line based on the velocity offset and dispersion. We construct a 2$^{nd}$ moment map using the fit's velocity dispersion from the Gaussian model. All moment maps are constructed from data cubes that were corrected for the primary beam's response. Figure \\ref{fig:all_sources_CO} showcases the integrated CO (3-2) or CO (4-3) maps for sources with detected extended emission.\\\\\n\nHerein we use the quasar redshifts that are reported from Keck\/OSIRIS observations \\citep{Vayner19b} where they were derived from the centroid of the spatially unresolved ($<1.5$ kpc) [O{\\sc III}]\\xspace\\ or H$\\alpha$\\xspace emission lines, effectively originating from the narrow-line-region (NLR) of the quasar. The [O{\\sc III}]\\xspace\\ and H$\\alpha$\\xspace\\ redshifts agree within the centroiding uncertainty. For 3C 298 the redshift derived from the molecular gas disk traced through CO (3-2) is within 50 \\kms\\ of the redshift derived from the quasar emission \\citep{Vayner17}.\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=7.0in]{all_sources_ALMA.eps}\n    \\caption{ALMA band 4 observations of the sources within our sample. We present ALMA band 4 integrated intensity maps of CO emission. In contours we present the ALMA band 4 continuum emission that is dominated by synchrotron emission from the quasar jets. The contours stretch from a peak flux of 0.013, 0.0037, 0.2, 0.003, 0.045, and 0.002 Jy\/beam down to 2$\\sigma$ sensitivity in linear steps, from top left to bottom right, respectively. The ellipse in the lower-left corner of each map represents the beam of the emission line data. The star in the center represents the quasar's location, and the bar represents 1 arcsecond. The maps are at at a position angle of 0$^{\\circ}$, where north is oriented up, the only exception is 3C 298 that is at a position angle of 103$^{\\circ}$ East of North.}\n    \\label{fig:all_sources_CO}\n\\end{figure*}\n\n\n\\begin{deluxetable*}{lcclll@{\\extracolsep{-10pt}}c}[!th]\n\\tiny\n\\tablecaption{Sample properties \\label{tab:sample}}\n\\tablehead{\n\\colhead{Name} & \n\\colhead{RA} & \n\\colhead{DEC} &\n\\colhead{z\\tablenotemark{a}} &\n\\colhead{L$_{\\rm bol}$} &\n\\colhead{L$_{\\rm 178 MHz}$} &\n\\colhead{M$_{\\rm BH}$}\\\\\n\\colhead{} & \n\\colhead{J2000} &\n\\colhead{J2000} & \n\\colhead{} & \n\\colhead{($10^{46}$ erg s$^{-1}$ )} & \n\\colhead{($10^{44}$ erg s$^{-1}$ )} & \n\\colhead{M$_{\\odot}$}}\n\\startdata\n4C 09.17 & 04:48:21.74  & +09:50:51.46 & 2.1170 & 2.88$\\pm$0.14 &2.6 & 9.11  \\\\\n7C 1354+2552 & \t13:57:06.54 & +25:37:24.49 & 2.0068 & 2.75$\\pm$0.11 & 1.4& 9.86 \\\\\n3C 298 & 14:19:08.18 & +06:28:34.76  & 1.439\\tablenotemark{b} & 7.80$\\pm$0.30 & 12 &9.51 \\\\\n3C 318 & 15:20:05.48  & +20:16:05.49 & 1.5723 & 0.79$\\pm$0.04 & 4.0 &9.30 \\\\\n4C 22.44 & 17:09:55.01 & +22:36:55.66 & 1.5492 & 0.491$\\pm$0.019 & 0.6 &9.64 \\\\\n4C 05.84 & 22:25:14.70  & +05:27:09.06 & 2.320 & 20.3$\\pm$1.00& 4.5 &9.75  \\\\\n\\enddata\n\\tablenotetext{a}{Redshift relative to narrow-line region emission of the quasar, derived from 5007 \\AA\\ [O{\\sc III}]\\xspace\\ emission.}\n\\tablenotetext{b}{Redshift derived from host galaxy CO (3-2) emission \\citep{Vayner17}}\n\\end{deluxetable*}\n\n\n\\section{Dynamics of the molecular gas}\\label{sec:dynamics}\n\n\\subsection{Systemic molecular gas}\n\nFor each source, we search for emission at the systemic redshift of the quasar host galaxy. Narrow, CO line emission is found in the host galaxies of 4C 09.17A, 4C 09.17B, 3C 298, and 7C 1354+2552. The narrow CO emission in 3C 298 resembles a rotating disk that we modeled in \\citet{Vayner17}. The narrow emitting gas in the other systems does not show a smooth velocity gradient that would be indicative of a rotating galactic disk. \n\n\\noindent We compute the emission line luminosity using the following equation:\n\n\\begin{equation}\n   L^{'}_{CO}=3.25 \\times 10^{7}S_{CO}\\Delta v \\frac{D^{2}_{L}}{(1+z)^{3}\\nu_{obs}^{2}} \\rm K~km~s^{-1}~pc^{2},  \n\\end{equation} \n\n\\noindent \nwhere $\\nu_{obs}$ is the observed CO transition frequency, $D_{L}$ is the luminosity distance, and $S_{CO}\\Delta v$ is the line integrated flux in units of Jy \\kms. We convert the observed CO transition luminosity into CO (1-0) luminosity (L$^{'}_{CO(1-0)}$) by assuming that the low-J CO transitions are thermalized and are optically thick, so ${L}_{\\mathrm{CO}\\ 4\\mbox{--}3}^{{\\prime} }={L}_{\\mathrm{CO}\\ 3\\mbox{--}2}^{{\\prime} } ={L}_{\\mathrm{CO}\\ 1\\mbox{--}0}^{{\\prime} }$. Using the ratios (${r}_{{\\rm{J}}1}={L}_{\\mathrm{CO}J\\to J-1}^{{\\prime} }\/{L}_{\\mathrm{CO}\\ 1\\mbox{--}0}^{{\\prime} }$) from \\cite{CarillinWalter13} with $r_{31}$=0.97 and $r_{41}$=0.87 did not significantly change our results. Furthermore in 3C 298 we found that the molecular gas is consistent with being thermalized and optically thick \\citep{Vayner17}. These physical conditions are consistent with what is found for Mrk 231 \\citep{Feruglio15}. Finally, we convert the ${L}_{\\mathrm{CO}\\ 1\\mbox{--}0}^{{\\prime}}$ line luminosity into molecular gas mass using the CO-to-H$_{2}$ conversion factor: $\\alpha_{\\rm CO}$ with units of (K \\kms$\\rm pc^{2}$)$^{-1}$. For sources where we do not detect any narrow CO emission at the systemic redshift, we place a limit on the molecular gas mass over an aperture equal to the beam size for an emission line with a velocity FWHM of 250 \\kms. The molecular gas mass limits can be linearly scaled with a different $\\alpha_{CO}$\\ value. For sources with detected molecular gas at the quasar's systemic redshift, we compute the radius of the molecular gas region, which allows us to measure the gas surface density. All radii are computed using a curve of growth method. Effective radii refer to a region which encloses 50\\% of the flux, while a ``maximum\" extent refers to a size scale which encloses 90\\% of the flux. In all cases, narrow emission at the systemic redshift of the quasar is spatially resolved by our observations. We deconvolve the size of the beam from all radius measurements. For sources with no detected CO emission, we use the molecular gas mass limit and the beam of the observations as a proxy for the radius. Values associated with the molecular gas at the systemic redshift are summarised in Table \\ref{tab:narrow-prop}.\n\n\\begin{deluxetable*}{lcccccr}[!th]\n\\tablecaption{Properties of molecular gas at the systemic redshift of each quasar. $S_{\\nu}\\Delta V$ is the spatially and line-integrated CO flux. $M_{H_{2}}$ is the mass of gas at the systemic redshift of each galaxy, assuming an $\\alpha_{CO}$ of 0.8. $R$ is the radius of the region. V$\\rm_{\\sigma}$ is the velocity dispersion. SFR is the expected star formation rate based on the currently available molecular gas reservoir based on the KS law.  \\label{tab:narrow-prop}}\n\n\n\n\\tablehead{\n\\colhead{Source}&\n\\colhead{$S_{\\nu}\\Delta V$}&\n\\colhead{$M_{H_{2}}$}&\n\\colhead{R}&\n\\colhead{V$\\rm_{\\sigma}$}&\n\\colhead{$\\Sigma_{molecular}$}&\n\\colhead{SFR}\\\\\n\\colhead{}&\n\\colhead{Jy \\kms}&\n\\colhead{$\\times10^{9}$M$_{\\odot}$}&\n\\colhead{kpc}&\n\\colhead{\\kms}&\n\\colhead{M$_{\\odot}$ pc$^{-2}$}&\n\\colhead{\\myr}}\n\\startdata\n4C 09.17 A RL & 0.29$\\pm$0.03 & 3$\\pm$0.3 & 2.8  & 158.0$\\pm$14 & 137$\\pm$13 & 6 \\\\\n4C 09.17 B RQ & 0.88$\\pm$0.09 & 10$\\pm$1 & 1.1 & 143.5$\\pm$12 & 2357$\\pm$235 & 56 \\\\\n7C 1354 & 0.028 $\\pm$0.003 & 0.3$\\pm$0.1 & 2.0 & 44.10$\\pm$11 & 23$\\pm$2 & 1\\\\\n3C 298 & 0.63$\\pm$0.07 & 6.6$\\pm$1 & 1.6 & 42.35$\\pm$12.8 & 820$\\pm$10 & 24\\\\\n3C 318 & $<0.05$ & $<0.6$ & -- & -- & -- & --\\\\\n4C 22.44 & $<$0.05 & $<$ 1 & -- & -- & -- & --\\\\\n4C 05.84 & $<0.07$ & $<0.8$ & -- & -- & -- & --\\\\\n\\enddata\n\\end{deluxetable*}\n\n\\subsection{Observed Molecular Outflows}\\label{sec:search_outflow}\n\nIonized gas outflows typically show a peak emission line offset  $>$ 300 \\kms\\ from the quasar redshift measured from the narrow emission line region, with a velocity dispersion $>$ 250-300 \\kms\\ over the outflow region for the sources within our sample \\citep{Vayner19b}. Based on these observed velocities of the ionized gas outflows, we define the criteria for a molecular outflow to be any molecular gas that has a peak emission line offset relative to the redshift of the quasar host galaxy $|v| >300$ \\kms\\ or a spaxel with a velocity dispersion greater than 250 \\kms. The selected outflow criteria are typical of molecular outflows found in nearby galaxies \\citep{Fluetsch19}. Using these criteria, we detect molecular gas outflows in four quasars in the ALMA sample. For 3C 318 and 4C 05.84, we select outflows based on both broad ($\\sigma>250$ \\kms) and offset ($|v| > 500$ \\kms) emission and for 4C 09.17, based on broad ($\\sigma>300$ \\kms) emission. We include the molecular gas outflow in 3C 298 that was previously detected in \\citet{Vayner17} based on broad CO (3-2) and CO (5-4) emission. For each source we extract a spectrum by integrating over all spaxels satisfying this outflow criteria. We present the extent of the molecular outflows along with the spectra in Figures \\ref{fig:3C318_spec}, \\ref{fig:4C0917_spec}, \\ref{fig:4C0584_spec}, and \\ref{fig:3C298_spec}. \n\nWe potentially may be missing dense outflowing gas moving at slower speeds since our high velocity criteria is based on atomic gas observations, which may lead us to underestimate the total molecular gas in the outflow for the 4C 09.17 and 3C 298 systems where we detect CO emission moving at speeds $<$ 300 \\kms. Observations of denser molecular gas tracers, such as CS or HCN, could provide a more complete picture of the outflow. \\\\\n\nWe calculate the molecular gas mass in the outflows based on flux associated with the broad or highly offset emission line component. We use an $\\alpha_{CO}$\\ value of 0.8 \\msun$\\rm(K~kms^{-1}pc^{2})^{-1}$ for consistency with other works at low and high redshift \\citep{Herrera-Camus19,Fluetsch19}, this value is commonly adopted for the molecular gas in the ISM of nearby Ultra Luminous Infrared Galaxies \\citep{Bolatto13}. However, in some well-studied molecular outflows in nearby galaxies, the conversion factor can be much higher, $\\alpha_{\\rm CO}\\sim 2$ \\citep{Cicone18}, so we may be conservative with our mass estimates.\n\nWe compute the molecular gas outflow rate using\n\n\\begin{equation}\\label{equation:outflow-thin-shell}\n    \\dot{M}_{H_{2}}=\\frac{M_{H_{2}}v_{out}}{R_{out}}.\n\\end{equation}\n\nWe select this equation since the molecular gas outflows in 3C 318, 4C 05.84, and 4C 09.17-A RL are seen as a single high-velocity offset component that spans either blue- or red-shifted velocities. The molecular gas outflows in 4C 09.17B and 3C 298 may be closer in geometry to a filled wide-angle cone since they span a broader velocity range. In these two sources, the estimates obtained from equation \\ref{equation:outflow-thin-shell} should be multiplied by a factor of 3 if the outflows are closer to a filled cone. Here $R_{out}$ is the extent of the outflow where 90\\% of the flux associated with the outflow emission accumulates. The velocity is computed as $v_{out} = \\left | v_{r} \\right |  + FWHM\/2$, where $\\left | v_{r} \\right |$ and $FWHM$  are the radial velocity and full-width-half-maximum in units of \\kms\\ of the emission relative to the systemic redshift of the emission associated with the outflow component.\n\nIn addition to the outflow rates we also compute the momentum flux of the outflow using: \n\n\\begin{equation}\n    \\dot{P}_{H_{2}} = \\dot{M_{H_{2}}} \\times v_{out}\n\\end{equation}\n\n\\noindent and the kinetic luminosity:\n\n\\begin{equation}\n    \\dot{E}_{H_{2}} = \\frac{1}{2}\\times\\dot{M_{H_{2}}}\\times v_{out}^{2}.\n\\end{equation}\n\nThe CO line luminosity used to compute the molecular gas mass along with the spatial extent, velocity, outflow rate, and energetics are summarized for each source in Table \\ref{tab:outflow-prop}.\n\n\n\\begin{deluxetable*}{lllccccccccl}\n\\tablecaption{Multi-phase outflow properties. $S_{\\nu}\\Delta V$ is the spatially and line-integrated CO flux associated with the molecular outflow. R$\\rm_{out}$ is the radial extent of the outflow. V$\\rm_{out}$ is the velocity of the outflow. dM\/dt$\\rm_{H_{2}}$ and dM\/dt$\\rm_{Ionized}$ are the molecular and ionized outflow rates. $\\dot{P}_{H_{2}}$ is the momentum flux of the molecular outflow (assuming $\\alpha_{CO}$$=0.8$), while $\\dot{P}_{Ionized}$ is the momentum flux of the ionized outflow, using the dM\/dt$\\rm_{H\\alpha}$ outflow rate from \\citep{Vayner19a}. $\\frac{\\dot{P}_{outflow}}{\\dot{P}_{Quasar}}$ is the ratio of the momentum flux of outflow to the momentum flux of the quasar accretion disk ($L_{bol}\/c$) using a sum of the ionized and molecular outflow momenta flux.\\label{tab:outflow-prop}}\n\n\\tablehead{\\colhead{Source}&\n\\colhead{$S_{\\nu}\\Delta V$}&\n\\colhead{M$\\rm_{H_{2}}$}&\n\\colhead{R$\\rm_{out}$}&\n\\colhead{V$\\rm_{out}$}&\n\\colhead{V$\\rm_{FWHM}$}&\n\\colhead{dM\/dt$\\rm_{H_{2}}$}&\n\\colhead{M$\\rm_{Ionized}$}&\n\\colhead{dM\/dt$\\rm_{Ionized}$}&\n\\colhead{$\\dot{P}_{H_{2}}$}&\n\\colhead{$\\dot{P}_{Ionized}$}&\n\\colhead{$\\frac{\\dot{P}_{outflow}}{\\dot{P}_{Quasar}}$}\\\\\n\\colhead{}&\n\\colhead{Jy \\kms}&\n\\colhead{$\\times10^{9}$M$_{\\odot}$}&\n\\colhead{kpc}&\n\\colhead{\\kms}&\n\\colhead{\\kms}&\n\\colhead{\\myr} &\n\\colhead{$\\times10^{9}$M$_{\\odot}$} &\n\\colhead{\\myr}&\n\\colhead{$10^{35}$dyne}&\n\\colhead{$10^{35}$dyne}&\n\\colhead{}\n}\n\\startdata\n4C 05.84 & 0.1$\\pm$0.03 & 1.4$\\pm$0.2 & 8.6 & 653$\\pm$30 & 382.2$\\pm$47.4 & 110$\\pm$12 & 0.4$\\pm$0.3 & 870$\\pm$600 & 4.4$\\pm$0.6  & 40$\\pm$30& 0.7$\\pm$0.4 \\\\\n3C 318 & 0.25$\\pm$0.03 & 3$\\pm$0.3 & 20.2 & 1132$\\pm$44 & 528.7$\\pm$67.4 & 168$\\pm$18 & 0.32$\\pm$0.2 & 220$\\pm$150 & 12$\\pm$2 & 10$\\pm$7  & 8$\\pm$3 \\\\\n3C 298 & 0.3$\\pm$0.03 & 3$\\pm$0.3 & 1.6 & 394$\\pm$64 & 624.0$\\pm$49.0 & 780$\\pm$150 & 0.6$\\pm$0.3 & 750$\\pm$400 & 20$\\pm$7 & 77$\\pm$40  & 4$\\pm$2 \\\\\n4C 09.17 A & 0.11$\\pm$0.01 & 1.3$\\pm$0.1 & 2.8 &852$\\pm$77& 439.1$\\pm$122.6 & 400$\\pm$50 & 0.05$\\pm$0.02 & 50$\\pm$20 & 21$\\pm$4 & 2.1$\\pm$1  & 2.4$\\pm$0.5 \\\\\n4C 09.17 B & 2.3$\\pm$0.2 & 27$\\pm$3 & 4.9 & 456$\\pm$26  & 870.6$\\pm$47.2 & 2500$\\pm$300 & -- & -- & 73$\\pm$11 & --  & -- \\\\\n\\enddata\n\\end{deluxetable*}\n\n\\section{Individual Objects}\\label{sec:indiv_obj}\n\nIn this section, we outline the known properties of each quasar within our sample, focusing on their radio jet morphology, far-infrared properties, morphologies, the extent of the ionized gas outflows, and a description of what we detect with our ALMA band 4 observations.\n\n\\subsection{3C 318 (z=1.5734)}\\label{sec:3c318}\n\n3C 318 is a luminous radio-loud quasar at $z=1.5734$. The radio jet is double-sided, stretching in the southwest and northeast, with a bright core emission associated with the quasar's optical emission location. Within our ALMA continuum observations, we only detect the jet's southwest component. The northeast component of the jet blends with the bright unresolved core. 3C 318 has been detected with the \\textit{Herschel Space Telescope} and is known to be a bright far-infrared emitting source \\citep{Podigachoski15}. Resolved band 7 ALMA observations of the dust emission reveal a ring-like structure on kpc scale centered on the quasar \\citep{Barthel19}.\\\\\n\nWith the Keck\/OSIRIS, we detected ionized gas emission in nebular emission lines [O{\\sc III}]\\xspace 5007 \\AA, H$\\alpha$\\xspace, [N{\\sc II}]\\xspace 6585 \\AA, and [S{\\sc II}]\\xspace 6717, 6731 \\AA\\ with an extent of 4 kpc \\citep{Vayner19b}. We detected an ionized outflow extending in the SW and NE direction with a maximum extent of 3.2 kpc. \\\\\n\n3C 318 was known to have molecular emission detected at the CO (2-1) transition with PdBI \\citep{willott07}. The CO (2-1) emission was known to be blueshifted by 400 \\kms\\ and spatially offset from the quasar continuum by 2.4\\arcsec\\ to the west and 0.5\\arcsec\\ to the north with considerable uncertainty due to a coarse beam of 8.05\\arcsec$\\times$4.32\\arcsec. \\\\\n\nOur ALMA band 4 observations reveal an extended CO (3-2) emission with one component offset 1.7 kpc to the west and a second component offset 17 kpc towards the south. The emission is divided between two regions that have widths of 4 and 8 kpc, that show similar highly blueshifted emission relative to the quasar. The spatially integrated emission is blueshifted (-936.0 \\kms) and relatively broad with an FWHM of 534 \\kms. The spectrum and integrated intensity map of the detected CO emission are shown in Figure \\ref{fig:3C318_spec}. Likely the ALMA and PdBI observations trace the same molecular gas components. However, the differences in the beams, maximum recoverable scales, and SNR play a role in the observed line shift and integrated line flux. 3C 318 has also been observed with VLA targeting the CO (1-0) transition \\citep{Heywood13}. Emission associated with the CO (1-0) emission line at the velocity offset of the PdBI observations was found 0.33\\arcsec\\ north from the quasar continuum. We do not detect any CO (3-2) emission at that location. Using typical ratios between the CO (1-0) and CO (3-2) line luminosity, we would have expected to detect this component at an SNR of 100, integrated over an emission line equivalent with a velocity dispersion of 250 \\kms. \\\\\n\nThe separation between the two high velocity clumps roughly matches the maximum recoverable scale of the interferometric observations. We have attempted to recover the more diffuse emission between the two clumps by smoothing the data in the UV plane with Gaussian kernel using the \\textit{uvtaper} option within \\textit{tclean} in CASA. Using a uv-taper parameter of 0.5\\arcsec\\ and 1\\arcsec\\ on the sky, the fainter emission between the two clumps seen in Figure \\ref{fig:3C318_spec} is below the noise in the uv-tapered data. The total integrated flux from the CO (3-2) emission is within 10\\% of the original data, within the statistical noise of the observations, hence no additional ``diffuse\" emission was recovered. A loss of baselines resulted in an increase in noise with larger uv-taper parameters. Observations in a more compact ALMA configuration are necessary to detect the more diffuse emission. \\\\\n\nALMA's higher angular resolution and sensitivity lead us to speculate that the molecular gas emission in 3C 318 is associated with a molecular outflow rather than a merging system. This is supported by more accurate redshift measurement from the narrow-line region, allowing us to do better kinematics and dynamics measurement of the molecular gas emission. The Dark Energy Survey \\citep{DES2021} shows no apparent optical detection of a companion galaxy down to an $r$-band magnitude of 24 at a significance of 10$\\sigma$. Infrared observations with Spitzer and archival ALMA band 7 observations do not show any evidence for a galaxy at the spatial locations of the highly blueshifted CO (3-2) emission \\citep{Barthel19}. There may be a possibility that high-velocity emission is associated with an obscured galaxy near the 3C 318 quasar host galaxy. In recent years there have been several galaxies detected with ALMA that have faint or no counterparts in very deep optical imaging and are referred to as ``optically dark ALMA galaxies\" \\citep{Williams19,Zhou20}. These galaxies show high far-infrared luminosities and are characterized as dusty star-forming galaxies at high redshift ($z>2$). Such galaxies are typically relatively massive, with stellar masses of 10$^{10.2-11.5}$ M$_{\\odot}$\\ and contain a substantial amount of molecular gas. If the emission in 3C 318 is associated with such a galaxy, then the associated dust continuum emission would have been easily detectable in the ALMA band 7 observations.\\\\ \n\nWe divide the molecular gas mass by the area of the emitting region using the effective radius as the scale size. Using the Milky Way's hydrogen column density - $V$-band extinction relationship \\citep{Guver09} we convert the column density into a $V$-band extinction value for the molecular gas traced by CO. We measure an $\\rm A_{v}$ value in the CO gas of 1-4 magnitudes, which is a factor of 100 lower than those found in the dusty star-forming galaxies. Furthermore, we use the molecular gas mass of individual clumps and convert them into expected dust continuum emission in band 7 based on the relationship between ISM mass and dust emission of \\citet{Scoville17}. The expected continuum flux density is 0.017 mJy\/beam, which would be undetected in the band 7 observations with a sensitivity of 0.0297 mJy\/beam. We assume the clumps have the same surface brightness profile in both bands and in the dust and molecular gas emissions. Based on these calculations, the clumps are unlikely to be associated with typical dust-obscured galaxies at this redshift. Band 4 continuum is not optimal for detecting the dust continuum at this redshift since the dust emission is expected to be fainter at the longer wavelength of the Rayleigh-Jeans tail. Another possibility is that the emission may be associated with a tidal tail feature. However the FWHM of 534 \\kms\\ and an offset of -936.0 \\kms\\ relative to the redshift of the quasar are both larger than what would be expected for a tidal feature. Based on morphological identification of tidal tail from HST imaging and Keck\/OSIRIS observations, in two other systems, we found that the velocity dispersion in both the ionized and molecular gas mass is $\\sim$ 150 \\kms\\ with velocity offsets of -250 \\kms\\ \\citep{Vayner19b}. \\\\\n\nCombining rest-frame optical and sub-mm observations, we find that the ionized and molecular gas outflows in 3C 318 show different morphologies, spatial extent, and kinematics. The molecular gas outflow is far more extended, with a maximum distance of 21 kpc from the quasar, while the ionized outflow shows a maximum extent of only 3.2 kpc. The molecular outflow is also faster moving with blueshifted velocities up to -1200 \\kms. In contrast, the ionized gas outflow has a velocity of 703 \\kms\\ with a bi-conal morphology that is both blue- and red-shifted relative to the quasar in the SW and NE directions. We find no evidence for CO (3-2) emission at the quasar's systemic redshift associated with narrow emission. We place a limit on the molecular gas reservoir at the quasar's location over an aperture matching the size of the beam of $<0.7\\times10^{9}$ M$_{\\odot}$\\ at 2$\\sigma$ confidence.\\\\\n\nThe extent of the molecular outflow is similar to the cold gas outflow detected in $z=6.4$ quasar SDSS J1148+5251, through the 158 \\micron\\ [C II] emission line \\citep{Cicone15}. The morphology and kinematics of the molecular outflow in 3C 318 are also similar to the outflow recently detected in zC400528 \\citep{Herrera-Camus19} through CO (3-2) observations, where they see an extended emission entirely redshifted from the galaxy with a relatively collimated morphology similar to the case of 3C 318. The clumpy morphology and high velocity of the outflowing molecular gas are also similar to recent molecular outflows detected in PDS 456 \\citep{Bischetti19} and in the lensed quasar HS 0810+2554 \\citep{Chartas20}.\n\n\n\\begin{figure*}[!th]\n    \\centering\n    \\includegraphics[width=7.0in]{3C318_spec.eps}\n    \\caption{ALMA band 4 observations of 3C 318. On the left we show optimally extracted intensity map of the molecular outflow in the 3C 318 system, detected in the CO (3-2) line. The white contours outline the molecular outflow region. On the right we show a spectrum integrated over the entire molecular outflow region shown in the white contours, along with fit to the CO (3-2) emission line. The systemic redshift of the quasar host galaxy is at 0 \\kms. The ellipse in the lower left corner on the right panel shows the beam of our ALMA band 4 observations. We detect no molecular CO (3-2) emission at the systemic redshift.}\n    \\label{fig:3C318_spec}\n\\end{figure*}\n\n\n\\subsection{4C 09.17 (z=2.117)}\\label{sec:4c0917}\n\n4C 09.17 is a radio-loud quasar at $z=2.117$. A one-sided jet extends towards the southwest, with a bright core emission associated with the quasar's optical emission location. The system is also bright at far-infrared wavelength \\citep{Podigachoski15}. \\\\\n\nWith Keck\/OSIRIS we detected ionized gas emission in nebular emission lines [O{\\sc III}]\\xspace\\ 5007 \\AA, H$\\alpha$\\xspace, and [N{\\sc II}]\\xspace\\ 6585 \\AA\\ \\citep{Vayner19b}. We found an ionized gas outflow extending towards the east with a maximum extent of 6 kpc.\\\\ \n\nWe detect broad, blueshifted emission resembling a molecular gas outflow in the host galaxy of the radio-loud (RL) quasar 4C 09.17. From here on, we refer to this object as 4C 09.17 A - RL. We also detect a very broad component in the merging radio-quiet (RQ) galaxy towards the northeast; from here on, we refer to this object as 4C 09.17 B - RQ. This galaxy is also detected in \\textit{K}-band imaging of \\citet{Armus97}, with a red optical to near-IR continuum color. In 4C 09.17 B - RQ, we detect very faint narrow [O{\\sc III}]\\xspace\\ emission in Keck\/OSIRIS, the ionized outflow is undetected. 4C 09.17 B - RQ also contains a narrower emission line component in CO (4-3) at a similar redshift to the narrow [O{\\sc III}]\\xspace\\ emission, which we use to calculate its redshift. The velocity offset between the 4C 09.17 A - RL and 4C 09.17 B - RQ is -593 \\kms. The majority of the narrow CO emission-line flux is found concentrated in 4C 09.17 B - RQ, within a 1 kpc radius region. The majority of the dust continuum detected at far-infrared wavelengths with the \\textit{Herschel Space Telescope} is likely associated with this galaxy. 4C 09.17B - RQ is highly obscured; the narrow CO emission component yields a line integrated gas column density of 3.4$\\rm \\times10^{24}~cm^{-2}$ computed by dividing the molecular gas mass by area of the emitting region. Using the Milky Way's hydrogen column density - $V$-band extinction relationship \\citep{Guver09}, we find a $V$-band extinction of 150 mag. The narrow CO emission is likely at the center of the merging galaxy because it roughly corresponds to the $K$-band continuum's peak location.\\\\\n\nFor the galaxies detected to the southwest and northwest of the quasar in \\citet{Armus97} and \\citet{Lehnert99} we detect narrow CO (4-3) emission near their optical locations. We do not detect any high velocity or broad molecular gas associated with these two systems. The detection of 3 galaxies found within 20 kpc of the quasar host galaxy from both ALMA and Keck\/OSIRIS observations makes the 4C 09.17 system likely a proto-group environment at z $\\sim 2.11$. \\\\\n\nWe find that the molecular gas outflow in 4C 09.17 A -RL is more compact than the ionized gas outflow. The ionized and molecular gas outflow show similar blueshifted velocities and velocity dispersion. The maximum extent of the molecular gas outflow is 2.8 kpc, while the ionized outflow extends to 6 kpc. We find that both the ionized and molecular outflow in this system are not along the path of the radio jet, but extend in the same eastern direction. Similar results have been found for a subset of nearby galaxies recently, where outflows appear to expand perpendicular to the path of the jet \\citep{Venturi21}. \\\\\n\nIn 4C 09.17 B - RQ, the molecular outflow extends 4.9 kpc from the narrow CO emission line component. The extent of the molecular outflow in 4C 09.17 B - RQ roughly matches the maximum extent of the $K$-band stellar continuum; hence the molecular outflow is occurring on galactic scales in this galaxy. Extinction within the outflowing gas can potentially prevent ionization by quasar photons and prevent the observer from detecting recombination photons. Spectra of the distinct regions detected in this system are shown in Figure \\ref{fig:4C0917_spec}.  \\\\\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=6.5 in]{4C0917_spec.eps}\n    \\caption{ALMA band 4 observations of 4C 09.17. On the left we show optimally extracted intensity map of CO emission in the 4C 09.17 system. The 4C 09.17 is a merger system with molecular outflows detected in both galaxies. The teal contours outline the molecular outflow in the radio-quiet galaxies 4C 09.17 B, while the purple contours outline the molecular outflow detected in the host galaxy of the radio loud quasar 4C 09.17. On the right we show the spectra extracted over the respective outflow regions along with the Gaussian fit model. Dashed lines represent the individual Gaussian components of the emission line fit, while the solid black line represents the sum of all components and a 0th order polynomial fit to any residual continuum. Component 1 (C1) in 4C 09.17 B is the fit to the narrow emission at the systemic redshift of the merging galaxy, that has a velocity offset of about 593 \\kms\\ relative to 4C 09.17A, while C2 is gas in the outflow. For 4C 09.17 A, C1 corresponds to the outflow gas while C2 is the narrow gas at the systemic redshift of the quasar host galaxy. The systemic redshift of the quasar host galaxy is at 0 \\kms. The ellipse in the lower left corner on the right panel shows the beam of our ALMA band 4 observations.}\n    \\label{fig:4C0917_spec}\n\\end{figure*}\n\\subsection{4C 22.44 (z=1.5492)}\\label{sec:4c2244}\n\n\n4C 22.44 is a radio-loud quasar at z=1.5492. The radio jet in this system extends along the east-west direction with a length of 7\\arcsec\\ and a bright core emission associated with the quasar's optical emission location. With Keck\/OSIRIS we detected extended ionized gas in the [O{\\sc III}]\\xspace, H$\\alpha$\\xspace, and the [N{\\sc II}]\\xspace emission lines with an extent of 2 kpc. An ionized gas outflow is detected on a spatial scale of $<1$ kpc. We detect no emission in the CO (3-2) line with ALMA. We place a limit on the CO (3-2) line luminosity of 0.08 Jy \\kms\\ for an aperture matching the beam with a line velocity dispersion of 250 \\kms, which converts to a molecular gas mass limit of $<$ 1$\\times10^{9}$ M$_{\\odot}$.\n\n\\subsection{4C 05.84 (z=2.323)}\\label{sec:4c0584}\n\n4C 05.84 is a radio-loud quasar at z=2.323. The one-sided radio jet in this system extends towards the southwest with a maximum extent of 12 kpc. There is a bright radio core component associated with the quasar's optical emission location.\n\nWith Keck\/OSIRIS observations, we detected extended ionized gas on an 8 kpc scale in the [O{\\sc III}]\\xspace, H$\\alpha$\\xspace, and [N{\\sc II}]\\xspace emission lines \\citep{Vayner19a}. We detected a bi-conical ionized outflow extending along the northeast and southwest direction.\n\nIn our ALMA band 4 observations, we detect extended CO (4-3) emission that is offset in the western direction consisting of multiple clumps. The individual clump components and the spatially integrated emission is highly blueshifted relative to the quasar's systemic redshift. This emission is not associated with any known galactic component. The spectrum and integrated intensity map of the detected CO emission are shown in Figure \\ref{fig:4C0584_spec}. \n\nThe ionized and molecular gas outflows in this system are on similar scales. The ionized outflow extending towards the southwest direction shows a similar blueshifted velocity offset as the molecular outflow. The ionized outflow detected on spatial a scale $<1$ kpc is similarly blueshifted to the more extended ionized and molecular outflow. The ionized outflow appears to be more turbulent with a larger velocity dispersion than the molecular outflow. We find no evidence for CO (4-3) emission at the quasar's systemic redshift associated with narrow emission. We place a limit on the molecular gas reservoir mass of $<0.8\\times10^{9}$ M$_{\\odot}$\\ at the quasar's location with an aperture the size of the beam and using a velocity dispersion of 250 \\kms.\n\n\n\\begin{figure*}[!th]\n    \\centering\n    \\includegraphics[width=7.5 in]{4C0584_spec.eps}\n    \\caption{ALMA band 4 observations of 4C 05.84. On the left we show optimally extracted intensity map of the molecular outflow in the 4C 05.84 system, detected in the CO (4-3) line. The white contours outline the molecular outflow region. On the right we show a spectrum integrated over the entire molecular outflow region, along with fit to the CO (4-3) emission line. The systemic redshift of the quasar host galaxy is at 0 \\kms. The ellipse in the lower left corner on the right panel shows the beam of our ALMA band 4 observations. We detect no molecular CO (4-3) emission at the systemic redshift.}\n    \\label{fig:4C0584_spec}\n\\end{figure*}\n\n\\subsection{3C 298 (z=1.439)}\\label{sec:3C298}\n\n3C 298 is a radio-loud quasar at z=1.439. The jets in the system extend in the east-west direction, with a length of about 16 kpc and a bright core emission associated with the quasar's optical emission location. The system is bright at far-infrared wavelengths. ALMA band 7 observations have revealed that the dust emission mostly comes from a kpc scale region centered on the quasar \\citep{Barthel18}.\n\nWith Keck\/OSIRIS, we detected extended ionized gas emission \\citep{Vayner19b} on scales up to 20 kpc from the quasar. A bi-conical ionized outflow is detected with a maximum extent of 3 kpc from the quasar along the jet's path. The redshifted cone is associated with the western jet component, while the blueshifted cone is associated with the eastern jet component. We also detected an ionized gas outflow in the merging system 8 kpc from the quasar.\n\nIn the ALMA band 4 observations, we detect a molecular gas emission in two distinct components, one centered on the quasar with a radius of 2.1 kpc and a second component 21 kpc from the quasar, offset by -250 \\kms. The component near the quasar shows both broad and narrow emission. The narrow emission is associated with a galactic disk \\citep{Vayner17}, while the broad component is associated with the outflow. The outflow emanates from the molecular disk centered on the quasar with a maximum extent of 1.6 kpc. The majority of the molecular gas in the outflow is on the blueshifted side of the disk and extends in the direction of the jet's western component. The ionized outflow is more extended than the molecular outflow, has a faster velocity, and appears to be more turbulent with a larger velocity dispersion. \n\n\n\n\\begin{figure*}[!th]\n    \\centering\n    \\includegraphics[width=7.0 in]{3C298_spec.eps}\n    \\caption{ALMA band 4 observations of 3C 298. On the left we show optimally extracted intensity map of CO emission in the 3C 298 system, detected in our 2017 study of this object \\citep{Vayner17}. The purple contours outline the molecular outflow region, white contours outline total emission from the molecular disk, while the teal contours outline a star forming\/tidal tail feature. On the right we show a spectrum integrated over each distinct region along with fit to the CO (3-2) emission line. Dashed lines represent the individual Gaussian components of the emission line fit, while the solid black line represents the sum of all components and a 0th order polynomial fit to any residual continuum. The systemic redshift of the quasar host galaxy is at 0 \\kms. The ellipse in the lower left corner on the right panel represents the beam of the ALMA band 4 observations.}\n    \\label{fig:3C298_spec}\n\\end{figure*}\n\n\n\\subsection{7C 1354+2552 (z=2.0064)}\\label{sec:7C1354}\n\n7C 1354+2552 is a radio-loud quasar at z=2.0064. The system contains two jets that are perpendicular to each other. The east-west jet has a length of about 24 kpc, while the north-south jet has a length of 86 kpc. Only the east-west jet is detected in the continuum in our ALMA observations due to limited sensitivity to low-surface brightness emission on scales $>$ 6\\arcsec.\n\nUsing the Keck\/OSIRIS observations, we detected extended emission in the nebular emission lines [O{\\sc III}]\\xspace and H$\\alpha$\\xspace on scales of 4-6 kpc. An ionized outflow is detected on a spatial scale of $<1$ kpc. \n\nIn the ALMA band 4 observations, we detect molecular gas emission towards the northeast. The narrow emission line resides near the quasar's systemic redshift. We do not detect any broad emission from a turbulent molecular gas outflow. We detect no highly offset emission consistent with our outflow criteria; hence there is no evidence for a cold molecular gas outflow in this system at our observations' sensitivity. Spectra of the distinct region detected in this system is shown in Figure \\ref{fig:7C1354_spec}. The detected emission may be associated with a merging galaxy. With the large (8 kpc) separation from the quasar and the fact that the motion of the gas does not appear to follow the kinematics of the galactic disk on the eastern side \\citep{Vayner19b}, there is a possibility that this emission is associated with a merging galaxy.\n\n\n\\begin{figure*}[!th]\n    \\centering\n    \\includegraphics[width=7.0 in]{7C1354_spec.eps}\n    \\caption{ALMA band 4 observations of 7C 1354+2552. On the left we show optimally extracted intensity map of the molecular gas in the 7C 1354+2552 system, detected in the CO (4-3) line. The gas is narrow and is consistent with gravitational motion. Yellow contours outline the CO emitting region towards the north-west that is slightly redshifted, the spectrum associated with it is shown on right. The systemic redshift of the quasar host galaxy is at 0 \\kms. The ellipse in the lower left corner on the right panel represents the beam of the ALMA band 4 observations.}\n    \\label{fig:7C1354_spec}\n\\end{figure*}\n\n\n\\section{Discussion}\\label{sec:discussion}\n\n\\subsection{What is driving the outflows?}\\label{sec:driving}\nSeveral powerful mechanisms can drive galactic-scale outflows. Quasars can reside in galaxies with powerful star formation activity \\citep{Duras17,Aird19,Circosta21,Bischetti21}, especially at redshifts near the peak of the star formation activity. Star formation can result in powerful galactic winds \\citep{Rupke18}. Radio-loud quasars that are optically luminous can drive galactic outflows by both jets \\citep{Wagner12,Mukherjee16} and radiation pressure \\citep{Murray95,Faucher12a,Zubovas12,Costa18} from the accretion disk. In this section, we explore primary mechanisms that are capable of driving the multi-phase galactic outflows. We combine the momentum flux and kinetic luminosities from the cold molecular and ionized gas phases to look at both the total impact of galactic winds on the quasar host galaxies and what mechanism may be responsible for driving the entirety of the outflowing gas in each system.\n\nTo understand the main driving mechanism of galactic outflows, we compare the momentum flux of the outflow ($\\dot{P}_{outflow}$ ) to the momentum input from the quasar accretion disk ($\\dot{P}_{Quasar}$ ) and to the momentum deposition from stellar feedback ($\\dot{P}_{SNe}$).\n\n\\subsection{Stellar feedback as potential driver of galactic outflows}\nTo explore whether star formation activity can drive the galactic scale outflows, we compare the momentum deposition from supernovae explosions, based on the star formation rate, to the energetics of the multi-phase gas outflow. To compute the energy deposition from supernovae, we use the results of recent simulations by \\citet{2015MNRAS.450..504M} and \\citet{2015ApJ...802...99K}, that predict the momentum deposition per unit of solar mass formed. We assume that one supernova explodes for every 100 M$_{\\odot}$\\ of stars. We assume an electron density of 100 cm$^{-3}$  in the interstellar medium with solar metallicity.\n\n\n\\begin{equation}\n    \\dot{P}_{SNe} = 7.5\\times 10^{33} \\frac{\\dot{M}_{SFR}}{1M_{\\odot}{\\rm yr}^{-1}} \\left(\\frac{n}{100\\,{\\rm cm}^{3}}\\right)^{-0.18}\n    \\left(\\frac{Z}{Z_\\odot}\\right)\n    \\rm dyne \\label{eq:SNe_mom}.\n\\end{equation} \n\nFor sources with resolved rest-frame far-infrared observations (3C 318 and 3C 298), we use the far-infrared luminosity and convert that into a star formation rate based on work by \\citet{Barthel18,Barthel19}. The total-infrared (8-1000 \\micron) derived star formation rates of 3C 298 and 3C 318 are 930\\myr\\ and 580 \\myr\\ derived using the \\citet{Kennicutt98} calibration \\citep{Podigachoski15}. Infrared-derived star formation rates have high uncertainties since it is not clear what fraction of the far-infrared emission comes from dust heating from massive stars and quasar heating on kpc scales \\citep{Symeonidis17,Symeonidis21}. In radio-loud objects synchrotron emission from the jets and the core of the quasar can also contribute to the far-infrared and mm-emission. In \\citet{Podigachoski15} correction for synchrotron emission was only applied to the 850 \\micron\\ flux of a handful of objects, including 3C 298 that is part of our study. We therefore use the infrared-derived star-formation rates as an upper limit on the actual star formation rates. The total infrared star formation rates are presented in Table \\ref{tab:SFR-rates}. \\\\\n\nAnother tracer of recent star formation comes from recombination lines of hydrogen. In Table \\ref{tab:SFR-rates} we present the star formation rates derived from the H$\\alpha$\\xspace line luminosity using the \\citep{Kennicutt98} calibration. The H$\\alpha$\\xspace derived star formation rates are derived from integrated H$\\alpha$\\xspace emission at the systemic redshift of the quasar host galaxy. The H$\\alpha$\\xspace and total infrared derived star formation rates show a significant discrepancy, with the far-infrared star formation rates being almost an order of magnitude higher. The discrepancy between these inferred star formation rates may be due to several reasons; dust extinction, different tracers of the star formation history, contamination of the far-infrared emission from quasar processes, or in addition a large difference between the resolution of the \\textit{Herschel Space Telescope} and Keck\/OSIRIS observations. Indeed, the far-infrared derived star formation rate in 4C 09.17 can be attributed to several galaxies falling within \\textit{Herschel Space Telescope} PSF leading to contamination of the far-infrared emission. The merging galaxy 4C 09.17 B-RQ contains higher molecular gas surface density, hence the larger fraction of the star formation rate among the galaxies. Contribution from nearby galaxies can also affect the far-infrared derived flux for 3C 298 and 3C 318, but we do not see such a strong over-density compared to 4C 09.17. Recent observations with ALMA of a high redshift ($z=4.4$) and luminous quasar have revealed multiple sources falling within 17 kpc from the quasar \\citep{Bischetti18}, which all contribute to the far-infrared emission detected with \\textit{Herschel Space Telescope} for this system. \n\nWe find that stellar processes can deposit a momentum flux of 7.5 - 10,000 $\\times10^{33}$ dyne, comparable to momentum fluxes of the multi-phase outflows. However, since these estimates rely on the maximum momentum flux from SNe and the maximum star formation rates, it is unlikely that star formation alone drives the outflows in these systems.\n\n\\begin{deluxetable*}{cccc}\n\\tiny\n\\tablecaption{Star formation rates based on H$\\alpha$\\xspace emission line \\citep{Vayner19b},infrared observations \\citep{Podigachoski15} and expected star formation rate using the molecular gas surface density based on the KS law. \\label{tab:SFR-rates}}\n\\tablehead{\n\\colhead{Name} & \n\\colhead{SFR [H$\\alpha$\\xspace]}&\n\\colhead{SFR [Total IR]}&\n\\colhead{SFR [KS]}\\\\\n\\colhead{} & \n\\colhead{\\myr}&\n\\colhead{\\myr}&\n\\colhead{\\myr}}\n\\startdata\n4C 09.17 A RL &  9$\\pm$1 & 1330 \\tablenotemark{a}  & 6  \\\\\n7C 1354+2552 & \t 29$\\pm$3 & -- & 1 \\\\\n3C 298 &  22$\\pm$2 & 930 & 24 \\\\\n3C 318 &  88$\\pm$9 & 580 & - \\\\\n4C 22.44 &  32$\\pm$3 & -- & - \\\\\n4C 05.84 &  11$\\pm$1 & $<$540 & - \\\\\n\\enddata\n\\tablenotetext{a}{Star formation rate likely contaminated by a merging galaxy}\n\\end{deluxetable*}\n\n\\subsection{Quasar as potential driver of galactic outflows}\nThe photon momentum fluxes of $10^{35-37}$ dyne and bolometric luminosities of $10^{46-47}$ erg s$^{-1}$ \\ for these quasars indicate that they have sufficient energy and momentum to drive the observed multi-phase gas outflows. To test which quasar mechanism is the primary driver of the observed galactic-scale outflows, we compare $\\dot{P}_{outflow}$  to $\\dot{P}_{Quasar}$  and the location and extent of the quasar jets. High ($>2$) $\\frac{\\dot{P}_{outflow}}{\\dot{P}_{Quasar}}$\\ on scales $>$ 1 kpc are generally attributed to energy-conserving outflows, where a radiatively-driven nuclear wind \\citep{Faucher12a} or a jet \\citep{Mukherjee16,Wagner12} drives a hot shock (T$>10^{7}$ K) in the interstellar medium that does not cool efficiently, maintaining nearly all of the kinetic energy provided to it. The swept-up material is shocked and is able to cool to produce a multi-phase outflow medium \\citep{Faucher12,Faucher12a,Zubovas12}, and may explain the presence of molecular gas moving at fast outflow velocities. To decipher whether the galactic-scale wind is ultimately powered by a jet or by radiation, we need to compare the location and extent of the quasar jet to the outflow and to search for evidence of a fast, radiatively driven wind in X-ray and UV spectrum of the quasar.\n\nHigh ($>2$) $\\frac{\\dot{P}_{outflow}}{\\dot{P}_{Quasar}}$ on scales $<$ 1 kpc can be attributed to outflows driven by radiation pressure in a high column density environment, where the ``momentum boost\" is provided by photons scattering multiple times off dust grains as they are driving the outflow \\citep{Thompson15}. Detecting and resolving the hot shocked gas produced by the jet or the radiatively-driven nuclear wind would be helpful in understanding what is driving the outflow. The shocked hot gas can be detected through the Sunyaev\u2013Zeldovich effect \\citep{Chatterjee07} and has recently been found in the host galaxy of a luminous quasar at z=1.71 \\citep{Lacy19}. Additionally it was detected by stacking analysis of luminous quasars from the Atacama Cosmology Telescope, \\textit{Herschel Space Telescope} and VLA observations \\citep{Hall19}.