diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzelho" "b/data_all_eng_slimpj/shuffled/split2/finalzzelho" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzelho" @@ -0,0 +1,5 @@ +{"text":"\n\n\\subsection{Retrieval Methodology}\nWe take a two-step approach for retrieving \\emph{evidence sentences}: given a statement, (1) constructing one query per sentence and retrieving relevant articles from Wikipedia, and (2) reranking paragraphs and then sentences to create the final set of evidence sentences. \nWikipedia is used as our evidence source mainly due to its objective perspective and broad coverage of topics. A dump of December 21, 2016 was downloaded. \nFor training, evidence sentences are retrieved with queries constructed from target user arguments. For test, queries are constructed from OP.\n\n\\smallskip\n\\noindent \\textbf{Article Retrieval.} \nWe first create an inverted index lookup table for Wikipedia as done in \\newcite{chen-EtAl:2017:Long4}. \nFor a given statement, we construct one query per sentence to broaden the diversity of retrieved articles. Therefore, multiple passes of retrieval will be conducted if more than one query is created. \nSpecifically, we first collect topic signature words of the post. Topic signatures~\\cite{lin2000automated} are terms strongly correlated with a given post, measured by log-likelihood ratio against a background corpus. We treat posts from other discussions in our dataset as background. \nFor each sentence, one query is constructed based on the noun phrases and verbs containing at least one topic signature word. \nFor instance, a query ``\\texttt{the government}, \\texttt{my e-mails}, \\texttt{national security}'' is constructed for the first sentence of OP in the motivating example (Figure~\\ref{fig:pipeline}). \nTop five retrieved articles with highest TF-IDF similarity scores are kept per query.\n\n\n\\smallskip\n\\noindent \\textbf{Sentence Reranking.} \nThe retrieved articles are first segmented into paragraphs, which are reranked by TF-IDF similarity to the given statement. Up to 100 top ranked paragraphs with positive scores are retained. These paragraphs are further segmented into sentences, and reranked according to TF-IDF similarity again. We only keep up to 10 top sentences with positive scores for inclusion in the evidence set. \n\n\n\\begin{table}[t]\n\\fontsize{10}{12}\\selectfont\n \\centering\n \\setlength{\\tabcolsep}{2mm}\n \\begin{tabular}{l|p{15mm} p{15mm}}\n \\hline\n & \\multicolumn{2}{|c}{\\textit{Queries Constructed from}}\\\\\n & \\textbf{OP} & \\textbf{Argument} \\\\ \n \\hline\n Avg \\# Topic Sig. & 17.2 & 9.8 \\\\\n Avg \\# Query\t& 6.7 &\t1.9 \\\\\n Avg \\# Article Retrieved \t & 26.1 &\t8.0 \\\\\n Avg \\# Sent. Retrieved \t & 67.3 &\t8.5 \\\\\n \\hline\n \\end{tabular}\n \n\n \\caption{\\fontsize{10}{12}\\selectfont \n Statistics for evidence sentence retrieval from Wikipedia. Considering query construction from either OP or target user arguments, we show the average numbers of topic signatures collected, queries constructed, and retrieved articles and sentences. \n }\n \\label{tab:retrieval-stats}\n\\end{table}\n\n\\subsection{Gold-Standard Keyphrase Construction}\nTo create training data for the keyphrase decoder, we use the following rules to identify keyphrases from evidence sentences that are reused by human writers for argument construction: \n\n\n{\n\\begin{itemize}\n\\vspace{-2mm}\n\\item Extract noun phrases and verb phrases from evidence sentences using Stanford CoreNLP~\\cite{manning-EtAl:2014:P14-5}.\n\\vspace{-2mm}\n\\item Keep phrases of length between 2 and 10 that overlap with content words in the argument.\n\\vspace{-2mm}\n\\item If there is span overlap between phrases, the longer one is kept if it has more content word coverage of the argument; otherwise the shorter one is retained.\n\\end{itemize}\n}\n\nThe resultant phrases are then concatenated with a special delimiter \\texttt{} and used as gold-standard generation for training.\n\n\n\n\\subsection{Final Dataset Statistics}\nEncoding the full set of evidence by our current decoder takes a huge amount of time. \nWe there propose a sampling strategy to allow the encoder to finish encoding within reasonable time by considering only a subset of the evidence: For each sentence in the statement, up to three evidence sentences are randomly sampled from the retrieved set; then the sampled sentences are concatenated. \nThis procedure is repeated three times per statement, where a statement is an user argument for training data and an OP for test set. In our experiments, we remove duplicates samples and the ones without any retrieved evidence sentence. \nFinally, we break down the augmented data into a training set of 224,553 examples (9,737 unique OPs), 13,911 for validation (640 OPs), and 30,417 retained for test (1,892 OPs). \n\n\n\\subsection{Training Setup}\nFor all models, we use a two-layer biLSTM as encoder and a two-layer unidirectional LSTM as decoder, with 200-dimensional hidden states in each layer. We apply dropout~\\cite{gal2016theoretically} on RNN cells with a keep probability of 0.8. \nWe use Adam~\\cite{kingma2014adam} with an initial learning rate of 0.001 to optimize the cross-entropy loss. Gradient clipping is also applied with the maximum norm of 2. \nThe input and output vocabulary sizes are both 50k. \n\n\\smallskip\n\\noindent \\textbf{Curriculum Training.} \nWe train the models in three stages where the truncated input and output lengths are gradually increased. Details are listed in Table \\ref{tab:staged-training}. Importantly, this strategy allows model training to make rapid progress during early stages. \nTraining each of our full models takes about 4 days on a Quadro P5000 GPU card with a batch size of 32. The model converges after about 10 epochs in total with pre-training initialization, which is described below.\n\n\\begin{table}[H]\n \\centering\n \\fontsize{10}{12}\\selectfont\n \\begin{tabular}{l|c|c|c}\n \\hline\n \\textbf{Component} & \\textbf{Stage 1} & \\textbf{Stage 2} & \\textbf{Stage 3} \\\\\n \\hline\n \\multicolumn{4}{l}{\\it {Encoder}}\\\\\n \\quad OP & 50 & 150 & 400 \\\\\n \\quad Evidence \t& 0 & 80 & 120 \\\\\n \\multicolumn{4}{l}{\\it {Decoder}}\\\\\n \\quad Keyphrases & 0 & 80 & 120 \\\\\n \\quad Target Argument \t& 30 & 80 & 120 \\\\\n \\hline\n \\end{tabular}\n \\caption{\\fontsize{10}{12}\\selectfont\n Truncation size (i.e., number of tokens including delimiters) for different stages during training. Note that in the first stage we do not include evidence and keyphrases.}\n \\label{tab:staged-training}\n\\end{table}\n\n\\smallskip\n\\noindent \\textbf{Adding Pre-training.} \nWe pre-train a two-layer seq2seq model with OP as input and target argument as output from our training set. After 20 epochs (before converging), parameters for the first layer are used to initialize the first layer of all comparison models and our models (except for the keyphrase decoder). Experimental results show that pre-training boosts all methods by roughly 2 METEOR~\\cite{denkowski-lavie:2014:W14-33} points. We describe more detailed results in the supplementary material.