\n\nLow ($\\ll1$) $\\frac{\\dot{P}_{outflow}}{\\dot{P}_{Quasar}}$\\ on scales $>$ 1 kpc can be attributed to outflows driven by a radiative shock produced where the shocked gas can cool efficiently, and kinetic energy is radiated away. Outflows driven through radiation pressure in a low column density environment can also produce an observed $\\frac{\\dot{P}_{outflow}}{\\dot{P}_{Quasar}}$ $\\ll$1. \n\nTo compare the energetics of winds of our sample with other massive galaxies with AGN, we collated data on molecular and ionized outflows in galaxies at $1<z<3$. We recomputed the ionized outflow rates, luminosities, and momentum fluxes using a consistent prescription; details can be found in \\citet{Vayner19a}. To differentiate AGN outflow mechanisms we compare the momentum flux of the outflow against the photon momentum flux from the accretion disk, as well as and the kinetic luminosity of the outflow against the bolometric luminosity of the quasar (Figure \\ref{fig:energetics}). On Figure \\ref{fig:energetics} we overlay theoretical predictions of the expected $\\frac{\\dot{P}_{outflow}}{\\dot{P}_{Quasar}}$ value based on energy-conserving outflows, radiative shock driving, and radiation pressure \\citep{Faucher12,Zubovas12,Zubovas14}. Minimum coupling efficiency between the quasar bolometric luminosity and outflow kinetic luminosity are prescribed in theoretical simulations when outflows directly impact the star-forming properties of the host galaxy. Based on theoretical models, minimum coupling efficiencies are presented in Figure \\ref{fig:energetics} \\citep{Hopkins10,Choi12,Costa18}. We present the molecular outflows and the total energetics from the multi-phase outflow in Figure \\ref{fig:energetics}. After combining the ionized and molecular outflows' energetics, the outflows exhibit a $\\frac{\\dot{P}_{outflow}}{\\dot{P}_{Quasar}}$ $>1$ on kpc scales. An energy-conserving shock is responsible for driving the outflow for 3C 318, 4C 09.17 A, and 3C 298. For 4C 05.84, it is still possible for an energy-conserving shock to drive the outflow; however, radiation pressure or a radiative shock model is still able to explain the observed $\\frac{\\dot{P}_{outflow}}{\\dot{P}_{Quasar}}$. We overlay the location of the ionized and molecular outflows in Figure \\ref{fig:CO_ionized_outflows}, to highlight the extent and morphologies of the multi-phase gas outflows.\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=7.0 in]{energetics_comparison.eps}\n    \\caption{A comparison of the momentum flux (left) and kinetic luminosity (right) of the outflow to the momentum flux of the accretion disk and the quasar's bolometric luminosity. On the left, we plot lines of constant $\\frac{\\dot{P}_{outflow}}{\\dot{P}_{Quasar}}$, points above the 2:1 line represent outflows that are likely driven by an energy-conserving shock on kpc scale or radiation pressure on small ($<1$ kpc) scales. On the right, we plot lines of constant coupling efficiency between the outflow's kinetic luminosity and the quasar's bolometric luminosity. Blue circles and squares represent molecular outflows detected through CO emission and OH absorption, respectively. Red circles represent ionized outflows at cosmic noon, recomputed in the exact same manner in \\citep{Vayner19a}. Black stars are the ionized outflows of the parent sample for this study. Blue stars are molecular outflows derived in this study through CO emission, and orange stars are the energetics from the total (molecular + ionized) momentum flux and kinetic luminosity. Combining the energetics of the multi-phase outflows indicates that they are likely driven by an energy-conserving shock and have coupling efficiencies between 0.1-1\\%.}\n    \\label{fig:energetics}\n\\end{figure*}\n\n\\subsection{The nature of the multi-phase outflows}\n\nFor 3C 318, 4C 05.84, and 4C 09.17 A, compared to the ionized outflowing gas the molecular gas shows a smaller velocity range single component that spans only blueshifted emission from -200 to -1200 \\kms. The morphology of the outflowing molecular gas is much clumpier and more confined compared to the ionized outflows. The ionized outflows span a broader range of velocities and fill a larger volume likely residing in a wide-angle cone. The high velocity and clumpy molecular outflows are unexpected, appearing to be out of pressure equilibrium with their surroundings at the larger separations observed in 3C 318 and 4C 05.84. Recently high velocity and clumpy outflows have also been detected in one low redshift radio-quiet quasar and a higher redshift lensed quasar, appearing to have similar velocity offset, dispersion, and morphology to 3C 318, 4C 05.84 and 4C09.17A \\citep{Bischetti19,Chartas20}. There is no consensus on the observed spatial morphology and velocity structure of high redshift molecular outflows in luminous quasars due to the small number of detections. The velocity dispersions and offsets for 3C 318, 4C 05.84, and 4C 09.17 A are consistent with molecular outflows detected through OH absorption \\citep{Veilleux13}, which have been hypothesized to have a thin shell geometry. For 3C 298 and 4C 09.17 B, the molecular outflows show a broader velocity range, which likely indicates that these molecular outflows are more volume filling, similar to the ionized outflows. In case these outflows are indeed filled cones, then their outflow rates and energetics would scale by a factor of 3. We are unable to fully rule out a shell geometry for these outflows. It would require higher angular resolution observations to distinguish if they reside in a shell or in a filled cone\/sphere. The velocities of the molecular and ionized outflows are consistent with recent theoretical predictions by \\citet{Richings20} for multi-phase gas outflows driven by a quasar. \\\\\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=7.0in]{all_sources_ALMA_OSIRIS_overlay.eps}\n    \\caption{We present the comparison between the ionized outflow and the molecular gas distribution, extent, and morphology in sources where we detect a multi-phase gas outflow. In the background, we show the line integrated CO intensity tracing the cold molecular gas. Violet contours outline the location of the molecular outflows in the quasar host galaxies, the teal contour in the 4C09.17 system represent the molecular outflow in the merging galaxy. The white contours show the ionized outflow traced through the [O{\\sc III}]\\xspace\\ 5007 \\AA\\ emission line for all systems except 4C 05.84 where it is traced with H$\\alpha$\\xspace. The white star represents the quasar's location, while the bar to the right of each source represents 1 arcsecond.}\n    \\label{fig:CO_ionized_outflows}\n\\end{figure*}\n\nThe energy-conserving shocks responsible for the galactic scale outflows are either powered by radiatively driven outflow or through shock heating of the gas by the radio jet. In 3C 298 and 3C 318, the jet's path is consistent with both the ionized and molecular outflows, while in 4C 05.84, the jet is only consistent with the path of the ionized outflow; however, the size of the jet is still consistent with both the extent of the ionized and molecular gas outflows. In 4C 09.17, the jet is not consistent with either the molecular or ionized outflow path. The lack of correlation between the jet and the galactic outflows does not indicate that the jet could not have driven the outflow. At later stages in their evolution, the hot bubble cocoons shocked by the jet could be relatively spherical and volume filling relative to the thin jet that we observe at radio wavelength \\citep{Wagner12,Mukherjee16}. The lack of path correlation between the multiple phases of the outflow, the general asymmetry, and spatial differences in ionized and molecular outflow can be caused by the non-uniform cooling properties of the galactic outflow, subject to gas and dust extinction and surface brightness sensitivity of our observations. Some nearby galaxies show fast-moving outflows that are not correlated with the axis of the jet \\citep{Venturi21} but are rather found to expand perpendicular to the jet's path. None of the quasars appear to show evidence for a broad absorption line wind, based on their rest-frame UV spectra from SDSS \\citep{Paris18}. The X-ray observations are either missing or are too shallow to search for radiatively-driven nuclear winds in Fe absorption lines. Future space-based X-ray telescopes with larger effective apertures and higher spectral resolving power will be crucial to search for ultra-fast outflows and compare their energetics with the galactic scale winds to decipher whether the jet or quasar disk winds are the primary driver.\n\nAt the current evolutionary stage of the quasar host galaxies in our study, the dominant outflow component, in terms of mass, is the molecular gas. In fact, for sources with both a molecular outflow and a molecular gas reservoir at the systemic redshift, between 20-60\\% of the cold molecular gas reservoir is in the galactic outflow. For sources with no detected molecular reservoir at the systemic redshift, we are catching these systems when the majority of the cold molecular gas is in an outflow. This suggests that we are observing these systems at a phase where the quasar is responsible for removing a significant fraction of the gas in these galaxies, and therefore may be directly responsible for removing the fuel for subsequent star formation, in turn impacting the stellar growth of these galaxies.\n\nFor the galactic outflows in 3C 298 and 4C 05.84, we find that most of the momentum flux is in the ionized gas phase. For 3C 318, the ionized outflow can have higher energetics within the errors that are dominated by the measurement of the electron density. For 4C 09.17 A, the molecular outflow has the higher energetics. In 4C 09.17 B, we do not detect an ionized outflow, likely due to obscuration effects since we measure a high concentration of molecular gas with a high $\\rm A_{V}$ value near the center of galaxy and 4C 09.17 B galaxy also has a red $V-K$ color of $>$5.35. Hence it is unclear how the energetics are distributed between the ionized and molecular gas phase of the outflow. For most of our objects, the distribution in energetics is consistent with recent theoretical predictions by \\citet{Richings20}. The coupling efficiency between the multi-phase outflow and the bolometric luminosity of the quasar is consistent with theoretical predictions by \\citet{Hopkins10,Choi12,Costa18}, where a minimum of 0.1-0.5$\\%$ of the quasar's bolometric luminosity is expected to transfer into the kinetic luminosity of the outflow for there to impact star formation processes.\n\n\\subsection{Missing mass in galactic outflows}\n\nWe do not have any measurement of the neutral atomic gas mass in the outflow. Yet the neutral atomic phase likely exists within each outflow since we observe ionized emission through the optical [O{\\sc III}]\\xspace and H$\\alpha$\\xspace lines. Theoretical work by \\citet{Dempsey18} suggests that in the absence of a substantial amount of neutral gas, the [O{\\sc III}]\\xspace transition would not be present, and most of the gas would be over-ionized (e.g., into [O IV]). At the same time, theoretical work by \\citet{Richings18} has shown that a substantial amount of the molecular gas in an outflow is expected to be in the warmer molecular gas phase, with temperatures on the order of 400 K. The CO transition that trace the molecular gas reservoir within our study are only capable of tracing the cold molecular gas phase at temperatures of 40-70 K. The actual total momentum flux ratios are still likely lower limits, so energy-conserving outflows driven through a quasar mechanism are the most likely scenario for explaining the driving mechanism behind these galactic-scale outflows. Future observations with the MIRI instrument aboard \\textit{JWST} will trace the warm molecular gas phase through the rest-frame mid-infrared rotational transitions of hydrogen. Observations with future 30-meter class telescopes will be able to probe the neutral gas phase through the Na D, [OI] 6300 \\AA\\ lines. Furthermore, the Square Kilometer Array (SKA) will enable us to probe the neutral gas phase directly through the 21 cm hydrogen line. \n\n\\subsection{Molecular gas depletion time scales}\n\nThis section we explore the molecular gas depletion time scale in the galaxies within our sample. The molecular gas depletion time scale is defined as $t_{depletion, SFR} = M_{molecular}\/SFR$. We will also compare the star formation depletion scale to the outflow depletion scale defined in a similar manner; $t_{depletion, outflow} = M_{molecular}\/\\dot{M}_{outflow}$. We compare the depletion time scale of the molecular gas due to star formation and from both the ionized and the molecular gas outflows. In all cases, we find a molecular gas depletion time scale of $<$ 31 Myr, with $t_{depletion, outflow}$ having the shorter timescale. The infrared-derived star formation rates are two orders of magnitude too high to be supported by the current molecular surface gas density, suggesting that the IR emission is likely contaminated by the quasar. While the maximum derived star formation rate from the total-infrared emission can be comparable to the multi-phase gas outflow rates, at present time, star formation rates expected from the Kennicutt-Schmidt (KS) law are two orders of magnitude smaller. The gas depletion timescale due to the KS-derived star formation rates is about two orders of magnitude higher than the depletion time scale due to outflows. Due to the expected low star formation rate from the low molecular gas surface density, in the next few Myr, the depletion time scale will be dominated by the galactic scale outflows. The depletion time scales from each source for the systems in our survey are presented in Table \\ref{tab:depletion}. Not only are we catching these systems when a substantial fraction of the gas is in an outflow state, the rate of molecular gas depletion is dominated by the quasar driven outflows.\n\n\\begin{deluxetable*}{ccccc}\n\\tablecaption{Measured depletion time scales based on the star formation activity and multi-phase outflows in our sample. t$_{depletion,SFR}$ is the depletion time scale of the current molecular reservoir using the highest star formation rate observed. t$_{depletion,outflow}$ is the depletion time scale due to the outflows, and t$_{depletion,KS}$ is the depletion time scale based on the KS law for the observed molecular gas surface density. t$_{depletion,MS}$ is the expected depletion time scale for a galaxy at the measured stellar mass on the galaxy main sequence. \\label{tab:depletion}}\n\n\n\n\\tablehead{\n\\colhead{Source}&\n\\colhead{t$_{depletion,SFR}$}&\n\\colhead{t$_{depletion,outflow}$}&\n\\colhead{t$_{depletion,KS}$} &\n\\colhead{t$_{depletion,MS}$}\\\\\n\\colhead{}&\n\\colhead{Myr}&\n\\colhead{Myr}&\n\\colhead{Myr}&\n\\colhead{Myr}}\n\\startdata\n4C 09.17 A RL &  & 6$\\pm$1 & 500$\\pm$50 & 700 \\\\\n4C 09.17 B RQ &  & 4$\\pm$1 & 178$\\pm$18 & 700 \\\\\n7C 1354 & 10$\\pm$4 & 6$\\pm$3 & 300$\\pm$100& 700 \\\\\n3C 298 & 7$\\pm$1 & 4$\\pm$1 & 275$\\pm$40 & 700 \\\\\n3C 318 & $<$ 1 & $<$ 1.7$\\pm$0.6 & & 800 \\\\\n4C 22.44 & $<$ 31$\\pm$3 & $<$3$\\pm$2 & & 800 \\\\\n4C 05.84 & $<$1.5 & $<$0.8$\\pm$0.5 & & 700 \\\\\n\\enddata\n\\end{deluxetable*}\n\nIn the last several years, there have been many molecular gas observations in massive galaxies near cosmic noon. We can compare the depletion time scales that we observe within our systems to those that do not have powerful quasars. We compare the expected depletion time scale due to star formation for an average galaxy in our stellar mass range to the observed depletion time scale due to the outflows. \\citet{Tacconi18} measured the depletion time scale for galaxies at an extensive range of redshifts and stellar masses. The dynamical mass \\citep{Vayner19b} of the galaxies within our sample range from $10^{10.5-11.5}$ M$_{\\odot}$, with star formation rates of 30-1330 \\myr. This places them on the massive end of the galaxy luminosity function and the galaxy main sequence (MS) at $z=2$ that relates the star formation rate of a galaxy to its stellar mass. For galaxies with a mass in the range of $10^{10.5-11.5}$ M$_{\\odot}$, the expected depletion time scale is around 700-800 Myr for galaxies on the star-formation main sequence at $z\\sim2$. The range in the molecular gas depletion time scale for galaxies in the mass range of $10^{10.5-11.5}$ M$_{\\odot}$\\ is $0.3-3.5$ Gyr. Our systems appear to show a depletion time scale of the molecular reservoir that is least 100 times faster, indicating that powerful quasars can play a role in having shorter depletion times compared to star formation in a typical massive galaxy. While the H$\\alpha$\\xspace derived star formation rates place the sample quasar hosts on or below the star forming main sequence at z$\\sim$2, the much larger FIR-derived star formation rates place them well above the MS.  We therefore estimate a range of depletion times in Table \\ref{tab:depletion} covering the range of estimated star formation rates for each source.\n\nThe rapid depletion time scales of the cold molecular gas reservoir and the present amount of molecular gas has a profound implication for the evolution of these galaxies from cosmic noon to the present day. From the Keck\/OSIRIS observations, we learned that these galaxies are offset from the local $M_{\\bullet}-\\sigma~$ and $M_{\\bullet}-M_{*}~$ relationships \\citep{Vayner19b}, indicating that the galaxies are under-massive for the mass of their SMBH. We estimated that they require a continuous stellar mass growth of approximately 100 \\myr\\ between z=2 and z=0 to land onto the local scaling relations. For each system, we have computed the molecular gas at the systemic redshift of the quasar within the radius where the dynamical masses are calculated in \\cite{Vayner19b}. We find molecular gas mass in the range of $0.3-10\\times10^{9}$ M$_{\\odot}$, while in systems without detection of CO emission at the systemic redshift, we have placed limits of $<1\\times10^{9}$ M$_{\\odot}$. None of the systems have the molecular gas mass necessary to increase their stellar mass to bring them close to the local scaling relations between the SMBH mass and the stellar mass of the galaxy.\n\nFurthermore, the majority of the molecular gas is in the galactic outflows rather than in the host galaxy; hence the overall stellar mass is still not going to increase by a substantial amount through star-formation. This study further indicates that a large new fresh reservoir of gas is necessary to accrete from the circumgalactic medium to replenish the fuel necessary for future star formation, and\/or a large number of mergers is expected between z=2 and z=0 \\citep{Burke13,Cooke19}. Our results further indicate that strong quasar feedback occurs before galaxies assemble onto the local scaling relations.\n\n\n\\section{Conclusions}\\label{sec:conc}\n\nWe have conducted a study of the molecular gas properties in 6 radio-loud quasar host galaxies at $1.4<z<2.3$ through ALMA band 4 observation of the CO (3-2) and CO (4-3) rotational transitions at sub-arcsecond resolution. The survey aimed to study the molecular gas morphologies and kinematics to address the energetics and evolution of these massive galaxies. We resolve the molecular gas reservoirs in 5 galaxies and detect molecular outflows in 4 systems. Coupling these observations with high spatial resolution observations with Keck\/OSIRIS we are able to study the ionized gas and molecular gas to gauge the impact of quasar activity on the star formation properties of the host galaxies.\n\n\\begin{itemize}\n    \\item We detect CO emission on kpc scale in 5\/6 quasar host galaxies.\n    \\item We detect high-velocity clumps in 3C 318, 4C 09.17A, and 4C 05.84, offset up to -1200 km\/s while in 3C 298 and 4C 09.17B we detect broad CO emission lines. We interpret these spectral features as molecular outflows. We derive outflow rates of 168-2500 \\myr, combining the energetics of the ionized to the molecular gas outflows, we find that the dominant outflow mechanism is through an energy-conserving shock from the quasar jets.\n    \\item We find that the outflowing gas to be predominantly in the molecular gas phase while the momentum flux and kinetic luminosity resides in the ionized gas phase for 3\/4 multi gas-phase outflows.\n    \\item While the maximum star-formation rate from total infrared luminosity show a similar depletion time scales to the outflows, due to a much lower molecular gas surface densities than expected, the outflow depletion time scales are at least 50 times faster. The depletion time scale compared to the expected value for an average galaxy on the galaxy star formation main sequence at the stellar mass of our survey is about 100 times faster.\n    \\item A substantial accretion of fresh gas from the circumgalactic medium, and\/or stellar growth through dry mergers is necessary for these galaxies to achieve the expected mass based on their central SMBH mass using local scaling relations. The current molecular gas reservoir is insufficient to provide a significant amount of fuel for star formation.\n    \\item In the 4C 09.17 system, we find a molecular gas outflow in its quasar host galaxy, as well as a nearby dust-obscured galaxy, indicating evidence for feedback occurring well before the coalescence phase of this merger system.\n\\end{itemize}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\\label{intro}\n\nThe formation process of low- to intermediate-mass stars is divided into several \nstages, ranging from the deeply embedded protostellar stage to the period when \na young star is dispersing its protoplanetary disk in which planets may have formed.\nDuring the protostellar phase, which is estimated to last $\\sim$ 0.5 Myr \n\\citep{evans09,dunham14}, the growing central source accretes dust and gas \nfrom a collapsing envelope. The material from the envelope is most likely accreted \nthrough a disk, feeding the growing star. A fraction of the mass is ejected in outflows, \nwhich carve openings into the envelope along the outflow axis. \nDespite our understanding of the basic processes operating in low-mass protostars, \nfundamental questions remain \\citep[e.g.,][]{dunham14}. In particular, it is not \nunderstood how the processes of infall, feedback from outflows, disk accretion, \nas well as the surrounding birth environment, affect mass accretion and determine the \nultimate stellar mass. The luminosity of protostars, which can be dominated by \naccretion, is observed to span more than three orders of magnitude, yet the underlying \nphysics of this luminosity range is also not understood \\citep{dunham10,offner11}. \nIt is in this protostellar phase that disks are formed, setting the stage for planet \nformation, yet how infall, feedback, accretion, and environment influence the \nproperties of disks and of planets that eventually form from them is unknown. \nThe large samples of well-characterized protostars identified from surveys with \n{\\it Spitzer} and {\\it Herschel} now provide the means to systematically study the \nprocesses controlling the formation of stars and disks; the goal of this work is to\nprovide such a characterization for the protostars found in the Orion A and B \nclouds, the largest population of protostars for any of the molecular clouds within \n500 pc of the Sun \\citep{kryukova12,dunham13,dunham15}.\n  \nIn protostars, dust in the disk and envelope reprocesses the shorter-wavelength \nradiation emitted by the central protostar and the accretion shock on the stellar surface \nand reemits it prominently at mid- to far-infrared wavelengths. As a result, the combined \nemission of most protostellar systems (consisting of protostar, disk, and envelope) peaks \nin the far-IR. \nYoung, deeply embedded protostars have spectral energy distributions (SEDs) \nwith steeply rising slopes in the infrared, peaking around 100 \\micron, and large \nfractional submillimeter luminosities \\citep[e.g.,][]{enoch09,stutz13}. Near 10 \n\\micron\\ and 18 \\micron, absorption by sub-micron-sized silicate grains causes broad \nabsorption features; in addition, there are several ice absorption features across \nthe infrared spectral range \\citep{boogert08, pontoppidan08}. These absorption \nfeatures are indicative of the amount of material along the line of sight, with the \ndeepest features found for the most embedded objects. In addition, due to the\nasymmetric radiation field, the orientation of a protostellar system to the line of \nsight, whether through a dense disk or a low-density cavity, plays a role in the\nappearance of the SED. It influences the near- to far-IR slope, the depth of the \nsilicate feature, the emission peak, and the fraction of light emitted at the longest \nwavelengths \\citep[see, e.g.,][]{whitney03a}. \n \nTo classify young stellar objects (YSOs) into observational classes, the near- to \nmid-infrared spectral index $n$ ($\\lambda F_{\\lambda} \\propto \\lambda^n$) \nfrom about 2 to 20 $\\mu$m has traditionally been used \\citep{lada87,adams87,\nandre94,evans09,dunham14}. This index is positive for a Class 0\/I protostar, \nbetween $-0.3$ and 0.3 for a flat-spectrum source, and between $-1.6$ and $\n-0.3$ for a Class II pre-main-sequence star. Class 0 protostars are distinguished \nfrom Class I protostars as having $L_{submm}\/L_{bol}$ ratios larger than 0.5\\%, \naccording to the original definition by \\citet{andre93}. Other values for this threshold \nthat have recently been used are 1\\% \\citep{sadavoy14} and even 3\\% \\citep{maury11}.\nAnother measure for the evolution of a young star is the bolometric temperature \n($T_{bol}$), which is the temperature of a blackbody with the same flux-weighted \nmean frequency as the observed SED \\citep{myers93}. A Class 0 protostar has \n$T_{bol}<70$ K, a Class I protostar 70 K $<T_{bol}<$ 650 K, and a Class II \npre-main-sequence star 650 K $<T_{bol}<$ 2800 K \\citep{chen95}. \nThese observational classes are inferred to reflect evolutionary stages, with the \ninclination angle to the line of sight being the major source of uncertainty in \ntranslating classes to ``stages'' \\citep{robitaille06,evans09}. Also the accretion\nhistory, which likely includes episodic accretion events and thus temporary \nincreases in luminosity, adds to this uncertainty \\citep{dunham10,dunham12}.\nProtostars with infalling envelopes of gas and dust correspond to Stages 0 and I, \nwith the transition from Stage 0 to I occurring when the stellar mass becomes \nlarger than the envelope mass \\citep{dunham14}. Young stars that have dispersed \ntheir envelopes and are surrounded by circumstellar disks correspond to Stage II. \n\nBy modeling the SEDs of protostars, properties of their envelopes, and to some\nextent of their disks, can be constrained. The near-IR is particularly sensitive to \nextinction and thus constrains the inclination angle and cavity opening angle, \nas well as the envelope density.  Mid-IR spectroscopy reveals the detailed emission \naround the silicate absorption feature and thus provides additional constraints for \nboth disk and envelope properties \\citep[see, e.g.,][]{furlan08}. At longer wavelengths, \nenvelope emission starts to dominate. Thus, photometry in the far-IR is necessary \nto determine the peak of the SED and constrain the total luminosity and envelope \nproperties.\n\nHere we present 1.2--870 $\\mu$m SEDs and radiative transfer model fits\nof 330 YSOs, most of them protostars, in the Orion star formation complex. \nThis is the largest sample of protostars studied in a single, nearby star-forming \nregion (distance of 420 pc; \\citealt{menten07,kim08}) and therefore significant \nfor advancing our understanding of protostellar structure and evolution.\nThese protostars were identified in {\\it Spitzer Space Telescope} \\citep{werner04} \ndata by \\citet{megeath12} and were observed at 70 and 160 \\micron\\ with \nthe Photodetector Array Camera and Spectrometer \\citep[PACS;][]{poglitsch10} \non the {\\it Herschel Space Observatory}\\footnote{{\\it Herschel} is an ESA \nspace observatory with science instruments provided by European-led \nPrincipal Investigator consortia and with important participation from NASA.}\n\\citep{pilbratt10} as part of the Herschel Orion Protostar Survey (HOPS), a \n{\\it Herschel} open-time key program \\citep[e.g.,][W. J. Fischer et al. 2016, \nin preparation; B. Ali et al. 2016, in preparation]{fischer10, stanke10, \nmanoj13, stutz13}. To extend the SEDs into the sub-mm, most of the\nYSOs were also observed in the continuum at 350 and 870~$\\mu$m with \nthe Atacama Pathfinder Experiment (APEX) telescope \\citep{stutz13}. \nOur sample also includes 16 new protostars identified in PACS data obtained \nby the HOPS program (\\citealt{stutz13,tobin15}; see section \\ref{sample}).\nWe use a grid of 30,400 protostellar model SEDs to find the best fit to the SED for each\nobject and constrain its protostellar properties. As mentioned above, the far-infrared \ndata add crucial constraints for the model fits, given that for most protostars the \nSED peaks in this wavelength region, and therefore, within the framework of the \nmodel grid, our SED fits yield the most reliable protostellar parameters to date \nfor these sources. \n\n\n\\section{Sample Description}\n\\label{sample}\n\nThe 488 protostars identified in {\\it Spitzer} data by \\citet{megeath12}\nrepresent the basis for the HOPS sample\\footnote{The selection of HOPS\ntargets is based on an earlier version of the {\\it Spitzer} Orion Survey,\nand in addition some objects likely in transition between Stages I and II\nwere included; thus, not all protostars in the HOPS sample are classified\nas protostars with envelopes in \\citet{megeath12}.} \\citep[see][]{fischer13,\nmanoj13,stutz13}. They have 3.6-24 $\\mu$m spectral indices $\\geq$ \n$-0.3$ and thus encompass flat-spectrum sources.  \nTo be included in the target list for the PACS observations, the predicted flux \nof a protostar in the 70 $\\mu$m PACS band had to be at least 42 mJy as\nextrapolated from the {\\it Spitzer} SED. Since targets were required to have \na 24 $\\mu$m detection, protostars in the Orion Nebula -- where the {\\it Spitzer} \n24 $\\mu$m data are saturated -- are excluded. In addition, after the PACS data \nwere obtained, several new point sources that were very faint or undetected \nin the {\\it Spitzer} bands were discovered in the {\\it Herschel} data \n\\citep{stutz13}. Fifteen of them were found to be reliable new protostars.\nOne more protostar, which was not included in the sample of \\citet{stutz13} due \nto its more spatially extended appearance at 70 $\\mu$m, was recently confirmed \nby \\citet{tobin15}. We have added these 16 protostars to the HOPS sample for \nthis work (see Table D\\ref{New_proto} in the Appendix). Most of these new \nprotostars have very red colors and are thus potentially the youngest protostars \nidentified in Orion \\citep[see][]{stutz13}. \n\nEach object in the target list was assigned a ``HOPS'' identification number, resulting\nin 410 objects with such numbers; HOPS 394 to 408 are the new protostars\nidentified by \\citet{stutz13}, and HOPS 409 is the new protostar from \\citet{tobin15}.\nFour of the 410 HOPS targets turned out to be duplicates, and 31 are likely \nextragalactic contaminants (see Appendix \\ref{exgal_not_modeled} for details). \nSome objects in the HOPS target list were not observed by PACS; of these \n33 objects, 16 are likely contaminants, while the remaining objects were \noriginally proposed but were not observed since they were too faint to have \nbeen detected with PACS in the awarded observing time. In addition, \n35 HOPS targets were not detected at 70 \\micron\\ (see Appendices\n\\ref{YSOs_not_modeled} and \\ref{exgal_not_modeled}); eight of these are \nconsidered extragalactic contaminants, while two of them (HOPS 349 and 381) \nhave only two measured flux values each, making their nature more uncertain. \nOne more target, HOPS 350, also has just two measured flux values (at 24 and \n70 $\\mu$m) and is therefore also excluded from the analysis of this paper.\nSimilarly, we excluded HOPS 352, since it was only tentatively detected at 24 \n\\micron\\ (it lies on the Airy ring of HOPS 84) and in none of the other data sets.\n\nTo summarize, starting from the sample of 410 HOPS targets, but excluding likely \ncontaminants and objects not observed or detected by PACS, there are 330 \nremaining objects that have {\\it Spitzer} and {\\it Herschel} data and are considered \nprotostars (based on their {\\it Spitzer} classification from \\citealt{megeath12}). \nThey form the sample studied in this work. Their SEDs are presented in the \nnext section, and in later sections we show and discuss the results of SED fits \nfor these targets. Their coordinates, SED properties, and classification, as well \nas their best-fit model parameter values, are listed in Table A\\ref{bestfit}. \nThe 41 likely protostars that lack PACS data (either not observed or not detected) \nare presented in Appendix \\ref{YSOs_not_modeled}.\n    \n\n\\section{Spectral Energy Distributions}\n\\label{SEDs}\n\n\\subsection{Data}\n\n\\begin{splitdeluxetable*}{lcccccccccccBlcccccc}\n\\tablewidth{0.9\\linewidth}\n\\tabletypesize{\\scriptsize}  \n\\tablecaption{SED Data for the HOPS targets\n\\label{SED_data}}\n\\tablehead{\n\\colhead{Object} & \\colhead{$J$ Flux} & \\colhead{$J$ Unc.} & \\colhead{$J$ Flag} &\n\\colhead{\\nodata} &\n\\colhead{[70] Flux} & \\colhead{[70] Unc.} & \\colhead{[70] Flag} &\n\\colhead{\\nodata} &\n\\colhead{[870] Flux} & \\colhead{[870] Unc.} & \\colhead{[870] Flag} &\n\\colhead{Object} & \\colhead{[5.4] Flux} & \\colhead{[5.4] Unc.} & \\colhead{\\nodata} & \n\\colhead{[35] Flux} & \\colhead{[35] Unc.} & \\colhead{IRS scaling}}\n\\startdata\n HOPS 1 &    0.000E+00 &    0.000E+00 &   3 & \\nodata &    3.697E+00 &    1.850E-01 &   1 & \\nodata &    6.354E-01 &    1.271E-01 &   2 &   HOPS 1 &  8.631E-03 &    6.069E-04 & \\nodata &    1.185E+00 &    6.460E-02 &  1.17 \\\\\n HOPS 2 &    2.770E-04 &    5.000E-05 &   1 & \\nodata &    5.188E-01 &    2.617E-02 &   1 & \\nodata &    3.865E-01 &    7.730E-02 &   2 &    HOPS 2 &  4.360E-02 &    3.757E-03 & \\nodata &    3.704E-01 &    3.035E-02 &  1.00 \\\\\n HOPS 3 &    2.198E-03 &    8.900E-05 &   1 & \\nodata &    3.187E-01 &    1.622E-02 &   1 & \\nodata &    1.201E-01 &    2.402E-02 &   1 &    HOPS 3 &  4.460E-02 &    7.925E-03 & \\nodata &    4.050E-01 &    2.443E-02 &  1.66 \\\\\n HOPS 4 &    3.820E-04 &    5.300E-05 &   1 & \\nodata &    6.116E-01 &    3.083E-02 &   1 & \\nodata &    1.840E-01 &    3.680E-02 &   2 &    HOPS 4 &  2.055E-02 &    1.240E-03 & \\nodata &    4.943E-01 &    3.484E-02 &  1.00 \\\\\n HOPS 5 &    0.000E+00 &    0.000E+00 &   3 & \\nodata &    7.103E-01 &    3.573E-02 &   1 & \\nodata &    6.973E-02 &    1.395E-02 &   2 &    HOPS 5 &  1.475E-02 &    2.695E-03 & \\nodata &    3.077E-01 &    4.484E-03 &  1.00 \\\\\n HOPS 6 &    0.000E+00 &    0.000E+00 &   3 & \\nodata &    9.110E-02 &    5.523E-03 &   1 & \\nodata &    2.311E-01 &    4.622E-02 &   2 &    HOPS 6 &  1.271E-03 &    3.935E-04 & \\nodata &    5.350E-02 &    5.244E-03 &  1.00 \\\\\n HOPS 7 &    0.000E+00 &    0.000E+00 &   3 & \\nodata &    1.342E+00 &    6.728E-02 &   1 & \\nodata &    3.577E-01 &    7.154E-02 &   2 &    HOPS 7 &  6.459E-04 &    3.090E-04 & \\nodata &    3.258E-01 &    2.114E-02 &  1.00 \\\\\n \\enddata\n\\tablecomments{\nEach object has up to 13 photometric data points and 16 IRS data points\n(see Section \\ref{method}). Here we only show some of the data points for\na few HOPS targets. For each measurement, we provide the measured flux\nin Jy, its uncertainty (also in Jy) and, for the photometry only, a flag value \n(0---not observed, 1---measured, 2---upper limit, 3---not detected). For those \nHOPS targets with IRS spectra, we also provide the scaling factor that was \napplied to all IRS fluxes in each spectrum to bring them in agreement \nwith the IRAC and MIPS fluxes (see Section \\ref{method} for details). \\\\\nTo convert the 2MASS magnitudes and the {\\it Spizter} magnitudes from\n\\citet{megeath12} to fluxes, we used the following zero points: 1594 Jy for\n$J$, 1024 Jy for $H$, 666.7 Jy for $K_s$, 280.9 Jy for [3.6], 179.7 Jy for\n[4.5], 115.0 Jy for [5.8], 64.1 Jy for [8], and 7.17 Jy for [24] (2MASS: \n\\citealt{cohen03}; IRAC: \\citealt{reach05}; MIPS: \\citealt{engelbracht07}). \\\\\n{\\it This table is published in its entirety in the machine-readable format on \nthe journal website. A portion is shown here for guidance regarding its \nform and content.}}\n\\end{splitdeluxetable*}\n\nIn order to construct SEDs for our sample of 330 YSOs, we combined \nour own observations with data from the literature and existing catalogs. \nFor the near-infrared photometry, we used $J$, $H$, and $K_s$ data from \nthe Two Micron All Sky Survey \\citep[2MASS;][]{skrutskie06}. For the \nmid-infrared spectral region, we used {\\it Spitzer} data from \\citet{kryukova12}\nand \\citet{megeath12}: the Infrared Array Camera \\citep[IRAC;][]{fazio04} \nprovided 3.6, 4.5, 5.8, and 8.0 \\micron\\ photometry, while the Multiband \nImaging Photometer for Spitzer \\citep[MIPS;][]{rieke04} provided 24 \\micron\\ \nphotometry. In addition, most of the YSOs in the HOPS sample \nwere also observed with the Infrared Spectrograph \\citep[IRS;][]{houck04} \non {\\it Spitzer} using the Short-Low (SL; 5.2-14 \\micron) and Long-Low \n(LL; 14-38 \\micron) modules, both with a spectral resolution of about 90\n(see, e.g., \\citet{kim16} for a description of IRS data reduction). \n{\\it Herschel} PACS data at 70, 100, and 160 \\micron\\ yielded far-infrared \nphotometric data points (B. Ali et al. 2016, in preparation; the 100 $\\mu$m \ndata are from the Gould Belt Survey; e.g., \\citealt{andre10}). Most YSOs \nwere also observed at 350 and 870 \\micron\\ (see \\citealt{stutz13}) by the \nAPEX telescope using the SABOCA and LABOCA instruments \n\\citep[][respectively]{siringo10, siringo09}. Thus, our SEDs have well-sampled\nwavelength coverage from 1.2 to 870 $\\mu$m; we did not include additional \ndata from the literature in order to preserve a homogeneous data set for \nall the objects in our sample.\n\nThe aperture radius used for the photometry varies depending on the\ninstrument and wave band. The photometry in the 2MASS catalog was\nderived from point-spread function (PSF) fits using data from 4\\arcsec\\ \napertures around each object (see the Explanatory Supplement to the 2MASS \nAll Sky Data Release and Extended Mission Products). \\citet{megeath12} used \nan aperture radius of 2{\\farcs}4 for IRAC and PSF photometry for MIPS\n24 $\\mu$m data. We used aperture radii of 9{\\farcs}6 and sky annuli \nof 9{\\farcs}6-19{\\farcs}2 for PACS 70 and 100 \\micron\\ images; we then applied \naperture correction factors of 0.7331 and 0.6944 to the 70 and 100 \\micron\\ fluxes,\nrespectively. For  PACS 160 \\micron, we used an aperture radius of \n12{\\farcs}8, a sky annulus of 12{\\farcs}8-25{\\farcs}6, and an aperture correction\nfactor of 0.6602. In some cases (background contamination, close companions)\nwe used PSF photometry at 70 and 160 $\\mu$m instead (see B. Ali et al. 2016, \nin preparation, for details). Finally, we adopted beam fluxes at 350 and 870 \n$\\mu$m (with FWHMs of 7{\\farcs}34 and 19\\arcsec, respectively).\nThe IRS SL module has a slit width of 3{\\farcs}6, while the LL module\nis wider, with a slit width of 10{\\farcs}5. Sometimes the flux level of the\ntwo segments did not match at 14 $\\mu$m (due to slight mispointings or \nmore extended emission from surrounding material measured in LL), \nand in these cases usually the SL spectrum was scaled by at most a \nfactor of $\\sim$ 1.4 (typically 1.1-1.2). In a few cases, especially when \nthe LL spectrum included substantial amounts of extended emission or \nflux from a nearby object, the LL spectrum was scaled down to match the \nflux level of the SL spectrum at 14 $\\mu$m, typically by a factor of 0.8-0.9.\nWe discuss how the different aperture sizes are accounted for in the model\nfluxes in section \\ref{model_ap}.\n\nThe SEDs of our HOPS sample are shown in Figure A\\ref{bestSEDs} together\nwith their best-fit models from our model grid (see sections below); the data\nare listed in Table \\ref{SED_data}.\nMany objects display a deep silicate absorption feature at 10 $\\mu$m and\nice features in the 5-8 $\\mu$m region, as expected for protostars. Those \nobjects with very deep 10 $\\mu$m features and steeply rising SEDs are likely \ndeeply embedded protostars, often seen at high inclination angles. \n\n\n\\subsection{Multiplicity and Variability}\n\nA large fraction (203 out of 330) of the young stars in our sample have at least one\n{\\it Spitzer}-detected source within a radius of 15\\arcsec; in most cases, this\n``companion'' is faint in the infrared and likely a background star or galaxy.\nThus, the emission at far-IR and sub-mm wavelengths is expected to be \ndominated by the protostar or pre-main-sequence star, and we can assume that \nthe SEDs are representative of the YSOs even if the nearby sources cannot be \nseparated at these wavelengths. There are a few YSOs that have objects \nseparated by just 1\\arcsec-3\\arcsec\\ and are only resolved in one or two IRAC \nbands (HOPS 22, 78, 108, 184, 203, 247, 293, 364); in these cases we used \nthe flux at the IRAC position that most closely matched those at longer wavelengths. \nWe note that some of these very close ``companions'' are likely outflow knots.\nThere are also unresolved binaries, which appear as single sources even in the \nIRAC observations \\citep{kounkel16}; in these cases our SEDs \nshow the combined flux in all wave bands. If two point sources are not fully resolved \nand the resulting blended source is elongated, no IRAC photometry was extracted.  \nIn such cases, a protostar may not have IRAC fluxes even though it was detected \nin the {\\it Spitzer} images.\n\nThere are also several protostars that lie close to other protostars:\nHOPS 66 and 370 ($d$=14.9\\arcsec), HOPS 76 and 78 ($d$=14.1\\arcsec),\nHOPS 86 and 87 ($d$=12.1\\arcsec), HOPS 117 and 118 ($d$=13.7\\arcsec),\nHOPS 121 and 123 ($d$=7.6\\arcsec), HOPS 124 and 125 ($d$=9.8\\arcsec),\nHOPS 165 and 203 ($d$=13.3\\arcsec), HOPS 175 and 176 ($d$=8.0\\arcsec), \nHOPS 181 and 182 ($d$=10.2\\arcsec), HOPS 225 and 226 ($d$=9.2\\arcsec), \nHOPS 239 and 241 ($d$=12.4\\arcsec), HOPS 262 and 263 ($d$=6.3\\arcsec), \nHOPS 316 and 358 ($d$=6.9\\arcsec), HOPS 332 and 390 ($d$=11.2\\arcsec), \nHOPS 340 and 341 ($d$=4.7\\arcsec), and HOPS 386 and 387 ($d$=9.9\\arcsec).\nHOPS 105 lies 8.7\\arcsec\\ to the north of an infrared-bright source, identified\nby \\citet{megeath12} as a young star with a protoplanetary disk. This source\nis brighter than HOPS 105 in all {\\it Spitzer} bands and at 70 $\\mu$m, but it is\nwell separated at all wavelengths. A similar situation applies to HOPS 128, which\nhas a disk-dominated source 6.3\\arcsec\\ to the southeast. \nHOPS 108 is 6.6\\arcsec\\ from HOPS 64, which is brighter than HOPS 108 out\nto 8 $\\mu$m, but not detected in the far-IR and sub-mm. HOPS 108 also lies\n16.6\\arcsec\\ from HOPS 369 and 28.2\\arcsec\\ from HOPS 370.\nHOPS 140 has two neighboring sources, at 9.6\\arcsec\\ and 13.9\\arcsec, that\nare likely surrounded by protoplanetary disks; they are both brighter than \nHOPS 140 out to 8 $\\mu$m, but at 70 $\\mu$m and beyond HOPS 140\ndominates.\nHOPS 144 lies 7.9\\arcsec\\ from HOPS 377; there is also a somewhat fainter,\nred source 11.7\\arcsec\\ to the northeast, which is not detected beyond 24 \n$\\mu$m. This source also lies 9.7\\arcsec\\ to the southwest of HOPS 143.\nHOPS 173 forms a small cluster with HOPS 174 (at 7.1\\arcsec) and HOPS 380\n(at 11.4\\arcsec); HOPS 174 is the brightest source out to 24 $\\mu$m, but at\n70 $\\mu$m HOPS 173 takes on this role. \nAlso HOPS 322, 323, and 389 form a group of protostars; HOPS 322 lies\n13.4\\arcsec\\ from HOPS 389 and 20.1\\arcsec\\ from HOPS 323, while\nHOPS 323 and 389 are 10.2\\arcsec\\ apart. HOPS 323 is the brightest source.\n\nThus, there are 45 targets in our sample that have an object within 15\\arcsec\\\nthat is bright in the mid- or far-IR and that is resolved with IRAC and MIPS.\nGiven that \\citet{megeath12} used PSF photometry for the MIPS 24 $\\mu$m \nobservations, they obtained reliable fluxes even for companions separated by \nless than 6\\arcsec, the typical PSF FWHM. For fluxes at 70 and 160 $\\mu$m, \nwe also used PSF photometry for objects that were point sources, but too close \nfor aperture photometry. In cases where the fluxes could not be determined \neven with PSF photometry, we had to adopt upper limits instead. Similarly, \nwe performed PSF photometry on protostars without companions, but \ncontaminated by extended or filamentary emission; if the PSF photometry\ndid not return a good fit, we used the flux value from aperture photometry \nas an upper limit.\n\nSince most of our targets have an IRS spectrum, in addition to data points from\nIRAC at 5.8 and 8 \\micron\\ and from MIPS at 24 \\micron, we can detect\ndiscrepancies if flux values at similar wavelengths, but from different instruments,\ndo not agree. They might be due to calibration or extraction problems in the \nIRS spectrum (for example, some extended emission around the target or a \nclose companion), but also to variability. We assumed the former scenario if \nthe flux deviations between IRS and IRAC and between IRS and MIPS were \nsimilar (and more than 10\\%, a conservative estimate for the typical calibration \nuncertainty), and in such cases scaled the IRS spectrum to the MIPS 24 \n\\micron\\ flux. Even though this scaling could mask actual variability, it \ncreates a representative SED for the YSO and yields an estimate of \nthe protostellar parameters from model fits of the SED.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[angle=90, scale=0.68]{HOPS_n_Tbol_Lbol_histogram.eps}\n\\caption{Histograms of the 4.5-24 $\\mu$m spectral indices ({\\it left}), bolometric \ntemperatures ({\\it middle}), and bolometric luminosities ({\\it right}) for the 330 \nYSOs in our sample.  \n\\label{HOPS_n_Tbol_Lbol_histo}}\n\\end{figure*}\n\nIn Appendix \\ref{variability} we identify potentially variable HOPS targets \nbased on their mid-IR fluxes and find that about 5\\% of the protostars \nwith IRS, IRAC, and MIPS data could be variable. The Young Stellar Object \nVariability (YSOVAR) program, which monitored large samples of protostars \nand pre-main-sequence stars in nearby star-forming regions with {\\it Spitzer} \nat 3.6 and 4.5 $\\mu$m \\citep{morales11, cody14, rebull14, guenther14, \npoppenhaeger15, wolk15, rebull15}, found that up to $\\sim$ 90\\% of flat-spectrum \nand Class I YSOs are variable on a timescale of days, with typical changes in \nbrightness of 10\\%-20\\%. On longer timescales (years as opposed to days), \n20\\%-40\\% of members of young clusters show long-term variability, with the \nhighest fraction for those clusters with more Class I protostars \\citep{rebull14}. \nIn Orion, the fraction of variable Class I protostars is $\\sim$ 85\\% \n\\citep{morales11}. Using a larger sample of protostars in Orion and IRAC \ndata at 3.6, 4.5, 5.8, and 8.0 $\\mu$m, \\citet{megeath12} found that, on a \ntimescale of about 6 months, 60\\%-70\\% of Orion protostars show brightness \nvariabilities of $\\sim$ 20\\%, with some as high as a factor of four.\nThus, given that our SEDs consist of noncontemporaneous data sets, small \nflux discrepancies should be common, but we also expect some protostars \nwith large mismatches.\n\nOne protostar with a large discrepancy between various data sets is HOPS 223.\nIt is an outbursting protostar (also known as V2775 Ori; \\citealt{caratti11}), \nand for its SED we had 2MASS, IRAC, and MIPS data from the pre-outburst \nphase available, while the IRS spectrum, PACS, and APEX data are from the \npost-outburst period. Thus, its SED does not represent an actual state of the \nobject, and the derived $T_{bol}$ and $L_{bol}$ values are unreliable. \nPre- and post-outburst SEDs and model fits for this protostar can be found\nin \\citet{fischer12}.\nHOPS 223 is the only protostar with an SED affected by extreme variability.\nA few more protostars, HOPS 71, 132, 143, 228, 274, and 299, show notable \ndiscrepancy between the IRAC and IRS fluxes, and to a minor extent between \nMIPS 24 $\\mu$m and IRS, and thus have somewhat unreliable SEDs and \nSED-derived parameters. \nHOPS 383, which was identified as an outbursting Class 0 protostar by\n\\citet{safron15}, does not appear variable in the SED presented here,\nsince we adopted post-outburst IRAC 3.6 and 4.5 $\\mu$m fluxes obtained \nby the YSOVAR program \\citep{morales11,rebull14} to construct a \nrepresentative post-outburst SED for this object.\n\n\\vspace{6ex}\n\n\\subsection{Determination of $L_{bol}$, $T_{bol}$, Spectral Index, and\nSED Classification}\n\nThe SEDs provide the means to determine $L_{bol}$, $T_{bol}$, and\nthe 4.5-24 $\\mu$m spectral indices for our sample of protostars. For measuring \nthe near- to mid-IR SED slope ($n= d \\log(\\lambda F_{\\lambda})\/d \\log(\\lambda)$), \nwe chose a spectral index between 4.5 and 24 $\\mu$m to minimize the effect of \nextinction on the short-wavelength data point; also, the IRAC 4.5 $\\mu$m fluxes \nfor our HOPS targets are more complete than the IRAC 3.6 $\\mu$m fluxes due \nto the lower extinction at this wavelength. \nFor calculating $L_{bol}$ and $T_{bol}$, we used all available fluxes for each \nobject, including the IRS spectrum, assumed a distance of 420 pc, and used trapezoidal \nsummation; for $T_{bol}$, we applied the equation from \\citet{myers93}.\nFigure \\ref{HOPS_n_Tbol_Lbol_histo} shows the distribution of $n_{4.5-24}$, \n$T_{bol}$, and $L_{bol}$ values for our targets. There is a peak in the distribution \nof spectral indices around 0, while the distribution of $T_{bol}$ values is relatively \nuniform from about 30 K to 800 K. The bolometric luminosities cover a wide range, \nwith a broad peak around 1 $L_{\\odot}$. The median $L_{bol}$, $T_{bol}$, and\n$n_{4.5-24}$ values are 1.1 $L_{\\odot}$, 146 K, and 0.68, respectively.\n\nOur distribution of $L_{bol}$ values is very similar to the observed luminosity function \nof Orion protostars presented in \\citet{kryukova12}; both distributions peak around\n1 $L_{\\odot}$\\ and include values from $\\sim$ 0.02 $L_{\\odot}$\\ up to several hundred $L_{\\odot}$.\nSome differences between the two distributions are expected, given that\n\\citet{kryukova12} only had {\\it Spitzer} 3.6-24 $\\mu$m data available and thus\nhad to extrapolate $L_{bol}$ from the measured near- to mid-infrared luminosity.\nThe main difference is a somewhat larger number of protostars with $L_{bol} \n\\lesssim$ 0.5 $L_{\\odot}$\\ for the \\citet{kryukova12} Orion sample; our median $L_{bol}$ \nvalue amounts to 1.1 $L_{\\odot}$, while their value is 0.8 $L_{\\odot}$. However, with the \ncontaminating sources removed from their sample (which tend to have lower \nluminosities; see \\citealt{kryukova12} for details), their median bolometric luminosity \nand our value match. \nOverall, given that Orion is considered a region of high-mass star formation, its \nluminosity function is similar to that of other regions where massive star forms \n\\citep{kryukova12,kryukova14}, and it is different from that of low-mass star-forming \nregions such as Taurus and Ophiuchus \\citep{kryukova12, dunham13}. Compared \nto the sample of 230 protostars in 18 different molecular clouds studied by \n\\citet{dunham13}, the observed (i.e., not extinction-corrected) $L_{bol}$ values \nof those protostars span from 0.01 to 69 $L_{\\odot}$, with a median value of 0.9 $L_{\\odot}$. \nHowever, given that almost half the protostars in the \\citet{dunham13} sample lack \nfar-IR and sub-mm data, the true luminosities are likely higher, which would bring the \nmedian closer to the Orion value. Finally, we note that the distribution of observed \n$T_{bol}$ values from \\citet{dunham13} is similar to our distribution for Orion protostars;  \nthe median $T_{bol}$ of their and our sample is 160 K and 146 K, respectively, and the \nbulk of their protostars also has $T_{bol}$ values between 30 K and 1000 K, with\na tail down to temperatures of $\\sim$ 10 K and another tail up to $T_{bol}$=2700~K.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[angle=90, scale=0.41]{HOPS_n_vs_Tbol.eps}\n\\caption{The 4.5-24 $\\mu$m spectral index versus the bolometric temperature\nfor the 330 YSOs in our sample. The dashed lines delineate \nthe regions that define the various SED classes (see text for details).\n\\label{HOPS_n_Tbol}}\n\\end{figure}\n\nTo separate our targets into Class 0, Class I, Class II, and flat-spectrum sources, \nwe used the 4.5 to 24 \\micron\\ spectral index ($n_{4.5-24}$) and\/or bolometric \ntemperature ($T_{bol}$): Class 0 protostars have $n_{4.5-24}>0.3$ and \n$T_{bol}<70$~K, Class I protostars have $n_{4.5-24}>0.3$ and $T_{bol}>70$~K, \nflat-spectrum sources have $-0.3 < n_{4.5-24} < 0.3$, and Class II pre-main-sequence \nstars have $n_{4.5-24}<-0.3$.\nBased on this, we identify 92 targets as Class 0 protostars, 125 as Class I protostars, \n102 as flat-spectrum sources, and 11 as Class II pre-main-sequence stars (see Table\nA\\ref{bestfit} and Figure \\ref{HOPS_n_Tbol}). \nThere are nine protostars with $T_{bol}$ values between 66.5 and 73.5 K (which \ncorresponds to a $\\pm$ 5\\% range around the Class 0--I boundary of 70 K); \nsix of them have $T_{bol}$ $>$ 70 K (HOPS 1, 18, 186, 256, 322, 370), and \nthe other three have $T_{bol}$ values just below 70 K (HOPS 75, 250, 361). \nThese protostars' classification is less firm than for the other HOPS targets. \nThere are also a few flat-spectrum sources whose classification is more \nuncertain: HOPS 45, 183, 192, 194, 210, 264, and 281 should be Class I \nprotostars based on their 4.5-24 $\\mu$m spectral index, but when considering \nthe IRS spectrum (specifically, the 5-25 $\\mu$m spectral index), they fall \ninto the flat-spectrum regime ($n_{5-25} < 0.3$). Also, for HOPS 45 and 194 \nthe $T_{bol}$ values are relatively high ($>$ 500 K).\nSimilarly, HOPS 33, 134, 242, 255, and 284 should be Class II pre-main-sequence\nstars based on their 4.5-24 $\\mu$m spectral index, but the spectral slope over \nthe IRS wavelength range suggests that they are flat-spectrum sources.\nIn these cases where the $n_{4.5-24}$ and $n_{5-25}$ spectral indices were \nsomewhat discrepant, we adopted the latter, and thus these objects were \nclassified as flat-spectrum sources. \n\nThere are five objects with $T_{bol}$ $<$ 70 K and $n_{4.5-24}$ $<$ 0\n(HOPS 164, 340, 341, 373, 405); despite their negative 4.5-24 $\\mu$m SED \nslopes, their SEDs either show or imply a deep silicate absorption feature at \n10 $\\mu$m, rise steeply in the mid- to far-IR, and their long-wavelength emission \nis strong. Thus, their $T_{bol}$ values are low, and we identify them as Class 0 \nprotostars, even though they have 4.5-24 $\\mu$m spectral indices not typical \nof embedded protostars. In particular, HOPS 341, 373, and 405 are likely young \nprotostars with dense envelopes (\\citealt{stutz13}; see also section \\ref{Class0}). \nIn the case of HOPS 373, the 4.5 $\\mu$m flux may be contaminated by bright\nH$_2$ emission from an outflow shock, rendering the $n_{4.5-24}$ value more \nunreliable. This might also explain the negative 4.5-24 $\\mu$m spectral index \nfor the other four protostars.\n\nFinally, the few Class II objects in our sample were thought to be potential\nprotostars prior to their observations with {\\it Herschel}. Their 4.5-24 \\micron\\ \nSED slopes are usually just slightly more negative than the cutoff for a \nflat-spectrum source ($-0.3$); three Class II pre-main-sequence stars \n(HOPS 22, 184, 201) have SEDs that are typical of disks with inner holes, \ndisplaying a 10 $\\mu$m silicate emission feature and a rising SED from \n12 to about 20 \\micron\\ \\citep[e.g.,][]{kim13}. The SEDs of the other\nClass II objects are similar to those of flat-spectrum sources; thus, they could\nhave (remnant) envelopes that contribute to their long-wavelength emission.\n\nOur HOPS sample is mostly complete in the number of Class 0, Class I, and \nflat-spectrum sources in the areas of Orion surveyed by {\\it Spitzer} excluding\nthe Orion Nebula \\citep[see][]{megeath12,stutz13}. Of the 357 unique YSOs \noriginally identified in {\\it Spitzer} data that were included in the HOPS sample and \nobserved with PACS, 322 were detected at least at 70 $\\mu$m, which amounts \nto a fraction of 90\\%. We removed likely contaminants and added 16 new\nprotostars discovered in PACS data to get to our sample of 330 YSOs,\nmost of which are protostars.\nOur lowest $L_{bol}$ source is HOPS 208, with $L_{bol}$= 0.017 $L_{\\odot}$. This \nprotostar also has the lowest PACS 70 $\\mu$m flux in our sample (8.2 mJy). \nOverall, our sample has 27 protostars with $L_{bol}<$ 0.1 $L_{\\odot}$, which places \nthem in the luminosity range of very low luminosity objects \n\\citep[VeLLOs;][]{diFrancesco07,dunham08}. The number of VeLLOs in our \nsample is likely larger, given that VeLLOs are defined as having internal \nluminosities less than 0.1 $L_{\\odot}$, and the bolometric luminosity has contributions \nfrom both the internal luminosity and that due to external heating \n\\citep[see][]{dunham08}. In addition, our sample could miss fainter flat-spectrum \nsources and Class 0 and Class I protostars. In fact, there are several faint YSOs \nwithout PACS data that were excluded from our sample, but do have {\\it Spitzer} \ndetections (see Appendix section \\ref{YSOs_not_modeled}).\n\n\n\\vspace{3ex}\n\n\\section{Model Grid}\n\\label{grid}\n\nTo characterize the SEDs of our HOPS sample in a uniform manner,\nwe fit the data to simple but physically plausible models. In this way we\ncan assess how well such simple models can fit the data, and how the\nquality of the fits changes with evolutionary class. We can also determine\nthe full range of physical parameters implied by the fits and the range of\nparameters for each protostellar class. There are degeneracies and biases \nin the fits, and the uncertainties in model parameters will vary from object \nto object, but our results represent a first step in estimating physical \nparameters that describe the protostars in our sample.\n\nWe use a large model grid calculated using the 2008 version of the\n\\citet{whitney03a,whitney03b} Monte Carlo radiative transfer code \n\\citep[see][]{stutz13}; an early version of the grid was presented in \n\\citet{ali10}. \nEach model consists of a central protostar, a circumstellar disk, and an envelope;\nthe radiation released by the star and the accretion is reprocessed by the\ndisk and envelope. The density in the disk is described by power laws in the \nradial and vertical directions, while the density distribution in the envelope \ncorresponds to that of a rotating, collapsing cloud core with constant infall\nrate (the so-called TSC model, after \\citealt{terebey84}; see also \n\\citealt{ulrich76,cassen81}). The envelope also contains an outflow cavity, \nwhose walls are assumed to follow a polynomial shape. At favorable inclination \nangles, this evacuated cavity allows radiation from the inner envelope and \ndisk regions to reach the observer directly. Also, radiation is scattered off the \ncavity walls and can increase the near-IR emission from a protostellar system.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.49]{Opacities_plot.eps}\n\\caption{Extinction opacities of the \\citet{ormel11} dust model ``icsgra3''\n({\\it black}) compared to other dust opacities from the literature: grains with\nthin ice mantles after 10$^5$ years of coagulation with a gas density of\n10$^6$ cm$^{-3}$ from \\citet{ossenkopf94} ({\\it orange});\ncase A model of carbon and silicate dust for R$_V$=5.5 from \\citet{draine03}\n({\\it green}); two extinction curves derived for star-forming regions by\n\\citet{mcclure09}, one for $0.76 < A_J < 2.53$ ({\\it blue}), and one\nfor $2.53 < A_J < 17.71$ ({\\it purple}).\n\\label{opacities}}\n\\end{figure}\n\nWe used dust opacities from \\citet{ormel11} to account for larger, icy grains\n(as opposed to the small grains made of amorphous silicates typically found in\nthe interstellar medium). We adopted their dust model that includes graphite \ngrains without ice coating and ice-coated silicates, with a size distribution that \nassumes growth of aggregates for $3 \\times 10^5$ years, when grains have \ngrown up to 3 $\\mu$m in size (``icsgra3''). Particle sizes range from 0.1 to \n3 $\\mu$m, with a number density that is roughly proportional to $a^{-2.3}$ \n(where $a$ is the particle radius).\nFigure \\ref{opacities} shows our adopted opacities compared to different \nones found in the literature. The opacities from \\citet{draine03} assume a \nmixture of small carbonaceous and amorphous silicate grains. Including larger \nand icy grains broadens the 10 $\\mu$m silicate feature (which is mostly due to \nthe libration mode of water ice) and causes additional absorption at 3 $\\mu$m \nand in the 40-60 $\\mu$m range (all mostly due to the presence of water ice). \nThe mid-IR opacities of the ``icsgra3'' dust model are similar to the ones \ndetermined by \\citet{mcclure09} for star-forming regions and also to those \nused by \\citet{tobin08} to model an edge-on Class 0 protostar; in the mid- to \nfar-IR, they resemble the opacities of \\citet{ossenkopf94}, which are often \nused to model embedded sources. In Figure \\ref{opacities}, we show model \n`OH5' from \\citet{ossenkopf94}, which is listed as the fifth model in their \nTable 1 and corresponds to grains with thin ice mantles after 10$^5$ years \nof coagulation and a gas density of 10$^6$ cm$^{-3}$. We could not use the \n`OH5' opacities for our model grid, since that opacity law does not include \nscattering properties (which are required by the Whitney Monte Carlo \nradiative transfer code). Other authors have modified the `OH5' dust  \nto include the scattering cross section and extend the opacities to shorter\nand longer wavelengths \\citep{young05,dunham10}.\n\n\\subsection{Model Parameters}\n\\label{model_parameters}\n\nThere are 3040 models in the grid; they cover 8 values for the total (i.e.,\nintrinsic) luminosity, 4 disk radii, 19 envelope infall rates (which correspond \nto envelope densities), and 5 cavity opening angles. Each model is calculated \nfor 10 different inclination angles, from 18.2\\degr\\ to 87.2\\degr, in equal steps \nin $\\cos(i)$ (starting at 0.95 and ending at 0.05), resulting in 30,400 different \nmodel SEDs. The values for the various model parameters are listed in Table \n\\ref{modelpars}. Since there are a large number of parameters that can be set \nin the Whitney radiative transfer models, we focused on varying those \nparameters that affect the SED the most, leaving the other parameters \nat some typical values. For example, we assumed a stellar mass of 0.5 $M_{\\odot}$, \na disk mass of 0.05 $M_{\\odot}$, and an envelope outer radius of 10,000 AU.\nThe stellar mass enters the code in two ways. First, it is needed to relate the \ndensity of the envelope to the infall rate (see Equation 1 below). Since we fit the \ndensity of the envelope, the infall rate plays no role in the best-fit envelope \nparameters; any stellar mass can be chosen to determine the infall rate for a \ngiven best-fit envelope density. Second, the stellar mass is combined with the \nstellar radius and disk accretion rate to set the disk accretion luminosity. Given \nthat the accretion luminosity is the actual parameter that influences the SED, \nit does not matter which of the three factors is varied. For simplicity and reasons \ndescribed below, we varied the disk accretion rate and the stellar radius, but\nleft the stellar mass constant, to achieve different values for this component \nof the luminosity.\n \nThe total luminosity for each system consists of the stellar luminosity \n(derived from a 4000 K stellar atmosphere model), the accretion luminosity \nresulting from material accreting through the disk down to the disk truncation \nradius, and the accretion luminosity from the hot spots on the stellar surface, \nwhere the accretion columns, which start at the magnetospheric truncation \nradius, land (these columns are not included in the modeled density distribution, \nsince they do not contain dust and do not have a source of opacity in the radiative \ntransfer models). Typically, the accretion luminosity from the hot spots is much \nlarger than the disk accretion luminosity; in our models, the former is about a factor \nof 9 larger than the latter.\nWe chose three different stellar radii, 0.67, 2.09, and 6.61 $R_{\\odot}$\\ (with the same \nstellar temperature), resulting in three different stellar luminosities. Since both \ncomponents of the accretion luminosity depend on the disk accretion rate, \nchoosing a total of eight different disk accretion rates (three for the 0.67 $R_{\\odot}$\\ \nstar, two for the 2.09 $R_{\\odot}$\\ star, and three for the 6.61 $R_{\\odot}$\\ star) results in \neight values for the total luminosity used in the grid (see Table \\ref{modelpars}). \nThe input spectrum produced by the central protostar depends on the relative \ncontributions from the intrinsic stellar luminosity (which peaks at 0.7~$\\mu$m) \nand the accretion luminosity (which is radiated primarily in the UV). In the \nmodels, it can be altered to some degree by choosing different combinations of \nthe disk accretion rate and stellar radius (the former affects only the accretion \nluminosity, while the latter affects both the stellar and accretion luminosity). \nHowever, the effect of the input spectrum on the output SED is negligible. \nConsequently, we cannot reliably measure the relative contributions of stellar \nand accretion luminosity through our SED fits. Instead, we adjusted the particular \nvalues for the stellar radius and disk accretion rate to set the values of the\ntotal luminosity.\n\nFor our model grid, we chose four values for the disk outer radius, which we\nset equal to the centrifugal radius ($R_c$). In a TSC model, the centrifugal \nradius is the position in the disk where material falling in from the envelope \naccumulates; due to envelope rotation, material from the envelope's\nequatorial plane lands at $R_c$, while material from higher latitudes falls\ncloser to the star. The disk could extend beyond $R_c$, but in our models \nit ends at $R_c$. In this work, we use the terms ``disk (outer) radius'' and \n``centrifugal radius'' interchangeably. The primary effect of $R_c$ is to set \nthe rotation rate of the infalling gas and thereby determine the density \nstructure of the envelope \\citep{kenyon93}.  \n \n\\begin{deluxetable*}{clcc}\n\\tablecaption{Model Parameters\n\\label{modelpars}}\n\\tablehead{\n\\colhead{Parameter} & \\colhead{Description} & \\colhead{Values} & \n\\colhead{Units}}\n\\startdata\n\\multicolumn{4}{c}{\\it{\\bf Stellar Properties}} \\\\ \n$M_{\\ast}$ &  Stellar mass & 0.5 & $M_{\\odot}$ \\\\\n$T_{\\ast}$ &  Stellar effective temperature & 4000 & K \\\\\n$R_{\\ast}$ &  Stellar radius & 0.67, 2.09, 6.61 & $R_{\\odot}$ \\\\ \n\\hline\n\\multicolumn{4}{c}{\\it{\\bf Disk Properties}} \\\\ \n$M_{disk}$ &  Disk mass & 0.05 & $M_{\\odot}$ \\\\\n$R_{disk}$ &  Disk outer radius & 5, 50, 100, 500 & AU \\\\\nA & Radial exponent in disk density law & 2.25 & \\nodata \\\\\nB & Vertical exponent in disk density law  & 1.25 & \\nodata \\\\\n$\\dot{M}_{disk,1}$ & Disk-to-star accretion rate for $R_{star}$=0.67 $R_{\\odot}$\\ & \n0, $1.14 \\times 10^{-8}$, $5.17 \\times 10^{-8}$  & $M_{\\odot}$\\ yr$^{-1}$ \\\\\n$\\dot{M}_{disk,2}$ & Disk-to-star accretion rate for $R_{star}$=2.09 $R_{\\odot}$\\ & \n$3.67 \\times 10^{-7}$, $1.63 \\times 10^{-6}$ & $M_{\\odot}$\\ yr$^{-1}$ \\\\\n$\\dot{M}_{disk,3}$ & Disk-to-star accretion rate for $R_{star}$=6.61 $R_{\\odot}$\\ & \n$1.14 \\times 10^{-5}$, $5.15 \\times 10^{-5}$,  $1.66 \\times 10^{-4}$ & \n$M_{\\odot}$\\ yr$^{-1}$ \\\\\n$R_{trunc}$ & Magnetospheric truncation radius$^a$ & 3 & $R_{\\ast}$ \\\\\n$f_{spot}$ & Fractional area of the hot spots on the star$^b$ & 0.01 & \\nodata \\\\\n\\hline\n\\multicolumn{4}{c}{\\it{\\bf Envelope Properties}} \\\\ \n$R_{env}$ &  Envelope outer radius$^c$ & 10,000 & AU \\\\\n$\\rho_{1000}$ & Envelope density at 1000 AU$^d$ & 0.0, 1.19 $\\times 10^{-20}$, \n1.78 $\\times 10^{-20}$, 2.38 $\\times 10^{-20}$, &  g cm$^{-3}$ \\\\\n& & 5.95 $\\times 10^{-20}$, 1.19 $\\times 10^{-19}$, 1.78 $\\times 10^{-19}$, \n&  g cm$^{-3}$ \\\\\n& & 2.38 $\\times 10^{-19}$, 5.95 $\\times 10^{-19}$, 1.19 $\\times 10^{-18}$,\n&  g cm$^{-3}$ \\\\\n& & 1.78 $\\times 10^{-18}$, 2.38 $\\times 10^{-18}$, 5.95 $\\times 10^{-18}$,\n&  g cm$^{-3}$ \\\\\n& & 1.19 $\\times 10^{-17}$, 1.78 $\\times 10^{-17}$, 2.38 $\\times 10^{-17}$,\n&  g cm$^{-3}$ \\\\\n& & 5.95 $\\times 10^{-17}$, 1.19 $\\times 10^{-16}$, 1.78 $\\times 10^{-16}$\n&  g cm$^{-3}$ \\\\\n$R_c$ & Centrifugal radius of TSC envelope & $= R_{disk}$ & AU \\\\ \n$\\theta$ & Cavity opening angle & 5, 15, 25, 35, 45 & degrees \\\\\n$b_{cav}$ & Exponent for cavity shape$^e$ (polynomial) & 1.5 & \\nodata \\\\\n$z_{cav}$ & Vertical offset of cavity wall & 0 & AU \\\\\n \\hline\n\\multicolumn{4}{c}{\\it{\\bf Derived Parameters}} \\\\ \n$L_{\\ast}$ &  Stellar luminosity$^f$ &  0.1, 1, 10 & $L_{\\odot}$ \\\\\n$L_{tot}$ & Total luminosity (star + accretion)$^g$ & 0.1, 0.3, 1.0, 3.1, \n         10.1, 30.2, 101, 303 & $L_{\\odot}$ \\\\\n\\hline\n\\multicolumn{4}{c}{\\it{\\bf Parameters for Model SEDs}} \\\\ \n$i$ & Inclination angle & 18.2, 31.8, 41.4, 49.5, 56.7, & degrees \\\\\n  &   &  63.3, 69.5, 75.6, 81.4, 87.2 & degrees \\\\\n  &  Aperture radii for model fluxes$^h$ &  420, 840, 1260, 1680, ..., 10080 & AU \\\\\n\\enddata\n\\tablecomments{\nThe dust opacities used for these models are those called ``icsgra3'' from\n\\citet{ormel11}.\\\\\n$^a$ This radius applies to the gas. The inner disk radius for the dust is equal to the \ndust destruction radius. The scale height of the disk at the dust sublimation radius is set \nto the hydrostatic equilibrium solution. \\\\\n$^b$ The hot spots are caused by the accretion columns that reach from the \nmagnetospheric truncation radius to the star. \\\\\n$^c$ The inner envelope radius is set to the dust destruction radius. \\\\\n$^d$ The actual input parameter for the Whitney code is the envelope infall\nrate, which can be derived from $\\rho_{1000}$ using Equation (2). The first six \n$\\rho_{1000}$ values correspond to envelope infall rates of 0, $5.0 \\times 10^{-8}$, \n$7.5 \\times 10^{-8}$, $1.0 \\times 10^{-7}$, $2.5 \\times 10^{-7}$, and $5.0 \\times 10^{-7}$  \n$M_{\\odot}$\\ yr$^{-1}$; the other values can be similarly deduced. \\\\\n$^e$ The cavity walls are assumed to have a polynomial shape; no material is assumed\nto lie inside the cavity. Also, the ambient density (outside the envelope) is 0. \\\\\n$^f$ The three values of $L_{\\ast}$ correspond to the three different stellar radii.\\\\\n$^g$ The total luminosities combine the stellar luminosities and the accretion luminosities\n(which depend on  $\\dot{M}_{disk}$). \\\\\n$^h$ For each model, the emitted fluxes are calculated for 24 apertures ranging from \n420 to 10080 AU, in steps of 420 AU. }\n\\end{deluxetable*}\n\nThe largest number of parameter values in our grid is for the envelope \ninfall rate. The envelope infall rate used as an input in the Whitney code \nsets the density of the envelope for a given mass of the protostar.  \nSince the SED depends on the density of the envelope (and not directly \non the infall rate, which is only inferred from the density and the acceleration \ndue to gravity from the central protostar), in this work we report a  \nreference envelope density instead of the envelope infall rate as one of our \nmodel parameters. For the TSC model, the envelope infall rate $\\dot{M}_{env}$ \nand the reference density at 1 AU in the limit of no rotation ($R_c$=0) are \nrelated as follows \\citep[see][]{kenyon93}:\n\\begin{equation}\n\\rho_1 = 5.318 \\times 10^{-14} \\left( \\frac{\\dot{M}_{env}}{10^{-5} \nM_{\\odot} \\, \\mathrm{yr}^{-1}} \\right) \\left(\\frac{M_{\\ast}}\n{1\\, M_{\\odot}}\\right)^{-1\/2} \\mathrm{g}\\, \\mathrm{cm}^{-3},\n\\end{equation} \nwhere $M_{\\ast}$ is the mass of the central protostar, which is assumed \nto be 0.5 $M_{\\odot}$\\ in our model grid. The density distribution in the\nenvelope follows a power law, $\\rho \\propto r^{-3\/2}$, at radii\nlarger than the centrifugal radius, $R_c$, but then flattens as a result\nof the rotation of the envelope. The density reported by $\\rho_1$ \nassumes a spherically symmetric envelope with a $-3\/2$ power-law \nexponent valid down to the smallest radii, and it is higher than the \nangle-averaged density of a rotating envelope at 1 AU. To quote \ndensities that are closer to actual values found in the modeled rotating \nenvelopes (which have $R_c$ values ranging from 5 to 500 AU), we \nreport $\\rho_{1000}$, the density at 1000 AU for a $\\rho \\propto r^{-3\/2}$ \nenvelope with a 0.5 $M_{\\odot}$\\ protostar:\n\\begin{eqnarray}\n\\rho_{1000} & = & \\rho_1 \\left(\\frac{1}{1000}\\right)^{3\/2} \\nonumber \\\\\n& = & 2.378 \\times 10^{-18} \\left( \n\\frac{\\dot{M}_{env}}{10^{-5} M_{\\odot} \\, yr^{-1}} \\right) \n\\mathrm{g}\\, \\mathrm{cm}^{-3}.\n\\end{eqnarray} \nThus, the range of reference densities probed in our model grid,\nfrom $1.2 \\times 10^{-20}$ to $1.8 \\times 10^{-16}$ g cm$^{-3}$ \n(see Table \\ref{modelpars}), would correspond to envelope\ninfall rates from $5.0 \\times 10^{-8}$ to $7.5 \\times 10^{-4}$ $M_{\\odot}$\\ yr$^{-1}$,\nassuming $M_{\\ast}$=0.5 $M_{\\odot}$\\ (this does not account for a reduction \nof the infalling mass due to clearing by outflow cavities).\nIn Figure \\ref{Rho_env_profiles}, we show the radial density profiles\nfor two TSC models with 5 AU and 500 AU centrifugal radii. The density \nprofiles are azimuthally symmetric and show the flattening of the density \ndistribution inside $R_c$ due to envelope rotation. These plots demonstrate\nthat the density $\\rho_1$ is much higher than the angle-averaged \ndensity at 1 AU; $\\rho_{1000}$ seems to yield more physical values for \nthe density in the envelope at 1000 AU, even for $R_c$ values of 500 AU.\n \n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.37,angle=90]{Envelope_rho_vs_radius_Rc5.eps}\n\\includegraphics[scale=0.37,angle=90]{Envelope_rho_vs_radius_Rc500.eps}\n\\caption{Envelope density versus radius for a model protostar with\n$\\dot{M}_{env}=1.0 \\times 10^{-6}$ $M_{\\odot}$\\ yr$^{-1}$,\n$M_{\\ast}$=0.5 $M_{\\odot}$, and $R_c$=5 AU ({\\it left}) and 500\nAU ({\\it right}) to show the difference between the reference densities\n$\\rho_1$ and $\\rho_{1000}$. The lines with different colors represent \nradial density profiles for different polar angles $\\theta$; the black line represents\nthe angle-averaged density profile (for equations see \\citealt{whitney03a,\nadams86}). The dashed line represents an $r^{-3\/2}$ power law. The vertical\ndotted line marks the location of the centrifugal radius.\n\\label{Rho_env_profiles}}\n\\end{figure*}\n \nAs can be seen from the values of the envelope density in Table \\ref{modelpars}, \nthere is one set of models with an envelope density of 0. These are models \nthat do not contain an envelope component; the entire excess emission \nis caused by the circumstellar disk. If an object is best fit by such a model, \nit would indicate that it is more evolved, having already dispersed its envelope.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.55]{Whitney_cavity_shapes.eps}\n\\caption{Schematic showing the shape of the cavity assumed in our models\nfor three cavity opening angles $\\theta$: 5\\degr, 25\\degr, and 45\\degr\\ \n(from left to right). The cavity walls are defined as a polynomial with exponent \n1.5 ($z \\propto \\tilde r^{1.5}$), with $\\tilde r_{max} = z_{max}\\, \\tan\\theta$, and \nare shown as solid lines. The outer envelope radius ($R_{env}$) at 10,000 AU \nis shown with a short-dashed line. The dotted lines show a different definition \nof the cavity size, where $\\tilde r_{max} = R_{env} \\sin\\theta$ \nand $z_{max} = R_{env} \\cos\\theta$.\n\\label{cavity_shape}}\n\\end{figure}\n\nThe cavities in our models range from 5\\degr\\ to 45\\degr\\ and are defined\nsuch that $z \\propto \\tilde r^{1.5}$, where $\\tilde r$ and $z$ are the cylindrical\ncoordinates for the radial and vertical direction, respectively, and $\\tilde r_{max} \n= z_{max}\\, \\tan\\theta$, with $\\theta$ defined as the cavity opening angle that \nis specified in the parameter file of the Whitney radiative transfer code. In this \ncode, $z_{max}$ is set to the envelope outer radius. Thus, a polynomial-shaped \ncavity, which is wider at smaller $\\tilde r$ values and then converges toward the \nspecified opening angle, is somewhat larger than this opening angle at the outer \nenvelope radius (see Figure \\ref{cavity_shape}). This effect is most noticeable at \nlarger cavity opening angles, but negligible for small cavities. A different definition \nof the cavity size, where $\\tilde r_{max} = R_{env} \\sin\\theta$ and \n$z_{max} = R_{env} \\cos\\theta$ (with $R_{env}$ as the envelope outer radius), \nresults in $z$ values that are a factor of $1\/\\cos\\theta$ larger, and thus the cavity \nreaches the specified opening angle at the outer envelope radius. For this \nwork, the adopted definition of the cavity opening angle is inconsequential, \nbut it becomes relevant when comparing the results of SED modeling to \nscattered light images that reveal the actual cavity shape and size. \nWe also note that in our models the cavities are evacuated of material,\nso there is no dust and gas inside the cavity; in reality, there might be\nsome low-density material left that would add to the scattered light\n\\citep[see][]{fischer14}.\n\nFigures \\ref{Models_inc} to \\ref{Models_Ltot} display a few examples\nof model SEDs from our grid to show the effect of changing those model \nparameters that influence the resulting SED the most. \nThe inclination angle has a strong effect on the near- and mid-infrared SED\n(Figure \\ref{Models_inc}). While a low inclination angle results in an overall \nflat SED in this wavelength region, increasing the inclination angle causes \na deeper silicate absorption feature at 10 $\\mu$m and a steep slope \nbeyond it. The far-infrared to millimeter SED is not affected by the \ninclination angle, since emission at these wavelengths does not suffer \nfrom extinction through the envelope.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.52]{Models_inc_example.eps}\n\\caption{A model from the grid seen at 10 different inclination\nangles to illustrate the effect of viewing angle on the SED. \nThe model has $L_{tot}$=10.1 $L_{\\odot}$, $R_c$=50 AU, \n$\\rho_{1000}$=$1.2 \\times 10^{-18}$ g cm$^{-3}$, $\\theta$=15\\degr, \nand is seen at inclination angles 18\\degr, 32\\degr, 41\\degr, 49\\degr, \n57\\degr, 63\\degr, 69\\degr, 76\\degr, 81\\degr, and 87\\degr\\ (from top to bottom). \n\\label{Models_inc}}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.52]{Models_cavity_example.eps}\n\\caption{Models from the grid to illustrate the effect of cavity opening\nangle on the SED. The models have $L_{tot}$=10.1 $L_{\\odot}$, \n$R_c$=50 AU, $\\rho_{1000}$=$1.2 \\times 10^{-18}$ g cm$^{-3}$, i=63\\degr, \nbut each has a different cavity opening angle: 5\\degr\\ ({\\it red}),\n15\\degr\\ ({\\it yellow}), 25\\degr\\ ({\\it green}), 35\\degr\\ \n({\\it blue}), 45\\degr\\ ({\\it purple}).\n\\label{Models_cavity}}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.52]{Models_Rdisk_example.eps}\n\\caption{Models from the grid to illustrate the effect of the centrifugal radius\n($=R_{disk}$) on the SED. The models have $L_{tot}$=10.1 $L_{\\odot}$, \n$\\rho_{1000}$=$1.2 \\times 10^{-18}$ g cm$^{-3}$,  $\\theta$=5\\degr, i=63\\degr, \nbut different disk radii: 5 AU ({\\it red}), 50 AU ({\\it yellow}), 100 AU ({\\it green}), \n500 AU ({\\it purple}).\n\\label{Models_Rdisk}}\n\\end{figure}\n\nThe cavity opening angle affects the SED shape at all wavelengths (Figure\n\\ref{Models_cavity}). A small cavity only minimally alters the SED compared\nto a case without a cavity; there is still a deep silicate absorption at 10 $\\mu$m \nand steep SED slope, but the cavity allows some scattered light to escape in\nthe near-IR. A larger cavity results in higher emission at near- and mid-infrared \nwavelengths and reduced emission in the far-infrared. \nThe effect of the cavity on the SED would change if a different shape\nfor the cavity walls were adopted. For example, cavities where the outer\nwall follows the streamlines of the infalling gas and dust evacuate less inner \nenvelope material than our polynomial-shaped cavities, resulting in deeper \nsilicate absorption features and steeper mid-infrared SED slopes for the \nsame cavity opening angle \\citep[see][]{furlan14}. Thus, our cavity \nopening angles are tied to our assumed cavity shape.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.52]{Models_density_example.eps}\n\\caption{Models from the grid to illustrate the effect of envelope\ndensity on the SED. The models have $L_{tot}$=10.1 $L_{\\odot}$, \n$R_c$=50 AU, $\\theta$=15\\degr, i=63\\degr, but different reference \ndensities $\\rho_{1000}$:\n0, $2.4 \\times 10^{-20}$, $1.2 \\times 10^{-19}$, $2.4 \\times 10^{-19}$, \n$1.2 \\times 10^{-18}$, $2.4 \\times 10^{-18}$, $1.2 \\times 10^{-17}$, \n$2.4 \\times 10^{-17}$,  and $1.2 \\times 10^{-16}$ g cm$^{-3}$ (the \npeak of the SED moves to longer wavelengths as $\\rho_{1000}$ \nincreases).\n\\label{Models_density}}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.52]{Models_Ltot_example.eps}\n\\caption{Models from the grid to illustrate the effect of the total\nluminosity on the SED. The models have $R_c$=50 AU, $\\rho_{1000}$=\n$1.2 \\times 10^{-18}$ g cm$^{-3}$, $\\theta$=15\\degr, i=63\\degr, but \ndifferent values for the total luminosity: 0.1, 0.3, 1.0, 3.1, 10.1, 30.2, 101, \nand 303 $L_{\\odot}$\\ (from bottom to top).\n\\label{Models_Ltot}}\n\\end{figure}\n\nThe effect of the centrifugal radius is somewhat similar to those of the cavity\nopening angle and inclination angle, but less pronounced (Figure \n\\ref{Models_Rdisk}). Small disk radii imply more slowly rotating, less flattened \nenvelopes and depress the near- and mid-infrared fluxes more than larger \ndisk radii, but even with large disk radii (and more flattened envelopes) there \nis still sufficient envelope material along the line of sight to cause a pronounced \n10 $\\mu$m absorption feature. Overall, our models do not directly constrain the \nsize of the disk; the opacity is dominated by the envelope. Furthermore, the \nflattening of the envelope that is determined by $R_c$ has a similar effect on \nthe SED as changing the outflow cavity opening angle.  \n\nChanging the envelope density causes shifts in the SED in terms of both\nwavelength and flux level: the higher the envelope density, the less \nflux is emitted at shorter wavelengths, and the more the peak of the \nSED shifts to longer wavelengths (Figure \\ref{Models_density}). Deeply \nembedded protostars have SEDs that peak at $\\lambda >$ 100 $\\mu$m, \nsteep mid-IR SED slopes, and deep silicate absorption features.\nThe effect of the envelope density on the SED is different from that of the\ninclination angle, especially in the far-IR: while the SED is not very\nsensitive to the inclination angle in this wavelength region, the ratio of, \ne.g., 70 and 160 $\\mu$m fluxes changes considerably depending on \nthe envelope density. \n\nThe total luminosity of the source has an effect on the overall emission\nlevel of the protostar, but does not strongly affect the SED shape. The\nmain effect is that the peak of the SED shifts to longer wavelengths as \nthe luminosity decreases ($\\lambda_{peak} \\propto L^{-1\/12}$; \n\\citealt{kenyon93}). Especially when comparing models with $L_{tot}$ \nvalues that differ by a factor of a few, the SED shapes are similar \n(Figure \\ref{Models_Ltot}). Thus, one could scale a particular model by \na factor between $\\sim$ 0.5 and 2 and get a good representation of a \nprotostar that is somewhat fainter or brighter, without having to rerun \nthe model calculation with the different input luminosity.\n\n\n\\subsection{Model Apertures}\n\\label{model_ap}\n\nThe model fluxes are computed for 24 different apertures, ranging from \n420 to 10,080 AU in steps of 420 AU (which corresponds to 1\\arcsec\\ at\nthe assumed distance of 420 pc to the Orion star-forming complex). \nFor these SED fluxes, no convolution with a PSF is done, and therefore\nthe spatial distribution of the flux is solely due to the extended nature \nof protostars. Since the envelope outer radius is chosen to be 10,000 AU, \nthe largest aperture encompasses the entire flux emitted by each protostellar\nsystem. However, most of the near- and mid-infrared emission comes \nfrom smaller spatial scales, so an aperture of about 5000 AU will already\ncapture most of the flux emitted at these wavelengths.  \n\nFor a more accurate comparison of observed and model fluxes, in each \ninfrared photometric band where we have data available, we interpolate \nmodel fluxes from the two apertures that bracket the aperture used in \nmeasuring the observed fluxes (4\\arcsec\\ for 2MASS, 2{\\farcs}4\nfor IRAC, PSF photometry for MIPS 24 \\micron, with a typical FWHM of\n6\\arcsec, 9{\\farcs}6 for PACS 70 and 100 \\micron, 12{\\farcs}8 \nfor PACS 160 \\micron). For the IRS data points, we use fluxes interpolated \nfor a 5{\\farcs}3 aperture, since the spectra are composed of two segments,\nSL (5.2-14 \\micron; slit width of 3{\\farcs}6) and LL (14-38 \\micron, slit \nwidth of 10{\\farcs}5), and, if any flux mismatches were present, the SL \nsegment was typically scaled to match the LL flux level at 14 \\micron\\ \n\\citep[see, e.g.,][]{furlan08}. So, fluxes measured in an aperture with \na radius of 5{\\farcs}3 roughly correspond to fluxes from a 10{\\farcs}6-wide \nslit.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.39,angle=90]{Model_fluxes_PACS160.eps}\n\\caption{PACS 160 $\\mu$m fluxes versus aperture radius derived for \na model ($L_{tot}=1.0$ $L_{\\odot}$, $R_c=100$ AU, $\\rho_{1000}=2.378 \n\\times 10^{-18}$ g cm$^{-3}$, $\\theta$=15\\degr, $i= 63$\\degr) using \ndifferent methods. \nThe black symbols represent fluxes from the model SED, the blue symbols \nfluxes derived using aperture photometry on the model image convolved with \nthe PACS 160 $\\mu$m PSF, and the red symbols fluxes derived from the \nconvolved model image and then corrected for PSF losses (see text for details). \nThe maximum flux from the model SED was used to normalize all other fluxes.\nThe dotted line indicates an aperture radius of 12{\\farcs}8.\n\\label{Model_fluxes_PACS160}}\n\\end{figure}\n\nGiven that our targets are typically extended and that the near- to mid-infrared \ndata have relatively high spatial resolution, measuring fluxes in small apertures \n(a few arcseconds in radius) will truncate some of the object's flux, so it is\nimportant to choose similar apertures for the model fluxes. \nFrom about 30 to 100 $\\mu$m, the model fluxes calculated for smaller apertures \nare not very different from the total flux (i.e., the flux from the largest aperture),\nwhich is a result of the emission profile in the envelope and the lower spatial \nresolution at longer wavelengths. \nTo check whether extended source emission in the far-infrared might affect \nthe flux we measure in our models, we calculated a small set of model\nimages at 160 $\\mu$m, convolved them with the PACS 160 $\\mu$m\nPSF, and compared the fluxes from the model images to those written\nout for the model SEDs (which we refer to as ``SED fluxes''; these are\nthe fluxes from the models in the grid).\nModel images would be the most observationally consistent way to measure\nthe flux densities, but they are too computationally expensive and would not\nrepresent a significant gain.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.39,angle=90]{Model_fluxes_SABOCA.eps}\n\\caption{SABOCA (350 $\\mu$m) fluxes versus aperture radius derived \nfor the same model as in Figure \\ref{Model_fluxes_PACS160} using different\nmethods. The black symbols represent fluxes from the model SED, the \nblue symbols  fluxes derived using aperture photometry on the model \nimage convolved with a Gaussian PSF, and the red dot-dashed line the \nbeam flux (assuming a beam with a FWHM of 7{\\farcs}3). The maximum \nflux from the model SED was used to normalize all other fluxes.\nThe dotted line indicates an aperture radius of 3{\\farcs}65.\n\\label{Model_fluxes_SABOCA}}\n\\end{figure}\n\nIn Figure \\ref{Model_fluxes_PACS160} we show the fluxes derived for\na particular model at 160 $\\mu$m using different methods. The fluxes \nmeasured in the convolved model image are lower than the SED fluxes; this \nis caused by the wide PACS 160 $\\mu$m PSF, which spreads flux to very\nlarge radii. Since the shape of the PSF is known, we can correct for these \nPSF losses (assuming a point source and using standard aperture\ncorrections). The fluxes corrected for these PSF losses are very similar to the \nSED fluxes, typically within $\\sim$ 5-10\\% at apertures larger than 5\\arcsec.\nSince our observed fluxes correspond to these PSF-corrected fluxes (we apply \naperture corrections to our fluxes measured in a 12{\\farcs}8 aperture to \naccount for PSF losses), adopting the SED fluxes from the largest aperture \nwould yield model fluxes that are somewhat too high. Thus, we chose to adopt \nthe SED flux measured in a 12{\\farcs}8 aperture as a good approximation for \nthe model flux we would get if we had model images available for all models in \nthe grid and measured aperture-corrected fluxes in these images. We note that \nin our PACS data, the 160 $\\mu$m sky annulus, which extends from 12{\\farcs}8 \nto 25{\\farcs}6 (see B. Ali et al. 2016, in preparation), can include extended emission\nfrom surrounding material and also some envelope emission. In these cases, we \noften used PSF photometry to minimize contamination from nearby sources and \nnebulosity; however, PSF fitting was not used for more isolated sources since \nthe envelopes can be marginally resolved at 160 $\\mu$m and thus deviate \nslightly from the adopted PSF shape.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.39,angle=90]{Model_fluxes_LABOCA.eps}\n\\caption{Similar to Figure \\ref{Model_fluxes_SABOCA}, but for the\nLABOCA (870 $\\mu$m) fluxes. The dotted line indicates an aperture \nradius of 9{\\farcs}5.\n\\label{Model_fluxes_LABOCA}}\n\\end{figure}\n\nFor the SABOCA and LABOCA data, beam fluxes were adopted; the\nFWHM of the SABOCA beam is 7{\\farcs}3, while for the LABOCA\nbeam it is 19\\arcsec. In order to determine which aperture radius\ncorresponds best to beam fluxes, we created a similar set of model \nimages as above at 350 and 870 $\\mu$m, convolved them with \nGaussian PSFs, and measured fluxes in the model images using \ndifferent apertures (see Figures \\ref{Model_fluxes_SABOCA} and\n\\ref{Model_fluxes_LABOCA}, where we show the results for one\nmodel). Fluxes measured in the convolved model image are smaller \nthan the SED fluxes, especially at aperture radii smaller than the \nFWHM of the beam. We find that the beam fluxes for SABOCA and \nLABOCA are best matched by SED fluxes from apertures with radii \nhalf the size of the FWHM of the beam, i.e., 3{\\farcs}65 for \nSABOCA and 9{\\farcs}5 for LABOCA (thus, the aperture sizes are\nthe same as the beam FWHM). \nThis is again an idealized situation, since the measured SABOCA\nand LABOCA beam fluxes also include extended emission (if the\nsource lies on top of background emission), and thus they could be \nhigher than those from the model.\n\n\\subsection{Effect of External Heating}\n\\label{model_ext_heat}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.65,angle=90]{Ext_heat_models.eps}\n\\caption{{\\it Left:} Comparison of models with $L_{tot}$=0.1 $L_{\\odot}$, $R_c$=100 AU, \n$\\theta$=15\\degr, $\\rho_{1000}$=2.4 $\\times 10^{-18}$ g cm$^{-3}$ ({\\it top}) or\n2.4 $\\times 10^{-20}$ g cm$^{-3}$ ({\\it bottom}), $i$=63\\degr, without external heating \n({\\it black}), with external heating by an ISRF equal to that in the solar neighborhood\n({\\it green, dashed line}), and with heating by an ISRF 10 times stronger ({\\it orange, \ndashed line}). {\\it Right:} Similar to the models in the left panels, but these models have \n$L_{tot}$=1.0 $L_{\\odot}$.\n\\label{models_ext_heat}}\n\\end{figure*}\n\nIn our models, the luminosity is determined by the central protostar and\nthe accretion; no external heating is included. The interstellar radiation\nfield (ISRF) could increase the temperature in the outer envelope regions,\nthus causing an increase in the longer-wavelength fluxes \\citep[e.g.,][]\n{evans01,shirley02,young03}.\nIt is expected that external heating has a noticeable effect only on \nlow-luminosity sources ($\\lesssim$ 1 $L_{\\odot}$), while objects with strong \ninternal heating are not affected by the ISRF. Moreover, the strength of \nthe ISRF varies spatially \\citep{mathis83}, and thus its effect on each \nindividual protostar is uncertain. Nonetheless, in the following we estimate\nthe effect of external heating on model fluxes by using a different set of\nmodels.\n \n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.7,angle=90]{Ext_heat_models_flux_ratios.eps}\n\\caption{Ratio of the excess emission due to external heating and the emission of \nthe protostar with external heating in different bands, for heating by an ISRF\nequal to that in the solar neighborhood ({\\it green diamonds}) and by an ISRF \n10 times stronger ({\\it orange squares}). The vertical lines show the range of\nflux excess ratios resulting from different viewing angles (inclination angles range \nfrom 18\\degr\\ to 87\\degr), while the symbols represent mean values. The top (bottom) \npanels are for models with $L_{tot}$=0.1 (1.0) $L_{\\odot}$. The four columns correspond \nto the four reference densities probed.\n\\label{models_ext_heat_ratios}}\n\\end{figure*}\n\nFor this model calculation, we used the 2012 version of the Whitney radiative \ntransfer code \\citep{whitney13}, which allows for the inclusion of external illumination \nby using the ISRF value in the solar neighborhood from \\citet{mathis83}; to vary \nthe ISRF strength, the adopted value can be scaled by a multiplicative factor \nand extinguished by a certain amount of foreground extinction. \nWe calculated a small number of models with and without external heating\nand then compared their far-infrared and submillimeter fluxes. One set of models \nhas $L_{tot}$=0.1 $L_{\\odot}$, $R_c$=100 AU, $\\theta$=15\\degr, and four different \nreference densities $\\rho_{1000}$, ranging from 2.4 $\\times 10^{-17}$ g cm$^{-3}$ \nto 2.4 $\\times 10^{-20}$ g cm$^{-3}$. The other set has the same parameters\nexcept for $L_{tot}$, which is 1.0 $L_{\\odot}$. We calculated models without external \nheating, with heating from an ISRF equal to that in the solar neighborhood, and \nwith ISRF heating 10 times the solar neighborhood value. For these models, we \ndid not include any foreground extinction for the ISRF; thus, the ISRF heating in \nthese models can be considered an upper limit -- especially the 10-fold increase \nover the ISRF in the solar neighborhood represents an extreme value.