\n\n\\subsection{Baseline and Comparisons}\n\nWe first consider a \\textsc{Retrieval}-based baseline, which concatenates retrieved evidence sentences to form the argument. \nWe further compare with three seq2seq-based generation models with different training data: \n(1) \\textsc{seq2seq}: training with OP as input and the argument as output; \n(2) \\textsc{seq2seq} + \\textit{encode evd}: augmenting input with evidence sentences as in our model; \n(3) \\textsc{seq2seq} + \\textit{encode KP}: augmenting input with gold-standard keyphrases, which assumes some of the talking points are known. \nAll seq2seq models use a regular beam search decoder with the same beam size as ours. \n\n\\smallskip\n\\noindent \\textbf{Variants of Our Models.} \nWe experiment with variants of our models based on the proposed separate decoder model (\\textsc{Dec-separate}) or using a shared decoder (\\textsc{Dec-shared}). For each, we further test whether adding keyphrase attention for argument decoding is helpful (+ \\textit{attend KP}).\n\n\n\n\n\\smallskip\n\\noindent \\textbf{System vs. Oracle Retrieval.} \nFor test time, evidence sentences are retrieved with queries constructed from OP (\\textit{System Retrieval}). We also experiment with an \\textit{Oracle Retrieval} setup, where the evidence is retrieved based on user arguments, to indicate how much gain can be expected with better retrieval results.\n\n\n\n\n\\section{Introduction}\n\\input{intro.tex}\n\n\\section{Framework}\n\\label{sec:framework}\n\\input{framework.tex}\n\n\\section{Data Collection and Processing}\n\\label{sec:data}\n\\input{data.tex}\n\n\\section{Model}\n\\label{sec:model}\n\\input{model.tex}\n\n\\section{Relevant Evidence Retrieval}\n\\label{sec:retrieval}\n\\input{context.tex}\n\n\\section{Experimental Setup}\n\\label{sec:experiments}\n\\input{experiments.tex}\n\n\\section{Results}\n\\label{sec:results}\n\\input{results.tex}\n\n\\section{Further Discussion}\n\\label{sec:discussion}\n\\input{discussion.tex}\n\n\\section{Related Work}\n\\label{sec:related}\n\\input{related.tex}\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\\input{conclusion.tex}\n\n\n\\section*{Acknowledgements}\nThis work was partly supported by National Science Foundation Grant IIS-1566382, and a GPU\ngift from Nvidia. \nWe thank three anonymous reviewers for their insightful suggestions on various aspects of this work.\n\n\n\\subsection{Model Formulation}\nOur model takes as input a sequence of tokens $\\bm{x} = \\{\\bm{x}^O; \\bm{x}^E\\}$, where $\\bm{x}^O$ is the statement sequence and $\\bm{x}^E$ contains relevant evidence that is extracted from Wikipedia based on a separate retrieval module. A special token \\texttt{} is inserted between $\\bm{x}^O$ and $\\bm{x}^E$. Our model then first generates a set of keyphrases as a sequence $\\bm{y}^p = \\{y^p_l\\}$, followed by an argument $\\bm{y}^a = \\{y^a_t\\}$, by maximizing $\\log P(\\bm{y}|\\bm{x})$, where $\\bm{y}=\\{\\bm{y}^p; \\bm{y}^a\\}$.