\nFigure \\ref{models_ext_heat} shows a few examples of model SEDs with and\nwithout external heating. External heating results in flux increases in the far-IR\nand sub-mm; as expected, it affects low-luminosity sources more, and its effects \nare also more noticeable for higher-density envelopes.\n\nFor a more quantitative comparison of model fluxes in the far-IR and sub-mm,\nwe computed the fluxes for each model in six different bands, those of MIPS \n24 $\\mu$m, PACS 70, 100, and 160 $\\mu$m, and SABOCA (350 $\\mu$m) \nand LABOCA (870 $\\mu$m), using apertures as described in section \n\\ref{model_ap}. The model fluxes are affected by poorer signal-to-noise \nratios at the longest wavelengths, so the 870 $\\mu$m fluxes are less reliable.\nWe subtracted the fluxes of the models without external heating ($F_{\\rm no.ext.heating}$)\nfrom those with external heating ($F_{\\rm ext.heating}$) to determine the flux \nexcess due to external heating. The ratios of these excess fluxes and the model fluxes \nwith external heating ($(F_{\\rm ext.heating}-F_{\\rm no.ext.heating})\/F_{\\rm ext.heating}$)\nare shown in Figure \\ref{models_ext_heat_ratios}. Given that these ratios depend \non the inclination angle to the line of sight, we show them as average values for all\n10 inclination angles as well as the range subtended by all inclination angles.\nWe note overall smaller flux ratios at 350 $\\mu$m due to the smaller aperture \nsize chosen in this wave band (see section \\ref{model_ap}).\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.65,angle=90]{Models_ext_heat_SED1.eps}  \n\\caption{{\\it Black and orange lines:} SEDs for models with $L_{tot}$=0.1 $L_{\\odot}$, \n$R_c$=100 AU,  $\\theta$=15\\degr, $i$=75\\degr, reference densities \n$\\rho_{1000}$=2.4 $\\times 10^{-18}$ g cm$^{-3}$ ({\\it left}) and 2.4 $\\times 10^{-19}$ \ng cm$^{-3}$ ({\\it right}), without external heating ({\\it black}) and with heating by an ISRF \nscaled by a factor of 10 ({\\it orange}). The purple dashed lines show SEDs from our \nmodel grid (which does not include external heating) with model parameters\nchanged as indicated in the figure label; these models were chosen to closely \nmatch the model SEDs with external heating.\n\\label{ext_heat_SED1}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.65,angle=90]{Models_ext_heat_SED2.eps}  \n\\caption{Similar to Figure \\ref{ext_heat_SED1}, but for model SEDs with\n$L_{tot}$=1.0 $L_{\\odot}$\\ ({\\it black and orange lines}). The light blue and purple \ndashed lines show SEDs from our model grid (no external heating) with the same \nmodel parameters as shown except for a reference density 2.5 times higher \n({\\it light blue}) and $\\theta$=25\\degr, $i=$81\\degr, and a higher luminosity \n({\\it purple}).\n\\label{ext_heat_SED2}}\n\\end{figure*}\n\nOur analysis shows that heating by the ISRF results in flux increases in the far-IR \nand sub-mm that are about a factor of 2-3 higher for envelopes of low-luminosity \nsources ($L_{tot}$=0.1 $L_{\\odot}$) than for those with higher luminosity. Also, the\neffect of external heating is more noticeable at longer wavelengths (where \napertures\/beams are also larger) than at shorter ones; given our chosen apertures, \nthe largest effect occurs at 160 and 870 $\\mu$m. We also note that the flux increases \ndue to heating by the ISRF are smallest for the lowest $\\rho_{1000}$ value probed, \n2.4 $\\times 10^{-20}$ g cm$^{-3}$; at 160 $\\mu$m, the flux increase is largest for \nintermediate envelope densities. Finally, the flux increases in the far-IR and sub-mm \nare far larger for a solar-neighborhood ISRF scaled by factor of 10 than for an \nunscaled ISRF; for the $L_{tot}$=0.1 $L_{\\odot}$\\ models, an unscaled ISRF increases \nthe fluxes from a few percent (at $\\lesssim$ 100 $\\mu$m) to 50\\% (at 870 $\\mu$m), \nwhile an ISRF scaled by a factor of 10 increases these fluxes by 30\\%-75\\%. Thus, \nfor low-luminosity protostars, up to $\\sim$ 75\\% of a protostar's 870 $\\mu$m flux \ncould be due to external heating, if the environment is dominated by an extremely \nstrong ISRF.\n\nTo estimate how the contribution of external heating would modify derived\nmodel parameters, in Figures \\ref{ext_heat_SED1} and \\ref{ext_heat_SED2} \nwe compare model SEDs that include external heating by an ISRF 10 times \nstronger than in the solar neighborhood and model SEDs without this additional \nheating. For the latter, we used models from our model grid and tried to reproduce\nthe SEDs with external heating. For the models with $L_{tot}$=0.1 $L_{\\odot}$, the\neffect of external heating can be reproduced by increasing the luminosity by\nfactors of a few, increasing $\\rho_{1000}$ by up to an order of magnitude,\nand increasing the cavity opening angle and inclination angle by a small\namount. For the $L_{tot}$=1.0 $L_{\\odot}$\\ models, just increasing the reference \ndensity by a factor of 2.5 results in a good match to the long-wavelength \nemission of our externally heated models; however, the shorter-wavelength \nflux is either under- or overestimated. A better match is achieved with models \nhaving the same reference density as the externally heated models, but with \nslightly larger cavity opening angles and inclination angles, and luminosities \nabout a factor of 2 larger.\nThus, if the far-IR and sub-mm fluxes were contaminated by emission resulting \nfrom extremely strong external heating, a model fit using models from our grid \n(which does not include external heating) could overestimate the envelope density \nby up to an order of magnitude and the luminosity by a factor of 2-5. The cavity \nopening and inclination angles would also be more uncertain, but not by much.  \nFor a more realistic scenario with more modest external heating (which would \nalso include the effect of local extinction), the effect on model parameters would\nbe smaller.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.4,angle=90]{Ext_heat_models_with_Av.eps}\n\\caption{Models with $L_{tot}$=0.1 $L_{\\odot}$, $R_c$=100 AU, $\\theta$=15\\degr, \n$\\rho_{1000}$=2.4 $\\times 10^{-18}$ g cm$^{-3}$, $i$=63\\degr, without external \nheating ({\\it black}), with external heating by an ISRF 10 times stronger than in the \nsolar neighborhood ({\\it orange} to {\\it brown}, {\\it dashed lines}) and different \namounts of extinction applied to the ISRF (from $A_V = 2.5$ to $A_V = 50$, \n{\\it top} to {\\it bottom}).\n\\label{models_ext_heat_Av}}\n\\end{figure}\n\nFor the latter point, we explored the effect of extinction on the ISRF by calculating \na few more models with $L_{tot}$=0.1 $L_{\\odot}$, $R_c$=100 AU, $\\theta$=15\\degr, \n$\\rho_{1000} = 2.4 \\times 10^{-18}$ g cm$^{-3}$, an ISRF 10 times stronger than \nthat in the solar neighborhood, and $A_V$ values for the ISRF of 2.5, 10, 20, and 50. \nThe model SEDs are shown in Figure \\ref{models_ext_heat_Av}. Compared to ISRF\nheating without any foreground extinction, already $A_V=2.5$ causes a decrease \nby a factor of 1.5-2 in the overall emission at far-IR wavelengths. With $A_V$ of\n10 and 20, the far-IR emission decreases by factors of up to $\\sim$ 3.5 and 4, \nrespectively, compared to a strong ISRF that is not extinguished. The fraction of\nexcess emission due to external heating at 160 $\\mu$m decreases from an\naverage of 0.8 for $A_V$=0 (see Figure \\ref{models_ext_heat_ratios}) to 0.6,\n0.3, and 0.2 for $A_V$=2.5, 10, and 20, respectively. Therefore, considering that \ntypical $A_V$ values in Orion are $\\sim$ 10-20 mag \\citep{stutz15}, it is likely\nthat the effect of external heating on model parameters of low-luminosity sources\ndoes not exceed a factor of $\\sim$ 2 in luminosity and $\\sim$ 5 in envelope\ndensity.\n\n\n\\section{Fitting Method}\n\\label{method}\n\nA customized fitting routine determines the best-fit model from the grid \nfor each object in our sample of 330 YSOs (see Sections \\ref{sample} \nand \\ref{SEDs}) using both photometry and, where available, IRS spectroscopy.\nIdeally, an object has 2MASS, IRAC, IRS, MIPS, PACS, and SABOCA and LABOCA \ndata; in many cases, no submillimeter data are available, and in a few cases the \nobject is too faint to be detected by 2MASS. Of the 330 modeled objects, 40 do \nnot have IRS spectra. As a minimum, objects have some {\\it Spitzer} photometry \nand a measured flux value in the PACS 70 $\\mu$m band. No additional data \nfrom the literature were included in the fits to keep them homogeneous.\n\nIn order to reduce the number of data points contained in the IRS spectral\nwavelength range (such that the spectrum does not dominate over the photometry) \nand to exclude ice absorption features in the 5-8 \\micron\\ region and at 15.2 \\micron\\\nthat are usually observed, but not included in the model opacities, we rebin each \nIRS spectrum to fluxes at 16 wavelengths. These data points trace the continuum \nemission and the 10 and 20 \\micron\\ silicate features. Also, when rebinning the \nspectrum, we smooth over its noisy regions, and we scale the whole spectrum \nto match the MIPS 24 \\micron\\ flux if a similar deviation is also seen at the \nIRAC 5.8 and 8 \\micron\\ bands and is larger than 10\\%. Figure \\ref{IRS_rebin} \nshows three examples of our IRS spectra with the rebinned fluxes overplotted.\nOur selection of 16 IRS data points in addition to at most 13 photometric points \nspread from 1.1 to 870 \\micron\\ puts more emphasis on the mid-IR spectral \nregion in the fits. This wavelength region is better sampled by observations, \nmost of the emission is thermal radiation from the protostellar envelope and\ndisk (as opposed to some possible inclusion of  scattered light or thermal \nemission from surrounding material at shorter and longer wavelengths, \nrespectively), and it contains the 10 \\micron\\ silicate feature, which crucially \nconstrains the SED fits. As a result, most models are expected to reproduce \nthe mid-IR fluxes well and might fit more poorly in the near-IR and sub-mm.\n\n\\begin{figure}[!]\n\\centering\n\\includegraphics[scale=0.63]{HOPS_IRS_spectra_rebin.eps}\n\\caption{Three IRS spectra, one for HOPS 32 (Class 0 protostar; {\\it top}),\none for HOPS 84 (Class I protostar; {\\it middle}), and one for HOPS 105\n(flat-spectrum source; {\\it bottom}), overlaid with the rebinned data points\n({\\it filled circles}) used by the fitting routine. Note the different flux ranges\non the y axis in the three panels and thus the big differences in slopes\namong the three spectra.\n\\label{IRS_rebin}}\n\\end{figure}\n\nTo directly compare observed and model fluxes, we create model SEDs \nwith data points that correspond to those obtained from observations,\nfrom both photometry and IRS spectroscopy. For the former, the model \nfluxes are not only derived from the same apertures as the data (see \nsection \\ref{model_ap}), but also integrated over the various filter \nbandpasses, thus yielding model photometry. For the latter, the model \nfluxes are interpolated at the same 16 wavelength values as the IRS spectra.\n\nSince the model grid contains a limited number of values for the total luminosity\n(eight), but the objects we intend to fit have luminosities that likely do not\ncorrespond precisely to these values, we include scaling factors for the luminosity \nwhen determining the best-fit model. As long as these scaling factors are not far\nfrom unity, they are expected to yield SEDs that are very similar to those obtained \nfrom models using the scaled luminosity value as one of the input parameters.\nThe scaling factor can also be related to the distance of the source; for all model \nfluxes, a distance of 420 pc is assumed, but in reality the protostars in our sample\nspan a certain (presumably small) range of distances along the line of sight.\nFor example, a 10\\% change in distance would result in a $\\sim$ 20\\% change \nin flux values (scaling factors of 0.83 or 1.23). Here we report luminosities \nassuming a distance of 420 pc.\n\nIn addition to scaling factors, each model SED can be extinguished to account\nfor interstellar extinction along the line of sight. We use two foreground extinction \nlaws from \\citet{mcclure09} that were derived for star-forming regions: one applies \nto $0.76 \\leq A_J < 2.53$ (or $0.3 \\leq A_K < 1$), and the other one to $A_J \\geq 2.53$\n(or $A_K \\geq 1$). For $A_J < 0.76$, we use a spline fit to the Mathis $R_V=5$ \ncurve \\citep{mathis90}. Since the three laws apply to different extinction environments, \nwe use a linear combination of them to achieve a smooth change in the extinction law \nfrom the diffuse interstellar medium to the dense regions within molecular clouds. \nThus, to find a best-fit model for a certain observed SED, the model fluxes \n$F_{mod}(\\lambda)$ are scaled and extinguished  as follows:\n\\begin{equation}\nF_{obs}(\\lambda) = s F_{mod}(\\lambda) 10^{-0.4 A_{\\lambda}},\n\\label{F_scaled_ext}\n\\end{equation}\nwhere  $F_{obs}(\\lambda)$ and $F_{mod}(\\lambda)$ are the observed and model \nfluxes, respectively, $s$ is the luminosity scaling factor, and $A_{\\lambda}$\nis the extinction at wavelength $\\lambda$. We use three reddening laws,\n$k_{\\lambda}=A_{\\lambda}\/A_J$; by denoting them with the subscripts 1, 2,\nand 3, $A_{\\lambda}$ in the above equation becomes\n\\begin{eqnarray}\nA_{\\lambda} = A_J k_{1,\\lambda} \\quad {\\rm for}\\; A_J < 0.76 \\nonumber \\\\\nA_{\\lambda} = 0.76 k_{1,\\lambda} + (A_J - 0.76) k_{2,\\lambda} \n\\nonumber \\\\ {\\rm for}\\; 0.76 < A_J < 2.53 \\nonumber \\\\\nA_{\\lambda} = 0.76 k_{1,\\lambda} + 2.53 k_{2,\\lambda} + \n(A_J - 2.53) k_{3,\\lambda} \\nonumber \\\\ {\\rm for}\\; A_J > 2.53\n\\end{eqnarray}\nThus, equation \\ref{F_scaled_ext} can be written as\n\\begin{eqnarray}\n2.5 \\log(F_{mod}(\\lambda)\/F_{obs}(\\lambda)) \n= A_J k_{1,\\lambda} -2.5 \\log(s) \\nonumber \\\\ \n{\\rm for}\\; A_J < 0.76 \\nonumber \\\\\n2.5 \\log(F_{mod}(\\lambda)\/F_{obs}(\\lambda)) \n- 0.76 (k_{1,\\lambda}-k_{2,\\lambda}) = \\nonumber \\\\ \nA_J k_{2,\\lambda} -2.5 \\log(s) \\quad {\\rm for}\\; 0.76 < A_J < 2.53 \n\\nonumber \\\\\n2.5 \\log(F_{mod}(\\lambda)\/F_{obs}(\\lambda)) - 0.76 k_{1,\\lambda} \n- 2.53 (k_{2,\\lambda}-k_{3,\\lambda}) \\nonumber \\\\\n=  A_J k_{3,\\lambda} -2.5 \\log(s) \\quad {\\rm for}\\; A_J > 2.53 \\qquad\n\\end{eqnarray}\nThese are linear equations in $A_J$, with the left-hand side of the \nequations as the dependent variables and $k_{\\lambda}$ as the \nindependent variable. For each regime of $A_J$ values, a best-fit\nline can be determined that yields $A_J$ and $-2.5 \\log(s)$ from the \nslope and intercept, respectively, for each model that is compared to \nthe observations.\n\nFor each set of model fluxes and observed fluxes, we calculate three linear fits \n(using linear combinations of the three different extinction laws, as explained above),\nthus yielding three values for scaling factors and three for the extinction value. \nIf each extinction value is within the bounds of the extinction law that was \nused and smaller than a certain maximum $A_J$ value (which will be discussed\nbelow), and the scaling factor is in the range from 0.5 to 2.0, then the result \nwith the best linear fit will be used. \nHowever, if some of the values are not within their boundaries, then combinations\nof their limiting values are explored, and the set of scaling factor and extinction\nwith the best fit is adopted. For example, if a model has fluxes that are\nmuch higher than all observed fluxes, the linear fit described above will likely\nyield very large extinction values and small scaling factors. In this case the fitter\nwould only accept the smallest possible scaling factor (0.5) and the maximum \nallowed $A_J$ value as a solution (which will still result in a poor fit). \n\nFor each object, we allowed the model fluxes to be extinguished up to a maximum\n$A_J$ value derived from column density maps of Orion (\\citealt{stutz15}; see also\n\\citealt{stutz10,stutz13,launhardt13} for the methodology of deriving N$_H$ from \n160-500 \\micron\\ maps). We converted the total hydrogen column \ndensity from these maps to $A_V$ values ($A_V$=3.55 $A_J$)\nby using a conversion factor of \n$1.0 \\times 10^{21}$ cm$^{-2}$ mag$^{-1}$ \\citep{winston10, pillitteri13}. \nFor objects for which no column density could be derived, we set the maximum \n$A_J$ value to 8.45 (which corresponds to $A_V=30$). \n\nAfter returning a best-fit scaling factor and extinction value for each model, \neach data point is assigned a weight, and the goodness of the fit is \nestimated with \n\\begin{equation}\n R = \\frac{\\sum_{i=1}^{N} w_i |\\ln \\left(\\frac{F_{obs}(\\lambda_i)}\n{F_{mod}(\\lambda_i)}\\right)|}{N},\n\\end{equation}\nwhere $w_i$ are the weights, $F_{obs}(\\lambda_i)$ and $F_{mod}(\\lambda_i)$\nare the observed and the scaled and extinguished model fluxes, respectively, \nand N is the number of data points \\citep[see][]{fischer12}. Thus, $R$ is a measure \nof the average, weighted, logarithmic deviation between the observed and model SED. \nIt was introduced by \\citet{fischer12} since the uncertainty of the fit is dominated by \nthe availability of models in the grid (i.e, the spacing of the models in SED space) \nand not by the measurement uncertainty of the data, making the standard $\\chi^2$ \nanalysis less useful. Also, a statistic that measures deviations between models and \ndata in log space more closely resembles the assessment done by eye when \ncomparing models and observed SEDs in log($\\lambda F_{\\lambda}$) vs.\\ $\\lambda$ \nplots.\nWe set the weights $w_i$ to the inverse of the estimated fractional uncertainty \nof each data point; so, for photometry at wavelengths below 3 \\micron\\ they \nare equal to 1\/0.1, between 3 and 60 \\micron\\ they are 1\/0.05, at 70 and \n100 \\micron\\ they are 1\/0.04, at 160 \\micron\\ the weight is 1\/0.07, and \nfor photometry at 350 and 870 \\micron\\ they are 1\/0.4 and 1\/0.2, respectively. \nFor fluxes from IRS spectra the weights are 1\/0.075 for wavelength ranges \n8-12 \\micron\\ and 18-38 \\micron, while they are 1\/0.1 for the 5-8 \\micron\\ \nand 12-18 \\micron\\ regions. These IRS weights are also multiplied by 1.5 for\n high signal-to-noise spectra and by 0.5 for noisy spectra. In this way those \nparts of the IRS spectrum that most constrain the SED, the 10 \\micron\\ silicate \nabsorption feature and slope beyond 18 $\\mu$m, are given more weight; for \nhigh-quality spectra, the weights in these wavelength regions are the same as \nfor the 3-60 \\micron\\ photometry.\n\nFor small values, $R$ measures the average distance between model and data\nin units of the fractional uncertainty.\nIn general, the smaller the $R$ value, the better the model fit, but protostars with fewer \ndata points can have small $R$ values, while protostars with some noisy data can have \nlarger $R$ values (but still an overall good fit). We find a best-fit model for each object, \nbut we also record all those models that lie within a certain range of $R$ values from the \nbest-fit $R$. These models give us an estimate on how well the various model parameters\nare constrained (see Section \\ref{deltaR}).\n\nOur model grid is used to characterize the parameters that best describe the \nobserved SED of each object; the $R$ values rank the models for each \nobject and thus can be used to derive best-fit parameters, as well as estimates of \nparameter ranges. In several instances, better fits could be achieved if the model \nparameters were further adjusted, for example by testing more values of cavity \nopening angle or shape, or even changing the opacities (see, e.g., HOPS 68 \n\\citep{poteet11}, HOPS 223 \\citep{fischer12}, HOPS 59, 60, 66, 108, 368, 369, 370\n\\citep{adams12}, HOPS 136 \\citep{fischer14}, and HOPS 108 \\citep{furlan14}).\nHowever, for protostars that are well fit with one of the models from the grid or for \nwhich the grid yields a narrow range of parameter values, it is unlikely that a more \nextended model grid would yield much different best-fit parameters. Overall, our \nmodel fits yield good estimates of envelope parameters for a majority of the sample, \nand thus we can analyze the protostellar properties of our HOPS targets in a statistical \nmanner. \n\n\n\\section{Results of the Model Fits}\n\nThe best-fit parameters resulting from our models can be found in Table \nA\\ref{bestfit}, and Figure A\\ref{bestSEDs} shows the SEDs and best fits for our sample.\nIn this section we give an overview of the quality of the fits, the distributions \nof the best-fit model parameters, both for the sample as a whole and separated \nby SED class, the parameter uncertainties, and the various degeneracies between \nmodel parameters.\n\n\\subsection{Quality of the Fits}\n\\label{fit_quality}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.55]{R_histogram.eps}\n\\caption{Histogram of the $R$ values of the best fits of the 330 \nYSOs in the HOPS sample that have {\\it Spitzer} and {\\it Herschel}\ndetections. \\label{R_histo}}\n\\end{figure}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.62,angle=90]{R_histogram_by_class.eps}\n\\caption{Histograms of the $R$ values of the best fits  shown separately\nfor the three classes of objects (Class 0, I, and flat-spectrum). The three \nfits with $R>8$ (two Class 0 protostars, one Class I protostar) are not shown.\n\\label{R_histo_by_class}}\n\\end{figure*}\n\nFigure \\ref{R_histo} displays the histogram of $R$ values of the best model\nfits for the 330 objects in our HOPS sample that have {\\it Spitzer} and \n{\\it Herschel} data (more than two data points at different wavelengths)\nand are not contaminants (see Section \\ref{sample}). The median $R$ \nvalue is 3.10, while the mean value is 3.29. Fitting a Gaussian to the \nhistogram at $R$ $\\leq$ 7 yields 3.00 and 2.24 as the center and FWHM \nof the Gaussian, respectively. \nThe distribution of $R$ values implies that, on average, the model deviates\nby about three times the average fractional uncertainty from the data.\nThis is not unexpected, given that we fit models from a grid to observed \nSEDs that span almost three orders of magnitude in wavelength range, \nwith up to 29 data points. The fewer the data points, the easier it is to \nachieve a good fit; in fact, the eight protostars with $R < 1$, HOPS 371, 391, \n398, 401, 402, 404, 406, and 409, have SEDs with measured flux values at \nonly 4-5 points.\nStarting at $R$ values of about 1, $R$ can be used as an indicator of the \ngoodness of fit. However, in some cases a noisy IRS spectrum can \nincrease the $R$ value of a fit that, judged by the photometry alone, does \nnot deviate much from the observed data points. In other cases, mismatches \nbetween different data sets, like offsets between the IRAC fluxes and the IRS\nspectrum, can result in larger $R$ values. These might be interesting protostars \naffected by variability and are thus ideal candidates for follow-up observations. \n\nWhen looking at the SED fits in Figure A\\ref{bestSEDs} (and the corresponding \n$R$ values in Table A\\ref{bestfit}), we estimate that an $R$ value of up to \n$\\sim$ 4 can identify a reliable fit (with some possible discrepancies between \ndata and model in certain wavelength regions). When $R$ gets larger than \nabout 5, the discrepancy between the fit and the observed data points usually\nbecomes noticeable; the fit might still reproduce the overall SED shape\nbut deviate substantially from most measured flux values. \n\nIn Figure \\ref{R_histo_by_class}, we show the histogram of $R$ values \nseparately for the three main protostellar classes in our sample. The \nmedian $R$ value decreases from 3.27 for the Class 0 protostars to 3.18 for \nthe Class I protostars to 2.58 for the flat-spectrum sources. There are \n4 Class 0 protostars and 4 Class I protostars with $R$ values between 1.0 \nand 2.0, but 17 flat-spectrum sources in this $R$ range. These numbers \ntranslate to 17\\% of the flat-spectrum sources in our sample, 4\\% of the \nClass 0 protostars, and 3\\% of the Class I protostars. When examining objects' \n$R$ values between 2.0 and 4.0, there are 51 Class 0 protostars (55\\% of \nClass 0 protostars in the sample), 91 Class I protostars (73\\% of the Class I \nsample), and 74 flat-spectrum sources (73\\% of the flat-spectrum sample).\n\nThus, close to 90\\% of flat-spectrum sources are fit reasonably well ($R$ \nvalues $<$ 4), representing the largest fraction among the different classes \nof objects in our sample. This could be a result of their source properties \nbeing well represented in our model grid, but also lack of substantial \nwavelength-dependent variability (see, e.g., \\citealt{guenther14}), which, \nif present, would make their SEDs more difficult to fit.  \nAbout three-quarters of Class I protostars also have best-fit models with $R < 4$; \nthis fraction drops to about two-thirds for the Class 0 protostars. The latter \ngroup of objects often suffers from more uncertain SEDs due to weak\nemission at shorter wavelengths (which, e.g., results in a noisy IRS\nspectrum); they might also be more embedded in extended emission, \nsuch as filaments, which can contaminate the far-IR to submillimeter fluxes. \nAnother factor that could contribute to poor fits is their presumably\nhigh envelope density, which places them closer to the limit in parameter\nspace probed by the model grid.\nOverall, 75\\% of the best-fit models of the protostars in our sample have\n$R < 4$.\n\nWhen examining the SED fits of objects with $R$ values larger than 5.0,\nseveral have very noisy IRS spectra (HOPS 19, 38, 40, 95, 164, 278, 316,\n322, 335, 359). In a few cases the measured PACS 100 and 160 $\\mu$m \nfluxes seem too high compared to the best-fit model (e.g., HOPS 189), \nwhich could be an indication of contamination by extended emission \nsurrounding the protostar. \n\nOf particular interest are objects where variability likely plays a role in \na poor fit. As mentioned in Section \\ref{SEDs}, variability among protostars\nis common; we found in Appendix \\ref{variability} that about 5\\% of our \ntargets display noticeable ($\\gtrsim$ 50\\%) mismatches between \nthe IRS, IRAC, and MIPS fluxes that could be due to intrinsic variability.\nThe SED fits of objects for which the flux mismatches between IRS and \nIRAC and between IRS and MIPS are different are particularly affected, \nsince in that case we did not scale the IRS spectrum to match the \nMIPS 24 $\\mu$m flux.\nHOPS 228 exemplifies such a case: there is a clear discrepancy\nbetween the IRAC and IRS fluxes (a factor of 2.1-2.7) and also\nbetween MIPS 24 $\\mu$m and IRS (a factor of 0.8); even though \nthe fit gives more weight to the IRS data, they are not fit well, especially \nthe silicate absorption feature. The $R$ value of 5.74 for the fit of\nHOPS 228 reflects the discrepant data sets and poor fit. \nHOPS 223 is another case where the IRS  fluxes do not match the \nshorter-wavelength data (they are more than an order of magnitude \nlarger); however, it is a known FU Ori source \\citep[see][]{fischer12},\nand the SED presented here contains both pre- and post-outburst data. \nThe model fit is very poor, which can also be gauged by the $R$ value \nof 8.41. \n\nThere are also objects with overall good fits whose SEDs show discrepancies\nthat may be signs of variability or contamination. \nFor example, for the Class I protostar HOPS 71 the IRAC fluxes are a factor \nof 1.8-2.4 lower than the IRS fluxes in the 5-8 $\\mu$m region, and also the \nMIPS flux is about 20\\% lower. The best-fit model ($R=3.63$) fits the SED \nextremely well beyond about 6 $\\mu$m, with some discrepancy at shorter \nwavelengths. There is a source just 11\\arcsec\\ from HOPS 71 that is detected\nin 2MASS and {\\it Spitzer} data, but not by PACS; this object, HOPS 72, is\nlikely an extragalactic object (see Appendix \\ref{exgal_not_modeled}) that\ncould contaminate the IRS fluxes. Thus, in this case, wavelength-dependent\ncontamination by a companion could explain the discrepancies observed in\nthe SED.\n\nAnother example is HOPS 124, which is a deeply embedded Class 0 protostar. \nFor this object, the mismatch between IRS and IRAC and MIPS fluxes \ndecreases with increasing wavelength (from a factor of 2.5 to a factor of 1.4); \nfor the SED fit, the IRS spectrum was scaled by 0.7 to match the MIPS 24 \n$\\mu$m flux. As with HOPS 71, there is a nearby source that could contaminate \nsome of the fluxes, especially at shorter wavelengths: HOPS 125, a flat-spectrum \nsource, lies 9.8\\arcsec\\ from HOPS 124 and is brighter than HOPS 124 out to \n$\\sim$ 20 $\\mu$m, but then much fainter at longer wavelengths. The best-fit \nmodel of HOPS 124 ($R=2.43$) matches the mid- to far-IR photometry and \nalso most of the IRS spectrum well.\n\nAs an example of a probably variable flat-spectrum source, HOPS 132 \nhas IRAC fluxes that lie a factor of 1.3-1.7 above those of IRS and a \nMIPS 24 $\\mu$m flux that is a factor of 0.6 lower. It does not have a close\ncompanion; the nearest HOPS source, HOPS 133, is 27\\arcsec\\ away.\nThe IRS spectrum was not scaled, and since the SED fitter gave more \nweight to the spectrum, it is fit well, but the IRAC photometry is underestimated\nand the MIPS photometry overestimated. Nonetheless, the $R$ value\nof the best fit is 2.87.\n\nOverall, the SED fits of objects that are likely variable or suffer from some\ncontamination are less reliable, but it is not always clear from the $R$ \nvalue of the best fit. The SED fitting procedures assume that the protostars \nare not variable, so when large mismatches between different data sets are \npresent, the fit will appear discrepant with at least some of the observed\ndata points, but the $R$ value would not end up particularly high \nif, e.g., the IRS spectrum was fit exceptionally well. However, given the\ndata sets we have for these protostars, our SED fits will still yield the best \npossible estimate for the protostellar parameters describing these systems.\n\n \n\\subsection{Overview of Derived Parameters}\n\\label{results_overview}\n\nThe histogram of best-fit $\\rho_{1000}$ values (which is the density of the \nenvelope at 1000 AU; see Section \\ref{model_parameters}) is shown in \nFigure \\ref{rho1000_histo}.\nThe median value of the distribution amounts to $5.9 \\times 10^{-19}$ \ng cm$^{-3}$; this corresponds to a $\\rho_1$ value of $1.9 \\times \n10^{-14}$ g cm$^{-3}$. There is a spread in values: 69 objects have \ndensities $\\rho_{1000}$ smaller than $5.0 \\times 10^{-20}$ g cm$^{-3}$\n(6 of them have actually no envelope), 89 fall in the $5.0 \\times\n10^{-20}$ to $5.0 \\times 10^{-19}$ g cm$^{-3}$ range, 96\nare between $5.0 \\times 10^{-19}$ and $5.0 \\times 10^{-18}$ g \ncm$^{-3}$, 60 between  $5.0 \\times 10^{-18}$ and $5.0 \\times \n10^{-17}$ g cm$^{-3}$, and 16 have $\\rho_{1000}$ values\nlarger than $5.0 \\times 10^{-17}$ g cm$^{-3}$.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.55]{Rho1000_histogram.eps}\n\\caption{Histogram of the envelope reference density $\\rho_{1000}$ \nof the best fits for the 330 targets in our sample. \n\\label{rho1000_histo}}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.55]{Menv_histogram.eps}\n\\caption{Histogram of the envelope mass within 2500 AU derived for\nthe best fits for the 330 targets in our sample.\n\\label{Menv_histo}}\n\\end{figure}\n\nWe also calculated the envelope mass ($M_{env}$) within 2500 AU for \nthe best-fit models (see Figure \\ref{Menv_histo} for their distribution). \nThe 2500 AU radius is close to half the FWHM of the PACS 160 $\\mu$m \nbeam at the distance of Orion (i.e., $\\sim$ 6\\arcsec), and thus roughly \nrepresents the spatial extent over which we measure the SEDs.\nThis envelope mass is determined from the integrated envelope density \nof our best-fit models, with allowances made for outflow cavities, and thus \nonly valid in the context of our models. The median envelope mass within \n2500 AU amounts to 0.029 $M_{\\odot}$. The majority of protostars have \nmodel-derived masses in the inner 2500 AU of their envelopes around \n0.1~$M_{\\odot}$; just 22 objects have $M_{env}$ ($<$ 2500 AU) larger than \n1.0~$M_{\\odot}$. Of the 330 modeled objects, 291 have $M_{env}$ ($<$ 2500 AU)\nsmaller than 0.5 $M_{\\odot}$\\ (6 of these 291 objects have no envelope).  \n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.55]{Ltot_histogram.eps}\n\\caption{Histogram of the total luminosities of the best fits for the 330 targets \nin our sample. \n\\label{Ltot_histo}}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.55]{Rdisk_histogram.eps}\n\\caption{Histogram of the disk radii of the best fits for the 330 targets \nin our sample. \n\\label{Rdisk_histo}}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.55]{Cavity_histogram.eps}\n\\caption{Histogram of the cavity opening angles of the best fits for the \n330 targets in our sample. \n\\label{Cav_histo}}\n\\end{figure}\n\n\\begin{figure*}[!]\n\\centering\n\\includegraphics[scale=0.7]{Rho1000_and_cavity_histogram.eps}\n\\includegraphics[scale=0.7]{Ltot_and_cavity_histogram.eps}\n\\includegraphics[scale=0.7]{Rdisk_and_cavity_histogram.eps}\n\\caption{Histograms of the envelope reference density $\\rho_{1000}$ \n({\\it left}), the total luminosity ({\\it middle}), and the disk radius ({\\it right})\nof the best fits grouped by cavity opening angles. \n\\label{Pars_cav_histo}}\n\\end{figure*}\n\nFigure \\ref{Ltot_histo} contains the histogram of the total luminosities\nderived from the best-fit models. These luminosities consist of the stellar,\ndisk accretion, and accretion shock components. The median total luminosity \namounts to 3.02 $L_{\\odot}$, while the values cover four orders of magnitude, \nfrom 0.06 $L_{\\odot}$\\ (for HOPS 336) to 607 $L_{\\odot}$\\ (for HOPS 288 and 361). \nSince the minimum and maximum values for the total luminosity \nin our grid amount to 0.1 and 303.5 $L_{\\odot}$, respectively, and our scaling \nfactors range from 0.5 to 2.0, our fitting procedure can return best-fit\nluminosities that range from 0.05 to 607 $L_{\\odot}$. Thus, two protostars are\nreaching the upper limit allowed for total luminosities in our grid; it is possible \nthat even better fits could be achieved by increasing the luminosity further. \n\nFrom the distribution of best-fit outer disk radii in Figure \\ref{Rdisk_histo}, it is \napparent that most protostars are fit by small disks whose radius is only 5 AU. \nSince the outer disk outer radius is the centrifugal radius in our models, \ninfalling material from the envelope tends to accumulate close to the star \nfor most sources. Thus, the disk radius is tied to the envelope structure; \na small centrifugal radius implies higher envelope densities at smaller radii\nand a less flattened envelope structure. The median disk radius is 50 AU, \nbut the number of objects with disk radii $\\geq$ 50 AU is roughly evenly \nsplit among the values of 50, 100, and 500 AU.\n\nThe distribution of best-fit cavity opening angles is displayed in Figure \n\\ref{Cav_histo}. Most protostars seem to have either very small (5\\degr) \nor very large (45\\degr) cavities; the median value is 25\\degr.\nWhen dividing the envelope densities by cavity opening angle (see Figure\n\\ref{Pars_cav_histo}, left column), differences emerge: the distributions of \n$\\rho_{1000}$ values are significantly different when comparing objects \nwith $\\theta$=5\\degr\\ and $\\theta \\geq$35\\degr, objects with \n$\\theta$=15\\degr\\ and $\\theta$ $\\geq$ 25\\degr, and objects with \n$\\theta$=25\\degr\\ and $\\theta$=45\\degr. The Kolmogorov-Smirnov (K-S)\ntests yield significance levels that these subsamples are drawn from the\nsame parent population of $\\lesssim$ 0.015. Thus, there seems to be a \ndifference in the distribution of envelope densities among the best-fit \nmodels with smaller cavity opening angles and those with larger cavities. \nProtostars with larger cavities ($\\geq$ 35\\degr) tend to have higher envelope \ndensities (their median $\\rho_{1000}$ values are about an order of magnitude \nlarger compared to objects with cavities $\\leq$ 15\\degr). \n\nFigure \\ref{Pars_cav_histo} (middle column) also shows the distribution \nof total luminosities for the different cavity opening angles. The only \nsignificant difference can be found for the $\\theta$=5\\degr\\ histogram \nas compared to the histograms for larger $\\theta$ values (K-S test\nsignificance level $\\lesssim$ 0.03); the luminosities of models with \n$\\theta$=5\\degr\\ have a different distribution, and also their median \nvalue is 1.45 $L_{\\odot}$, as compared to $\\sim$ 3-5 $L_{\\odot}$\\ for the models\nwith larger cavities. So, protostars with small cavities seem to have lower\ntotal luminosities.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.55]{Inc_histogram.eps}\n\\caption{Histogram of the inclination angles of the best fits for the 330 \ntargets in our sample. The green dashed histogram represents the \ndistribution of uniformly (randomly) distributed inclination angles.\n\\label{Inc_histo}}\n\\end{figure}\n\nThe distribution of centrifugal radii for different cavity opening angles \n(right column in Figure \\ref{Pars_cav_histo}) shows that, independent of \ncavity size, most objects have $R_{disk}$ = 5 AU. However, the\ndistribution among the four different disk radii becomes flatter for \nthe largest cavity opening angles; the histograms for $\\theta$=35\\degr\\ \nand $\\theta$=45\\degr\\ are very similar (K-S test significance level of\n0.98). There is also no significant difference (K-S test values $>$ 0.075)\nbetween the $\\theta$=15\\degr\\ and $\\theta$=25\\degr\\ histograms and \nbetween the $\\theta$=5\\degr\\ and $\\theta \\geq $ 35\\degr\\ histograms. \nThe distributions of disk radii for the other cavity opening angles are all \ndifferent from one another (K-S test significance levels $<$ 0.015). \nOverall, Figure \\ref{Pars_cav_histo} shows that protostars best fit by \nmodels with large cavity opening angles are also fit by models with \nhigher envelope densities and larger centrifugal radii.  \n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.54]{Cumulative_inc.eps}\n\\caption{Cumulative distribution of the inclination angles of the best fits,\nnormalized by the total number of fits ({\\it solid line}), compared to the \ncumulative probability of finding an inclination angle below a given value\nfor randomly distributed inclinations ({\\it green dashed line}). \n\\label{Inc_CDF}}\n\\end{figure}\n\nIn Figure \\ref{Inc_histo}, we show the distribution of the inclination angles\nfor the best-fit models. There is a clear concentration of models in the \n60\\degr$-$70\\degr\\ range; the median inclination angle is 63\\degr.\nThis median value is close to 60\\degr, which is where the probability for \nisotropically distributed inclination angles reaches 50\\% (i.e., the probability \nof observing an inclination angle less than 60\\degr\\ is the same as the \nprobability of observing i$>$60\\degr). However, the details of the\ndistributions differ.\nThe cumulative probability of finding an inclination angle less than a certain value, \n$i_c$, is $1-\\cos(i_c)$, assuming a random distribution of inclination angles. For\ninclination angles $i_1$ and $i_2$, the probability for $i_1 < i < i_2$ is\n$\\cos(i_1)-\\cos(i_2)$. Thus, since the inclination angles in our model grid were \nchosen to be equally spaced in $\\cos(i)$ (there are five values $<$60\\degr\\ and \nfive values $>$60\\degr), one would expect a flat distribution in Figure \\ref{Inc_histo} \nif the best-fit inclination angles were randomly distributed (see the green dashed\nhistogram). However, we find a distribution peaked at 63\\degr\\ and 70\\degr. \nThis can also be seen in Figure \\ref{Inc_CDF}, where we compare our observed \ncumulative distribution of inclination angles to that of randomly distributed ones. \nOur distribution shows a deficit at inclination angles below 60\\degr\\ and \nis just slightly higher at large inclination angles. A K-S test of the two \ndistributions yields a 5.6\\% chance that they are drawn from the same parent\ndistribution.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.59]{Rho1000_and_inc_histogram.eps} \n\\includegraphics[scale=0.59]{Ltot_and_inc_histogram.eps}\n\\includegraphics[scale=0.59]{Cavity_and_inc_histogram.eps}\n\\caption{Histograms of the envelope reference density $\\rho_{1000}$\n({\\it left}), total luminosity ({\\it middle}), and cavity opening angles \n({\\it right}) of the best fits divided by bins of inclination angles. \n\\label{Pars_inc_histo}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.37,angle=90]{Ltot_vs_Lbol_and_inc.eps}\n\\includegraphics[scale=0.37,angle=90]{Ltot_vs_Lbol_and_Av.eps}\n\\caption{Ratio of the total luminosity from the best fits and the bolometric \nluminosity derived from the observed SEDs versus the inclination angle ({\\it \nleft}) and foreground extinction ({\\it right}) of the best fits. In the left\npanel, the open stars represent the median ratios at each inclination angle.\nIn the right panel, the open circles represent the median ratios for eight\nbins in $A_V$ values, represented by the horizontal lines bisecting\neach circle.\n\\label{Ltot_Lbol}}\n\\end{figure*}\n\nTo examine whether the distribution of envelope parameters changes with\ninclination angle (which could imply a degeneracy), Figure \\ref{Pars_inc_histo}\nshows the reference envelope density $\\rho_{1000}$, the total luminosity, \nand the cavity opening angle binned by three ranges of inclination angles.\nNone of the three model parameters show a significantly different distribution \nfor any of the inclination bins (K-S test significance levels are $\\gtrsim$ 0.1,\nexcept for the cavity opening angles for the lowest and middle inclination\nrange, for which the K-S test significance value is 0.02). \nThe median $\\rho_{1000}$ values for the $i=$18\\degr--41\\degr, 49\\degr--63\\degr, \nand 69\\degr--87\\degr\\ inclination bins are all $5.9 \\times 10^{-19}$ g cm$^{-3}$. \nEven though not shown in Figure \\ref{Pars_inc_histo}, the objects whose best-fit\nmodel does not include an envelope are only found at $i \\geq$ 49\\degr. It is\nnoteworthy that protostars with the highest envelope densities do not have \ninclination angles in the 69\\degr--87\\degr\\ range; it is not clear whether this \nis an observational bias, whether our observed sample does not contain high-density, \nedge-on protostars, or whether this is due to biases in the fitting procedure and\/or \nmodel grid.\nThe median values for the total luminosity do not differ by much for the \ndifferent bins of inclination angle, increasing from 2.9 to 4.1 $L_{\\odot}$\\ from\nthe lowest to the middle inclination range and then decreasing to 2.0 $L_{\\odot}$\\\nfor the highest inclination angles. The few protostars with very high $L_{tot}$\nvalues have large inclination angles ($i \\geq$ 49\\degr).\nFinally, the distribution of cavity opening angles is quite similar for different\nranges in inclination, except for a somewhat larger number of $\\theta =$ 45\\degr\\\nvalues at intermediate inclination angles. Half the objects in the $i=$18\\degr--41\\degr\\ \nand 69\\degr--87\\degr\\ inclination bins have $\\theta \\leq $ 15\\degr\\ (with the most\ncommon value 5\\degr), while almost half the objects at intermediate inclination \nangles have $\\theta \\geq $ 35\\degr\\ (the most common value is 45\\degr). \n\nIn Figure \\ref{Ltot_Lbol}, we show ratios of the total and bolometric luminosities\nas a function of inclination angle and foreground extinction ($i$ and $A_V$ are\nadopted from the best model fits). The total luminosity is the intrinsic luminosity \nfrom the best-fit model of each object, while the bolometric luminosity is derived \nby integrating the fluxes of the observed SED. It is expected that $L_{tot}$ is \nhigher than $L_{bol}$ for objects seen at higher inclination angles, since for \nthese objects a large fraction of the emitted flux is not directed toward the \nobserver (and thus deriving bolometric luminosities from observed fluxes will \nunderestimate the intrinsic source luminosity). Conversely, objects seen more \nface-on should have lower $L_{tot}$ values compared to $L_{bol}$. Our data \nand model fits yield $L_{tot}$ values that are usually higher than the $L_{bol}$\nvalues measured from the SED; the discrepancy is larger for the more \nhighly inclined sources. The median $L_{tot}\/L_{bol}$ ratio is 1.5 for \nprotostars with inclination angles in the 18\\degr--41\\degr\\ range, 2.5\nfor the i=49\\degr--63\\degr\\ range, and 3.5 for inclination angles \n$\\geq$ 69\\degr. \nThe fact that $L_{tot}>L_{bol}$ even for $i=$18\\degr--41\\degr\\ could\nbe related to the typically smaller cavity opening angles for this range of\ninclination angles (see Figure \\ref{Pars_inc_histo}); less flux, especially at\nshorter wavelengths, is detected since the opacity along the line of sight \nis still high due to the small cavities. \n\nForeground extinction also plays a role in increasing the $L_{tot}$\/$L_{bol}$\nratio. The median ratio of these luminosities increases from 1.8 for the $A_V$=\n0-5 mag range to 5.0 for $A_V$=25-30; it decreases somewhat for the next\n$A_V$ bin, but reaches 5.9 at $A_V$=40-50 (the 23 objects with $A_V >$ 50, \nnot shown in Figure \\ref{Ltot_Lbol}, have a median $L_{tot}$\/$L_{bol}$ ratio of 8.2). \nAmong the 22 objects with best-fit $A_V$ values of 0-5 mag and inclination angles \n$\\leq$ 50\\degr, only four have $L_{tot}\/L_{bol}$ ratios that are larger than \n1.5 (they are HOPS 57, 147, 199, and 201; in most cases the model overestimates \nthe near-IR emission).   \n\n\n\\subsection{Envelope Parameters for Different SED Classes}\n\\label{par_classes}\n\nFigures \\ref{Rho_class_histo}--\\ref{AV_class_histo} divide the histograms of the \nbest-fit reference density $\\rho_{1000}$, inclination angle, cavity opening angle, \ntotal luminosity, disk radius, and foreground extinction, respectively, by protostar \nclass. As explained in Section \\ref{SEDs}, we divided our targets into Class 0, \nClass I, flat-spectrum, and Class II objects based on their mid-infrared (4.5-24 \\micron) \nspectral index and bolometric temperature (see also Table A\\ref{bestfit}). Thus, \nClass 0 and I protostars have a spectral index $>$ 0.3, and Class 0 protostars have \n$T_{bol}$ values $<$ 70 K, but, as mentioned in Section \\ref{SEDs}, there are \na few protostars whose spectral index or $T_{bol}$ value places them very close \nto the transition region between Class 0 and I or between Class I and flat spectrum. \nGiven that our sample contains just eleven Class II pre-main-sequence stars, we did \nnot include them in the following histograms; they will be discussed in section \n\\ref{disk_sources}.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.65,angle=90]{Rho1000_histogram_by_class.eps}\n\\caption{Histograms of the envelope reference density $\\rho_{1000}$ \nof the best fits for the different SED classes. \n\\label{Rho_class_histo}}\n\\end{figure*}\n\nThe distributions of reference densities (Figure \\ref{Rho_class_histo}) are \ndifferent for all SED classes; none are consistent with being drawn from the \nsame parent population (K-S test significance level $<$ 0.01). Overall, Class 0 \nprotostars have higher envelope densities than Class I and flat-spectrum sources; \nthe median $\\rho_{1000}$ values decrease from 5.9 $\\times 10^{-18}$ g cm$^{-3}$ \nto 2.4 $\\times 10^{-19}$ g cm$^{-3}$ to 1.2 $\\times 10^{-19}$ g cm$^{-3}$ for \nthese three groups. The lower and upper quartiles for $\\rho_{1000}$ are \n1.8 $\\times 10^{-18}$ and 1.8 $\\times 10^{-17}$ g cm$^{-3}$ for the Class 0\nprotostars, and 2.4 $\\times 10^{-20}$ and 1.2 $\\times 10^{-18}$ g cm$^{-3}$ for the \nClass I and flat-spectrum objects.\nWe will discuss some implications of these differences in derived envelope\ndensities in section \\ref{SED_class_properties}.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.65,angle=90]{Inc_histogram_by_class.eps}\n\\caption{Histograms of the inclination angles of the best fits for the different \nSED classes. \n\\label{Inc_class_histo}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.65, angle=90]{Cavity_histogram_by_class.eps}\n\\caption{Histograms of the cavity opening angles of the best fits for the different \nSED classes. \n\\label{Cav_class_histo}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.65, angle=90]{Ltot_histogram_by_class.eps}\n\\caption{Histograms of the total luminosity of the best fits for the different \nSED classes. \n\\label{Ltot_class_histo}}\n\\end{figure*}\n\nFor the inclination angles (Figure \\ref{Inc_class_histo}), the distributions \nare significantly different for all protostellar classes, too (K-S test significance \nlevel $\\ll$ 0.01). As was shown in Figure \\ref{Inc_histo}, a random distribution of \ninclination angles would result in equal numbers of protostars at each value; \nthere is a deficit of Class 0 and Class I protostars at $i \\lesssim$ 60\\degr, and \nthere are also few Class I protostars and hardly any flat-spectrum sources \nat the highest inclination angles. \nThe median inclination angle is highest for Class 0 protostars (70\\degr), then \ndecreases somewhat for Class I protostars (63\\degr) and even more for \nflat-spectrum sources (57\\degr). Similar to the envelope density, \nthe median inclination angle decreases as one progresses from Class 0 to \nflat-spectrum sources. \n\nIn the distributions of cavity opening angles (Figure \\ref{Cav_class_histo}), \nsignificant differences can be found between Class 0 and Class I protostars \nand between Class I protostars and flat-spectrum sources (K-S test significance \nlevel $\\ll$ 0.01). The median cavity opening angle is 15\\degr\\ for the Class I \nprotostars, but 25\\degr\\ for the other two classes. About 40\\% of Class I protostars \nhave $\\theta$=5\\degr, while the distribution among the different cavity opening \nangles is flatter for the other two object classes. The large fraction of Class I\nprotostars with small cavities could be the result of degeneracy in model parameters \n(see section \\ref{SED_class_properties}) or our assumptions on envelope geometry \n(see section \\ref{Model_problems}). There are notably few flat-spectrum sources \nwith a 5\\degr\\ cavity opening angle; most of them have cavity opening angles of \n15\\degr\\ or 45\\degr.\n\nWhen comparing the total luminosities for the different SED classes\n(Figure \\ref{Ltot_class_histo}), the distribution of $L_{tot}$ values\nis different for the Class 0 protostars when compared to the other two\nclasses (K-S test significance level $<$ 0.015), but similar for Class I \nprotostars and flat-spectrum sources. The median total luminosity for \nClass 0 protostars is 5.5 $L_{\\odot}$, compared to 2.0 $L_{\\odot}$\\ for Class I \nprotostars and 3.0 $L_{\\odot}$\\ for flat-spectrum sources. Both Class 0 and I \nprotostars cover close to the whole range of $L_{tot}$ values in the model grid \n($\\sim$ 0.06-600 $L_{\\odot}$), while flat-spectrum sources span a more limited \nrange, from 0.1 to 316 $L_{\\odot}$.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.65, angle=90]{Rdisk_histogram_by_class.eps}\n\\caption{Histograms of the disk radii of the best fits for the different \nSED classes. \n\\label{Rdisk_class_histo}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.65, angle=90]{AV_histogram_by_class.eps}\n\\caption{Histograms of the foreground extinction of the best fits for the different \nSED classes. \n\\label{AV_class_histo}}\n\\end{figure*}\n\nThe distribution of centrifugal radii for the whole sample showed a preference\nfor 5 AU (see Figure \\ref{Rdisk_histo}). When separating the best-fit disk\nradii by protostellar class (Figure \\ref{Rdisk_class_histo}), it is clear that the\ntrend for small centrifugal radii is driven by the flat-spectrum sources \nand also Class I protostars. The fraction of Class 0 protostars with $R_{disk}$=\n5 AU is 17\\%; it increases to 46\\% and 73\\% for Class I protostars and \nflat-spectrum sources, respectively. The median disk radius decreases \nfrom 100 AU for Class 0 protostars to 50 AU for Class I protostars to 5 AU for \nflat-spectrum sources. All three histograms are significantly different from \none another (K-S test significance level $\\ll$ 0.001). The unexpectedly small\ncentrifugal radii for Class I protostars and flat-spectrum sources could point\nto parameter degeneracies (see section \\ref{SED_class_properties}) or the\nneed to revise certain model assumptions (see section \\ref{Model_problems}).\n\nFinally, the distribution of best-fit foreground extinction values (Figure \n\\ref{AV_class_histo}) is similar for all three object classes (K-S test significance \nlevel $>$ 0.03). Even the median values are close: $A_V$=9.2 for Class 0 \nprotostars, $A_V$=8.9 for Class I protostars, and $A_V$=10.1 for flat-spectrum\nsources. Most objects are fit with relatively low foreground extinction values.\nAs can be seen from Figure \\ref{AV_models_maps}, the majority of protostars\nhave best-fit $A_V$ values well below the maximum $A_V$ values determined\nfrom column density maps, which were used as the largest allowed $A_V$\nvalues for the SED fitter. The ratio of model-derived $A_V$ to observationally\nconstrained maximum $A_V$ is lower than 0.5 for about 60\\% of the sample.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.53]{AV_model_AV_maps.eps}\n\\caption{Foreground extinction values $A_V$ from the best-fit models versus\nthe maximum $A_V$ value determined from column density maps of Orion.\nThe dashed line indicates where the two $A_V$ values are equal.\n\\label{AV_models_maps}}\n\\end{figure}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.65, angle=90]{Rho1000_vs_AV_by_class.eps}\n\\caption{Best-fit $\\rho_{1000}$ values versus the foreground extinction for the different \nSED classes. Note that there are a few objects at $A_V > 75$, but they are not shown\nfor overall clarity of the figure.\n\\label{Rho_AV_class}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.65, angle=90]{Rho1000_vs_inc_by_class.eps}\n\\caption{Best-fit $\\rho_{1000}$ values versus inclination angle for the different \nSED classes. The size of the plotting symbol increases with the number of objects\nhaving the same ($i$, $\\rho_{1000}$) combination; the legend in the leftmost panel\nshows which symbol size corresponds to which number of objects. \n\\label{Rho_inc_class}}\n\\end{figure*}\n\nIn Figure \\ref{Rho_AV_class}, we plot the reference densities $\\rho_{1000}$\nversus the foreground extinction for Class 0, Class I, and flat-spectrum sources.\nAs was already seen in Figure \\ref{AV_class_histo}, the extinction along the line\nof sight is similar for all three classes, with most objects in the $A_V \\sim$ 0-30\nregime. Class 0 protostars, which have higher envelope densities, tend to \nhave lower $A_V$ values from foreground extinction; the highest-density\nenvelopes are spread among a wide range of $A_V$ values. The result is\nsimilar for Class I protostars. Flat-spectrum sources display a range in \nenvelope densities at various foreground extinction values; the lowest-density\nenvelopes typically have $A_V < 20$.\nThus, foreground extinction does not seem to affect the classification of\nprotostars. This result is also supported by the statistical analysis of \\citet{stutz15},\nwho found that, for $A_V$ values up to 35, the misclassification of a Class I\nprotostar as a Class 0 protostar due to foreground extinction (which results in a\nlower $T_{bol}$) is low.\n\n\\begin{deluxetable*}{l|ccc}\n\\tablewidth{0.9\\linewidth}\n\\tablecaption{Median Best-Fit Parameter Values for the Three Protostellar Classes\n\\label{Median_par}}\n\\tablehead{\n\\colhead{Parameter} & \\colhead{Class 0} & \\colhead{Class I} & \n\\colhead{Flat-spectrum}}\n\\startdata\n$L_{tot}$ & 5.5 $L_{\\odot}$ & 2.0 $L_{\\odot}$ & 3.0 $L_{\\odot}$ \\\\\n$\\rho_{1000}$ & 5.9 $\\times 10^{-18}$ g cm$^{-3}$ & \n2.4 $\\times 10^{-19}$ g cm$^{-3}$ & 1.2 $\\times 10^{-19}$ g cm$^{-3}$ \\\\\n$\\theta$ & 25\\degr & 15\\degr & 25\\degr \\\\\n$R_{disk}$ & 100 AU & 50 AU & 5 AU \\\\\n$i$ & 70\\degr & 63\\degr & 57\\degr \\\\\n$A_V$ & 9.2 & 8.9 & 10.1 \\\\\n\\enddata\n\\end{deluxetable*}\n\n\\begin{figure*}[!]\n\\centering\n\\includegraphics[scale=0.5, angle=90]{Median_model_SEDs.eps}\n\\caption{Model SEDs for Class 0 protostars ({\\it red}), Class I protostars\n({\\it green}), and flat-spectrum sources ({\\it blue}) with parameter values \nequal to the median values for each SED class (see Table \\ref{Median_par}).\n\\label{median_SEDs}}\n\\end{figure*}\n\nWe found differences in the best-fit envelope densities and inclination angles\nfor the various protostellar classes.\nThe result that Class 0 protostars tend to have larger inclination angles and \nenvelope densities compared to Class I and flat-spectrum objects can also\nbe seen in Figure \\ref{Rho_inc_class}. There are very few Class 0 protostars\nwith low inclination angles; most have relatively high density and $i>$60\\degr. \nClass I protostars are best fit by somewhat lower inclination angles than Class 0 \nprotostars and also lower $\\rho_{1000}$ values. The best-fit reference density\nfor Class I protostars decreases as the inclination angle increases; thus, higher-density\nprotostars are typically classified as Class I protostars only if they are not seen at\nclose to edge-on orientations.\nFlat-spectrum sources are spread out in density--inclination space, but intermediate \ninclination angles and low envelope densities are common. There is a relatively\nlarge number of objects at $i=$18\\degr\\ and a deficit of objects at high inclination\nangles. The highest-density flat-spectrum sources are seen at inclination angles \n$<$ 50\\degr, while the lower-density objects cover almost the full range of \ninclination angles.\n\nThe median parameter values we determined from the best fits for the \nClass 0, Class I, and flat-spectrum sources (see Table \\ref{Median_par}) \ncan be used to show representative median SEDs for each protostellar class. \nIn Figure \\ref{median_SEDs}, we show model SEDs whose parameter values \nare equal to the median values found for each of the three protostellar classes. \nIt is apparent that the large envelope density and higher inclination angle for \nClass 0 protostars cause a deep absorption feature at 10 $\\mu$m and a steeply \nrising SED in the mid- and far-IR, with a peak close to 100 $\\mu$m. In Class I \nprotostars, the SED is less steep and peaks at a shorter wavelength than \nthe median SED of Class 0 protostars. Flat-spectrum sources show the \nstrongest near-IR emission of the three protostellar classes; their median\nSED is very flat out to 70 $\\mu$m, but at longer wavelengths it is very \nsimilar in shape and flux level to that of Class I protostars.\n\n\n\\subsection{Estimating Parameter Uncertainties}\n\\label{deltaR}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.69, angle=90]{Inc_modes.eps}\n\\caption{Mode of the inclination angle of all models that lie within \n0.5, 1.0, 1.5, and 2 of the best-fit $R$ value (from left to right) versus \nthe best-fit inclination angle for all 330 objects in our sample. \nNote that for each data point, small random offsets in the x and y direction \nhave been applied to avoid overlap. Also, when two or more parameter \nvalues had the same frequency within a $\\Delta R$ bin (i.e., not a unique \nmode value), we computed the average of these values and used it for \nthe mode. The dashed line indicates where the mode and best-fit value \nare equal.\n\\label{inc_modes}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.69, angle=90]{Ltot_modes.eps}\n\\caption{Mode of the total luminosity of all models that lie within \n0.5, 1.0, 1.5, and 2 of the best-fit $R$ value (from left to right) versus \nthe best-fit total luminosity for all 330 objects in our sample. \n\\label{lum_modes}}\n\\end{figure*}\n\nGiven that the $R$ values are a measure of the goodness of fit in units of\nthe fractional uncertainty, we can use models that lie within a certain range\nof the best-fit $R$ value to estimate uncertainties for the various model \nparameters. For each modeled HOPS target, we tabulated the model\nparameters for all those models that lie within a difference of 0.5, 1.0, \n1.5, and 2.0 of the best-fit $R$. We then computed the mode (i.e., the \nvalue with the highest frequency) for the inclination angle, total luminosity, \n$\\rho_{1000}$, cavity opening angle, outer disk radius, and $A_V$ in \neach of the $\\Delta R$ bins for each object. \nFor any given protostar, the models in each $\\Delta R$ bin span certain \nranges in parameter values; while the modes do not capture the full extent of \nthese ranges, they convey the most common value within each parameter \nrange. The farther away a mode is from the best-fit value, the more poorly \nconstrained the model parameter. Conversely, if a mode of a certain\nparameter is close to or matches the best-fit value, especially for $\\Delta R=$ \n1.5 or 2, that particular model parameter is well constrained.\nIn Figures \\ref{inc_modes} to \\ref{AV_modes} we plot the mode versus the best-fit \nvalue for six model parameters and four $\\Delta R$ bins for all 330 targets \nin our sample. The larger $\\Delta R$, the larger the spread in modes is expected \nto be for each parameter value. \n\nFor example, Figure \\ref{inc_modes} shows that even when considering \nall models with an $R$ value of up to 2 larger than the best-fit $R$ \n($\\Delta R=$ 2), the inclination angle for objects with a best-fit $i$ of \n18\\degr\\ is well constrained; most modes lie at $i=$ 18\\degr, too, and \nonly a few modes can be found at larger inclination angles. However, \nobjects with best-fit $i$ values of 32\\degr\\ or 41\\degr\\ typically can also \nbe fit by models with lower inclination angles (the majority of modes lies \nbelow the line where mode and best-fit value are equal). Inclination \nangles $\\gtrsim$ 63\\degr\\ are better constrained, since their modes \nmostly lie at high inclination angle values, but there are protostars \nwith modes of $i=$18\\degr, too.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.69, angle=90]{Rho1000_modes.eps}\n\\caption{Similar to Figure \\ref{inc_modes}, but for the reference density\n$\\rho_{1000}$.\n\\label{rho1000_modes}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.69, angle=90]{Cavity_modes.eps}\n\\caption{Similar to Figure \\ref{inc_modes}, but for the cavity opening\nangle.\n\\label{cavity_modes}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.69, angle=90]{Rdisk_modes.eps}\n\\caption{Similar to Figure \\ref{inc_modes}, but for the outer disk radius ($=R_c$).\n\\label{Rdisk_modes}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.69, angle=90]{AV_modes.eps}\n\\caption{Similar to Figure \\ref{lum_modes}, but for the foreground\nextinction.\n\\label{AV_modes}}\n\\end{figure*}\n\nThe modes for the total luminosity (Figure \\ref{lum_modes}) show a small \nspread for models within $\\Delta R$=0.5, but the spread increases as $R$ \nincreases, with some objects displaying up to an order of magnitude in \nvariation of $L_{tot}$.\nAs illustrated in Figure \\ref{rho1000_modes}, the reference density \n$\\rho_{1000}$ is usually well constrained; however, as $R$ increases, \nthe modes of the $\\rho_{1000}$ values are often lower than the best-fit \nvalues.\nFor the cavity opening angle (Figure \\ref{cavity_modes}), many models up \nto $\\Delta R$=2 have modes of $\\theta$=45\\degr, independent of the \nbest-fit value. Similarly for the centrifugal radius (Figure \\ref{Rdisk_modes}), \n$R_{disk}$=500 AU is a common mode. For all four disk radii, the modes \ntend to be larger than the best-fit values; in particular, objects with a best-fit \n$R_{disk}$ of 5 AU have a very uncertain disk radius. In general, it looks like \nour models do not constrain the disk radius and cavity opening angle well.\nThe foreground extinction (Figure \\ref{AV_modes}) displays a certain range \nof modes for each best-fit value, but objects with $A_V \\lesssim$ 20 typically\nhave more reliable $A_V$ values from their model fits. \n\nFigures \\ref{inc_modes} to \\ref{AV_modes} allow us to gauge general trends \nbetween best-fit values and modes for different model parameters. For results \non individual objects, we refer to Appendix \\ref{models_unc}, where we show \nplots of the difference between the modes and the best-fit values of the major \nmodel parameters for all modeled HOPS targets. In this way it is possible to \nestimate which models are better constrained and thus which objects have \nmore reliable SED fits. \nIn addition, in Appendix  \\ref{models_unc} we also include contour plots of \n$R$ values for different pairs of model parameters for a few targets to \nillustrate typical parameter degeneracies, which also contribute to parameter\nuncertainties.\n\n\n\\section{Discussion} \n  \n\\subsection{Deriving Envelope Parameters from a Model Grid}\n\nWe compared a grid of TSC models to each target by ranking the models \nusing a statistic, $R$, which measures the deviation between observed and  \nmodel fluxes in logarithmic space. We did not model each source by further \nadjusting the model parameters, but instead identified the best-fit SED from \nour model grid. Thus, we are bound by the range and sampling of parameters \nchosen for the grid, and while we constructed the grid with the aim of covering \nthe typical parameter space for protostars, it is limited to discrete values. It is \nlikely that many protostars have best-fit parameters that would fall between those \nsampled by the grid, and a few objects could have parameter values that lie \nbeyond the limits set by the grid. In addition, TSC models are axisymmetric \nand have mostly smooth density and temperature profiles, and they do not\ninclude external heating. They assume a rotating, infalling envelope with \nconstant infall rate and with the gravitational force dominated by the central\nprotostar, but the true envelope structure is likely more complex. \nThe models would not apply to the collapse of a cloud in an initial filamentary \nor sheet-like geometry or to multiple systems with, e.g., more than one \noutflow cavity \\citep[e.g.,][]{hartmann96,tobin12}.\n\nDespite the relatively simple models that we use, many of the observed SEDs are \nfit remarkably well: 75\\% of the fits have $R<4$. In those cases, the continuum \ntraced by the IRS spectrum, the silicate absorption feature at 10 $\\mu$m, and \nthe PACS fluxes are all accurately reproduced by the model. Even many \nflat-spectrum sources, which often do not display any spectral features in \nthe mid-infrared and have an overall flat SED out to 30 or 70 $\\mu$m, often \nhave models that fit them very well. About 75\\% of Class I protostars and  $\\sim$ \n70\\% of Class 0 protostars have $R < 4$, while close to 90\\% of flat-spectrum \nsources have $R$ values in this range. This validates the choice of parameter \nvalues for our model grid.  \nAdditional constraints, like limits on foreground extinction or information on the \ninclination and cavity opening angles from scattered light images or mapping of \noutflows, would allow us to further test and refine the models.  We have used limits \non the extinction in our analysis.  Although {\\it Hubble Space Telescope (HST)} \nscattered light images have been used to constrain models for one HOPS \nprotostar \\citep{fischer14}, scattered light images are not available for many \nof our targets. We therefore chose to focus on fitting the SEDs of all of our \ntargets in a uniform way to a well-defined set of models. Future studies will \nincorporate scattered light images and compare the results to those from \nthe SED fits (J. Booker et al. 2016, in preparation).\n\nThe best-fit models from our grid for the HOPS targets both reproduce the SEDs \nand yield estimates for their protostellar parameters, mostly envelope properties.  \nHowever, these are not necessarily unique fits to the data for three reasons. \nFirst, there are degeneracies in the model parameters; increasing the envelope \ndensity or inclination angle, or decreasing the cavity opening angle or disk radius, \nresults in a steeper mid-IR SED slope and deeper silicate features. Each of these \nparameters affects the SED differently (just the general trends are the same), and \nthe best fit for each object tries optimizing them. The next best fit, however, could \nbe a different combination of these parameters, especially if the SED is not \nwell constrained by observations (see Section \\ref{deltaR}).  \nSecond, although the TSC models reproduce the observed SEDs, other models \nwith different envelope geometries may also be able to fit the same SEDs. The \nmodeling presented here is only valid in the context of the TSC models of single \nstars, and the resulting derived properties are only valid within that framework.   \nLast, neglecting external heating could result in overestimated envelope densities\nand luminosities, with the most noticeable effects ($\\rho_{1000}$ and $L_{tot}$ too \nlarge by factors of a few) on low-luminosity sources exposed to strong radiation \nfields (see Section \\ref{model_ext_heat}). From the distribution of best-fit $L_{tot}$ \nvalues, we estimate that $\\sim$ 20\\% of HOPS targets in our sample could be \naffected by external heating. Even though we do not know the strength of \nexternal heating for each protostar, it is likely that external heating would only \nresult in relatively small changes in the derived envelope parameters for these \nprotostars.\n\n\n\\subsection{Envelope Properties and SED Classes}\n\\label{SED_class_properties}\n\nWhen comparing envelope parameters sorted by SED classes, we found that \nenvelope densities and inclination angles decrease from the sample of\nClass 0 protostars through that of Class I protostars to that of flat-spectrum objects. \nThe former is likely an evolutionary effect, while the latter confirms the results of \nprevious work \\citep[e.g.,][]{evans09} that  the inclination angle has an important \neffect on the SED and that the evolutionary state of an object cannot be derived \nfrom SED slopes alone. Thus, there is a difference between the ``stage'' and \n``class'' of an object \\citep{robitaille06}; Stage 0 and I objects are characterized \nby substantial envelopes, Stage II objects are surrounded by optically thick disks, \nwith possibly some remnant infalling envelopes, and Stage III objects have \noptically thin disks. \n\nIn general, the trends we see among model parameters are a consequence \nof the definition of a protostar based on its SED: in order to be classified \nas a Class 0 or I object, a protostar is required to have a near- to mid-infrared \nSED slope larger than 0.3. A protostellar  model with a small cavity opening \nangle, small centrifugal radius, and\/or high inclination angle will generate \nsuch an SED, since it increases the  optical depth along the line of sight. \nModels with a large cavity will only yield a rising SED in the 2$-$40 $\\mu$m \nspectral range if their  envelope density is large or the inclination angle is \nrelatively high. \n\nWe find that Class 0 protostars can be best fit not only by very high envelope \ndensities but also moderately high envelope densities and large inclination \nangles. The bolometric temperature, which is used to separate \nClass~0 from Class~I protostars, is inclination dependent; some Class~I \nprotostars are shifted to the Class~0 regime if they are viewed more edge-on. \nThe higher-density Class~I protostars tend to have lower inclination angles (but \nstill $>$ 50\\degr); thus, their evolutionary stage could be similar to more \nembedded protostars that are seen edge-on and classified as Class 0 protostars. \nConversely, some Class~0 objects with large inclination angles, but lower \nenvelope densities, could be in a later evolutionary stage than typical Class~0 \nprotostars. Similarly, Class I protostars with large $i$ and low $\\rho_{1000}$ \nvalues could be edge-on Stage II objects (whose infrared emission is dominated \nby a disk). Finally, low-inclination Stage 0 and I protostars can appear \nas a flat-spectrum sources \\citep{calvet94}.\n\n\\begin{figure*}[!]\n\\centering\n\\includegraphics[scale=0.58, angle=90]{Rho1000_vs_inc_cavity.eps}\n\\caption{Best-fit $\\rho_{1000}$ values versus inclination angle with the cavity\nsize indicated by the different symbol sizes and gray shades: symbols become\nlarger and lighter colored with increasing cavity size (5\\degr, 15\\degr, 25\\degr,\n35\\degr, 45\\degr). A small random offset in the x direction has been applied to \neach data point to prevent too much overlap.\n\\label{Rho1000_inc_cavity}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.58, angle=90]{Rho1000_vs_inc_Rdisk.eps}\n\\caption{Similar to Figure \\ref{Rho1000_inc_cavity}, but with the outer disk \nradius indicated by the different symbol sizes and gray shades: symbols \nbecome larger and lighter colored with increasing disk radius (5, 50, 100,\n500 AU).\n\\label{Rho1000_inc_Rdisk}}\n\\end{figure*}\n\nNevertheless, the observed trend in envelope densities suggests that the variations \nin the observed SEDs track, in great part, an evolution toward lower envelope \ndensities and lower infall rates. Assuming a certain mass for the central \nstar, the reference density in our models can be used to infer an envelope infall rate\n($\\dot{M}_{env} \\propto \\rho_{1000} \\sqrt{M_{\\ast}}$). \nAs mentioned in section \\ref{model_parameters}, this infall rate is model dependent \nand therefore tied to the assumptions of the models. With this in mind, the median \n$\\rho_{1000}$ values for the Class 0, Class I, and \nflat-spectrum protostars in our sample correspond to envelope infall rates of \n$2.5 \\times 10^{-5}$, $1.0 \\times 10^{-6}$, and $5.0 \\times 10^{-7}$ $M_{\\odot}$\\ yr$^{-1}$, \nrespectively, for a 0.5 $M_{\\odot}$\\ star. \nUsing a more realistic assumption of larger stellar mass for more evolved protostars, \nthe infall rates for Class I and flat-spectrum protostars would be larger than these \nvalues by a factor of a few. However, just larger stellar masses cannot explain the\nlarge decrease of a factor of 50 in the median envelope density from Class 0 to \nflat-spectrum protostars; to achieve such a decrease with a constant infall rate\nof $2.5 \\times 10^{-5}$ $M_{\\odot}$\\ yr$^{-1}$, the stellar mass would have to increase\nby a factor of 2500. Thus, within the context of our model fits, we can conclude that,\nas envelopes become more tenuous, the infall rates also decrease.\n\nOther trends are also apparent. Class 0 protostars and flat-spectrum sources show \na relatively flat distribution of cavity opening angles. On the other hand, the best \nfit for a large fraction of Class I protostars (40\\%) results in $\\theta$=5\\degr. \nThis could point to a  degeneracy in the models, since protostars with small cavity \nopening angles tend to have lower envelope densities (and also lower total \nluminosities); thus, the smaller cavity partly compensates for the lower opacity \nresulting from the lower envelope density (see also Figure \\ref{Rho1000_inc_cavity}). \n\nEven though our models do not yield reliable disk properties, we can make a\nstatement about the difference in the best-fit centrifugal radii (or $R_{disk}$),\nwhich are tied to the structure of the rotating envelope given by the model fits.  \nIt should be noted that the centrifugal radii set a lower limit to the disk radii, \nsince disks may spread outward due to viscous accretion.  Most Class I protostars \nand flat-spectrum sources are fit with a centrifugal radius of just 5 AU. Since \nthe smallest centrifugal radius in our model grid is 5 AU and the next value is 50 AU, \nwe can state that, except for Class 0 protostars, most protostars in our sample \nhave $R_{disk} < 50$~AU, and some might even have $R_{disk} <$ 5~AU.\n\nSmall disks of those sizes have been observed; radio interferometry  of the \nmultiple protostellar system L1551 IRS 5 shows disks whose semi-major axes \nare $\\lesssim$~20~AU \\citep{rodriguez98,lim06}. However, there is a degeneracy \nbetween the centrifugal radius and the envelope density; for protostars with low \nenvelope densities, the small centrifugal radius can compensate the decrease in \nopacity by concentrating more material closer to the star. As can be seen in Figure \n\\ref{Rho1000_inc_Rdisk}, most protostars with $R_{disk}=$ 5 AU also have lower \nenvelope densities. Inclination angle also plays a role; protostars seen at \n$i>$ 80\\degr\\ typically have larger centrifugal radii. In addition, our envelope\nmodels include outflow cavities, which allow some of the mid-IR radiation to\nescape. In order to generate model SEDs with silicate absorption features \nand steep mid-IR slopes at low to intermediate inclination angles, a lower \n$R_{disk}$ value is needed.  \nWe will discuss the potential implications of the small cavity sizes and $R_c$ \nvalues for Class I protostars and flat-spectrum sources in Section \n\\ref{Model_problems}.\n\n\n\\subsubsection{The Most Embedded Protostars}\n\\label{Class0}\n\nAmong the Class 0 protostars, there are protostars in the earliest evolutionary \nstages, when the envelope is massive and the protostar still has to accrete \nmost of its mass. \n\\citet{stutz13} identified 18 protostars with very red mid- to far-infrared\ncolors ($\\log(\\lambda F_{\\lambda}(70)\/\\lambda F_{\\lambda}(24))$\n$>$ 1.65), of which 11 were newly identified (see Table D\\ref{New_proto}). \n\\citet{tobin15} added an additional object.\nThese protostars were named PACS Bright Red sources (PBRs) by \\citet{stutz13}; \nthey are HOPS 169, 341, 354, 358, 359, 372, 373, 394, 397-405, 407, and 409. \nBased on their steep 24-70 $\\mu$m SEDs and large submillimeter luminosities, \nthey were interpreted as the youngest protostars in Orion with very dense \nenvelopes.\n\nFrom our best-fit models to the SEDs of the PBRs, we derive a median \n$\\rho_{1000}$ value of 1.2 $\\times$ 10$^{-17}$ g cm$^{-3}$, \nwhich is twice as high as the median value of all the Class 0 protostars in \nour sample. These fits also result in a median envelope mass within 2500 AU \nof 0.66 $M_{\\odot}$\\ for the PBRs, but the individual objects cover a large \nrange, from 0.07 to 1.83 $M_{\\odot}$. The median total luminosity amounts to \n5.6 $L_{\\odot}$\\ (with a range from 0.6 to 71.0 $L_{\\odot}$), which is very similar\nto the median $L_{tot}$ value for the Class 0 protostars in our sample. \nMost PBRs (14 out of 19 protostars) are fit by models with large inclination \nangles ($i \\geq$ 70\\degr), but, as shown in \\citet{stutz13}, high inclination\nalone cannot explain the redness of the PBRs. \nThus, our models confirm the results of \\citet{stutz13} that the PBRs\nare deeply embedded and thus likely among the youngest protostars\nin Orion.\n\n\n\\subsubsection{Flat-spectrum Sources}\n\\label{flat-spectrum}\n\nA particularly interesting group of protostars that are not easy to categorize\nare the flat-spectrum sources. They are thought to include objects in transition \nbetween Stages I and II, when the envelope is being dispersed \\citep{greene94}. \nIn particular, those with low envelope densities could be more evolved protostars, \nor they could be protostars that started out with more tenuous envelopes. \nOn the other hand, flat-spectrum sources could also be highly inclined \ndisk sources \\citep[see][]{crapsi08}, or protostars surrounded by dense \nenvelopes, but seen close to face-on \\citep{calvet94}. This type of \nmisclassification could have a large effect on the lifetimes of the earlier \nprotostellar stages and thus on the timeline of envelope dispersal. \nAmong the 330 HOPS targets in our sample, we identified 102 flat-spectrum \nsources based on their flat ($-0.3$ to $+0.3$) spectral index from 4.5 to 24 \n$\\mu$m (or 5-25 $\\mu$m in a few cases). Thus, they compose a fairly large \nfraction of our protostellar sample. Of these 102 objects, 47 have a negative \nspectral index and 55 have one between 0 and $+0.3$; 41 have a spectral \nindex between $-0.1$ and 0.1, which results in a very flat mid-infrared SED. \n\nDespite a flat SED slope between 4.5 and 24 $\\mu$m, many flat-spectrum\nsources display a weak silicate emission or absorption feature at 10 $\\mu$m,\nwhich may indicate the presence of a very tenuous infalling envelope or\nmay be the result of the viewing geometry. Some SEDs are very flat out to \n100 $\\mu$m, others have negative spectral slopes beyond 40 $\\mu$m, \nand again others a rising SED from the mid- to the far-IR.\nThere are also objects with more pronounced absorption features due to \nnot only silicates but also ices, as are typically found in Class 0 and I protostars,\nbut also edge-on disks (see HOPS 82, 85, 89, 90, 92, 129, 150, 200, 210, 211, \n281, 304, 331, and 363). Only two flat-spectrum sources have prominent silicate \nemission features, and their SEDs are reminiscent of protoplanetary disks \n(see HOPS 187 and 199). Thus, flat-spectrum sources likely include objects of a \nvariety of evolutionary stages.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.39, angle=90]{Median_Rho1000_vs_inc.eps}\n\\caption{Median best-fit $\\rho_{1000}$ values at each inclination angle \nfor the Class 0 and I protostars ({\\it squares}) and the flat-spectrum sources\n({\\it circles}) in our sample. \n\\label{Rho_inc_flat-spectrum}}\n\\end{figure}\n\nThe latter conclusion can also be drawn when analyzing the distribution\nof envelope reference densities and inclination angles for flat-spectrum\nsources. In Figure \\ref{Rho_inc_class}, we showed that flat-spectrum\nsources typically have intermediate inclination angles and lower envelope\ndensities. To compare their properties more directly to Class 0 and I protostars,\nin Figure \\ref{Rho_inc_flat-spectrum} we show the median best-fit \n$\\rho_{1000}$ value at each best-fit inclination angle; it is larger for Class 0\nand I protostars than for flat-spectrum sources at all inclination angles.\nFor Class 0 and I protostars, the median $\\rho_{1000}$ value is highest\nat intermediate inclination angles, decreases at larger inclination angles,\nand then increases again for $i>$ 80\\degr. For flat-spectrum sources, the \nmedian $\\rho_{1000}$ value is relatively flat over the 18\\degr--63\\degr\\\nregion but has its peak value at $i=$41\\degr; it decreases for larger \ninclination angles. The only flat-spectrum source with a best-fit inclination \nangle of 81\\degr, HOPS 357, has a very low envelope density (the lowest \nvalue for this parameter in the model grid), and its spectrum displays a \ndeep silicate absorption feature.\n\nOverall, this shows that, while a range of envelope densities and inclination \nangles can explain flat-spectrum sources, their envelope densities are typically\nlower than for Class 0 and I protostars. The higher-density objects are seen at low\nto intermediate inclination angles, while only the lowest-density objects\nare seen closer to edge-on. Some of the high-density flat-spectrum sources\ncould actually be more embedded protostars (Stage 0 objects) seen\nface-on (which would be classified as Class 0 objects if seen at larger\ninclination angles). Thus, in terms of envelope evolution, they include a\ndiverse group of objects. \n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.71, angle=90]{Rho1000_vs_F350.eps}\n\\caption{Best-fit $\\rho_{1000}$ values versus the 350 $\\mu$m fluxes\nfor the Class 0 and I protostars ({\\it left}) and the flat-spectrum sources\n({\\it right}) in our sample. Detections at 350 $\\mu$m are shown with \ndiamonds, while upper limits are shown with arrows. The histograms \nshow the distribution of best-fit $\\rho_{1000}$ values for sources with \na 350 $\\mu$m flux measurement ({\\it thick solid line}) and with 350 \n$\\mu$m upper limits ({\\it shaded area}). \n\\label{Rho_F350}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.71, angle=90]{Rho1000_vs_F870.eps}\n\\caption{Similar to Figure \\ref{Rho_F350}, but for the 870 $\\mu$m fluxes.\n\\label{Rho_F870}}\n\\end{figure*}\n\nWe note that even though we find that flat-spectrum sources have in general\nlower envelope densities than Class 0 and Class I objects, their best fit does\ninclude an envelope in almost all cases; just 3 of the 102 flat-spectrum \nsources are best fit without an envelope. This seems to contradict recent\nfindings by \\citet{heiderman15}, who found that only about 50\\% of flat-spectrum\nsources were actually protostars surrounded by envelopes. This could be partly\nexplained by different criteria used to select flat-spectrum sources; in the \n\\citet{heiderman15} sample, flat-spectrum sources are selected by their\nextinction-corrected 2-24 $\\mu$m spectral index (see also \\citealt{evans09,\ndunham13}), while our sample uses a flat 4.5-24 $\\mu$m spectral index.\nMoreover, in their study \\citet{heiderman15} detected the presence of an \nenvelope via HCO$^+$ emission, and they found that almost all sources \ndetected in the sub-mm are also detected in HCO$^+$ (but the opposite \ndoes not always hold). For our sample of Orion protostars, we find that \n75\\% of Class 0+I protostars observed with SABOCA (350 $\\mu$m) are \ndetected, while only 47\\% of flat-spectrum sources have detections. For \nLABOCA observations (870 $\\mu$m), these two fractions amount to \n41\\% and 21\\%, respectively. Thus, we find that flat-spectrum sources \nhave a $\\sim$ 50\\% lower sub-mm detection rate than Class 0+I protostars. \nFlat-spectrum sources without sub-mm detections would likely also not \ndisplay HCO$^+$ emission and thus would be considered as protostars \nwithout envelopes by \\citet{heiderman15}.\n\nTo compare how our submillimeter detections correlate with the presence\nof an envelope, in Figures \\ref{Rho_F350} and \\ref{Rho_F870} we show the\nderived best-fit reference envelope densities as a function of 350 or 870 $\\mu$m\nfluxes for the combined Class 0+I sample and the flat-spectrum sources. We\nalso differentiate the distribution of envelope densities between measured\nflux values and upper limits; at 870 $\\mu$m, the upper limits are often cases\nwhere the sources are not detected due to confusion with bright, spatially\nvarying emission. We find that even protostars with upper limits at 350 and \n870 $\\mu$m are best fit with an envelope; however, the envelope density \nis lower for objects with upper limits in the sub-mm. This is especially evident \nfor Class 0+I protostars; for flat-spectrum sources, the distributions of envelope \ndensities for sub-mm detections and upper limits show  significant overlap. Four \ntimes as many flat-spectrum sources have upper limits instead of detections \nat 870 $\\mu$m, but their derived $\\rho_{1000}$ values span almost the full \nrange of values. Furthermore, the median $\\rho_{1000}$ value of 1.19 $\\times \n10^{-19}$ g cm$^{-3}$ for sources without detections is relatively close to the \nmedian value of 5.95 $\\times 10^{-19}$ g cm$^{-3}$ for the sources with \n870 $\\mu$m detections. Thus, our model fits do not rely on sub-mm detections \nto yield a best fit with an envelope; in most cases the near- to far-IR SED is \nsufficient to constrain the properties of the envelope.\n\n\n\\subsubsection{Sources without an Envelope and Class II Objects}\n\\label{disk_sources}\n\nAmong the six objects whose best-fit SED required no envelope ($\\rho_{1000}$\nvalue of 0), three are flat-spectrum sources (HOPS 47, 187, 265), two are Class \nII pre-main-sequence stars (HOPS 113, 293), and one is a Class I protostar (HOPS 232).  \nThe low 70 $\\mu$m fluxes of HOPS 47 and 265 constrained the best model to \none without an envelope. The SED of HOPS 187 looks like that of a transitional \ndisk, which are disks with gaps or holes in their inner regions (see \\citealt{espaillat14} \nand references therein). If HOPS 187 were a transitional disk, it would not have \nan envelope. HOPS 232 has a rising SED over the mid-IR spectral range; its best \nfit requires no envelope, but an edge-on disk with a high accretion luminosity.\n\nIt would be expected that the SEDs of Class II objects can be best fit by a \nmodel that does not include an envelope. This is the case for HOPS 113 and \n293. Of the nine remaining Class II objects in our sample, four have very \nlow envelope densities ($\\rho_{1000} \\sim (1-2.5) \\times 10^{-20}$ g cm$^{-3}$; \nHOPS 22, 26, 98, 283), while five have $\\rho_{1000}$ between $6.0 \\times \n10^{-20}$ and $1.8 \\times 10^{-19}$ g cm$^{-3}$ (HOPS 184, 201, 222, 272, 277).\nThe SEDs of HOPS 22, 184, and 201 are similar to those of transitional disks,\nwith some silicate emission at 10 $\\mu$m and a rising SED between about\n13 and 20 $\\mu$m. The best-fit models require some envelope emission\nto fit the long-wavelength data. \nHOPS 222, 272, and 277 lie close to the border between a Class II \npre-main-sequence star and a flat-spectrum source based on their \n4.5-24 $\\mu$m spectral index, and therefore they could have some \nenvelope material left, despite being classified as Class II objects.\n\nOverall, of the 330 YSOs in our sample, 319 were classified as\neither Class 0, Class I, or flat-spectrum protostars based on their SEDs. \nHowever, four of them are best fit without an envelope. Conversely, of the\n11 Class II objects in our sample, nine are best fit with an envelope; however,\nthree of these might be transitional disks. Thus, based on our model fits and\nSEDs, 321 of our 330 YSOs are protostars with envelopes, and nine are likely\npre-main-sequence stars with disks.\n\n\\clearpage\n\n\\subsection{The Total Luminosities of Protostars}\n\nThe luminosity distribution of protostars is a significant constraint on \nprotostellar evolution, and it is important to understand the effect of the \nenvelope on the observed luminosity \\citep[e.g.,][]{offner11}. The bolometric \nluminosity distribution of the HOPS protostars is very similar to that determined \nfor the {\\it Spitzer}-identified protostars by \\citet{kryukova12} with a \npeak near 1~$L_{\\odot}$\\ (Fig.\\ \\ref{HOPS_n_Tbol_Lbol_histo}). In contrast, the \ndistribution of the total luminosities from the models shows a peak near \n2.5~$L_{\\odot}$\\ (Fig.\\ \\ref{Ltot_class_histo}), indicating that the luminosities of \nprotostars may be systematically underestimated by the bolometric \nluminosities, which do not take into account the inclination angle (and thus\nbeaming of the radiation along the outflow cavities) as well as foreground\nextinction (see Fig.\\ \\ref{Ltot_Lbol} in section \\ref{results_overview}).\n\nHigher intrinsic luminosities for protostars could help address the \n``luminosity problem'' first pointed out by \\citet{kenyon90}, who found \nthat the luminosities of protostars are lower by about an order of magnitude \nthan a simple estimate of the expected accretion luminosity. However, \nan increase in the luminosity by a factor of 2.5-3 would not solve the \nproblem; solutions proposed by other authors, such as mass-dependent\naccretion rates \\citep{offner11} or episodic accretion events \\citep{dunham12}, \nare still needed.\n\nOur best-fit models also suggest that Class 0 protostars have a \ndifferent distribution of $L_{tot}$ values compared to Class I protostars or \nflat-spectrum sources. Their median total luminosity is higher, which could \nbe an indication of larger accretion luminosities for younger protostars. \nWe must bear in mind the caveats and degeneracies mentioned above; \nin particular, in some cases the higher luminosity could be related to the \nadoption of an overly large inclination angle, which results in most \nof the emitted radiation not reaching the observer. Nevertheless, these \ndifferences have potentially important implications for protostellar evolution, \nwhich will be discussed in a future publication (W. Fischer et al. 2016, in \npreparation).\n\n\n\\subsection{Potential Problems with TSC Models}\n\\label{Model_problems}\n\nAlthough the TSC models provide impressive fits to the SEDs, some of the \nobserved trends suggest problems with the models.  First, the distribution \nof inclination angles (Fig.\\ \\ref{Inc_histo}) deviates from what we expect from \na randomly oriented sample of protostars.  Although this could result from \nunintentional selection biases in our sample of protostars, it may also be the \neffect of applying the wrong envelope model to the data. \n\nFurthermore, our data show flat distributions in cavity opening angles \nfor Class~0 and flat-spectrum sources, but an excess of small cavities for the \nClass~I protostars (Figure \\ref{Cav_class_histo}). We also find that protostars \nwith large cavities often have high envelope densities (Figure \\ref{Rho1000_inc_cavity}).\nFor example, models with high envelope densities viewed more edge-on require \nlarge cavity opening angles and high $L_{tot}$ values to \ngenerate sufficient mid-IR flux; this is the case for a few of our highest-luminosity \nobjects (HOPS 87, 108, and 178). These trends do not support the notion of \nincreasing cavity size with later evolutionary stage, which would be expected if \noutflows play a major role in dispersing envelopes \\citep{arce06}. This may \nsuggest that cavity sizes are not growing with time; however, this may also imply  \na deviation from spherical symmetry for the initial configuration of the collapsing \nenvelopes.  Such a deviation may result if the envelope collapses from the \nfragmentation of a flattened sheet or elongated filament.  \n\nFinally, we find an excess of small values of $R_{disk}$, and therefore small \ncentrifugal radii, for Class I and flat-spectrum protostars (Figure \n\\ref{Rdisk_class_histo}). This is contrary to the expectation from the TSC \nmodel, in which the late stages of protostellar evolution are characterized \nby the infall of high angular momentum material from large radii and hence \nlarger values of $R_c$.  This may imply that disks sizes are small, but it \nmay also be the result of incorrect assumptions about the distribution of \nangular momentum in the TSC model.  \n\nIn total, these ``conundrums'' that arise from our model fits hint that the \ncurrent models do not realistically reproduce the structure of collapsing \nenvelopes. Future high-resolution observations at submillimeter and longer\nwavelengths that resolve the structure and motions of envelopes may \nprovide the means to develop more refined models that can fit the SEDs \nwith more realistic envelope configurations. \n\n\n\\vspace{2ex}\n\n\\section{Conclusions}\n\nWe have presented SEDs and model fits for 330 young stellar objects\nin the Orion A and B molecular  clouds. The SEDs include data from \n1.2 to 870 $\\mu$m, with near-infrared photometry from 2MASS, mid-infrared \nphotometry and spectra from the {\\it Spitzer Space Telescope}, far-infrared \nphotometry at 70, 100, and 160 $\\mu$m from the {\\it Herschel Space Observatory}, \nand submillimeter photometry from the APEX telescope. \nWe calculated bolometric luminosities ($L_{bol}$), bolometric temperatures\n($T_{bol}$), and 4.5-24 $\\mu$m spectral indices ($n_{4.5-24}$) for all 330 sources \nin our sample. From the distributions of these three parameters, we find that $L_{bol}$\nhas a broad peak near 1 $L_{\\odot}$\\ and extends from 0.02 to several hundred $L_{\\odot}$,\nwhile the distribution of $T_{bol}$ values is broad and flat from about 30 K to 800 K,\nwith a median value of 146 K. The 4.5-24 $\\mu$m spectral indices range from \n-0.75 to 2.6, with a peak near 0.\n\nBased on traditional classification schemes involving $n_{4.5-24}$\nand $T_{bol}$, we have identified 92 sources as Class 0 protostars \n($n_{4.5-24} > 0.3$ and $T_{bol} < 70$~K), 125 as Class I protostars \n($n_{4.5-24} > 0.3$ and $T_{bol} > 70$~K), and 102 as flat-spectrum sources \n($-0.3 < n_{4.5-24} < 0.3$). The remaining 11 sources are Class II pre-main-sequence \nstars with $n_{4.5-24} < -0.3$; most of them just missed the flat-spectrum cutoff,\nand three have SEDs typical of disks with inner holes. Considering these\ntransitional disks and YSOs whose best fit does not require an envelope, we\nfind that 321 of the 330 HOPS targets in our sample are protostars with\nenvelopes. Class 0 and I protostars often display a deep silicate absorption \nfeature at 10 $\\mu$m due to the presence of the envelope, while many \nflat-spectrum sources have a weak silicate emission or absorption feature \nat that wavelength. \n\nWe have used a grid of 30,400 protostellar model SEDs, calculated using the\n2008 version of the \\citet{whitney03a,whitney03b} Monte Carlo radiative \ntransfer code, to find the best-fit models for each observed SED. The grid \nis limited to discrete values for protostellar parameters, and their ranges \nwere chosen to represent typical protostars. Within the framework of these \nmodels, we find the following:\n\n\\begin{itemize}\n\\item{About 70\\% of Class 0 protostars, 75\\% of Class I protostars, and close to \n90\\% of flat-spectrum sources have reliable SED fits ($R < 4$, where $R$\nis a measure of the average distance between model and data in units\nof the fractional uncertainty). Thus, our model grid can reproduce most of the \nobserved SEDs of Orion protostars.}\n\\item{Our results show a clear trend of decreasing envelope densities as we \nprogress from Class 0 to Class I and then to flat-spectrum sources: we find that \nthe median $\\rho_{1000}$ values decrease from 5.9 $\\times 10^{-18}$ g cm$^{-3}$ \nto 2.4 $\\times 10^{-19}$ g cm$^{-3}$ to 1.2 $\\times 10^{-19}$ g cm$^{-3}$.\nThe decrease in densities implies a decrease in the infall rates of the protostars \nas they evolve.\nWe find that the PACS Bright Red sources (PBRs) have median $\\rho_{1000}$ \nvalues twice as high as the median value of the Class 0 protostars in our sample, \nsupporting the interpretation that they are likely the youngest protostars in Orion.}\n\\item{There are degeneracies in the parameters for models that reproduce the \nobserved SEDs. For example, increasing the mid-IR SED slope and \ndeepening the silicate absorption feature at 10 $\\mu$m of a model protostar \ncan be done by increasing the envelope density or inclination angle, \ndecreasing the cavity opening angle or centrifugal radius, or even increasing \nthe foreground extinction. Hence, the properties of a specific source may be fit \nby a wide range of parameters. The best-fit model parameters are particularly \nuncertain for objects whose SED is not well constrained by observations. \nBecause of these degeneracies, the observed classes contain a mixture \nof evolutionary stages.}\n\\item{We find that flat-spectrum sources are particularly well fit by our models.\nThey have, on average, lower envelope densities and intermediate inclination \nangles, so many flat-spectrum sources are likely more evolved protostars, \nbut this group also includes protostars with higher envelope densities (and\nsometimes larger cavity opening angles) seen at lower inclination angles. \nFlat-spectrum sources seen at $i>$ 65\\degr\\ have very tenuous envelopes. \nThus, the sample of flat-spectrum sources includes protostars \nat different stages in their envelope evolution. All but three of the flat-spectrum \nsources in our sample have envelopes in their best-fit models, indicating that, \nwith a small number of exceptions, these objects are protostars with infalling \ngas.}\n\\item{The luminosity function for the model luminosities peaks at a higher \nluminosity than that for the observed bolometric luminosities as a result of \nbeaming along the outflow cavities.  Furthermore, the total luminosity \ndetermined by the models is higher for Class 0 protostars: the median \ntotal luminosities are 5.5, 2.0, and 3.0 $L_{\\odot}$\\ for Class 0, Class I, and \nflat-spectrum sources, respectively.}\n\\item{Since heating by external radiation fields is not included \nin our model grid, we assessed its influence by adding an interstellar radiation \nfield to a set of models. We find that an ISRF ten times that typical of the solar \nneighborhood can substantially change the SEDs of  sources with internal \nluminosities of 0.1 $L_{\\odot}$. However, when we incorporate the effect of \nextinction on the external radiation field, the effect on the protostellar\nSEDs is smaller; the best-fit luminosities and envelope densities would be \noverestimated by factors of a few for $\\sim$ 0.1 $L_{\\odot}$\\ prototars and much\nless for higher-luminosity protostars. We estimate that the best-fit parameters\n(in particular, $L_{tot}$, $\\rho_{1000}$) of $\\sim$ 20\\% of the HOPS sources \ncould be affected by external heating.}\n\\item{Although the adopted TSC models reproduce the observed SEDs well, \nthere are trends that suggest inadequacies with these models.  First, the \ndistribution of best-fit inclination angles does not reproduce that expected \nfor randomly oriented protostars.  Second, although the distribution of outflow \ncavity sizes for flat-spectrum and Class~0 sources is flat, there is an excess \nof small cavities for Class I sources. This is in contradiction to the typical \npicture that outflow cavities grow as protostars evolve. Finally, the distribution \nof outer disk radii set by the rotation of the envelope is concentrated at small \nvalues ($<$ 50 AU) for the Class I and flat-spectrum sources but is slightly \ntilted toward large values ($>$ 50 AU) for Class 0 protostars. Again, this trend \ncontradicts the expected growth of disks as the infall region in protostellar \nenvelopes expands. These findings suggest that either the envelope structure \nof the adopted models is incorrect, or our understanding of the evolution of \nprotostars needs to be revised substantially.}\n\\end{itemize}\n\nOur work provides a large sample of protostars in one molecular cloud\ncomplex for future, more detailed studies of protostellar evolution. For example,\nusing additional constraints, such as from scattered light imaging, the \nstructure of envelope cavities and thus the role of outflows can be better \nunderstood. In addition, the detailed structure of the envelope and the disk \nembedded within, as well as multiplicity of the central source, can be studied \nwith high spatial resolution imaging such as ALMA can provide. With the \nanalysis of their SEDs presented in this work, the HOPS protostars constitute \nan ideal sample to derive a better understanding of the early evolution of \nyoung stars, when the assembly of the stellar mass and the initial \nstages of planet formation likely take place.\n\n\\vspace{1ex}\n\n\\acknowledgments\nSupport for this work was provided by NASA through awards issued by\nJPL\/Caltech.\nThe work of W.J.F. was supported in part by an appointment to the NASA \nPostdoctoral Program at Goddard Space Flight Center, administered by \nOak Ridge Associated Universities through a contract with NASA.\nJ.J.T. acknowledges support provided by NASA through Hubble Fellowship grant \n\\#HST-HF-51300.01-A awarded by the Space Telescope Science Institute, which is \noperated by the Association of Universities for Research in Astronomy, Inc., \nfor NASA, under contract NAS 5-26555. J.J.T acknowledges further support from \ngrant 639.041.439 from the Netherlands Organisation for Scientific Research (NWO).\nThe work of A.M.S. was supported by the Deutsche Forschungsgemeinschaft\npriority program 1573 (``Physics of the Interstellar Medium'').\nM.O. acknowledges support from MINECO (Spain) AYA2011-3O228-CO3-01 \nand AYA2014-57369-C3-3-P grants (co-funded with FEDER funds). \nWe thank Thomas Robitaille for helpful discussions regarding the model grid\nand model parameters.\nThis work is based on observations made with the {\\it Spitzer Space Telescope}, \nwhich is operated by the Jet Propulsion Laboratory (JPL), California Institute of \nTechnology (Caltech), under a contract with NASA; it is also based on\nobservations made with the {\\it Herschel Space Observatory}, a European Space\nAgency Cornerstone Mission with significant participation by NASA. \nThe {\\it Herschel} spacecraft was designed, built, tested, and launched under \na contract to ESA managed by the {\\it Herschel\/Planck} Project team by an industrial \nconsortium under the overall responsibility of the prime contractor Thales Alenia Space \n(Cannes), and including Astrium (Friedrichshafen) responsible for the payload module \nand for system testing at spacecraft level, Thales Alenia Space (Turin) responsible \nfor the service module, and Astrium (Toulouse) responsible for the telescope, with \nin excess of a hundred subcontractors.\nWe also include data from the Atacama Pathfinder Experiment, a collaboration \nbetween the Max-Planck Institut f\\\"ur Radioastronomie, the European Southern \nObservatory, and the Onsala Space Observatory. \nThis publication makes use of data products from the Two Micron All Sky Survey, \nwhich is a joint project of the University of Massachusetts and the Infrared Processing \nand Analysis Center\/Caltech, funded by NASA and the NSF. \n\n\\vspace{2ex}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Observations and reductions}\n\\subsection{Spectroscopy}\nNew electronic spectra were obtained at four observatories: with\nthe Ond\\v{r}ejov 2~m reflector (OND) and a~coud\\'e\nspectrograph, with the Dominion Astrophysical Observatory 1.22~m reflector (DAO)\nand a coud\\'e spectrograph, with the Cerro Armazones 1.5~m Hexapod Telescope (HPT),\nthe Bochum Echelle Spectroscopic Observer (BESO)\nspectrograph, which is similar to FEROS \\citep{beso}, and with the Cerro Tololo Inter-American\nObservatory (CTIO) 1.5~m reflector with the CHIRON echelle spectrograph\n\\citep{toko2013}.\nWe also used one archival ESO FEROS echelle spectrum \\citep{feros,feros2},\nthe medium-resolution CCD  spectra obtained and published by Christian Buil,\n\\footnote{For the description of his instrumentation and data reduction, see\n\\url{http:\/\/www.astrosurf.com\/buil\/us\/bestar.htm}\\,.} and a selection of\namateur CCD spectra with resolutions better than 10000 from the BeSS\nspectroscopic database \\citep{neiner2011}.\nTable~\\ref{jourv} lists a journal of all spectral observations used.\n\nThe initial reduction of all Ond\\v{r}ejov and DAO spectra (bias subtraction,\nflat-fielding, creation of 1D spectra, and wavelength calibration) was carried\nout in {\\tt IRAF}. Initial reduction of the HTP and CTIO spectra\nwas carried out at the respective observatories. Rectification, removal\nof residual cosmics and flaws, and RV measurements of all spectra were carried\nout with the Pascal program {\\tt SPEFO}  \\citep{sef0,spefo}, namely the latest\nversion 2.63 developed by J.~Krpata \\citep{spefo3}.\n{\\tt SPEFO}  displays direct and flipped traces of the line\nprofiles superimposed on the computer screen that the user can slide\nto achieve a precise overlapping of the parts of the profile for which the RV\nis to be measured.\nAll RVs measurements were carried out independently by PH and also by AH, who\nstudied the spectra as a~part of his student's research project. In addition to the wings of the H$\\alpha$  and \\ion{He}{i}~6678  emission, we also measured the bottom of the\n(sometimes asymmetric) cores of the H$\\alpha$, \\ion{He}{i}~6678, \\ion{Si}{ii}, and \\ion{Fe}{ii}\nshell absorptions. Both sets of measurements were\nintercompared, the larger deviations were carefully\nchecked, and the mean of the measurements for each line was then used here.\nOnly after these RV measurements were completed was a new program\nfor spectral reduction {\\tt reSPEFO}, a~modern replacement of {\\tt SPEFO} , written\nin JAVA and running on different platforms (Linux, Windows) developed\nby A.~Harmanec.\\footnote{\\url{https:\/\/astro.troja.mff.cuni.cz\/projects\/respefo}}\nIt can, among other things, import the spectra that were originally reduced in {\\tt SPEFO} \nand treat spectra stored as FITS files.\nWe used this new program to measure line intensities, equivalent widths, and\nthe $V\/R$ ratio of the double H$\\alpha$  emission.\\\\\n\n\\subsection{Photometry}\nWe attempted to collect all available observations with known\ndates of observations. Basic information about all data sets\ncan be found in Table~\\ref{jouphot}, and the details of the photometric\nreductions and standardisation are described in Appendix~\\ref{apb}.\n\nFor the convenience of other investigators, we also publish all our individual\nobservations together with their HJDs. All measured RVs are listed in Table~3,\nspectrophotometric quantities are provided in Table~4, and photometric observations\nare collected in Table~5.  These three tables are available in electronic form only.\n\n\\section{Long-term spectral, light, and colour changes of V1294~Aql}\nAs mentioned above, we attempted to collect and homogenise all available\nphotometric observations and measurements of RVs and line strengths from\nthe records with known dates of observations. Long-term behaviour of these\nquantities and their mutual correlations are discussed in the following\nsub-sections.\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{vis-v.pdf}}\n\\caption{Time plot of the Otero visual brightness estimates\n(empty red circles) along with Johnson $V$ photometry from Hvar (blue dots),\n        SPM (green), \\c{C}anakkale (cyan), and TUG (magenta).}\n\\label{vis-v}\n\\end{figure}\n\n\\subsection{Value of visual estimates of brightness}\nThe visual estimates of brightness by skilful amateur observers are commonly\nused to determine the times of minima (or maxima) of periodic\nvariables and to monitor light changes of variables with a large\namplitude (over several magnitudes). The scatter band of\nvisual estimates is typically about 0\\m1 to 0\\m15, but it can be pushed down by\ntalented individuals who, moreover, follow a few principles:\n(a) to perform only one visual estimate per night (without recalling the previous\none), and (b) to reduce the estimates to Johnson $V$ magnitudes of the\ncomparison stars (known to one thousandth of a magnitude), not to\nthe Harvard scale of magnitudes, which is only accurate to 0\\m1.\nOne of us (SO) contacted the first author of this study back in 2003\nto inform him that he had observed another light decrease of V1294~Aql  for\nabout 0\\m3. We then agreed to test his ability to obtain accurate visual\nestimates via parallel photoelectric observations at Hvar,\nSan Pedro M\\'artir (SPM), Tubitak National Observatory\n(TUG), and \\c{C}anakkale. Figure~\\ref{vis-v}\nshows the comparison of visual estimates and Johnson $V$ photoelectric\nphotometry from several stations over the time interval covered by visual\nestimates. The visual estimates agree very well with\nthe general trend of variations that were recorded via photoelectric photometry, but\nthe deep light minimum appears broader than that recorded by photoelectric\nphotometry. One possible reason is that the minimum was observed\nclose to the end of visibility of the star in the sky and the visual\nestimates were not corrected for the differential extinction.\n\n\\subsection{Correlation between the long-term light and spectral changes\nin time}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=-90]{all.pdf}}\n\\caption{Time plot of available observations over the whole time interval\nof about 25000~d covered by the data. Top panel: Yellow brightness observations,\nwhich could be transformed into Johnson $V$ magnitude. Second and third panel:\nAvailable \\hbox{$B\\!-\\!V$}\\  and \\hbox{$U\\!-\\!B$}\\ colour index observations. Bottom panel: $V\/R$ changes in the peaks of the double H$\\alpha$  emission.\nIn the three panels with photometry, the differential observations\nare shown as blue dots, and all-sky observations are shown as black dots. In the bottom panel,\nblue circles denote the DAO spectra, red circles show the OND spectra, green circles represent BESO spectra,\nmagenta circles show BeSS spectra, black circles show Castanet Tolosan spectra, and the black crosses plot the data\nfrom the literature.}\n\\label{all}\n\\end{figure}\n\n\\begin{figure}[t]\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=-90]{all-n.pdf}}\n\\caption{Time plot showing the correlation between the secular brightness\nand colour variations in the $V$ -band magnitudes and \\hbox{$B\\!-\\!V$}\\ and \\hbox{$U\\!-\\!B$}\\ colour\nindices (black shows all-sky and blue shows differential observations),\nand the $V\/R$ changes, EW, and strength of the\nH$\\alpha$  emission for the more recent electronic spectra. The colour symbols for \nspectra from different sources are the same as in Fig. 2.}\n\\label{all-n}\n\\end{figure}\n\n\\begin{figure}[t]\n\\resizebox{\\hsize}{!}{\\includegraphics{si1vr.pdf}}\n\\caption{Time plot of the $V\/R$ changes recorded for the electronic spectra\nin the \\ion{Si}{ii}~6347~\\AA\\ line. The colour symbols for spectra from different\nsources are the same as in Fig.~\\ref{all}.}\n\\label{si1vr}\n\\end{figure}\n\n\\begin{figure}[t]\n\\resizebox{\\hsize}{!}{\\includegraphics{rvhe.pdf}}\n\\resizebox{\\hsize}{!}{\\includegraphics{rvh3.pdf}}\n\\caption{Time plot of radial velocities. Top panel: RV of the \\ion{He}{i}\nshell lines. Bottom panel: RV measured on the steep wings of the H$\\alpha$  emission.\nData from individual instruments are shown by different symbols:\nthe circles are the same as in Fig. 2, and\nblack crosses plot data from \\citet{balle89}. The ranges on the two\naxes in the two plots are different.}\n\\label{rvtime}\n\\end{figure}\n\nFigure~\\ref{all} is a time plot of all available observations secured in\nor transformed into the Johnson \\hbox{$U\\!B{}V$}\\ magnitudes. The top panel\nshows that the usual brightness level in $V$ is occasionally\ndisturbed by rapid light decreases of different durations.\nMoreover, we note that there is also a secular, steady slow light decrease\nof the undisturbed brightness of the star outside the more rapid light\ndecreases until about HJD~2455000, when it suddenly changed to\na~steeper secular light increase. We return to this new phenomenon\nin a separate section below. The second panel of Fig.~\\ref{all} shows\nthat the \\hbox{$B\\!-\\!V$}\\ index followed the brightness changes, but with a small amplitude,\nwhile the \\hbox{$U\\!-\\!B$}\\ index showed a similar pattern of changes, but with a~larger\namplitude and with a~more or less steady secular reddening.\n\nAn enlarged Fig.~\\ref{all-n} covers only the more recent time interval,\nwhen electronic spectra became available. It shows that in the time intervals\nthat are sufficiently densely covered by the data, the sharp light decreases are\naccompanied by similarly sharp strengthening in the H$\\alpha$  emission.\n\nFigure~\\ref{si1vr} shows the $V\/R$ changes of \\ion{Si}{ii}~6347~\\AA\\ line.\nIt reveals a~pattern similar to that observed for H$\\alpha$, but the time coverage\nis less dense because these variations could not be measured in time intervals\nwhen the emission was faint.\n\nFigure~\\ref{rvtime} shows the variation of the shell absorption RVs\n\\citep[characterised by \\ion{He}{i} RVs, for which data are also\npublished by][]{balle89} and emission-line RVs measured on the wings of\nthe H$\\alpha$  line in electronic spectra.  We note that while the shell RVs\nshows large cyclic changes that have also been observed for a number of other Be stars,\nthe RVs measured on the wings of the H$\\alpha$  emission are secularly stable and\nshow only mild changes on a shorter timescale.\n\n\\subsection{Unusual colour variations}\n\n\\begin{figure*} [t]\n\\includegraphics[angle=0,scale=0.95]{ubbvc.pdf}\n\\includegraphics[angle=0,scale=0.95]{ubbvd.pdf}\n\\includegraphics[angle=0,scale=0.95]{ubbva.pdf}\n\\includegraphics[angle=0,scale=0.95]{ubbvb.pdf}\n\\caption{ \\hbox{$U\\!-\\!B$}\\ vs. \\hbox{$B\\!-\\!V$}\\ diagram for several distinct data subsets.\nTop panels: Older data until JD 2450000 (left). All-sky observations\nare shown as black circles, and data from stations defined\nin Table~\\ref{jouphot} are denoted as follows:\n01 (blue), 04 (green), 12 (red), and 26 (magenta).\nMore recent data from the interval of secular\nlight brightening, all from station 01 (blue) (right).\nThe bottom panels show \\hbox{$U\\!B{}V$}\\ observations from\nthe two sharp increases in the emission-line strength accompanied\nby light decreases. Time interval JD~2452741 -- 2453309, which\ncovers the first sharp light decrease (left, cf. Figs.~\\ref{vis-v} and \\ref{all-n}.)\nData from the time interval JD~2455357 -- 2456094 corresponding\nto the second sharp increase in the emission-line strength (right).\nData from stations 1, 30, 66, and 89 of Table~\\ref{jouphot} are shown\nby blue, red, green, and magenta dots, respectively. The main sequence\nand the supergiant sequence based on data from \\citet{golay74} (pp. 79-80)\nare shown, as is the reddening line.}\n\\label{ubbv}\n\\end{figure*}\n\n\n Many systematically studied Be stars are known to exhibit always the same\nand a rather clear type of either a positive or an inverse correlation\nbetween the long-term brightness variations, a~characteristic type of\nbehaviour in the colour-colour diagram, and the Balmer emission-line strength\nas defined by \\citet{hvar83,hec2000}. He identified these two types\nof correlation as an aspect effect.\nFor Be stars with an~inverse type of correlation, light decreases are followed\nby the rise of the Balmer emission-line strength and by a~shift along\nthe main sequence towards later spectral subclasses in the \\hbox{$U\\!-\\!B$}\\ versus \\hbox{$B\\!-\\!V$}\\\ndiagram. For a~positive type of correlation, the brightenings are followed\nby the rise of the emission strength and a~shift from the main sequence\ntowards the supergiant sequence in the \\hbox{$U\\!-\\!B$}\\ versus \\hbox{$B\\!-\\!V$}\\ diagram. The inverse\ncorrelation is observed for stars that are seen more or less equator-on (a growing\ngaseous envelope is attenuating the light of the central object), while\nthe positive correlation is observed for stars that are seen more pole-on (inner\noptically thick parts of the growing envelope mimic an~apparent increase in\nthe stellar radius). Several examples of both types of correlation in\nthe colour-colour diagram can be found, for instance, in Fig.~2 of\n\\citet{bozic2013}.\n\n  The situation is dramatically different for V1294~Aql. The \\hbox{$U\\!B{}V$}\\ observations\naccumulated over several decades cover a large part of the whole\ncolour-colour diagram, with a single clear pattern.\n\n  To understand better what is going on, we investigated\nthe colour-colour diagrams for different segments of the long-term\nchanges. Figure~\\ref{ubbv} shows the colour changes separately for\nthe old data secured before JD~2450000, for more recent data from the secular\nbrightness increase (observations after JD~2457000), and for\nobservations covering two episodes of a rapid increase and decrease in the H$\\alpha$  emission\nassociated with sharp light decreases. The\npattern is remarkably similar for both these episodes. Formally, it looks like\na~positive correlation. However, the phases of minimum brightness and\nmaximum strength of the emission correspond to data that are close\nto the main sequence, even below it when the reddening is considered.\nThe older data are all clustered above the supergiant sequence, while\nthe recent data lie along the supergiant sequence for late-B and early-A\nspectral classes.\nAll this indicates that we observe a combination of several different types\nof long-term changes.\n\n\\section{Duplicity of V1294~Aql}\n\n\\setcounter{table}{5}\n\\begin{table}\n\\begin{flushleft}\n\\caption{Orbital solutions based on the H$\\alpha$  emission RVs.}\n\\label{sol}\n\\begin{tabular}{lccrrccc}\n\\hline\\hline\\noalign{\\smallskip}\n Element                   &All RVs        &Hi-res. spectra only\\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n$P$ (d)                    &$192.91\\pm0.18$& 192.91 fixed       \\\\\n$T_{\\rm super.conj.} ^*$   &$56318.5\\pm2.2$&$56316.2\\pm2.9$     \\\\\n $e$                       &0.0 fixed      & 0.0 fixed          \\\\\n$\\gamma$ (km~s$^{-1}$)             & $-6.27\\pm0.31$& $-5.52\\pm0.44$     \\\\\n$K_1$    (km~s$^{-1}$)             & $6.33\\pm0.41$ & $6.26\\pm0.61$      \\\\\nNo. of RVs                 & 172           & 38                 \\\\\nrms (km~s$^{-1}$)                  & 3.92          &  2.63              \\\\\n\\hline\\noalign{\\smallskip}\n\\end{tabular}\n\\end{flushleft}\n\\tablefoot{$^*$) All epochs are in HJD-2400000.}\n\\end{table}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{orbit.pdf}}\n\\resizebox{\\hsize}{!}{\\includegraphics{orbithe.pdf}}\n\\caption{Radial-velocity curve corresponding to the orbital solution, based on\nRVs measured on the steep wings of the H$\\alpha$  emission, plotted for\nphases from ephemeris (top; \\ref{efem}).\nOrbital curve based on RV of the \\ion{He}{i}~6678~\\AA\\ shell line\nprewhitened for the long-term changes, plotted for the same ephemeris (bottom).\nData from individual instruments are shown by different symbols. The circles \nare the same as in Fig. 2, and black triangle shows CTIO.}\n\\label{orbit}\n\\end{figure}\n\nThe idea that duplicity can be an important factor for the very existence\nof the Be phenomenon is not new. \\citet{plahor69,kriz69} and\n\\citet{plavec70} have argued that at least some Be stars could be binaries\nthat are observed in the later phases of mass exchange between the binary components.\n\\citet{krizhec75} and \\citet{hk76} formulated the general hypothesis that\nBe stars are mass-accreting components of binaries and showed that this idea \ncan also explain several types of time variations observed for Be\nstars.  Additional arguments were provided by \\citet{plavec76a} and\n\\citet{peters76}. However, as pointed out already by \\citet{plavec76b},\nif all Be stars have Roche-lobe filling secondaries,\nmore eclipsing binaries should be observed among them. Later investigations also led to the\nfinding that the presence of Roche-lobe filling secondaries can be excluded\nfor some Be stars that were found to be spectroscopic binaries, such as V744~Her = 88~Her\n\\citep{zarf7a,zarf7b} or V439~Her = 4~Her \\citep{zarf5,zarf6}.\nThis led \\citet{pols91} to suggest that many\nBe stars might be objects created by large-scale mass transfer that were\nobserved in phases after the mass transfer ceased. The expected\nsecondaries of such objects would be hot compact stars, white dwarfs\nin some cases. These are the most easily detectable sources in far-UV spectra. Evidence\nfor a hot secondary to the well-known Be binary $\\varphi$~Per was found\nfirst from the antiphase variation in the \\ion{He}{ii}~4686~\\AA\\ emission\nseen in the photographic spectra \\citep{poeckert81} and later from\n\\ion{He}{i}~6678~\\AA\\ emission in the electronic spectra obtained\nby \\citet{gies93}. Its ultimate direct detection as an O~VI subdwarf\ncame from the study of the far-UV spectra from the Hubble Space Telescope by\n\\citet{gies98}. The secondary was then resolved with optical spectro-interferometry\nby \\citet{mourard2015}. Detections for several other systems followed.\n\\citet{wang2018} carried out a systematic search for the presence of hot\nsecondaries and summarised our knowledge of already known cases. \\citet{wang2021}\ndetected nine new Be+sdO binaries from analyses of the Hubble Space Observatory\nspectra, and \\citet{klement2022} reported the first interferometric detection and\nsignatures of the orbital motion for three known Be+sdO systems. On the other\nhand, \\citet{boden2020} carried out a systematic search for Be stars with\nmain-sequence secondaries, with a completely null result. This constitutes\nindirect evidence that the mass exchange is or was behind the formation\nof binaries with Be primaries. \\citet{hast2021} carried out evolutionary\ncalculations of mass exchange in binaries in an effort to set some limits\non the fraction of Be stars produced by binary interaction. They found\nthat under certain conditions, this fraction can be quite high.\n\n\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{haprofs.pdf}}\n\\caption{Three Ond\\v{r}ejov H$\\alpha$  profiles. They are mutually shifted in\nordinate by 1.0 of the continuum level for better clarity.\nProfiles from HJD~2457128.5976 and 2457137.5762 have anomalously positive\nRVs of the emission wings, which stem from the episode of a large strengthening\nof the emission. The next profile, from HJD~2457154.5435, has a~normal\norbital RV.}\n\\label{haprofs}\n\\end{figure}\n\n\\begin{table}\n\\begin{flushleft}\n\\caption{Possible properties of the binary system:\nMass of the secondary $M_2$, mass ratio $M_2\/M_1$, semi-amplitude of\nthe RV curve of the secondary $K_2$, and the semi-major axis $a$ for\nseveral possible orbital inclinations $i$. The mass of the primary was\nassumed to be $M_1=16.9$~\\hbox{$\\mathcal{M}^{\\mathrm N}_\\odot$}\\ after \\citet{zorec2016}.}\n\\label{secmass}\n\\begin{tabular}{lccrrccc}\n\\hline\\hline\\noalign{\\smallskip}\n  $i$     & $ M_2$ &$M_2\/M_1$& $K_2$  & $a$\\\\\n($^\\circ$)& (\\hbox{$\\mathcal{M}^{\\mathrm N}_\\odot$})   &      & (km~s$^{-1}$)  & (\\hbox{$\\mathcal{R}^{\\mathrm N}_\\odot$}) \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n 90.0  & 1.171 &  0.0693 & 90.43  &    368.69   \\\\\n 85.0  & 1.175 &  0.0695 & 90.07  &    368.72   \\\\\n 80.0  & 1.189 &  0.0704 & 88.99  &    368.82   \\\\\n 70.0  & 1.249 &  0.0739 & 84.73  &    369.23   \\\\\n 60.0  & 1.361 &  0.0805 & 77.77  &    369.98   \\\\\n\\hline\\noalign{\\smallskip}\n\\end{tabular}\n\\end{flushleft}\n\\tablefoot{We express the values of masses and radii\nin the nominal values \\hbox{$\\mathcal{M}^{\\mathrm N}_\\odot$}, and \\hbox{$\\mathcal{R}^{\\mathrm N}_\\odot$}\\ as defined by \\citet{prsa2016}.}\n\\end{table}\n\n\nOne always has to be cautious when analysing binaries with clear signatures\nof the presence of circumstellar matter in the system. The experience from\nour previous studies of individual Be stars \\citep{bozic95,zarf18,zarf20,\nzarf21,zarf24,zarf26} shows that the binary nature of particular Be stars\nis most easily detected via periodic RV variations of the steep\nemission wings of the H$\\alpha$  line and often also via the periodic changes\nin the $V\/R$ ratio of the double Balmer emission lines.\n\nPeriod analyses of all H$\\alpha$  emission-line RVs of V1294~Aql, using\nboth the \\citet{deeming75} and \\citet{stelling78} methods, revealed that\nthe RV of the H$\\alpha$ \\ emission wings varies with a period of 193~d\nand a~semi-amplitude of $\\sim 5$ km~s$^{-1}$.\nThe same periodicity is also detected in the RV of the H$\\alpha$  absorption core\nand in the absorption RVs of \\ion{Si}{ii} doublet at 6347 and 6371~\\AA\\,\nand \\ion{Fe}{ii}~6456~\\AA\\  after long-term changes are removed.\n\nUsing the program {\\tt FOTEL}  \\citep{fotel1,fotel2}, we derived the circular-orbit\norbital elements for all H$\\alpha$  emission-wing RVs and for those from the\nhigh-resolution spectra alone. The mutual agreement of the two solutions is\nvery satisfactory. They are presented in Table~\\ref{sol}, and\nthe corresponding RV curve is plotted in Fig.~\\ref{orbit}.\nIn the rest of this study, we adopt the following linear ephemeris:\n\n\\begin{equation}\nT_{\\rm super.conj.}={\\rm HJD}\\,2456318.5+192\\fd91\\times E \\label{efem}\n\\end{equation}\n\n\\noindent based on the solution for all spectra.\n\n To be fair, we note that some deviations from the mean RV curve in the upper\npanel of Fig.~\\ref{orbit} are rather large. This is, for instance,\nthe case of two Ond\\v{r}ejov spectra taken on HJD~2457128.6 and 2457137.6,\nwhen a very steep rise of the emission strength\nhad occurred. We remeasured these spectra several times, but the result was the\nsame. Their RVs are almost in anti-phase to the orbital RV curve near phase 0.3.\nWe show the corresponding line profiles in Fig.~\\ref{haprofs} together with\nanother profile, taken about two weeks later, which already gives a~RV in accord\nwith the orbital motion. The two peculiar RVs were given zero weight in the\norbital solution.\n\nWe tentatively adopted the mass of the Be primary after \\citet{zorec2016} and\nestimated the basic properties of the system for several\npossible orbital inclinations. Because no lines of the secondary were detected\nin the optical spectra and because no companions to Be stars were ever found\namong main-sequence objects \\citep{boden2020}, we conclude that the secondary\nis not a Roche-lobe filling object, but most probably a hot subdwarf star or\nwhite dwarf. It should be looked for in the far-UV spectral region.\nFor the Gaia DR2 parallax of 0\\farcs0007059, the projected angular separation\nof the binary components is 0\\farcs0048, which might be resolved with \npresent-day optical interferometers such as the currently tested interferometer\nSPICA \\citep{spica2020}.\n\n\n\\section{Correlations between orbital and long-term changes}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{v57500.pdf}}\n\\resizebox{\\hsize}{!}{\\includegraphics{v57900.pdf}}\n\\resizebox{\\hsize}{!}{\\includegraphics{v58300.pdf}}\n\\resizebox{\\hsize}{!}{\\includegraphics{v58600.pdf}}\n\\caption{Phase plots of $V$ magnitude for several seasons of Hvar observations\nfor binary ephemeris (\\ref{efem}). From top to bottom: Data from\nJD~$2457568-57657$,\nJD~$2457931-58079$,\nJD~$2458318-58392$, and\nJD~$2458664-58740$.}\n\\label{vorb}\n\\end{figure}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{haemisph.pdf}}\n\\caption{Phase plots of the strength of the H$\\alpha$  emission\nfor binary ephemeris (\\ref{efem}). Data from individual sources are\ndenoted as follows: Circles in blue show DAO spectra, red circles show OND spectra, green circles represent BESO spectra,\nmagenta circles show BeSS, and black circles show Castanet spectra. The black crosses show data from the literature.}\n\\label{haemisph}\n\\end{figure}\n\nInspection of the light, colour, and emission-line strength variations seems to\nindicate that the rapid episodes of large changes such as those near JD~2452900\nor JD~2457300 occurred during one binary orbital period. To investigate the problem,\nwe plot phase diagrams of the variability of the $V$ magnitude for several\nmore recent shorter time intervals in Fig.~\\ref{vorb}. The two\nlarge light decreases that are accompanied by strong increases in the H$\\alpha$  emission-line\nstrength apparently occurred around phases of elongations, with the Be primary receding from us.\nAt the same time, the plot shows that in another observing season, brightenings\nwere observed around similar orbital phases. The same is also confirmed by\na phase plot of the H$\\alpha$  emission-line strength; see Fig.~\\ref{haemisph}.\n\n  We also investigated the time behaviour of the $V\/R$ ratio of the double\nH$\\alpha$  emission. In this case as well, V1294~Aql  appears to be quite unusual.\nAs Figs.~\\ref{all} and \\ref{all-n} show, the $V\/R$ variations are different\nin different time intervals and do not recall either the long-term cyclic\nchanges known for Be stars with one-armed global oscillations or phase-locked\nchanges. In several panels of Fig.~\\ref{vrsets} we show enlarged plots of\nthe $V\/R$ changes. Instants of expected phase-locked $V\/R$ maxima predicted\nby the orbital ephemeris (\\ref{efem}) are shown by vertical lines.\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{vrsets.pdf}}\n\\caption{Enlarged subsets of time variability of the H$\\alpha$  $V\/R$ ratio\nwith the instants of the expected phase-locked maxima predicted by the\norbital ephemeris (\\ref{efem}).}\n\\label{vrsets}\n\\end{figure}\n\n\\section{Rapid changes}\nAlthough we collected a large number of photometric observations of V1294~Aql,\ntheir time distribution is not suitable for a search for rapid periodic changes. Perhaps the only observations suitable for\nsuch a search are the early $V$ observations by \\citet{lynds59}.\nHe himself concluded that his observations definitively indicate\nvariations in brightness, which appear to be somewhat erratic, however, and\nno period could be found. The observations were secured within one month\nduring a~time interval that was  not affected by secular variations. Our period analysis\nrevealed sinusoidal variations with a semi-amplitude of 0\\m0139(29).\nA~least-squares fit led to ephemeris (\\ref{efelynds}), the rms of one\nobservation being 0\\m0067. The corresponding phase plot is shown in Fig.~\\ref{rapid}.\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=0]{lynds.pdf}}\n\\caption{Possibly periodic rapid light changes based on \\citet{lynds59}\n$V$ magnitude photometry and plotted for ephemeris (\\ref{efelynds}).}\n\\label{rapid}\n\\end{figure}\n\n\\begin{equation}\nT_{\\rm light\\, min.}={\\rm HJD}\\,2436450.8053(92)+0\\fd64827(50)\\times E. \\label{efelynds}\n\\end{equation}\n\nThis indicates that a scatter of at least 0\\m03 is to be expected in individual\nobservations on longer timescales.\n\n\\citet{lefe2009} carried out an~automatic period search in the Hipparcos \\hbox{$H_{\\rm p}$}\\\nphotometry to find new periodic variables among OB stars. They identified\nV1294~Aql  as a~possible slowly pulsating B star (SPB) with a period of 7\\fd752.\nWe cannot confirm their result. They apparently did not take the secular\nlight change in \\hbox{$H_{\\rm p}$}\\ photometry into account; see the upper panel of Fig.~\\ref{all} here.\n\n\\citet{zorec2016} estimated the following physical properties of the Be component:\\\\\n$T_{\\rm eff}$  = ($30120\\pm2540$)~K, {\\rm log}~$g$  = ($4.08\\pm0.40$) [cgs],\nmass M = ($16.9\\pm2.7$)~M$_{\\odot}$, $v$~sin~$i$  = ($207\\pm18$)~km~s$^{-1}$,\ncritical rotational velocity $v=(517\\pm64)$~km~s$^{-1}$, and the\ninclination of the rotational axis $i=(37^\\circ\\pm9^\\circ)$.\nFor these values, the period 0\\fd648 appears as a reasonable rotation period of the Be component.\nIn passing we note that at the time of writing, the star has not been observed\nby the TESS satellite.\n\n\\section{Fourth timescale}\n\\citet{hec98} has called attention to the fact that the brightness of the\nBe star $\\omega$~CMa outside of the episodes of brightenings accompanied\nby the growth of emission-line strength (typical of the positive correlation\ndiscussed above) has been decreasing secularly. His observation was later\nconfirmed with more recent photometry \\citep{ghore2018, ghore2021}. These\nauthors and also \\citet{marr2021}, who studied another Be star, V2048~Oph = 66~Oph,\nmodelled the secular variability and the episodes of brightening and\nincreases in the Balmer emission-line strength with some success, estimating\nthe required viscosity values for individual episodes, and also discussing\nsome limitations of their effort. The yellow light curve of V2048~Oph\nis shown in the upper panel of Fig.~1 of \\citet{marr2021}. It shows a~secular\nlight decrease between 1980 and 2000, occasionally interrupted by brightenings\nreminiscent of a positive correlation. However, the strength of\nthe H$\\alpha$  emission is near its maximum over the same time interval of about\n20 years, and only then does it gradually decrease. No emission has been\nobserved since about 2010. However, as \\citet{marr2021} pointed out,\nthe outer parts of the disk are still seen in the radio wavelengths.\n\nWe collected and homogenised the $V$ photometry of V2048~Oph\nfrom the archive of \\hbox{$U\\!B{}V$}\\ photometry provided by J.R.~Percy, from Hvar,\nSPM, \\citet{john66}, \\citet{haupt74} and \\citet{kozok85} and transformed the Hipparcos\n\\hbox{$H_{\\rm p}$}\\ \\citep{esa97} photometry into Johnson $V$ and the observations of\n\\cite{hill76} secured in the DAO photometric system into \\hbox{$U\\!B{}V$} \\ using the transformation\nformul\\ae\\ provided by \\citet{hecboz01}.  The $V$ light curve of V2048~Oph\nbased on the above-mentioned data sets is shown in the upper panel\nof Fig.~\\ref{fourth}.\n\nA similar secular light decrease has also been reported for V744~Her = 88~Her,\na Be star with an inverse type of correlation \\citep{hecboz2013}. We show\nits light curve complemented by more recent observations, adapted from\nBo\\v{z}i\\'c et al. (in prep.) in Fig.~\\ref{fourth} as well. In the same figure,\nwe also show the $V$ and $B$ light curves of the Be star EW~Lac = HD~217050\nfrom another study in preparation. In this case, a secular increase\nin brightness is observed. We note that the large scatter around the mean trend\nis related to the known rapid light variability of EW~Lac on a timescale shorter than\none day. The fourth panel of Fig.~\\ref{fourth} shows the plot of Hvar $V$ photometry\nof $\\varphi$~And, another Be star with a positive type of correlation.\nA mild light decrease over several decades of observations is visible.\n\nSearching the literature, we found a few more examples.\nA secular light increase has also been observed for the Be star with a~positive\ncorrelation $\\gamma$~Cas \\citep[][Fig.~5]{gcas2002} over nearly 30000~days.\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=0]{66ophv.pdf}}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=0]{ewlacv.pdf}}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=0]{ewlacb.pdf}}\n\\resizebox{\\hsize}{!}{\\includegraphics{phiandv.pdf}}\n\\resizebox{\\hsize}{!}{\\includegraphics{v744herv.pdf}}\n\\caption{Secular photometric changes of several well-observed Be stars.}\n\\label{fourth}\n\\end{figure}\n\nFinally, as we have shown here, the secular light decrease of V1294~Aql \nhas recently changed to a secular light increase.\nAll this shows the large variety of different possible evolutions\nof the circumstellar disk, its replenishment, and a~gradual dissipation.\n\n\n\n\\section{Discussion}\nIn spite of the effort of several generations of stellar astronomers,\nthe engine leading to the occasional formation of circumstellar disks\naround Be stars has not been firmly identified so far.\nOne  possible explanation is based on the idea that Be stars are\nrapidly rotating non-radial pulsators (NRP) and that the additional\nforce needed to facilitate the outflow of gas and angular momentum transfer\nfrom the stellar equator arises from a constructive interference of two\nor more NRP modes\n\\citep{rivi98a,rivi98b,baade2017a,baade2017b,baade2020,borre2020,bartz2021}.\nEspecially the systematic photometries from space observatories were\nused and analysed to support this conjecture. Confirmation of this\nscenario would require the creation of new, self-consistent models,\nhowever, which would show that Be stars are indeed pulsationally unstable over the whole\narea that they occupy in the Hertzsprung-Russell (HR) diagram.\nIt should also be mentioned that \\citet{baade2017b} warned that evidence\nfor constructive interference of pulsational modes for a larger number of Be stars\nis lacking and pointed out additional problems such as the rotational splitting of modes\nand the presence of rapid changes in the circumstellar envelopes during active phases.\nAn alternative view was suggested by \\citet{hec98}, who argued that the dominant\nperiod of rapid changes undergoes small cyclic changes. Modelling such\na situation, he found that a standard period analysis of a~corresponding series\nof observations returns a multiperiodicity with several close periods.\nYet another possibility was suggested by \\citet{bebin2002}, who argued that\nthe presence of a secondary can facilitate outflow from the equatorial parts\nof the gaseous disk of the Be primary even in systems that are not filling\ntheir Roche lobes. This idea is problematic, however, in that the effect is\nrather small in the majority of cases.\n\n For a long time, various other suggestions have been made that\nthe Be phenomenon and observed variability patterns of Be stars might be causally\nrelated to their binary nature\n\\citep[e.g. ][among others]{plahor69,krizhec75,hk76,bebin87,pols91,\n  pano2018,boden2018,boden2020,langer2020,boden2021}. \nThe approach of \\citet{klement2019} is worth mentioning. These authors studied \nthe spectral energy distribution of several Be stars and provided arguments that \ntheir disks had to be truncated by the Roche lobes. This constitutes \nan~indirect argument for the presence of companions to these objects.\n\n  The phase-locked $V\/R$ changes observed for several Be binaries represent another\ninteresting phenomenon. As already noted above, they are usually\nobserved roughly in phase with the orbital RV changes\nof the Be stars in question \\citep{zarf6,zarf7b,zarf16,zarf21,stefl2007}. A phase-locked\nemission-line variation with a single maximum and minimum per orbital period\nwas also found by \\citet{borre2020} for $\\gamma$~Cas. These authors used\nan interesting detection technique. They analysed local pixel-per-pixel line\nfluxes across the H$\\alpha$  profile in a~series of higher-resolution BeSS spectra.\n\\citet{pano2018} modelled the phase-locked $V\/R$ changes as global oscillations\nin the circumstellar disks with two spiral patterns and concluded that the\nphase-locked $V\/R$ changes should exhibit two maxima and minima during one\norbital period. This clearly disagrees with the available observations\nmentioned above. The only case for which a double-wave $V\/R$ curve was detected is\nV696~Mon = HR~2142 \\citep{peters72,peters76}. There is a natural explanation\nof roughly sinusoidal phase-locked $V\/R$ changes in our view, as discussed\nin Appendix~C of \\citet{wolf2021}. The circumstellar disk probably\noccupies almost the whole volume of the Roche lobe near\nthe orbital plane. This causes there to be more gaseous material in the part\nof the disk facing the secondary than on the opposite side. Because the disk\nrotates, more emission power is available on the side facing the secondary,\nand this naturally leads to phase-locked $V\/R$ changes that are in phase\nwith the RV changes of the Be component.\n\n  To show how confusing the interpretation of $V\/R$ changes can be,\nwe compare the $V\/R$ changes of H$\\alpha$  and \\ion{He}{i}~6678~\\AA\\ for\ntwo shorter time intervals in Fig.~\\ref{fales}. In the first interval,\nthe variations for both lines are in phase, while in the second interval,\nantiphase variation is observed. We note that this is a consequence\nof the fact that the He shell RV becomes very negative (a~temporarily\nelongated envelope?) and apparently weakens the $V$ peak of a~relatively\nfaint He emission. This is best illustrated in Fig.~\\ref{vrdva}, where\nthe H$\\alpha$  and \\ion{He}{i}~6678  profiles for two dates are shown.\n\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=0]{heh3vra.pdf}}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=0]{heh3vrb.pdf}}\n\\caption{Apparent $V\/R$ changes observed for the H$\\alpha$  and \\ion{He}{i}~6678 \nemission lines intercompared for two time segments. The variation\nin the shell He RV is also shown. An~apparently anti-phase behavior\nis observed in the second time interval, when the shell RV becomes\nquite negative and the He shell line blends with the $V$ peak of the\nfaint He emission. The same colours as in the previous time plots\nare used to distinguish spectra from individual observatories.}\n\\label{fales}\n\\end{figure}\n\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=0]{havrline.pdf}}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=0]{hevrline.pdf}}\n\\caption{Apparent $V\/R$ changes observed for the H$\\alpha$  and \\ion{He}{i}~6678 \nemission lines intercompared for two dates. An~apparently anti-phase\nbehaviour is observed for the later date, when the \\ion{He}{i}~6678  shell RV becomes\nquite negative and the He shell line blends with the $V$ peak of the\nfaint He emission.}\n\\label{vrdva}\n\\end{figure}\n\nThis study of V1294~Aql  demonstrates very clearly how hard it is to identify\nand understand mutually coexisting and overlapping variability patterns\ngoverning the observed spectral, light, and colour changes. Attempts at modelling them\nquantitatively, planned for a~continuation of this study, are expected to shed\nmore light on the mysterious Be phenomenon. We also suggest that further\nmonitoring of the object with systematic photometry, high-resolution\nspectroscopy, and especially with the optical interferometry could help to\nreveal the secrets of this intriguing Be binary, or possibly a multiple system,\nas indicated by the analysis of the astrometric data \\citep{brandt2021}.\n\n\\begin{acknowledgements}\nWe gratefully acknowledge the use of the latest publicly available version\nof the program {\\tt FOTEL}  written by P.~Hadrava.\nWe thank A.~Aret, A.~Budovi\\v{c}ov\\'a, P.~Chadima, M.~Dov\\v{c}iak,\nJ.~Fuchs, P.~Hadrava, J.~Jury\\v{s}ek, E.~Kiran, L.~Kotkov\\'a,\nR.~K\\v{r}i\\v{c}ek, J.~Libich, J.~Nemravov\\'a, P.~Rutsch, S.~Saad, P.~\\v{S}koda,\nS.~\\v{S}tefl, and V.~Votruba, who obtained some of the Ond\\v{r}ejov spectra\nused in this study. J.R.~Percy kindly put the archive of his systematic\n\\hbox{$U\\!B{}V$}\\ observations of bright Be stars in three observatories at our disposal.\nWe also acknowledge the constructive suggestions of an anonymous referee\nto the first version of this study.\nOver the years, this long-term project was supported by the grants 205\/06\/0304,\n205\/08\/H005, P209\/10\/0715, and GA15-02112S  of the Czech Science Foundation,\nby the grants 678212 and 250015 of the Grant Agency of the Charles University\nin Prague, from the research project AV0Z10030501 of the Academy of Sciences\nof the Czech Republic, and from the Research Program MSM0021620860\n{\\sl Physical study of objects and processes in the solar system and\nin astrophysics} of the Ministry of Education of the Czech Republic.\nThe research of PK was supported by the ESA PECS grant 98058.\nHB, DR, and DS acknowledge financial support\nfrom the Croatian Science Foundation\nunder the project 6212 ``Solar and Stellar Variability\".\nThis work has made use of data from\nthe European Space Agency (ESA) mission Gaia\n(\\url{https:\/\/www.cosmos.esa.int\/gaia}), processed by the Gaia\nData Processing and Analysis Consortium (DPAC;\n\\url{https:\/\/www.cosmos.esa.int\/web\/gaia\/dpac\/consortium}).\nFunding for the DPAC has been provided by national institutions,\nin particular the institutions participating in\nthe Gaia Multilateral Agreement. We also used some spectra\nof the BeSS database, operated at LESIA,\nObservatoire de Meudon, France: \\url{http:\/\/basebe.obspm.fr}.\nFinally, we acknowledge the use of the electronic database from\nthe CDS, Strasbourg, and the electronic bibliography maintained by\nthe NASA\/ADS system.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\n\\section{Introduction}\nAlthough the fifth-generation (5G) wireless network is still under deployment, researchers have moved forward to define the next-generation or sixth-generation (6G) wireless network, with the aim for achieving more stringent performance, such as unprecedentedly high throughput, super-high reliability, ultra-low latency, extremely low power consumption, etc \\cite{9040264,8766143}.\nHowever, these targets may not be fully achieved by only relying on the existing technologies, such as massive multi-input multi-output (MIMO) and millimeter wave (mmWave) communications, which can attain enhanced performance but generally incur more substantial energy consumption and hardware cost.\nOn the other hand, wireless communication performance is fundamentally constrained by the wireless channel impairments such as path-loss, shadowing, and small-scale fading, which can be partially mitigated by conventional wireless communication techniques such as power control, adaptive modulation, diversity, dynamic beamforming, etc., but still remain random and uncontrolled at large.\nRecently, \\emph{intelligent reflecting surface} (IRS) has emerged as a promising technology to address the above issues by leveraging massive low-cost reflecting elements to flexibly and dynamically control the radio signal propagation environment in favor of wireless communications\/sensing, thus achieving substantially improved communication spectral\/energy efficiency and sensing accuracy cost-effectively \\cite{9326394,9724202}.\n\nThe existing works on IRS have mainly considered the wireless systems aided by \\emph{passive IRS}.\nSpecifically, as illustrated in Fig. \\ref{p-IRS}, the passive IRS is composed of a large number of passive reflecting elements with positive resistances (e.g., positive-intrinsic-negative (PIN) diodes, field-effect transistors (FETs), micro-electromechanical system (MEMS) switches) \\cite{9326394}.\nAs such, each passive element can reflect the incident signal with a desired phase shift, while it has no signal processing\/amplification capability due to the lack of transmit\/receive radio frequency (RF) chains. Moreover, compared with the conventional half-duplex active relay, the passive IRS operates in a full-duplex mode and hence is free of amplification\/processing noise as well as self-interference \\cite{9119122}.\nBy properly adjusting the individual phase shifts of all passive reflecting elements with the reflection amplitude no larger than one, the reflected signal by IRS can be added constructively with that from the other propagation paths for enhancing the signal power at the intended receiver \\cite{9362274,9241706} or destructively for suppressing the undesired interference \\cite{9171881}.\nRemarkably, it has been shown in \\cite{8811733} that the passive IRS beamforming can achieve a \\emph{squared power scaling order}, i.e., $\\mathcal{O}(N^2)$ with $N$ denoting the number of reflecting elements, which is even higher than that of the massive MIMO with active arrays.\nExtensive research has been conducted recently to efficiently incorporate passive IRS into wireless systems for various purposes, e.g., enhancing the communication throughput \\cite{9714463}, reducing the outage probability \\cite{9205879}, saving the transmit power \\cite{8741198}, and extending the range of active relays \\cite{9464248,9586067}, among others.\nHowever, the performance gain of passive IRS is fundamentally constrained by the severe \\emph{product-distance} path-loss of the reflected channel by IRS \\cite{8888223}.\nTwo practical approaches to dealing with this problem are, respectively, deploying more passive elements at each IRS to enhance its aperture\/beamforming gain and placing the passive IRSs closer to the transmitter and\/or receiver for reducing the reflected channel product-distance \\cite{8982186}. However, these solutions may not be suitable for practical scenarios when the space of IRS site is limited and\/or its location cannot be freely selected.\n\\begin{figure}[t] \\centering    \n{\\subfigure[{Passive IRS.}] {\n\\label{p-IRS}\n\\includegraphics[width=2in]{p-IRS.pdf}  \n}}     \n{\\subfigure[{Active IRS.}] {\\label{a-IRS}\n\\includegraphics[width=2in]{a-IRS.pdf}     \n}}\n{\\subfigure[{Hybrid active-passive IRS.}] {\\label{h-IRS}\n\\includegraphics[width=2in]{h-IRS.pdf}     \n}}\n{\\caption{Different types of IRS architecture.}}\n\\end{figure}\n\nTo tackle the above issues, a new type of IRS, called \\emph{active IRS}, has been recently proposed (see, e.g., \\cite{9377648,9734027,9530750,9568854}). \nSpecifically, the active IRS comprises a number of active reflecting elements, each equipped with an active load (or called negative resistance) such as the tunnel diode and negative impedance converter.\nAs illustrated in Fig. \\ref{a-IRS}, each active load is connected to an additional power supply for signal amplification \\cite{8403249,9219017}.\nTherefore, the active IRS not only enables adjustable phase shifts as the passive IRS, but also allows the amplitude amplification (i.e., larger than one) of incident signals in a full-duplex mode, albeit at modestly higher hardware and energy cost than the passive IRS \\cite{9377648}.\nOn the other hand, compared to the active relay that attaches RF chains to the antennas, the active IRS does not entail costly and power-hungry RF chain components \\cite{9758764}.\nThe performance comparison between the active and passive IRSs has been recently studied in the literature.\nFor example, given the active-IRS location and power budget, it has been shown that the active IRS can achieve higher spectral efficiency \\cite{9377648}, energy efficiency \\cite{9568854}, and reliability \\cite{9530403} than the passive IRS.\nBesides, the authors in \\cite{9530750} further optimized the IRS placement for both the passive- and active-IRS aided systems for rate maximization. It was shown that the passive IRS achieves higher rate performance than the active IRS with their respectively optimized placement when the number of reflecting elements is large and\/or the active-element amplification power is small. \nMoreover, the authors in \\cite{9734027} considered the same power budget constraint for both the passive- and active-IRS aided systems, where both the base station's (BS's) transmit power and active IRS's amplification power are considered in the case of active IRS. It was revealed that the active IRS outperforms the passive IRS only when the number of reflecting elements is small and\/or the amplification power of the active IRS is sufficiently large.\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=5in]{sysmod.pdf}}\n\\caption{A hybrid active-passive IRS aided wireless communication system.}\\label{sysmod}\n\\end{figure}\nTo summarize, the existing works on IRS have shown that passive and active IRSs have complementary advantages.\nSpecifically, the passive IRS has a higher asymptotic beamforming gain than the active IRS (i.e., $\\mathcal{O}(N^2)$ versus $\\mathcal{O}(N)$), thus is more appealing when the number of reflecting elements $N$ is large \\cite{9530750,9377648}. In contrast, the active IRS provides additional power amplification gain, which leads to a much higher signal-to-noise ratio (SNR) than the passive IRS when $N$ is relatively small \\cite{9723093,9716895,8403249}.\nBesides, the active IRS generally incurs higher cost and power consumption than the passive IRS. These thus indicate that given a total budget on the IRS deployment cost (or equivalently the number of active\/passive reflecting elements to be deployed), the conventional IRS architectures with either passive or active elements only in general may not achieve the optimum communication performance.\n\nMotivated by the above, we propose in this paper a new \\emph{hybrid active-passive IRS}\\footnote{We use the term hybrid IRS to denote this new architecture hereafter for brevity.} architecture as shown in Fig. \\ref{h-IRS} to achieve the advantages of both passive and active IRSs for further improving the performance over that with active or passive IRS alone.\nSpecifically, the hybrid IRS is composed of two co-located sub-surfaces, each consisting of a certain number of passive and active reflecting elements, respectively.  \nIn particular, we design the active versus passive reflecting elements allocation at the hybrid IRS under a given deployment budget, for optimally balancing the trade-off between the unique power amplification gain of active IRS and the higher beamforming gain of passive IRS than active IRS.\nTo this end, we consider a hybrid active-passive IRS aided multi-user communication system as shown in Fig. \\ref{sysmod}, where a BS transmits independent data to a cluster of users.\nA hybrid IRS is properly deployed at the edge of this user cluster to serve the users in its half-space reflection region over different time slots.\nWe consider a given deployment budget for the hybrid IRS with different costs of each active and passive reflecting element based on practical models, where an active element generally incurs higher cost than its passive counterpart.\nTo reduce the real-time channel estimation overhead and avoid frequent IRS reflection adjustment, we consider a practical approach that designs the hybrid IRS beamforming based on the \\emph{statistical} channel state information (CSI) only, assuming the practical Rician fading channel model (i.e., only the channel path-loss parameters and Rician fading factors are assumed to be known), instead of requiring the knowledge of the instantaneous CSI of all links involved.\nIn the following, we summarize the main contributions of this paper.\n\n\\begin{itemize}\n\\item First, to guarantee the achievable rate performance of all users, we formulate an optimization problem to maximize the ergodic capacity of the worst-case user located at the boundary of the IRS reflection region.\nSpecifically, we assume the statistical CSI available only and jointly optimize the active\/passive reflecting elements allocation, their phase shifts, and the amplification factors of active elements, subject to various practical constraints on the active-element amplification factor and amplification power consumption, as well as the total active and passive elements deployment budget.\nThis problem, however, is shown to be non-convex, which is difficult to be optimally solved in general.\nTo address this difficulty, we approximate the ergodic capacity of the worst-case user with high accuracy and thereby reformulate the original problem in a simpler form.\n\n\\item Next, we propose an efficient algorithm to solve the reformulated problem. First, we jointly optimize all elements' phase shifts and the amplification factors of active elements based on the statistical CSI, and obtain a closed-form expression for the achievable ergodic capacity with a given elements allocation.\nThen, we apply the one-dimensional search to find the optimal active\/passive elements allocation to maximize the ergodic capacity. \nTo obtain useful insight into the optimal elements allocation, we further consider two special cases where the involved channels are line-of-sight (LoS) and follow Rayleigh fading, respectively. \nIt is shown that in the former case with LoS paths, only active elements need to be deployed when the total deployment budget is sufficiently small, while both active and passive elements should be deployed with a decreasing number ratio when the budget increases and exceeds a certain threshold.\n\n\\item Last, we present extensive numerical results to evaluate the effectiveness of our proposed hybrid IRS architecture and its optimized design for rate maximization.\nWe show that the hybrid IRS with optimized elements allocation outperforms the conventional active\/passive-only IRS architecture as well as other benchmarks. \nMoreover, the optimal active\/passive elements allocation under the general Rician fading channel and that under the LoS channel are presented, which are in accordance with our theoretical analysis.\nThe effects of several key parameters such as the Rician fading factor, active-element amplification power, and active\/passive-element deployment cost on the capacity performance and optimal active\/passive elements allocation are also investigated.\n\\end{itemize}\n\nIt is worth noting that the authors in \\cite{48550,9733238} considered a hybrid relay-reflecting intelligent surface architecture with a few relaying elements connected to power amplifiers and RF chains, which, however, significantly differs from our proposed hybrid IRS architecture with both active and passive reflecting elements. \nFor example, in \\cite{48550,9733238}, the relaying elements forward the signals with power-hungry RF chains, while the active reflecting elements in our proposed hybrid IRS architecture adopt the negative resistance components of much lower power consumption for signal amplification.\nThe remainder of this paper is organized as follows. The system model is first introduced in Section \\ref{sec_sysmod}, based on which we formulate an optimization problem and approximate the worst-case user's ergodic capacity in Section \\ref{sec_prob_approx}. \nIn Section \\ref{sec_design}, we elaborate the algorithm to solve the reformulated optimization problem, and present theoretical results that provide useful insight into the optimal IRS active\/passive elements allocation.\nSimulation results and pertinent discussions are presented in Section \\ref{sec_sim}. Finally, the conclusions are drawn in Section \\ref{sec_conclu}.\n\n\\emph{Notations}: \nSuperscript $(\\cdot)^H$ stands for the Hermitian transpose operation. $\\mathbb{C}^{a \\times b}$ denotes the space of $a \\times b$ complex-valued matrices, $\\mathbb{N}$ denotes the set of natural numbers, and $\\mathbb{R}^+$ denotes the set of positive real numbers. \nThe operation $\\mathbb{E}\\left\\{\\cdot\\right\\}$ returns the expected value of a random variable, $\\arg(\\cdot)$ returns the angle of a complex number, and ${\\operatorname{diag}}{(\\boldsymbol{x})}$ returns a diagonal matrix with the elements in $\\boldsymbol{x}$ on its main diagonal. The notation $\\jmath$ represents the imaginary unit, $\\otimes$ denotes the Kronecker product, $\\mathcal{C} \\mathcal{N}(\\mu,\\sigma^2)$ denotes the circularly symmetric complex Gaussian (CSCG) random variable with mean of $\\mu$ and variance of $\\sigma^2$,\n$[\\cdot]_{m,n}$ denotes the $(m,n)$-th entry of a matrix, $[\\cdot]_{m}$ denotes the $m$-th entry of a vector, and $\\lfloor\nx\\rfloor$ denotes the largest integer that does not exceed the real number $x$.\n\n\\section{System Model}\\label{sec_sysmod}\nAs illustrated in Fig.~\\ref{sysmod}, we consider a hybrid active-passive IRS aided wireless communication system, where a single-antenna BS transmits independent data to a cluster of single-antenna users\\footnote{The proposed hybrid IRS and its results obtained in this paper can be extended to the case with multi-antenna BS and multiple IRSs\/user clusters, which will be investigated in our future work.}.\nWe assume that the BS and its served users are separated by long distance and the direct links between them are negligible due to the large channel path-loss and\/or severe blockage. \nAs such, a hybrid IRS comprising both active and passive reflecting elements is properly placed to serve the users in its half-space reflection region.\nMoreover, we consider the time division multiple access (TDMA) scheme, where the users are served by the BS over orthogonal time slots\\footnote{Under this setup, the TDMA scheme has been shown to outperform the non-orthogonal multiple access (NOMA) and orthogonal frequency division multiple access (OFDMA) schemes in terms of energy and spectral efficiency \\cite{8970580}.}. \nTo guarantee the system performance among all users, we consider the worst-case user performance at the boundary of the IRS reflection region with the IRS-user distance $d_{\\mathrm{IU}}$ and BS-IRS distance $D_{\\mathrm{BI}}$ (see Fig. \\ref{sysmod}).\n\nFor ease of implementation, we assume that the hybrid IRS comprises two co-located sub-surfaces with $N_{\\mathrm{pas}}$ passive and $N_{\\mathrm{act}}$ active reflecting elements, respectively (see Fig. \\ref{sysmod}).\nSpecifically, we denote by \n$\\mathbf{\\Psi}^{\\mathrm{pas}}\\triangleq {\\operatorname{diag}}{(e^{\\jmath\\varphi^{\\mathrm{pas}}_1},\\cdots,e^{\\jmath\\varphi^{\\mathrm{pas}}_{N_{\\mathrm{pas}}}})}$\nthe reflection matrix of the passive sub-surface, where $\\varphi^{\\mathrm{pas}}_n$ denotes the phase shift of the $n$-th passive element with $n\\in\\mathcal{N}_{\\mathrm{pas}}\\triangleq \\{1,\\cdots,N_{\\mathrm{pas}}\\}$, and the reflection amplitude of each passive element is set as one (i.e., its maximum value). The cost of each passive element is denoted by $W_{\\mathrm{pas}}$.\nOn the other hand, as the active sub-surface can simultaneously amplify the signal and tune its phase shift, we denote by $\\mathbf{\\Psi}^{\\mathrm{act}}\\triangleq\\mathbf{A}^{\\mathrm{act}}\\mathbf{\\Phi}^{\\mathrm{act}}$ the reflection matrix of the active sub-surface, where $\\mathbf{A}^{\\mathrm{act}}\\triangleq{\\operatorname{diag}}{(\\alpha_1,\\cdots,\\alpha_{N_{\\mathrm{act}}})}$ and $\\mathbf{\\Phi}^{\\mathrm{act}}\\triangleq\n{\\operatorname{diag}}{(e^{\\jmath\\varphi^{\\mathrm{act}}_1},\\cdots,e^{\\jmath\\varphi^{\\mathrm{act}}_{N_{\\mathrm{act}}}})}$\ndenote respectively its reflection amplification matrix and phase-shift matrix with $\\alpha_n$ and $\\varphi^{\\mathrm{act}}_n$ representing the amplification factor and phase shift of each active element $n\\in\\mathcal{N}_{\\mathrm{act}}\\triangleq\\{1,\\cdots,N_{\\mathrm{act}}\\}$. \nTo ensure that each active reflecting element amplifies the signal, we impose a constraint on the amplification factor for each active element as $\\alpha_n\\geq \\alpha_{\\min},\\forall n\\in\\mathcal{N}_{\\mathrm{act}}$ with $\\alpha_{\\min}\\geq 1$ \\cite{9377648}. Moreover, the limited load of each active element \\cite{8403249} leads to the following constraint on its maximum amplification factor: $\\alpha_n\\leq\\alpha_{\\max},\\forall n\\in\\mathcal{N}_{\\mathrm{act}}$ with $\\alpha_{\\max}>\\alpha_{\\min}$.\nLet $W_{\\mathrm{act}}$ represent the deployment cost of each active element, which in general is larger than that of each passive element (i.e., $W_{\\mathrm{act}}>W_{\\mathrm{pas}}$) due to its more sophisticated hardware (i.e., additional power amplifier and amplification control circuit \\cite{9377648}) and higher static operation power (e.g., 6-20 mW for active element \\cite{7920385} versus 5 mW for passive element \\cite{8888223}). \nMoreover, we denote $W_0$ as the total deployment budget for the hybrid IRS such that $N_{\\mathrm{act}}W_{\\mathrm{act}}+N_{\\mathrm{pas}}W_{\\mathrm{pas}}\\leq W_0$.\n\nWe assume the practical Rician fading channel model for all involved links. As such, the baseband equivalent channel from the BS to the active IRS sub-surface, denoted by $\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{act}}\\in\\mathbb{C}^{N_{\\mathrm{act}}\\times 1}$, can be modeled as \n\\begin{equation}\n    \\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{act}}=\\sqrt{\\frac{K_{1}}{K_{1}+1}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}+\\sqrt{\\frac{1}{K_{1}+1}}\\tilde{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}},\n\\end{equation}\nwhere $K_{1}$ is the Rician fading factor of the BS$\\to$IRS link, $\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}\\in\\mathbb{C}^{N_{\\mathrm{act}} \\times 1}$ is the LoS component, and $\\tilde{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}\\in\\mathbb{C}^{N_{\\mathrm{act}} \\times 1}$ is the non-LoS (NLoS) component.\nSpecifically, the LoS component can be modeled as $\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}={h}_{\\mathrm{BI}}^{\\mathrm{act}}\\boldsymbol{a}_{\\mathrm{r}}\\left(\\vartheta_{\\mathrm{BI}}^{\\mathrm{r}},\\vartheta_{\\mathrm{BI}}^{\\mathrm{r}}, N_{\\mathrm{act}}\\right)$, where ${h}_{\\mathrm{BI}}^{\\mathrm{act}}\\triangleq\\sqrt{\\beta}\/D_{\\mathrm{BI}}e^{-\\jmath\\frac{2\\pi}{\\lambda}D_{\\mathrm{BI}}}$ denotes the complex channel gain with $\\lambda$ and $\\beta$ representing respectively the carrier wavelength and the reference channel gain at a distance of 1 meter (m); and $\\vartheta_{\\mathrm{BI}}^{\\mathrm{r}}\\left(\\vartheta_{\\mathrm{BI}}^{\\mathrm{r}}\\right) \\in[0, \\pi]$ represents the azimuth (elevation) angle-of-arrival (AoA) at the IRS.\nMoreover, $\\boldsymbol{a}_{\\mathrm{r}}\\left(\\vartheta_{\\mathrm{BI}}^{\\mathrm{r}}, \\vartheta_{\\mathrm{BI}}^{\\mathrm{r}}, N_{\\mathrm{act}}\\right)$ denotes the receive response vector, which is given by $\\boldsymbol{a}_{\\mathrm{r}}\\left(\\vartheta_{\\mathrm{BI}}^{\\mathrm{r}}, \\vartheta_{\\mathrm{BI}}^{\\mathrm{r}}, N_{\\mathrm{act}}\\right)=\\boldsymbol{u}\\left(\\frac{2 d_{\\mathrm{I}}}{\\lambda} \\cos \\left(\\vartheta_{\\mathrm{BI}}^{\\mathrm{r}}\\right) \\sin \\left(\\vartheta_{\\mathrm{BI}}^{\\mathrm{r}}\\right), N_{\\mathrm{x}}\\right) \\otimes$ $\\boldsymbol{u}\\left(\\frac{2 d_{\\mathrm{I}}}{\\lambda} \\sin \\left(\\vartheta_{\\mathrm{BI}}^{\\mathrm{r}}\\right) \\sin \\left(\\vartheta_{\\mathrm{BI}}^{\\mathrm{r}}\\right), N_{\\mathrm{y}}\\right)$, with $\\boldsymbol{u}(\\varsigma, M)=[1, e^{-\\jmath \\pi \\varsigma},$ $\\ldots, $ $e^{-(M-1) \\jmath \\pi \\varsigma}]^{T}$ representing the steering vector function, $d_{\\mathrm{I}}$, $N_{\\mathrm{x}}$, and $N_{\\mathrm{y}}$ denoting the distance between adjacent reflecting elements and the number of active reflecting elements along the $x$- and $y$-axis on IRS surface, respectively.\nBesides, the NLoS component, $\\tilde{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}$, follows the complex Gaussian distribution with each entry $[\\tilde{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}]_{n}\\sim\\frac{\\sqrt{\\beta}}{D_{\\mathrm{BI}}}\\mathcal{C} \\mathcal{N}(0,1), \\forall n\\in\\mathcal{N}_{\\mathrm{act}}$.\nSimilarly, the baseband equivalent channel from the active IRS sub-surface to the worst-case user, $\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}}\\in\\mathbb{C}^{N_{\\mathrm{act}}\\times 1}$, can be modeled as\n\\begin{equation}\n    \\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}}=\\sqrt{\\frac{K_{2}}{K_{2}+1}}\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}}+\\sqrt{\\frac{1}{K_{2}+1}}\\tilde{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}},\n\\end{equation}\nwhere $K_{2}$ denotes the Rician fading factor of the IRS$\\to$user link, $\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}}\\in\\mathbb{C}^{N_{\\mathrm{act}} \\times 1}$ denotes the LoS component, and $\\tilde{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}}\\in\\mathbb{C}^{N_{\\mathrm{act}} \\times 1}$ denotes the NLoS component.\nThe baseband equivalent channel from the BS to IRS passive sub-surface, $\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{pas}}$, and that from the IRS passive sub-surface to the worst-case user, $\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{pas}}$, can be defined similarly as $\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{act}}$ and $\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}}$, with the details omitted for brevity. \n\nBased on the above, the received signal at the worst-case user aided by the hybrid IRS is given by\n\\begin{equation}\n    y = (\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{act}}s+(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{pas}}s+(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\mathbf{z}_{\\mathrm{I}}+z_0,\n   \n\\end{equation}\nwhere $s\\in\\mathbb{C}$ denotes the information symbol with $\\mathbb{E}\\left\\{\\left|s\\right|^{2}\\right\\}=P_{\\mathrm{B}}$, and $P_{\\mathrm{B}}$ denotes the transmit power of the BS. \nMoreover, $\\mathbf{z}_{\\rm{I}}\\in\\mathbb{C}^{N_{\\mathrm{act}} \\times 1}$ is the thermal noise introduced by the active elements due to signal amplification, which is assumed to follow the independent CSCG distribution, i.e., $\\mathbf{z}_{\\rm{I}} \\sim \\mathcal{C} \\mathcal{N}\\left(\\mathbf{0}_{N_{\\mathrm{act}}}, \\sigma_{\\mathrm{I}}^{2} \\mathbf{I}_{N_{\\mathrm{act}}}\\right)$ with $\\sigma_{\\mathrm{I}}^{2}$ denoting the amplification noise power, and $z_0\\sim\\mathcal{C} \\mathcal{N}\\left(0, \\sigma_0^{2}\\right)$ is the additive white Gaussian noise (AWGN) at the user. \nNote that at the user's receiver, the desired signal is superposed by the reflected signals over both active and passive elements, while the noise is due to the amplification noise at active elements and the thermal noise at the receiver.\n\nAs such, the receiver SNR at the worst-case user is given by\n\\begin{equation}\n    \\gamma = \\frac{P_{\\mathrm{B}}|(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{act}}+(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{pas}}|^2}{\\sigma_{\\mathrm{I}}^2\\|(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\|^2+\\sigma_0^2}.\\label{snr_hybrid}\n\\end{equation}\nThus, the ergodic capacity achieved by the worst-case user in the hybrid IRS aided wireless communication system is given by\n\\begin{equation}\n    C=\\mathbb{E}\\left\\{\\log _{2}(1+\\gamma)\\right\\},\\label{ergo_capa}\n   \n\\end{equation}\nwhere the expectation is taken over the random NLoS components in all channels involved.\n\\section{Problem Formulation and Ergodic Capacity Analysis}\\label{sec_prob_approx}\nWe aim to maximize the ergodic capacity of the worst-case user subject to a total deployment budget of $W_0$ by optimizing the numbers of active and passive elements,\n$N_{\\mathrm{act}}$ and $N_{\\mathrm{pas}}$, the IRS phase shifts, \\{$\\mathbf{\\Phi}^{\\mathrm{act}}$, $\\mathbf{\\Psi}^{\\mathrm{pas}}$\\}, and the active-element amplification matrix, $\\mathbf{A}^{\\mathrm{act}}$. This problem can be formulated as follows.\n\\begin{align}\n    &\\hspace{-0.2cm}\\mathrm{(P1)}~~~~~\\max_{\\mathbf{\\Phi}^{\\mathrm{act}},\\mathbf{\\Psi}^{\\mathrm{pas}},\\mathbf{A}^{\\mathrm{act}},N_{\\mathrm{act}},N_{\\mathrm{pas}}}\n    \\quad~~\\mathbb{E}\\left\\{\\log _{2}\\left(1\\!+\\!\\frac{P_{\\mathrm{B}}|(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{act}}\\!+\\!(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{pas}}|^2}{\\sigma_{\\mathrm{I}}^2\\|(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\|^2\\!+\\!\\sigma_0^2}\\right)\\right\\} \\label{obj_func_orig}\\\\\n    &\\qquad\\qquad~~~~\\!~\\quad~\\text{s.t.} \\qquad\\qquad\\qquad~~0<\\varphi_n^{\\mathrm{act}}\\leq 2\\pi,\\forall n\\in\\mathcal{N}_{\\mathrm{act}},\\label{cons_phase_act}\\\\\n    &\\qquad\\qquad~~~~~~~~\\quad~\\qquad\\qquad\\qquad~~0<\\varphi_n^{\\mathrm{pas}}\\leq 2\\pi,\\forall n\\in\\mathcal{N}_{\\mathrm{pas}},\\\\\n    &\\qquad\\qquad~~~~~~~~\\quad~\\qquad\\qquad\\qquad~~\\alpha_{\\min}\\leq \\alpha_n\\leq \\alpha_{\\max},\\forall n\\in\\mathcal{N}_{\\mathrm{act}},\\label{cons_alpha}\\\\\n    &\\qquad\\qquad~~~~~~~~\\quad~\\qquad\\qquad\\qquad~~ \\mathbb{E}\\left\\{P_{\\mathrm{B}}\\|\\mathbf{\\Psi}^{\\mathrm{act}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{act}}\\|^2+\\sigma_{\\mathrm{I}}^2\\|\\mathbf{\\Psi}^{\\mathrm{act}}\\|^2\\right\\}\\leq P_{\\mathrm{I}},\\label{cstr_power_HI}\\\\\n    &\\qquad\\qquad~~~~~~~~\\quad~\\qquad\\qquad\\qquad~~N_{\\mathrm{act}}W_{\\mathrm{act}}+N_{\\mathrm{pas}}W_{\\mathrm{pas}}\\leq W_0,\\label{cons_C}\\\\\n    &\\qquad\\qquad~~~~~~~~\\quad~\\qquad\\qquad\\qquad~~N_{\\mathrm{act}}\\in\\mathbb{N},N_{\\mathrm{pas}}\\in\\mathbb{N}\\label{cons_C_AnP},\n\\end{align}\nwhere the constraint \\eqref{cstr_power_HI} indicates that the average amplification power of all active elements over the Rician fading channels is constrained by the average amplification power budget, $P_{\\mathrm{I}}$.\nNote that different from the existing works on IRS-aided wireless communications that generally require the instantaneous CSI knowledge (e.g., \\cite{9133142}), we consider a practical scenario where only the statistical CSI, i.e., $\\{\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}},\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}},$ $\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{pas}},$ $\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{pas}},$ $K_1,$ $K_2\\}$ is known \\textit{a priori}, which suffices for the design of active\/passive elements allocation for maximizing the ergodic capacity (in the worst case).\nThis approach also reduces the real-time channel estimation overhead and avoids frequent IRS reflection adjustment for each user. \n\nFor problem (P1), note that it includes the conventional IRS architectures with all-passive and all-active elements as\ntwo special cases. Specifically, when $N_{\\mathrm{act}}=0$, the hybrid IRS reduces to the conventional passive IRS and we have $\\mathbf{\\Psi}^{\\mathrm{act}}=\\mathbf{0}_{N_{\\mathrm{act}}\\times N_{\\mathrm{act}}}$. On the other hand, when $N_{\\mathrm{pas}}=0$, it reduces to the conventional active IRS and thus $\\mathbf{\\Psi}^{\\mathrm{pas}}=\\mathbf{0}_{N_{\\mathrm{pas}}\\times N_{\\mathrm{pas}}}$.\nHowever, problem (P1) is challenging to solve even in the above two special cases, since the phase shifts are coupled with the amplification factors in the function of the ergodic capacity (see \\eqref{snr_hybrid} and \\eqref{ergo_capa}). Moreover, the numbers of active and passive elements are discrete, rendering the design objective a complicated function and the constraints in \\eqref{cstr_power_HI}--\\eqref{cons_C_AnP} non-convex.\n\nTo address the above issues, we first analyze the ergodic capacity to approximate the objective function of problem (P1) in a simpler form.\n\n{\\color{black}\\begin{lemma}\\label{lem_C_approx}\n\\textbf{\\emph{(Ergodic Capacity Approximation)}} \\emph{The ergodic capacity in \\eqref{obj_func_orig} can be approximated by}\n\\begin{align\n    C\\approx \\tilde{C}\\triangleq\\log _{2}\\left(1+\\frac{x_{\\mathrm{L}}+x_{\\mathrm{NL,act}}+x_{\\mathrm{NL,pas}}}{z_{\\mathrm{L,act}}+z_{\\mathrm{NL,act}}+\\sigma_0^2}\\right),\\label{sig_approx}\n\\end{align}\n\\emph{where}\n\\begin{align}\n    &x_{\\mathrm{L}} \\triangleq \\frac{K_1K_2P_{\\mathrm{B}}}{(K_1\\!+\\!1)(K_2\\!+\\!1)}\\left|(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}\\!+\\!(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{pas}}\\right|^2,\\label{x_1}\\\\\n    &x_{\\mathrm{NL,act}}\\triangleq\\frac{P_{\\mathrm{B}}}{(K_1\\!+\\!1)(K_2\\!+\\!1)}\\Big(\\frac{K_1\\beta}{d_{\\mathrm{IU}}^2}\\|\\mathbf{\\Psi}^{\\mathrm{act}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}\\|^2\\!+\\!\\frac{K_2\\beta}{D_{\\mathrm{BI}}^2}\\|(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\|^2\\!+\\!\\frac{\\beta^2}{D_{\\mathrm{BI}}^2d_{\\mathrm{IU}}^2}\\sum_{n=1}^{N_{\\mathrm{act}}}\\alpha^2_{n}\\Big),\\label{x_2}\\\\\n    &x_{\\mathrm{NL,pas}}\\triangleq\\frac{P_{\\mathrm{B}}}{(K_1\\!+\\!1)(K_2\\!+\\!1)}\\Big(\\frac{K_1\\beta}{d_{\\mathrm{IU}}^2}\\|\\mathbf{\\Psi}^{\\mathrm{pas}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{pas}}\\|^2\\!+\\!\\frac{K_2\\beta}{D_{\\mathrm{BI}}^2}\\|(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\|^2\\!+\\!\\frac{\\beta^2N_{\\mathrm{pas}}}{D_{\\mathrm{BI}}^2d_{\\mathrm{IU}}^2}\\Big),\\label{x_3}\\\\\n    &{z_{\\mathrm{L,act}}}\\triangleq{\\frac{K_2\\sigma_{\\rm I}^2}{K_2\\!+\\!1}\\|(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\|^2},\\quad\n    {z_{\\mathrm{NL,act}}}\\triangleq{\\frac{\\sigma_{\\rm I}^2}{K_2\\!+\\!1}\\|(\\tilde{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\|^2}.\n \\end{align}\n\\end{lemma}}\n\\begin{proof}\nSee Appendix \\ref{proof_lem1}.\n\\end{proof}\nThe accuracy of the approximation in Lemma \\ref{lem_C_approx} will be numerically verified in Section \\ref{sec_sim}.\nSimilarly, the average amplification power consumption can be expressed as\n\\begin{align}\n    \\!\\!\\!\\!\\!\\mathbb{E}\\left\\{P_{\\mathrm{B}}\\|\\mathbf{\\Psi}^{\\mathrm{act}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{act}}\\|^2\\!+\\!\\sigma_{\\mathrm{I}}^2\\|\\mathbf{\\Psi}^{\\mathrm{act}}\\|^2\\right\\}\\!=\n    \\!\\frac{P_{\\mathrm{B}}}{K_2\\!+\\!1}\\left(K_2\\|\\mathbf{\\Psi}^{\\mathrm{act}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}\\|^2\\!+\\!\\mathbb{E}\\left\\{\\|\\mathbf{\\Psi}^{\\mathrm{act}}\\tilde{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}\\|^2\\right\\}\\right)\\!+\\!\\sum_{n=1}^{N_{\\mathrm{act}}}\\sigma_{\\mathrm{I}}^2\\alpha^2_{n}\\label{power_decomp}.\n\\end{align}\nAs such, problem (P1) is reformatted as follows by using Lemma \\ref{lem_C_approx} and substituting \\eqref{power_decomp} into \\eqref{cstr_power_HI},\n\\begin{align}\n    &\\mathrm{(P2)}\\max_{\\mathbf{\\Phi}^{\\mathrm{act}},\\mathbf{\\Psi}^{\\mathrm{pas}},\\mathbf{A}^{\\mathrm{act}},N_{\\mathrm{act}},N_{\\mathrm{pas}}}\n    \\quad~~\\log _{2}\\left(1+\\frac{x_{\\mathrm{L}}+x_{\\mathrm{NL,act}}+x_{\\mathrm{NL,pas}}}{z_{\\mathrm{L,act}}+z_{\\mathrm{NL,act}}+\\sigma_0^2}\\right) \\\\\n    &\\qquad\\qquad~~~~\\!~\\text{s.t.} \\qquad\\qquad~~\\eqref{cons_phase_act}-\\eqref{cons_alpha}, \\eqref{cons_C},\\eqref{cons_C_AnP}\\nonumber,\\\\\n    &\\qquad~~~~~~~\\qquad\\qquad\\qquad~~ \\frac{P_{\\mathrm{B}}}{K_2\\!+\\!1}\\left(K_2\\|\\mathbf{\\Psi}^{\\mathrm{act}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}\\|^2\\!+\\!\\mathbb{E}\\left\\{\\|\\mathbf{\\Psi}^{\\mathrm{act}}\\tilde{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}\\|^2\\right\\}\\right)\\!+\\!\\sum_{n=1}^{N_{\\mathrm{act}}}\\sigma_{\\mathrm{I}}^2\\alpha^2_{n}\\leq\\! P_{\\mathrm{I}}.\\label{cstr_power_HI_decomp}\n\\end{align}\n\n\\section{Optimal Solution to Problem (P2)}\\label{sec_design}\nIn this section, we aim to optimally solve problem (P2) and gain useful insight into the optimal active\/passive elements allocation at the hybrid IRS.\nSpecifically, we first obtain the optimal hybrid IRS beamforming and active\/passive elements allocation based on the statistical CSI under the general Rician fading channel model. Then, we consider two special channel setups, i.e., the LoS and Rayleigh fading channel models, to gain useful insight into the optimal active\/passive elements allocation.\n\n\\subsection{IRS Phase Shift Optimization}\nGiven any feasible active\/passive elements allocation and active-element amplification factors, it can be easily shown that the approximated ergodic capacity in \\eqref{sig_approx} is maximized when the LoS components of the IRS-associated channels, i.e., $(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}$ and $(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{pas}}$, are phase-aligned. The optimal IRS phase shifts are thus given by\n\\begin{align}\n    &\\varphi_{n}^{\\mathrm{act}} = \\arg([\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}}]_n)-\\arg([\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}]_n),\\forall n\\in\\mathcal{N}_{\\mathrm{act}},\\label{opt_phase_1}\\\\\n    &\\varphi_{n}^{\\mathrm{pas}} = \\arg([\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{pas}}]_n)-\\arg([\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{pas}}]_n),\\forall n\\in \\mathcal{N}_{\\mathrm{pas}},\\label{opt_phase_2}\n\\end{align}\nwhere the optimal phase shifts of the active and passive sub-surfaces have the similar form.\nThen, by substituting the optimal phase shifts in \\eqref{opt_phase_1} and \\eqref{opt_phase_2} into \\eqref{sig_approx}, we have\n\\begin{align}\n     &x_{\\mathrm{L}}=\\gamma_1\\left(\\sum_{n=1}^{N_{\\mathrm{act}}}\\alpha_n+N_{\\mathrm{pas}}\\right)^2P_{\\mathrm{B}}\\beta^2\/D^2_{\\mathrm{BI}}d^2_{\\mathrm{IU}},\\\\\n     &x_{\\mathrm{NL,act}}+x_{\\mathrm{NL,pas}}=\\gamma_2\\left(\\sum_{n=1}^{N_{\\mathrm{act}}}\\alpha_n^2+N_{\\mathrm{pas}}\\right)P_{\\mathrm{B}}\\beta^2(K_1+K_2+1)\/D^2_{\\mathrm{BI}}d^2_{\\mathrm{IU}},\\label{x_NLoS}\\\\\n     &z_{\\mathrm{L,act}}+z_{\\mathrm{NL,act}}=\\sum_{n=1}^{N_{\\mathrm{act}}}\\alpha_n^2\\sigma_{\\mathrm{I}}^2\\beta\/d_{\\mathrm{IU}}^{2},\\label{n_act}\n\\end{align}\nwhere \n\\begin{equation}\n    \\gamma_1 \\triangleq \\frac{K_1K_2}{(K_1+1)(K_2+1)}, \\gamma_2 \\triangleq \\frac{1}{(K_1+1)(K_2+1)}.\\label{gamma_approx}\n\\end{equation}\nNote that the considered hybrid IRS beamforming with the statistical CSI only is designed based on the LoS components while ignoring the NLoS components of the involved channels.\n\n\\subsection{Amplification Factor Optimization for Active Elements}\n\nIn this subsection, we optimize the amplification factor given any feasible active\/passive elements allocation and optimal IRS phase shifts (see \\eqref{opt_phase_1} and \\eqref{opt_phase_2}). To this end, we first present an important lemma as follows.\n\\begin{lemma}\\label{lem_min}\n\\textbf{\\emph{(Minimum amplification power for active elements)}}\n\\emph{Given the total deployment budget $W_0$, the hybrid IRS should employ passive elements only (i.e., $N_{\\mathrm{act}}=0$) when}\n\\begin{equation\n    P_{\\mathrm{I}} < \\alpha^2_{\\min}\\left(P_{\\mathrm{B}}\\beta\/D^2_{\\mathrm{BI}}+\\sigma^2_{\\mathrm{I}}\\right),\\label{lem_pas\n\\end{equation}\n\\emph{and employ active elements (i.e., $N_{\\mathrm{act}}>0$) otherwise.}\n\\end{lemma}\n\\begin{proof}\n{It can be shown that when \\eqref{lem_pas} holds, the maximum amplification factor that $P_{\\mathrm{I}}$ can support is smaller than its allowable minimum value, i.e., $\\alpha_{n}<\\alpha_{\\min},\\forall n\\in \\mathcal{N}_{\\mathrm{act}}$, thus making it infeasible to satisfy $\\alpha_{n}\\geq\\alpha_{\\min}$.}\n\\end{proof}\n\nNext, we consider the non-trivial case with $P_{\\mathrm{I}} \\geq \\alpha^2_{\\min}\\left(P_{\\mathrm{B}}\\beta\/D^2_{\\mathrm{BI}}+\\sigma^2_{\\mathrm{I}}\\right)$ and hence $N_{\\mathrm{act}}>0$. Given the optimal phase shift design in \\eqref{opt_phase_1} and \\eqref{opt_phase_2}, problem (P2) reduces to the following problem for optimizing the amplification factors of the $N_{\\mathrm{act}}$ active elements.\n\\begin{align}\n    &\\mathrm{(P3)}~~~~~\\max_{\\mathbf{A}^{\\mathrm{act}}}\n    \\quad~~\\log _{2}\\left(1+\\frac{x_{\\mathrm{L}}+x_{\\mathrm{NL,act}}+x_{\\mathrm{NL,pas}}}{z_{\\mathrm{L,act}}+z_{\\mathrm{NL,act}}+\\sigma_0^2}\\right) \\\\\n    &\\qquad\\!~~~~\\quad~\\text{s.t.} \\qquad~~\\eqref{cons_alpha}\\nonumber,\\\\\n    &\\qquad~\\qquad\\qquad\\qquad\\sum_{n=1}^{N_{\\mathrm{act}}}\\alpha^2_{n}\\left(P_{\\mathrm{B}}\\beta\/D^2_{\\mathrm{BI}}+\\sigma^2_{\\mathrm{I}}\\right)\\leq P_{\\mathrm{I}}.\\label{cstr_power_HI_decomp_2}\n\\end{align}\nThe constraints in \\eqref{cons_alpha} and \\eqref{cstr_power_HI_decomp_2} indicate that the amplification factors are bounded by both the hardware limitation and the power budget for all active elements. \nTo simplify the analysis, we have the following lemma.\n\\begin{lemma}\\label{lem_P_I_range}\n\\textbf{\\emph{(Favorable amplification power condition)}}\n\\emph{For problem (P3), the constraint in \\eqref{cons_alpha} is always satisfied if}\n\\begin{equation}\n\\alpha^2_{\\min}\\left(P_{\\mathrm{B}}\\beta\/D^2_{\\mathrm{BI}}+\\sigma^2_{\\mathrm{I}}\\right)\\leq P_{\\mathrm{I}}\\leq W_0\\alpha^2_{\\max}\\left(P_{\\mathrm{B}}\\beta\/D^2_{\\mathrm{BI}}+\\sigma^2_{\\mathrm{I}}\\right)\/W_{\\mathrm{act}}.\\label{cons_pb\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFirst, it can be shown that when \n\\begin{equation}\n    P_{\\mathrm{I}}\\geq W_0\\alpha^2_{\\max}\\left(P_{\\mathrm{B}}\\beta\/D^2_{\\mathrm{BI}}+\\sigma^2_{\\mathrm{I}}\\right)\/W_{\\mathrm{act}},\\label{lem_max\n\\end{equation}\nthe amplification factors of $N_{\\mathrm{act}}$ active elements are equal to their maximum value, i.e., $\\alpha_{n}=\\alpha_{\\max},\\forall n\\in\\mathcal{N}_{\\mathrm{act}}$, where the power amplification is constrained by the maximum load.\nThen, combining \\eqref{lem_max} and Lemma \\ref{lem_min}, there always exists a feasible $\\alpha_{n}\\in[\\alpha_{\\min},\\alpha_{\\max}]$ such that the constraint in \\eqref{cons_alpha} of problem (P3) is satisfied if \\eqref{cons_pb} holds by choosing an appropriate $N_{\\mathrm{act}}\\in[0,W_0\/W_{\\mathrm{act}}]$.\n\\end{proof}\n\nLemma \\ref{lem_P_I_range} gives the favorable amplification power condition for active elements that are able to operate in the amplification mode (i.e., $\\alpha_{n}\\geq\\alpha_{\\min}$) and make full use of the amplification power (i.e., $\\alpha_{n}\\leq\\alpha_{\\max}$).\nIn contrast, if the amplification power is too large, i.e., satisfying \\eqref{lem_max}, the amplification factor of each active element is constrained by its hardware limitation, i.e., $\\alpha_n=\\alpha_{\\max},\\forall n\\in\\mathcal{N}_{\\mathrm{act}}$. On the other hand, if the amplification power is insufficient, i.e., when \\eqref{lem_pas} holds, the active elements cannot operate in the amplification mode, i.e., $\\alpha_n< \\alpha_{\\min},\\exists n\\in \\mathcal{N}_{\\mathrm{act}}$, thus making problem (P3) infeasible. \nTherefore, in the sequel, we only consider the case where $P_{\\mathrm{I}}$ satisfies the favorable amplification power condition given in \\eqref{cons_pb} in order to draw useful insight into the optimal IRS active\/passive elements allocation.\n\\begin{lemma}\\label{lem1}\n\\textbf{\\emph{(Optimal amplification factors)}}\n\\emph{Given \\eqref{cons_pb} and $N_{\\mathrm{act}}>0$, the optimal amplification factor for each active element in problem (P3) is given by}\n\\begin{align}\n    \\alpha_{n}=\\alpha^* \\triangleq \\sqrt{\\frac{P_{\\mathrm{I}}\/N_{\\mathrm{act}}}{P_{\\mathrm{B}}\\beta\/D^2_{\\mathrm{BI}}+\\sigma^2_{\\mathrm{I}}}}, \\forall n \\in\\mathcal{N}_{\\mathrm{act}}.\\label{opt_a_n}\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nSee Appendix \\ref{proof_lem2}.\n\\end{proof}\nLemma \\ref{lem1} shows that all active reflecting elements should adopt a common amplification factor due to the same path-loss over the LoS channels associated with the active elements.\nMoreover, it is observed from \\eqref{opt_a_n} that given the amplification power $P_{\\mathrm{I}}$, the optimal amplification factor, $\\alpha^*$, decreases with the number of active elements or signal power.\n\\begin{remark}\n\\textbf{\\emph{(Amplification noise power)}}\n\\emph{By substituting $\\alpha^*$ in \\eqref{opt_a_n} into \\eqref{n_act}, the amplification noise power is given by}\n\\begin{equation}\n    z_{\\mathrm{L,act}}+z_{\\mathrm{NL,act}}=\\frac{\\mathcal{I}_{\\mathbb{R}^+}(N_{\\mathrm{act}})P_{\\mathrm{I}}\\sigma_{\\mathrm{I}}^2\\beta\/d_{\\mathrm{IU}}^{2}}{P_{\\mathrm{B}}\\beta\/D_{\\mathrm{BI}}^{2}+\\sigma_{\\mathrm{I}}^{2}},\\label{noise_term}\n\\end{equation}\n\\emph{where the indicator function $\\mathcal{I}_{\\mathbb{R}^+}(N_{\\mathrm{act}})=1$ when $N_{\\mathrm{act}}>0$ and $\\mathcal{I}_{\\mathbb{R}^+}(N_{\\mathrm{act}})=0$ otherwise, i.e., the noise term exists only when the hybrid IRS consists of active elements.\nNote that the amplification noise power in \\eqref{noise_term} is a constant that depends on the total amplification power, $P_{\\mathrm{I}}$, and the path-loss of the BS-IRS and IRS-user channels.\n\n\\end{remark}\nBased on Lemma \\ref{lem1}, the ergodic capacity of the worst-case user with the optimal IRS phase shifts in \\eqref{opt_phase_1} and \\eqref{opt_phase_2} and amplification factors of active elements in \\eqref{opt_a_n} is given by\n\\begin{equation}\n\\tilde C^*\\!=\\!\\log _{2}\\left(1+\\frac{\\frac{P_{\\mathrm{B}} \\beta^{2}}{D^2_{\\mathrm{BI}}d^2_{\\mathrm{IU}}}\\left(\\gamma_{1} \\left(\\sqrt{A_{\\mathrm{sum}}N_{\\mathrm{act}}}+N_{\\mathrm{pas}}\\right)^{2}+\\gamma_{2}\\left(A_{\\mathrm{sum}}\\!+\\!N_{\\mathrm{pas}}\\right)\\right)}{A_{\\mathrm{sum}} \\sigma_{\\mathrm{I}}^{2}\\beta \/ d_{\\mathrm{IU}}^{2}\\!+\\!\\sigma_{0}^{2}}\\right),\\label{capa_approx2}\n\\end{equation}\nwhere $A_{\\mathrm{sum}}\\triangleq\\frac{\\mathcal{I}_{\\mathbb{R}^+}(N_{\\mathrm{act}})P_{\\mathrm{I}}}{P_{\\mathrm{B}}\\beta\/D_{\\mathrm{BI}}^{2}+\\sigma_{\\mathrm{I}}^{2}}$.\n\n\\subsection{Active\/Passive Elements Allocation Optimization}\\label{ele_alloc}\nGiven the optimal IRS phase shifts and amplification factors of active elements, in the next, we focus on optimizing the active\/passive elements allocation for maximizing the ergodic capacity subject to the total deployment budget constraint, which is formulated as follows.\n\\begin{align}\n    &\\mathrm{(P4)}~~~~~\\max_{N_{\\mathrm{act}},N_{\\mathrm{pas}}}\n    \\quad~~\\log _{2}\\left(1+\\frac{\\frac{P_{\\mathrm{B}} \\beta^{2}}{D^2_{\\mathrm{BI}}d^2_{\\mathrm{IU}}}\\left(\\gamma_{1} \\left(\\sqrt{A_{\\mathrm{sum}}N_{\\mathrm{act}}}+N_{\\mathrm{pas}}\\right)^{2}+\\gamma_{2}\\left(A_{\\mathrm{sum}}\\!+\\!N_{\\mathrm{pas}}\\right)\\right)}{A_{\\mathrm{sum}} \\sigma_{\\mathrm{I}}^{2}\\beta \/ d_{\\mathrm{IU}}^{2}\\!+\\!\\sigma_{0}^{2}}\\right) \\\\\n    &\\qquad~~~\\!~\\quad~\\text{s.t.}\\qquad\\quad \\eqref{cons_C},\\eqref{cons_C_AnP}\\nonumber.\n\\end{align}\n\nProblem (P4) is a non-convex optimization problem due to the discrete active\/passive-element number and the non-concave objective function, making it difficult to obtain a closed-form expression for the optimal solution in general.\nAlthough the optimal number of active (passive) elements can be numerically obtained by applying the one-dimensional search over $N_{\\mathrm{act}}=\\{0,1,\\cdots,\\lfloor\\frac{W_0}{W_{\\mathrm{act}}}\\rfloor\\}$ and hence $N_{\\mathrm{pas}} = \\lfloor\\frac{W_0-N_{\\mathrm{act}}W_{\\mathrm{act}}}{W_{\\mathrm{pas}}}\\rfloor$, it yields little useful insight into the optimal IRS active\/passive elements allocation. \nThus, we consider two special channel setups in the following to obtain closed-form expressions for their corresponding optimal active\/passive elements allocation.\n\n\\subsubsection{LoS channel model}\nFor the LoS channels with $K_1\\to\\infty$ and $K_2\\to\\infty$, we obtain from \\eqref{gamma_approx} that $\\gamma_1\\to 1$ and $\\gamma_2\\to 0$. Then, the approximated ergodic capacity, $\\tilde C$ in \\eqref{sig_approx}, is equal to the exact capacity, $C$ in \\eqref{ergo_capa}.\nNote that the term $A_{\\mathrm{sum}}$ is a positive constant, i.e., $A_{\\mathrm{sum}}=\\frac{P_{\\mathrm{I}}}{P_{\\mathrm{B}}\\beta \/ D_{\\mathrm{BI}}^{2}+\\sigma_{\\mathrm{I}}^{2}}$ when $N_{\\mathrm{act}}>0$, and $A_{\\mathrm{sum}}=0$ when $N_{\\mathrm{act}}=0$.\nTherefore, in the following, we solve problem (P4) in two cases, corresponding to the case of passive IRS with $N_{\\mathrm{act}}=0$ and the case of hybrid IRS with $N_{\\mathrm{act}}>0$ and $N_{\\mathrm{pas}}>0$, respectively. \nTo address the discrete active\/passive elements deployment cost in constraints \\eqref{cons_C} and \\eqref{cons_C_AnP}, we first relax the integer values $N_{\\mathrm{act}}$ and $N_{\\mathrm{pas}}$ into continuous values, $\\tilde N_{\\mathrm{act}}$ and $\\tilde N_{\\mathrm{pas}}$, and then apply the integer rounding technique to reconstruct them based on their optimal continuous values.\nMoreover, it can be shown that the equality in \\eqref{cons_C} holds in the optimal solution to problem (P4), i.e., $\\tilde N_{\\mathrm{pas}}=\\frac{W_0-W_{\\mathrm{act}}\\tilde N_{\\mathrm{act}}}{W_{\\mathrm{pas}}}$.\n\nFirst, consider the case of $\\tilde N_{\\mathrm{act}}=0$. By substituting $\\tilde N_{\\mathrm{pas}}=\\frac{W_0}{W_{\\mathrm{pas}}}$, $\\gamma_1\\to 1$, and $\\gamma_2\\to 0$ into \\eqref{capa_approx2}, the hybrid IRS reduces to the passive IRS for which the achievable capacity under the LoS channels is given by\n\\begin{equation}\n   C = C^*_{\\text {L,pas}} \\triangleq  \\log _{2}\\left(1+\\frac{W_0^{2} P_{\\mathrm{B}} \\beta^{2}}{W_{\\mathrm{pas}}^{2} D_{\\mathrm{BI}}^{2} d_{\\mathrm{IU}}^{2} \\sigma_{0}^{2}}\\right).\\label{C_p}\n\\end{equation} \nNext, when $\\tilde N_{\\mathrm{act}}>0$ under the LoS channel model, the achievable capacity in \\eqref{capa_approx2} reduces to\n\\begin{equation}\n    C = C_{\\text {L,hyb}} \\triangleq \\log _{2}\\left(1+\\frac{P_{\\mathrm{B}}\\beta^2(\\sqrt{A_{\\mathrm{sum}}\\tilde N_{\\mathrm{act}}}+\\tilde N_{\\mathrm{pas}})^2\/D_{\\mathrm{BI}}^2d_{\\mathrm{IU}}^2}{A_{\\mathrm{sum}}\\sigma_{\\mathrm{I}}^2\\beta\/d_{\\mathrm{IU}}^2+\\sigma_0^2}\\right).\\label{C_h}\n\\end{equation}\nBy substituting $\\tilde N_{\\mathrm{pas}}=\\frac{W_0-W_{\\mathrm{act}}\\tilde N_{\\mathrm{act}}}{W_{\\mathrm{pas}}}$ into \\eqref{C_h}, the optimal solution to problem (P4) given $\\tilde N_{\\mathrm{act}}>0$ can be obtained by solving the following equivalent maximization problem.\n\\begin{align}\n    &\\mathrm{(P5)}~~~~~\\max_{\\tilde N_{\\mathrm{act}}}\n    \\quad~~\\xi_1\\left(-\\tilde N_{\\mathrm{act}}+\\xi_2\\sqrt{\\tilde N_{\\mathrm{act}}}+\\xi_3\\right)^2\\label{obj_p3}\\\\\n    &\\qquad~~~\\!~\\quad~\\text{s.t.} ~~~~~0< \\tilde N_{\\mathrm{act}}\\leq\\frac{W_0}{W_{\\mathrm{act}}},\n\\end{align}\nwhere\n\\begin{align}\n    \\xi_1=\\frac{P_{\\mathrm{B}}\\beta^2W_{\\mathrm{act}}^2\/D_{\\mathrm{BI}}^2d_{\\mathrm{IU}}^2W_{\\mathrm{pas}}^2}{A_{\\mathrm{sum}}\\sigma_{\\mathrm{I}}^2\\beta\/d_{\\mathrm{IU}}^2+\\sigma_0^2},\n    \\qquad \\xi_2 = \\frac{\\sqrt{A_{\\mathrm{sum}}}W_{\\mathrm{pas}}}{W_{\\mathrm{act}}},\n    \\qquad \\text{and } \\xi_3 = \\frac{W_0}{W_{\\mathrm{act}}}.\n\\end{align}\n\\begin{theorem}\\label{the_opt_N}\n\\textbf{\\emph{(Optimal active\/passive elements allocation)}} \\emph{Under the LoS channel model and given $\\tilde N_{\\mathrm{act}}>0$, the optimal solution to problem (P4) is}\n\\begin{equation}\\label{opt_na}\n\\begin{cases}\n&\\tilde N^*_{\\mathrm{act}}=\\frac{W_0}{W_{\\mathrm{act}}},\\tilde N^*_{\\mathrm{pas}}=0, \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad~~~\\emph{if } W_0< \\frac{W_{\\mathrm{pas}}^{2} P_{\\mathrm{I}} \/ W_{\\mathrm{act}}}{4 P_{\\mathrm{B}} \\beta \/ D_{\\mathrm{BI}}^{2}+4 \\sigma_{\\mathrm{I}}^{2}},\\\\\n&\\tilde N^*_{\\mathrm{act}}=\\frac{P_{\\mathrm{I}}W_{\\mathrm{pas}}^2\/W_{\\mathrm{act}}^2}{4P_{\\mathrm{B}}\\beta\/D_{\\mathrm{BI}}^2+4\\sigma_{\\mathrm{I}}^2},\\tilde N^*_{\\mathrm{pas}}=\\frac{W_0}{W_{\\mathrm{pas}}}-\\frac{P_{\\mathrm{I}}W_{\\mathrm{pas}}\/W_{\\mathrm{act}}}{4P_{\\mathrm{B}}\\beta\/D_{\\mathrm{BI}}^2+4\\sigma_{\\mathrm{I}}^2},\\qquad \\emph{otherwise}.\n\\end{cases}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nSee Appendix \\ref{proof_lem3}.\n\\end{proof}\n\\begin{remark}\n\\emph{Theorem \\ref{the_opt_N} shows that when the total deployment budget is small, i.e., $W_0< \\frac{W_{\\mathrm{pas}}^{2} P_{\\mathrm{I}} \/ W_{\\mathrm{act}}}{4 P_{\\mathrm{B}} \\beta \/ D_{\\mathrm{BI}}^{2}+4 \\sigma_{\\mathrm{I}}^{2}}$, only active elements should be employed since they can provide a signal amplification gain and thus a higher rate than passive elements. In contrast, if the total deployment budget is sufficiently large, i.e., $W_0\\geq \\frac{W_{\\mathrm{pas}}^{2} P_{\\mathrm{I}} \/ W_{\\mathrm{act}}}{4 P_{\\mathrm{B}} \\beta \/ D_{\\mathrm{BI}}^{2}+4 \\sigma_{\\mathrm{I}}^{2}}$, the optimal number of active elements should balance the amplification gain of active elements and the beamforming gain of passive elements, which is independent of $W_0$ but determined by other parameters, such as $P_{\\mathrm{I}}$, $W_{\\mathrm{act}}$ and $W_{\\mathrm{pas}}$.\nThis is because the performance bottleneck of active elements becomes the limited amplification power when $W_0$ is large, where the limited power can only support partial active elements to operate with the optimal amplification factor.}\n\\end{remark\n\nBased on Theorem \\ref{the_opt_N} and substituting the optimal number of active elements, $\\tilde N_{\\mathrm{act}}=\\frac{P_{\\mathrm{I}} W_{\\mathrm{pas}}^{2} \/ W_{\\mathrm{act}}^{2}}{4 P_{\\mathrm{B}} \\beta \/ D_{\\mathrm{BI}}^{2}+4 \\sigma_{\\mathrm{I}}^{2}}$, into \\eqref{C_h}, the achievable capacity of the worst-case user is obtained as\n\\begin{equation}\nC^*_{\\text {L,hyb}} = \\log _{2}\\left(1+\\frac{P_{\\mathrm{B}} \\beta^{2}\\left(\\frac{A_{\\text {sum }} W_{\\mathrm{pas}}}{4 W_{\\mathrm{act}}}+\\frac{W_0}{W_{\\mathrm{pas}}}\\right)^{2} \/ D_{\\mathrm{BI}}^{2} d_{\\mathrm{IU}}^{2}}{A_{\\text {sum }} \\sigma_{\\mathrm{I}}^{2} \\beta \/ d_{\\mathrm{IU}}^{2}+\\sigma_{0}^{2}}\\right).\\label{C_h_2}\n\\end{equation}\n\n\\begin{corollary}\n\\emph{For the case of LoS channels, given the optimal IRS phase shifts in \\eqref{opt_phase_1} and \\eqref{opt_phase_2} and active\/passive elements allocation in \\eqref{opt_na}, the optimal amplification factor of each active element is given by}\n\\begin{equation\n    \\alpha_{n}=\\frac{2 W_{\\mathrm{act}}}{W_{\\mathrm{pas}}} ,\\forall  n \\in\\mathcal{N}_{\\mathrm{act}}\n\\end{equation}\n\\emph{for achieving the capacity in \\eqref{C_h_2}.}\n\\end{corollary}\n\nMoreover, when $W_0<\\frac{W_{\\mathrm{pas}}^2P_{\\mathrm{I}}\/W_{\\mathrm{act}}}{4P_{\\mathrm{B}}\\beta\/D_{\\mathrm{BI}}^2+4\\sigma_{\\mathrm{I}}^2}$ and hence $N^*_{\\mathrm{act}}=W_0\/W_{\\mathrm{act}}$, the achievable capacity of the worst-case user aided by the active IRS under the LoS channel model is given by\n\\begin{equation}\n    C^*_{\\text {L,act}} \\triangleq \\log _{2}\\left(1+\\frac{W_0A_{\\mathrm{sum}} P_{\\mathrm{B}} \\beta^{2} \/W_{\\mathrm{act}} D_{\\mathrm{BI}}^{2} d_{\\mathrm{IU}}^{2}}{A_{\\mathrm{sum}} \\sigma_{\\mathrm{I}}^{2} \\beta \/ d_{\\mathrm{IU}}^{2}+\\sigma_{0}^{2}}\\right).\\label{C_a}\n\\end{equation}\n\nNext, we compare the achievable capacity of the hybrid IRS with that of the fully-active and fully-passive IRSs with respect to (w.r.t.) different values of the deployment budget under the LoS channels. To this end, we define the following active\/passive elements allocation thresholds for the deployment budget.\n\\begin{align}\n    W_{\\mathrm{A-H}}&\\triangleq\\frac{W_{\\mathrm{pas}}^2P_{\\mathrm{I}}\/W_{\\mathrm{act}}}{4P_{\\mathrm{B}}\\beta\/D_{\\mathrm{BI}}^2+4\\sigma_{\\mathrm{I}}^2},\\label{B_H-A}\\\\\n    W_{\\mathrm{H-P}}&\\triangleq\\frac{W_{\\mathrm{pas}}^2\\sigma_0^2d_{\\mathrm{IU}}^2}{4W_{\\mathrm{act}}\\sigma_{\\mathrm{I}}^2\\beta}+\\frac{W_{\\mathrm{pas}}^2\\sigma_0d_{\\mathrm{IU}}}{4W_{\\mathrm{act}}\\sigma_{\\mathrm{I}}}\\sqrt{\\frac{\\sigma_0^2d_{\\mathrm{IU}}^2}{\\sigma_{\\mathrm{I}}^2\\beta^2}+\\frac{P_{\\mathrm{I}}}{P_{\\mathrm{B}}\\beta^2\/D_{\\mathrm{BI}}^2+\\sigma_{\\mathrm{I}}^2\\beta}}.\\label{condition_H-P}\n\\end{align}\nThen, we have the following result.\n\\begin{theorem}\\label{lem_thres}\n\\textbf{\\emph{(Capacity comparison among different IRS architectures)}}\n\\emph{Under the LoS channel model, the capacity comparison among different IRS architectures is given as follows.}\n\n\\emph{1) When $0\\leq W_0< W_{\\mathrm{A-H}}$, $C^*_{\\text {L,act}}=C^*_{\\text {L,hyb}}>C^*_{\\text {L,pas}}$;}\n\n\\emph{2) When $W_{\\mathrm{A-H}}\\leq W_0\\leq W_{\\mathrm{H-P}}$, $C^*_{\\text {L,hyb}}>C^*_{\\text {L,act}}$ and $C^*_{\\text {L,hyb}}>C^*_{\\text {L,pas}}$;}\n\n\\emph{3) When $W_0>W_{\\mathrm{H-P}}$, $C^*_{\\text {L,pas}}=C^*_{\\text {L,hyb}}>C^*_{\\text {L,act}}$.}\n\\end{theorem}\n\\begin{proof}\nSee Appendix \\ref{proof_lem4}.\n\\end{proof}\n\\begin{remark}\n\\textbf{\\emph{(IRS architecture selection)}}\n\\emph{Theorem \\ref{lem_thres} shows that the optimal architecture selection for the IRS (i.e., passive, active, or hybrid) is determined by the total deployment budget. Specifically, when $W_0$ is small, the hybrid IRS reduces to the fully-active IRS, which is the optimal architecture to achieve the maximum capacity. This is because when $N_{\\mathrm{pas}}$ is small, the active IRS can provide a high power amplification gain while passive elements only have limited beamforming gain. In contrast, when $W_0$ is sufficiently large, the hybrid IRS reduces to the fully-passive IRS. This is expected since in this case, the amplification factor for the active elements is limited, which may not be able to mitigate the amplification noise. However, when $W_{\\mathrm{A-H}}\\leq W_0\\leq W_{\\mathrm{H-P}}$, there in general exists a trade-off between increasing the power amplification gain (i.e., assigning more active elements) and beamforming gain (i.e., assigning more passive elements); hence, it is necessary to design the optimal active\/passive elements allocation for the hybrid IRS to maximize the capacity.}\n\\end{remark}\n\\begin{remark}\n\\textbf{\\emph{(Deployment budget thresholds)}}\n\\emph{It is observed from \\eqref{B_H-A} and \\eqref{condition_H-P} that $W_{\\mathrm{A-H}}$ and $W_{\\mathrm{H-P}}$ both increase with the amplification power, $P_{\\mathrm{I}}$, and the passive-element cost, $W_{\\mathrm{pas}}$, while they decrease with the active-element cost, $W_{\\mathrm{act}}$. \nThis can be explained as follows. \nFirst, a higher amplification power, $P_{\\mathrm{I}}$, allows more active elements to operate with sufficiently large amplification factors. \nSecond, higher passive-element cost, $W_{\\mathrm{pas}}$, and\/or lower active-element cost, $W_{\\mathrm{act}}$, make it more desirable to deploy active elements.}\n\\end{remark}\n\n\\subsubsection{Rayleigh fading channel model}\nFor the case of Rayleigh fading channels with $K_1=K_2=0$, we obtain from \\eqref{gamma_approx} that $\\gamma_1=0$ and $\\gamma_2=1$. As such, the ergodic capacity under the Rayleigh fading channels is given by\n\\begin{equation}\n    \\tilde C=C_{\\mathrm{NL}}\\triangleq\\log _{2}\\left(1+\\frac{\\left(A_{\\mathrm{sum}}+{N}_{\\mathrm{pas}}\\right) \\frac{P_{\\mathrm{B}} \\beta^{2}}{D_{\\mathrm{BI}}^{2} d_{\\mathrm{IU}}^{2}}}{A_{\\mathrm{sum}} \\sigma_{\\mathrm{I}}^{2} \\beta \/ d_{\\mathrm{IU}}^{2}+\\sigma_{0}^{2}}\\right).\n\\end{equation}\n\n\\begin{lemma}\\label{lem_opt_ea_NLoS}\n\\textbf{\\emph{(Optimal active\/passive elements allocation for Rayleigh fading channels)}}\n\\emph{Under the Rayleigh fading channel model and favorable amplification power condition in \\eqref{cons_pb}, the optimal number of active and passive elements are given by}\n\\begin{equation}\\label{opt_na_NLoS}\n\\begin{cases}\n& N^*_{\\mathrm{act}}=1, N^*_{\\mathrm{pas}}=\\lfloor\\frac{W_0-W_{\\mathrm{act}}}{W_{\\mathrm{pas}}}\\rfloor, \\qquad~~~\\emph{if } W_0> \\frac{A_{\\mathrm{sum}}W_{\\mathrm{pas}}\\sigma_0^2-W_{\\mathrm{act}}\\sigma_0^2}{A_{\\mathrm{sum}}\\sigma_{\\mathrm{I}}^2\\beta\/d_{\\mathrm{IU}}^2},\\\\\n& N^*_{\\mathrm{act}}=0, N^*_{\\mathrm{pas}}=\\lfloor\\frac{W_0}{W_{\\mathrm{pas}}}\\rfloor,\\qquad~~~~~~~\\emph{otherwise}.\n\\end{cases}\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFirst, when $N_{\\mathrm{act}}>0$, it can be shown that\n\\begin{equation}\n    C_{\\mathrm{NL}}\\leq C^*_{\\mathrm{NL,hyb}}\\triangleq\\log _{2}\\left(1+\\frac{\\left(A_{\\mathrm{sum}}+{N}_{\\mathrm{pas,max}}\\right) \\frac{P_{\\mathrm{B}} \\beta^{2}}{D_{\\mathrm{BI}}^{2} d_{\\mathrm{IU}}^{2}}}{A_{\\mathrm{sum}} \\sigma_{\\mathrm{I}}^{2} \\beta \/ d_{\\mathrm{IU}}^{2}+\\sigma_{0}^{2}}\\right),\\label{C_NLoS_hyb}\n\\end{equation}\nwhere ${N}_{\\mathrm{pas,max}}=\\lfloor\\frac{W_0-W_{\\mathrm{act}}}{W_{\\mathrm{pas}}}\\rfloor$ and $N_{\\mathrm{act}}=1$.\nSecond, when $N_{\\mathrm{act}}=0$, we obtain that $A_{\\mathrm{sum}}=0$ and the ergodic capacity achieved by passive IRS is given by\n\\begin{equation}\n    C_{\\mathrm{NL}}=C^*_{\\mathrm{NL,pas}}\\triangleq\\log _{2}\\left(1+\\frac{W_0P_{\\mathrm{B}} \\beta^{2}}{W_{\\mathrm{pas}} \\sigma_{0}^{2} D_{\\mathrm{BI}}^{2} d_{\\mathrm{IU}}^{2}}\\right). \\label{C_NLoS_pas}\n\\end{equation}\nThen, by comparing $C^*_{\\mathrm{NL,hyb}}$ in \\eqref{C_NLoS_hyb} and $C^*_{\\mathrm{NL,pas}}$ in \\eqref{C_NLoS_pas}, it can be shown that when $W_0> \\frac{A_{\\mathrm{sum}}W_{\\mathrm{pas}}\\sigma_0^2-W_{\\mathrm{act}}\\sigma_0^2}{A_{\\mathrm{sum}}\\sigma_{\\mathrm{I}}^2\\beta\/d_{\\mathrm{IU}}^2}$, we have $C^*_{\\mathrm{NL,hyb}}>C^*_{\\mathrm{NL,pas}}$. Based on the above, the optimal active\/passive elements allocation under the Rayleigh fading channel model is given in \\eqref{opt_na_NLoS}, thus completing the proof.\n\\end{proof}\n\\begin{remark}\n\\textbf{\\emph{(Active\/passive elements allocation under the Rayleigh fading channel model)}}\n\\emph{Lemma \\ref{lem_opt_ea_NLoS} shows that under the Rayleigh fading channel model (or rich scattering environment), most of the deployment budget should be assigned to passive elements, while at most one active element needs to be deployed.\nSpecifically, when the amplification power is sufficiently small and\/or the deployment budget is sufficiently large, all the deployment budget should be assigned to passive elements. This is because the power amplification gain provided by active elements cannot mitigate their amplification noise and\/or the beamforming gain of passive elements (which prevails active-element power amplification).\nOn the other hand, when the amplification power is large and\/or  the deployment budget is small, one active element is enough to achieve the amplification power gain because it has no active-element beamforming gain under the Rayleigh fading channels, thus making it desirable to assign most of the deployment budget to passive elements.\n}\n\\end{remark}\n\n\\section{Simulation Results}\\label{sec_sim}\nSimulation results are presented in this section to evaluate the effectiveness of the proposed hybrid IRS architecture and active\/passive elements allocation design. \nIf not specified otherwise, the system setups are as follows.\nWe consider a two-dimensional (2D) Cartesian coordinate system, where the reference points of the BS, the hybrid IRS, and the worst-case user are located at $(0,0)$ m, $(60,0)$ m, and $(60,20)$ m, respectively. \n\nThe deployment costs of each active\/passive element are set as $W_{\\mathrm{act}} = 5$ and $W_{\\mathrm{pas}} = 1$, respectively, by taking into account the fact that the active element in general incurs higher static operation power and hardware cost.\nFor each active element, the feasible region of the amplification factor is in the range of $[\\alpha_{\\min},\\alpha_{\\max}]=[0,14]$ dB \\cite{8403249}.\nThe system operates at a carrier frequency of 6 GHz with the wavelength of $\\lambda=0.05 \\mathrm{~m}$. \nWe consider the same Rician fading factor for the BS-IRS and IRS-user channels, i.e., $K_1=K_2=K$.\nOther parameters are set as $d_{\\mathrm{I}}=\\frac{\\lambda}{4}$, $\\beta=-30$ dB, $\\sigma_{\\mathrm{I}}=\\sigma_0=-80$ dBm, $P_{\\mathrm{B}}=15$ dBm, and $P_{\\mathrm{I}} = 5$ dBm \\cite{9377648}.\n\\subsection{Accuracy of Ergodic Capacity Approximation}\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=2.7in]{fig1.pdf}}\n\\caption{Accuracy of the ergodic capacity approximation given in \\eqref{sig_approx}.}\\label{eval_approx}\n\\end{figure}\n\nFirst, we evaluate in Fig. \\ref{eval_approx} the accuracy of the ergodic capacity approximation presented in Lemma \\ref{lem_C_approx} (see \\eqref{sig_approx}). \nBy setting equal active\/passive elements allocation, i.e., $N_{\\mathrm{act}}=N_{\\mathrm{pas}}$, the actual ergodic capacity in \\eqref{ergo_capa} is obtained based on 1000 independent channel realizations under different Rician factors of $K\\in\\{0$ dB, $10$ dB, $+\\infty\\}$ ($K\\to +\\infty$ corresponds to the LoS channels).\nIt is observed that the approximated ergodic capacity in \\eqref{sig_approx} is close to the exact capacity in \\eqref{ergo_capa} under different Rician factors. This is because when the number of reflecting elements is large, the variance of random channels is averaged due to the law of large numbers.\n\\subsection{Effect of Active\/passive Elements Allocation on Ergodic Capacity}\n\\begin{figure}[ht]\n\\centerline{\\includegraphics[width=2.7in]{fig2.pdf}}\n\\caption{Capacity performance versus budget percentage of active elements.}\\label{rate_ratio}\n\\end{figure}\n\nNext, we investigate the effect of the active\/passive elements allocation at the hybrid IRS on the capacity performance.\nWe set the total deployment budget $W_0=3000$, the amplification power $P_{\\mathrm{I}}=15$ dBm, and denote $\\rho$ as the percentage of the budget assigned to active elements ($0\\leq \\rho\\leq 1$).\nIn Fig. \\ref{rate_ratio}, we plot the ergodic capacity versus $\\rho$ with different Rician factors $K\\in\\{0$, $15$ dB, $+\\infty\\}$.\nIt is observed that for the cases of $K\\to\\infty$ and $K=15$ dB (with LoS channel components), the ergodic capacity first increases with $\\rho$, then decreases after it exceeds a threshold (i.e., $\\rho^*=0.35$ for $K\\to\\infty$, and $\\rho^*=0.18$ for $K=15$ dB). This shows that for the general Rician fading channels, the active\/passive elements allocation has a significant effect on the capacity maximization.\nThis is expected because if the budget assigned to active elements is too small, the active-element power amplification gain is not fully exploited due to the small number of active elements. However, if the budget assigned to active elements is too large, the performance bottleneck of active elements becomes the limited amplification power that cannot support all active elements to operate in the amplification mode, thus making it desirable to assign partial budget to passive elements for achieving higher beamforming gain.\nMoreover, the optimal budget assigned to active elements increases with the Rician factor because the active elements can achieve both the power amplification gain and beamforming gain over the LoS paths but have no beamforming gain over NLoS paths.\nLast, when $K=0$ (corresponding to the Rayleigh fading channels), the total deployment budget should be assigned to passive elements to maximize the ergodic capacity since the beamforming gain of passive elements is larger than the amplification gain of active elements.\n\\subsection{Effect of Total Deployment Budget}\n\\begin{figure}[t] \\centering    \n{\\subfigure[{LoS channels with $K\\to +\\infty$.}] {\n\\label{fig_3_LoS}\n\\includegraphics[width=2.7in]{fig3_LoS.pdf}  \n}}     \n{\\subfigure[{Rician fading channels with $K=10$ dB.}] {\\label{fig_3_Rician}\n\\includegraphics[width=2.7in]{fig3_Rician.pdf}  \n}}\n{\\caption{The ergodic capacity achieved by hybrid IRS, fully-active IRS, and fully-passive IRS versus deployment budget.}}\n\\end{figure}\n\nNext, we compare the proposed optimal active\/passive elements allocation (EA) design against three benchmarks: 1) Fully-active IRS with all the deployment budget assigned to active elements; 2) Fully-passive IRS with all the deployment budget assigned to active elements; 3) Hybrid IRS under equal EA for which the deployment budget is equally assigned to active and passive elements. We apply the optimal IRS beamforming design in \\eqref{opt_phase_1}, \\eqref{opt_phase_2}, and \\eqref{opt_a_n} for the proposed hybrid IRS architecture and the benchmarks.\n\nFigs. \\ref{fig_3_LoS} and \\ref{fig_3_Rician} show the ergodic capacity of the worst-case user versus the total deployment budget under the LoS and Rician fading channel models, respectively.\nFirst, it is observed that the ergodic capacity achieved by the hybrid IRS architecture with optimal active\/passive elements allocation is always larger than or equal to that achieved by the fully-active or fully-passive IRSs. This is expected because the hybrid IRS provides an extra degree of freedom for elements allocation to balance the trade-off between the power amplification and beamforming gains.\nSecond, the hybrid IRS with the optimal active\/passive elements allocation reduces to the fully-active IRS when the deployment budget is sufficiently small (i.e., $W_0<1250$ for the LoS channels, and $W_0<600$ for the Rician fading channels with $K=10$ dB). \nThis is because the active elements has a higher power amplification gain as compared to the passive-element beamforming gain when the budget is small.\nThird, one can observe that the hybrid IRS with the optimal active\/passive elements allocation outperforms that with equal elements allocation, which shows the effectiveness of elements allocation optimization for the hybrid IRS.\nLast, it is also observed that given the same deployment budget, the achievable capacity under the LoS channels is higher than that under Rician fading channels since the IRS phase shifts are designed based on the LoS channel components or statistical CSI only.\n\n\\begin{figure}[t] \\centering    \n{\\subfigure[{LoS channels with $K\\to +\\infty$.}] {\n\\label{fig_4_LoS}\n\\includegraphics[width=2.7in]{fig4_LoS.pdf}  \n}}     \n{\\subfigure[{Rician fading channels with $K=10$ dB.}] {\\label{fig_4_Rician}\n\\includegraphics[width=2.7in]{fig4_Rician.pdf}     \n}}\n{\\caption{The optimal numbers of active and passive elements versus total deployment budget.}}\n\\end{figure}\n\nFigs. \\ref{fig_4_LoS} and \\ref{fig_4_Rician} present the optimal numbers of active and passive elements for maximizing the ergodic capacity of the worst-case user under different total deployment budgets with $W_{\\mathrm{act}}=5$ and $W_{\\mathrm{pas}}=1$.\nIt is observed that as $K$ increases, the deployment budget is first assigned to active elements only and then assigned to more passive elements after it exceeds a threshold.\nIn addition, we observe that given the same deployment budget, \nthe optimal number of active elements for the LoS channels, i.e., $N^*_{\\mathrm{act}}=250$ is higher than that for the Rician fading channels with $K=10$ dB, i.e., $N^*_{\\mathrm{act}}=120$.\n\\subsection{Effect of Rician Factor}\nIn Figs. \\ref{fig5_capa_vs_K} and \\ref{fig5_ea_vs_K}, we investigate the effect of the Rician factor of the BS-IRS and IRS-user channels on the achievable capacity and active\/passive elements allocation. The IRS beamforming design and the active-element amplification factors are given by \\eqref{opt_phase_1}, \\eqref{opt_phase_2}, and \\eqref{opt_a_n}, respectively.\n\n\\begin{figure}[t] \\centering    \n{\\subfigure[{Ergodic capacity versus Rician factor.}] {\n\\label{fig5_capa_vs_K}\n\\includegraphics[width=2.63in]{fig5_capa_vs_K.pdf}  \n}}     \n{\\subfigure[Optimal number of active and passive elements versus Rician factor.] {\\label{fig5_ea_vs_K}\n\\includegraphics[width=2.93in]{fig5_ea_vs_K.pdf}     \n}}\n{\\caption{Effect of the Rician factor on ergodic capacity and active\/passive elements allocation.}}\n\\end{figure}\n\nSpecifically, Fig. \\ref{fig5_capa_vs_K} presents the ergodic capacity of the worst-case user under different Rician factors. It is observed that the proposed hybrid IRS architecture with the optimal active\/passive elements allocation outperforms other benchmarks under different channel conditions, while the active IRS achieves a higher capacity than the passive IRS only when the channels are LoS-dominant (i.e., Rician factor $K$ is large). \nFig. \\ref{fig5_ea_vs_K} plots the optimal numbers of active and passive elements of the hybrid IRS under different Rician factors. \nIt is observed that with an increasing Rician factor, the optimal number of active elements first increases and then keeps unchanged after it exceeds a threshold, where the NLoS channel components are negligible as compared to the LoS channel components. \nThe reason is that for the NLoS paths, the active elements have no beamforming gain but can achieve power amplification gain due to the high amplification power, while for the LoS paths, active elements can achieve both the power amplification and beamforming gains with multiple elements. \nTherefore, as $K$ increases, more deployment budget should be assigned to the active elements so as to achieve a higher beamforming gain of the active elements.\n\n\\subsection{Effect of Amplification Power}\nMoreover, we evaluate the effect of the amplification power of active elements on the ergodic capacity and active\/passive elements allocation of the proposed hybrid IRS architecture under the general Rician fading channel model with $K=15$ dB.\n\n\\begin{figure}[t] \\centering    \n{\\subfigure[{Ergodic capacity versus amplification power.}] {\n\\label{fig6_capa_vs_P_I}\n\\includegraphics[width=2.63in]{fig6_capa_vs_P_I.pdf}  \n}}     \n{\\subfigure[Optimal numbers of active and passive elements versus amplification power.] {\\label{fig6_ea_vs_P_I.pdf}\n\\includegraphics[width=2.93in]{fig6_ea_vs_P_I.pdf}     \n}}\n{\\caption{Effect of the amplification power on ergodic capacity and elements allocation.}}\n\\end{figure}\n\nIn Fig. \\ref{fig6_capa_vs_P_I}, we plot the ergodic capacity versus the amplification power $P_{\\mathrm{I}}$ for different schemes.\nFirst, it is observed that the ergodic capacity of the worst-case user aided by the fully-active or hybrid IRSs monotonically increases with the amplification power. Second, we observe that the hybrid IRS with the optimal active\/passive elements allocation achieves a much higher capacity than the fully-active IRS when $P_{\\mathrm{I}}$ is small and significantly outperforms the fully-passive IRS when $P_{\\mathrm{I}}$ is large. This shows the flexibility and advantages of the hybrid IRS under different amplification powers.\n\nIn Fig. \\ref{fig6_ea_vs_P_I.pdf}, we plot the optimal numbers of active and passive elements for the hybrid IRS architecture versus the amplification power. \nOne can observe that the optimal number of active elements increases with the amplification power, which is expected because a higher amplification power can support more active elements to operate in the amplification mode with optimal amplification factors. \n\\subsection{Effect of Active\/Passive-element Deployment Cost Ratio}\n\\begin{figure}[t] \\centering    \n{\\subfigure[{Ergodic capacity versus active\/passive-element cost ratio.}] {\n\\label{fig7_capa_vs_c_ratio}\n\\includegraphics[width=2.63in]{fig7_capa_vs_c_ratio.pdf}  \n}}     \n{\\subfigure[Optimal numbers of active and passive elements versus deployment cost ratio.] {\\label{fig7_ea_vs_c_ratio}\n\\includegraphics[width=2.93in]{fig7_ea_vs_c_ratio.pdf}     \n}}\n{\\caption{Effect of the deployment cost ratio on ergodic capacity and elements allocation.}}\n\\end{figure}\nIn Fig. \\ref{fig7_capa_vs_c_ratio}, we compare the ergodic capacity of the worst-case user versus different ratios of active-over-passive deployment cost, i.e., $W_{\\mathrm{act}}\/W_{\\mathrm{pas}}$, where we set $W_{\\mathrm{pas}}=1$, $K=15$ dB, $W_0=1500$, and $P_{\\mathrm{B}}=10$ dBm.\nFirst, the hybrid IRS with the optimal active\/passive elements allocation has the highest capacity as compared to other benchmarks. Besides, one can observe that as $W_{\\mathrm{act}}$ increases, the ergodic capacity decreases when $W_{\\mathrm{act}}$ is small, then remains unchanged when $W_{\\mathrm{act}}$ is sufficiently large, which can be explained as follows.\nWhen the deployment budget is small, a larger $W_{\\mathrm{act}}$ means that a smaller number of active elements can be deployed, thus resulting in reduced ergodic capacity.\nWhen $W_{\\mathrm{act}}$ exceeds a threshold, all the deployment budget is assigned to passive elements for maximizing the ergodic capacity. As such, increasing $W_{\\mathrm{act}}$ will not change the ergodic capacity and elements allocation.\n\\section{Conclusions}\\label{sec_conclu}\nIn this paper, we proposed a new hybrid active-passive IRS architecture and studied its optimal active\/passive elements allocation in a hybrid IRS aided wireless system based on the statistical CSI only.\nSpecifically, under the general Rician fading channel model, we formulated an optimization problem to maximize the ergodic capacity of the worst-case user, by jointly optimizing the active\/passive elements allocation, their phase shifts, and the amplification factors of active elements, subject to various practical constraints on the active-element amplification factor and amplification power consumption, as well as the total active and passive elements deployment budget.\nTo solve this problem, we first approximated the ergodic capacity in a simpler form and then proposed an efficient algorithm to obtain the optimal hybrid IRS beamforming and active\/passive elements allocation.\nMoreover, it was shown that when all channels are of LoS, only active elements need to be deployed when the total deployment budget is sufficiently small, while both active and passive elements should be deployed with a decreasing number ratio when the budget increases and exceeds a certain threshold.\nLast, numerical results demonstrated the performance gains achieved by the proposed hybrid IRS architecture with the optimal active\/passive elements allocation against the benchmarks of the fully-active and fully-passive IRSs, as well as the hybrid IRS with equal active\/passive elements allocation. This validated that the hybrid IRS can flexibly balance the trade-off between the peculiar power amplification gain of active IRS and superior beamforming gain of passive IRS.\n\n\\appendices\n\\section{Proof of Lemma \\ref{lem_C_approx}}\\label{proof_lem1}\nFirst, based on the Lemma 1 of \\cite{6816003}, we obtain the following approximation for the ergodic capacity. \n\\begin{align}\n    C&=\\mathbb{E}\\left\\{\\log _{2}\\left(1+\\frac{P_{\\mathrm{B}}|(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{act}}+(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{pas}}|^2}{\\sigma_{\\mathrm{I}}^2\\|(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\|^2+\\sigma_0^2}\\right)\\right\\} \\\\\n    &\\approx\\log_2\\left(1+\\frac{\\mathbb{E}\\left\\{P_{\\mathrm{B}}|(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{act}}+(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{pas}}|^2\\right\\}}{\\mathbb{E}\\left\\{\\sigma_{\\mathrm{I}}^2\\|(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\|^2+\\sigma_0^2\\right\\}}\\right).\\label{C_approx}\n\\end{align}\nThen, we focus on the derivation of the desired signals at the worst-case user, which can be expressed as\n\\begin{align}\n    &\\mathbb{E}\\left\\{P_{\\mathrm{B}}|(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{act}}+(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{pas}}|^2\\right\\}\\\\\n    &=\\frac{P_{\\mathrm{B}}}{(K_1+1)(K_2+1)}\\mathbb{E}\\Big\\{\\Big|\\underbrace{\\sqrt{K_1K_2}(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}}_{x_{1}}+\\underbrace{\\sqrt{K_1}(\\tilde{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}}_{x_{2}}\\nonumber\\\\\n    &+\\underbrace{\\sqrt{K_2}(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\tilde{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}}_{x_{3}}+\\underbrace{(\\tilde{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\tilde{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}}_{x_{4}}+\\underbrace{\\sqrt{K_1K_2}(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{pas}}}_{x_{5}}\\\\\n    &+\\underbrace{\\sqrt{K_1}(\\tilde{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{pas}}}_{x_{6}}+\\underbrace{\\sqrt{K_2}(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\tilde{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{pas}}}_{x_{7}}+\\underbrace{(\\tilde{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\tilde{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{pas}}}_{x_{8}}\\Big|^2\\Big\\}\\nonumber\\\\\n    &=\\frac{P_{\\mathrm{B}}}{(K_1\\!+\\!1)(K_2\\!+\\!1)}\\Big(\\mathbb{E}\\left\\{\\left|x_{1}\\!+\\!x_{5}\\right|^2\\right\\}\\!+\\!\\mathbb{E}\\Big\\{\\left|x_{2}\\right|^2\\!+\\!\\left|x_{3}\\right|^2\\!+\\!\\left|x_{4}\\right|^2\\!+\\!\\left|x_{6}\\right|^2\\!+\\!\\left|x_{7}\\right|^2\\!+\\!\\left|x_{8}\\right|^2\\Big\\} \\Big)\\\\\n    &=x_{\\mathrm{L}}+x_{\\mathrm{NL,act}}+x_{\\mathrm{NL,pas}},\\label{lem_C_approx_1}\n\\end{align}\nwhere $x_{\\mathrm{L}}$, $x_{\\mathrm{NL,act}}$, and $x_{\\mathrm{NL,pas}}$ are defined in \\eqref{x_1}, \\eqref{x_2}, and \\eqref{x_3}, respectively.\nNext, the noise introduced by active reflecting elements and that at the receiver can be expressed as\n\\begin{align}\n    &\\mathbb{E}\\left\\{\\sigma_{\\rm I}^2\\left\\|({\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\right\\|^2+\\sigma_0^2\\right\\}\\\\\n    &=\\mathbb{E}\\left\\{\\sigma_{\\rm I}^2\\left\\|\\sqrt{\\frac{K_2}{K_2+1}}(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}+\\sqrt{\\frac{1}{K_2+1}}(\\tilde{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\right\\|^2\\right\\}+\\sigma_0^2\\\\\n    &=\\underbrace{\\frac{K_2\\sigma_{\\rm I}^2}{K_2+1}\\|(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\|^2}_{z_{\\mathrm{L,act}}}+\\underbrace{\\frac{\\sigma_{\\rm I}^2}{K_2+1}\\|(\\tilde{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\|^2}_{z_{\\mathrm{NL,act}}}+\\sigma_0^2.\\label{noise_approx}\n\\end{align}\nLast, the proof is completed by substituting \\eqref{lem_C_approx_1} and \\eqref{noise_approx} into \\eqref{C_approx}.\n\\section{Proof of Lemma \\ref{lem1}}\\label{proof_lem2}\nFirst, given favorable amplification power condition in \\eqref{cons_pb}, it can be verified that at the optimal solution to problem (P3), the constraint \\eqref{cstr_power_HI_decomp_2} is always active, i.e., $\\sum_{n=1}^{N_{\\mathrm{act}}}\\alpha^2_{n}=A_{\\mathrm{sum}}$. Then, by substituting $A_{\\mathrm{sum}}$ into \\eqref{x_NLoS} and \\eqref{n_act}, we can show that the NLoS part of the desired signal, $x_{\\mathrm{NL,act}}+x_{\\mathrm{NL,pas}}$, and the amplification noise , $z_{\\mathrm{L,act}}+z_{\\mathrm{NL,act}}$, are constants. Thus, problem (P3) can be solved by maximizing the ergodic capacity due to the LoS channel component, $x_{\\mathrm{L}}$. \nBy using the Cauchy-Schwarz inequality, we have\n\\begin{align}\n    x_{\\mathrm{L}}&=\\gamma_1\\left(\\sum_{n=1}^{N_{\\mathrm{act}}}\\alpha_n+N_{\\mathrm{pas}}\\right)^2P_{\\mathrm{B}}\\beta^2\/D^2_{\\mathrm{BI}}d^2_{\\mathrm{IU}}\\\\\n    &=\\gamma_1\\left(\\sum_{n=1}^{N_{\\mathrm{act}}}\\left(\\alpha_n+N_{\\mathrm{pas}}\/N_{\\mathrm{act}}\\right)\\right)^2P_{\\mathrm{B}}\\beta^2\/D^2_{\\mathrm{BI}}d^2_{\\mathrm{IU}}\\\\\n    &\\leq \\gamma_1N_{\\mathrm{act}}^2\\left(\\alpha^*+N_{\\mathrm{pas}}\/N_{\\mathrm{act}}\\right)^2P_{\\mathrm{B}}\\beta^2\/D^2_{\\mathrm{BI}}d^2_{\\mathrm{IU}},\n\\end{align}\nwhere the equality holds if and only if $\\alpha_{n}=\\alpha^*,\\forall n \\in\\mathcal{N}_{\\mathrm{act}}$ with $\\alpha^*=\\sqrt{\\frac{P_{\\mathrm{I}}\/N_{\\mathrm{act}}}{P_{\\mathrm{B}}\\beta\/D^2_{\\mathrm{BI}}+\\sigma^2_{\\mathrm{I}}}}$ in \\eqref{opt_a_n}, which thus completes the proof.\n\\section{Proof of Theorem \\ref{the_opt_N}}\\label{proof_lem3}\n\\begin{table}[t]\\centering\n\\caption{Variations of the Receiver SNR under Different Conditions.}\\label{opt_N_A}\n\\begin{center}\n\\begin{tabular}{|c|c|}\n\\hline \nCondition & Variations of $\\gamma_{\\mathrm{hyb}}(\\tilde N_{\\mathrm{act}})$ \\\\\n\\hline\n$0<\\sqrt{\\frac{W_0}{W_{\\mathrm{act}}}}<n_{\\mathrm{A},2}$ & Increase for $\\tilde N_{\\mathrm{act}}\\in(0,\\frac{W_0}{W_{\\mathrm{act}}}]$\\\\\n\\hline \n$n_{\\mathrm{A},2}\\leq \\sqrt{\\frac{W_0}{W_{\\mathrm{act}}}}\\leq n_{\\mathrm{A},3}$ & Increase for  $\\tilde N_{\\mathrm{act}}\\in(0,n^2_{\\mathrm{A},2}]$, decrease for $\\tilde N_{\\mathrm{act}}\\in(n^2_{\\mathrm{A},2},\\frac{W_0}{W_{\\mathrm{act}}}]$\\\\\n\\hline \n$\\sqrt{\\frac{W_0}{W_{\\mathrm{act}}}}>n_{\\mathrm{A},3}$ &\\!\\! Increase for $\\tilde N_{\\mathrm{act}}\\in(0,n^2_{\\mathrm{A},2}]\\cup(n^2_{\\mathrm{A},3},\\frac{W_0}{W_{\\mathrm{act}}}]$, decrease for $\\tilde N_{\\mathrm{act}}\\in(n^2_{\\mathrm{A},2},n^2_{\\mathrm{A},3}]$ \\\\\n\\hline \n\\end{tabular}\n\\end{center\n\\end{table}\nFor the receiver SNR, $\\gamma_{\\mathrm{hyb}}(\\tilde N_{\\mathrm{act}}) = \\xi_1\\left(-\\tilde N_{\\mathrm{act}}+\\xi_2\\sqrt{\\tilde N_{\\mathrm{act}}}+\\xi_3\\right)^2$, its first-order derivative over $\\sqrt{\\tilde N_{\\mathrm{act}}}$ can be expressed as\n\\begin{equation}\n    \\frac{\\partial \\gamma_{\\mathrm{hyb}}(\\tilde N_{\\mathrm{act}})}{\\partial \\sqrt{\\tilde N_{\\mathrm{act}}}}=2\\xi_1\\left(-\\tilde N_{\\mathrm{act}}+\\xi_2 \\sqrt{\\tilde N_{\\mathrm{act}}}+\\xi_3\\right)(-2  \\sqrt{\\tilde N_{\\mathrm{act}}}+\\xi_2).\\label{deri_snr}\n\\end{equation}\n\nFirst, it can be shown that \\eqref{deri_snr} equals to 0 for $\\sqrt{\\tilde N_{\\mathrm{act}}}=n_{\\mathrm{A},1}\\triangleq\\frac{\\xi_2-\\sqrt{\\xi_2^2+4\\xi_3}}{2}$, $\\sqrt{\\tilde N_{\\mathrm{act}}}=n_{\\mathrm{A},2}=\\frac{\\xi_2}{2}$, and $\\sqrt{\\tilde N_{\\mathrm{act}}}=n_{\\mathrm{A},3}\\triangleq\\frac{\\xi_2+\\sqrt{\\xi_2^2+4\\xi_3}}{2}$ with $n_{\\mathrm{A},1}<0<n_{\\mathrm{A},2}<n_{\\mathrm{A},3}$.\nThen, we summarize in Table \\ref{opt_N_A} the optimal number of active elements under different conditions of $\\tilde N_{\\mathrm{act}}$.\n\nBased on Table \\ref{opt_N_A}, when $n_{\\mathrm{A},2}\\leq \\sqrt{\\frac{W_0}{W_{\\mathrm{act}}}}\\leq n_{\\mathrm{A},3}$, it is obtained that $\\gamma_{\\mathrm{hyb}}(\\tilde N_{\\mathrm{act}})$ increases for $0<\\tilde N_{\\mathrm{act}}\\leq n_{\\mathrm{A},2}^2$ and decreases for $n_{\\mathrm{A},2}^2<\\tilde N_{\\mathrm{act}}\\leq\\frac{W_0}{W_{\\mathrm{act}}}$, thus leading to the optimal number of active elements, $\\tilde N^*_{\\mathrm{act}}=n_{\\mathrm{A},2}^2$, for maximizing $\\gamma_{\\mathrm{hyb}}(\\tilde N_{\\mathrm{act}})$. Following the similar procedure, the optimal number of active elements under different conditions can be obtained.\nNote that when $\\sqrt{\\frac{W_0}{W_{\\mathrm{act}}}}>n_{\\mathrm{A},3}$, we have\n\\begin{equation}\n    \\gamma_{\\mathrm{hyb}}(n_{\\mathrm{A},2}^2)-\\gamma_{\\mathrm{hyb}}(\\frac{W_0}{W_{\\mathrm{act}}})=\\frac{\\xi_2^2}{4}+\\xi_3-\\xi_2\\sqrt{\\xi_3}=(\\frac{\\xi_2}{2}-\\xi_3)^2\\geq 0,\n\\end{equation}\nsuch that $\\gamma_{\\mathrm{hyb}}(n_{\\mathrm{A},2}^2)\\geq \\gamma_{\\mathrm{hyb}}(\\frac{W_0}{W_{\\mathrm{act}}})$.\nMoreover, when $0<\\sqrt{\\frac{W_0}{W_{\\mathrm{act}}}}<n_{\\mathrm{A},2}$, i.e., the deployment cost $W_0 \\leq \\frac{W_{\\mathrm{pas}}^{2} P_{\\mathrm{I}} \/ W_{\\mathrm{act}}}{4 P_{\\mathrm{B}} \\beta \/ D_{\\mathrm{BI}}^{2}+4 \\sigma_{\\mathrm{I}}^{2}}$, all the deployment budget should be assigned to active elements, i.e., $\\tilde N^*_{\\mathrm{act}}=\\frac{W_0}{W_{\\mathrm{act}}}$.\nBased on the above, the optimal active\/passive elements allocation for the hybrid IRS is given by \\eqref{opt_na}, thus completing the proof.\n\\section{Proof of Theorem \\ref{lem_thres}}\\label{proof_lem4}\nFirst, we compare the achievable capacity by the fully-passive and fully-active IRSs in \\eqref{C_p} and \\eqref{C_a}, respectively. By solving\n\\begin{equation}\n    \\frac{W_0A_{\\mathrm{sum}} P_{\\mathrm{B}} \\beta^{2} \/W_{\\mathrm{act}} D_{\\mathrm{BI}}^{2} d_{\\mathrm{IU}}^{2}}{A_{\\mathrm{sum}} \\sigma_{\\mathrm{I}}^{2} \\beta \/ d_{\\mathrm{IU}}^{2}+\\sigma_{0}^{2}}>\\frac{W_0^2P_{\\mathrm{B}}\\beta^2}{W_{\\mathrm{pas}}^2D_{\\mathrm{BI}}^2d_{\\mathrm{IU}}^2\\sigma_0^2},\n\\end{equation}\nit is obtained that when\n\\begin{equation}\n    W_0 > W_{\\mathrm{A-P}}\\triangleq\\frac{W_{\\mathrm{pas}}^2\/W_{\\mathrm{act}}}{\\frac{\\beta\\sigma_{\\mathrm{I}}^2}{d_{\\mathrm{IU}}^2\\sigma_0^2}+\\frac{P_{\\mathrm{B}} \\beta \/ D_{\\mathrm{BI}}^{2}+\\sigma_{\\mathrm{I}}^{2}}{P_{\\mathrm{I}}}},\n\\end{equation}\nthe fully-passive IRS outperforms the fully-active IRS in term of the capacity.\nSecond, it is obtained from \\eqref{deri_snr} that when $W_0>W_{\\mathrm{A-H}}$, the hybrid IRS outperforms the fully-active IRS, and achieves its maximum.\nThird, we compare the achievable capacity by the hybrid IRS and passive IRS. By comparing \\eqref{C_p} and \\eqref{C_h_2}, it can be shown that when\n\\begin{equation}\n    \\frac{W_0^2P_{\\mathrm{B}}\\beta^2}{W_{\\mathrm{pas}}^2D_{\\mathrm{BI}}^2d_{\\mathrm{IU}}^2\\sigma_0^2}>\\frac{P_{\\mathrm{B}}\\beta^2\\left(\\frac{A_{\\mathrm{sum}}W_{\\mathrm{pas}}}{4W_{\\mathrm{act}}}+\\frac{W_0}{W_{\\mathrm{pas}}}\\right)^2\/D_{\\mathrm{BI}}^2d_{\\mathrm{IU}}^2}{A_{\\mathrm{sum}}\\sigma_{\\mathrm{I}}^2\\beta\/d_{\\mathrm{IU}}^2+\\sigma_0^2},\\label{con_hp}\n\\end{equation}\nthe fully-passive IRS achieves the maximum capacity.\nThe condition in \\eqref{con_hp} can be re-expressed as\n\\begin{equation}\\label{ineq_H-P}\n    \\frac{\\sigma_{\\mathrm{I}}^2\\beta}{d_{\\mathrm{IU}}^2}W_0^2-\\frac{W_{\\mathrm{pas}}^2\\sigma_0^2}{2W_{\\mathrm{act}}}W_0-\\frac{A_{\\mathrm{sum}}W_{\\mathrm{pas}}^4\\sigma_0^2}{16W_{\\mathrm{act}}^2}>0,\n   \n\\end{equation}\nwhich leads to\n\\begin{equation}\n    W_0<\\frac{W_{\\mathrm{pas}}^2\\sigma_0^2d_{\\mathrm{IU}}^2}{4W_{\\mathrm{act}}\\sigma_{\\mathrm{I}}^2\\beta}-\\frac{W_{\\mathrm{pas}}^2\\sigma_0d_{\\mathrm{IU}}}{4W_{\\mathrm{act}}\\sigma_{\\mathrm{I}}}\\sqrt{\\frac{\\sigma_0^2d_{\\mathrm{IU}}^2}{\\sigma_{\\mathrm{I}}^2\\beta^2}+\\frac{P_{\\mathrm{I}}}{P_{\\mathrm{B}}\\beta^2\/D_{\\mathrm{BI}}^2+\\sigma_{\\mathrm{I}}^2\\beta}}<0,\n   \n\\end{equation}\nand $W_0>W_{\\mathrm{H-P}}$. \nLast, we discuss the relations of $W_{\\mathrm{A-P}},W_{\\mathrm{A-H}}$ and $W_{\\mathrm{H-P}}$\nto facilitate our analysis, we list all possible permutations of the above thresholds in Table \\ref{per_thres}.\n\\begin{table}[t]\\centering\n\\caption{Possible relations for budget thresholds.}\\label{per_thres}\n\\begin{tabular}{|c|c|}\n\\hline\n\\begin{tabular}[c]{@{}c@{}}Permutations\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}$W_{\\mathrm{A-H}}<W_{\\mathrm{A-P}}<W_{\\mathrm{H-P}},W_{\\mathrm{A-H}}<W_{\\mathrm{H-P}}<W_{\\mathrm{A-P}},W_{\\mathrm{A-P}}<W_{\\mathrm{A-H}}<W_{\\mathrm{H-P}},$\\\\ $W_{\\mathrm{H-P}}<W_{\\mathrm{A-P}}<W_{\\mathrm{A-H}},W_{\\mathrm{A-P}}<W_{\\mathrm{H-P}}<W_{\\mathrm{A-H}},W_{\\mathrm{H-P}}<W_{\\mathrm{A-H}}<W_{\\mathrm{A-P}}.$\\end{tabular} \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\\noindent It can be shown that only one permutation of the above thresholds is feasible, i.e., $W_{\\mathrm{A-H}}<W_{\\mathrm{A-P}}<W_{\\mathrm{H-P}}$. \nSpecifically, the hybrid IRS reduces to the fully-active IRS when $W_0<W_{\\mathrm{A-H}}$; it reduces to the fully-passive IRS when $W_0>W_{\\mathrm{H-P}}$; and it outperforms the fully-active and fully-passive IRSs with the optimal number of active elements of $\\tilde N_{\\mathrm{act}}=\\tilde N^*_{\\mathrm{act}}$ otherwise. \nMoreover, all other permutations can be verified to be infeasible by contradiction. For example, for the permutation $W_{\\mathrm{A-H}}<W_{\\mathrm{H-P}}<W_{\\mathrm{A-P}}$, when $W_{\\mathrm{A-H}}<W_{\\mathrm{H-P}}<W_0<W_{\\mathrm{A-P}}$, we obtain that the hybrid IRS outperforms the fully-active IRS when $W_0>W_{\\mathrm{A-H}}$, the fully-passive IRS outperforms hybrid IRS when $W_0>W_{\\mathrm{H-P}}$, and the fully-active IRS outperforms the fully-passive IRS when $W_0<W_{\\mathrm{A-P}}$, which contradicts each other and thus is infeasible.\nCombining the above leads to the desired result.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}