\n\n\nThe objective is further decomposed into $\\sum_{t} \\log P(y_t|y_{1:t-1},\\bm{x})$, with each term estimated by a softmax function over a non-linear transformation of decoder hidden states $\\bm{s}^a_t$ and $\\bm{s}^p_t$, for argument decoder and keyphrase decoder, respectively. The hidden states are computed as done in \\newcite{bahdanau2014neural} with attention: \n\n\\vspace{-2mm}\n{\\fontsize{10}{11}\\selectfont\n\\setlength{\\abovedisplayskip}{2pt}\n\\setlength{\\belowdisplayskip}{2pt}\n\\begin{align}\n\\fontsize{10}{11}\\selectfont\n \t & \\bm{s}_t = g(\\bm{s}_{t-1}, \\bm{c}_t, y_t) \\label{eq:attn_0}\\\\\n &\\bm{c}_t = \\sum_{j=1}^T \\alpha_{tj} \\bm{h}_j \\label{eq:attn_1} \\\\\n &\\alpha_{tj} = \\frac{\\textnormal{exp}(e_{tj})}{\\sum_{k=1}^T \\textnormal{exp}(e_{tk})} \\label{eq:attn_2} \\\\\n &e_{tj} = \\bm{v}^T\\tanh (\\bm{W_h} \\bm{h}_j + \\bm{W_s} \\bm{s}_t + \\bm{b}_{attn}) \\label{eq:attn_3}\n\\end{align}\n}\n\nNotice that two sets of parameters and different state update functions $g(\\cdot)$ are learned for separate decoders: $\\{\\bm{W}^a_h$, $\\bm{W}^a_s$, $\\bm{b}^a_{attn}, g^a(\\cdot) \\}$ for the argument decoder; $\\{\\bm{W}^p_h$, $\\bm{W}^p_s$, $\\bm{b}^p_{attn}, g^p(\\cdot) \\}$ for the keyphrase decoder. \n\n\\smallskip\n\\noindent \\textbf{Encoder.} \nA two-layer bidirectional LSTM (bi-LSTM) is used to obtain the encoder hidden states $\\bm{h}_i$ for each time step $i$. For biLSTM, the hidden state is the concatenation of forward and backward hidden states: $\\bm{h}_i=[\\overrightarrow{\\bm{ h}_i}; \\overleftarrow{\\bm{h}_i}]$. Word representations are initialized with 200-dimensional pre-trained GloVe embeddings~\\cite{pennington-socher-manning:2014:EMNLP2014}, and updated during training. \nThe last hidden state of encoder is used to initialize both decoders. In our model the encoder is shared by argument and keyphrase decoders.\n\n\n\\smallskip\n\\noindent \\textbf{Decoders.}\nOur model is equipped with two decoders: \\textit{keyphrase decoder} and \\textit{argument decoder}, each is implemented with a separate two-layer unidirectional LSTM, in a similar spirit with one-to-many multi-task sequence-to-sequence learning~\\cite{luong2015multi}. The distinction is that our training objective is the sum of two loss functions:\n\n\\vspace{-3mm}\n{\\fontsize{10}{11}\\selectfont\n\\begin{align}\n\\begin{split}\n \\mathcal{L}(\\theta) = &-\\frac{\\alpha}{T_p}{\\sum_{(\\bm{x},\\bm{y}^p)\\in D}} \\log P(\\bm{y}^p|\\bm{x};\\theta) \\\\ \n &- \\frac{(1-\\alpha)}{T_a}{\\sum_{(\\bm{x},\\bm{y}^a)\\in D}} \\log P(\\bm{y}^{a}|\\bm{x};\\theta) \\label{eq:loss_overall} \\\\\n\\end{split}\n\\end{align}\n}\n\n\\noindent where $T_p$ and $T_a$ denote the lengths of reference keyphrase sequence and argument sequence. \n$\\alpha$ is a weighting parameter, and it is set as $0.5$ in our experiments.\n\n\\smallskip\n\\noindent \\textbf{Attention over Both Input and Keyphrases.} \nIntuitively, the argument decoder should consider the generated keyphrases as talking points during the generation process. We therefore propose an attention mechanism that can attend both encoder hidden states and the keyphrase decoder hidden states. Additional context vector $\\bm{c}'_t$ is then computed over keyphrase decoder hidden states $\\bm{s}^p_j$, which is used for computing the new argument decoder state:\n\n\n\\vspace{-2mm}\n{\\fontsize{10}{11}\\selectfont\n\\setlength{\\abovedisplayskip}{2pt}\n\\setlength{\\belowdisplayskip}{2pt}\n\\begin{align}\n&\\bm{s}^a_t = g'(\\bm{s}^a_{t-1}, [\\bm{c}_t; \\bm{c}'_t], {y}^a_t) \\label{eq:kattn_1} \\\\\n& \\bm{c}'_t = \\sum_{j=1}^{T_p} \\alpha'_{tj}\\bm{s}^p_j \\label{eq:kattn_3}\\\\\n& \\alpha'_{tj} = \\frac{\\textnormal{exp}(e'_{tj})}{\\sum_{k=1}^{T_p} \\textnormal{exp}{(e'_{tk})}} \\label{eq:kattn_4}\\\\\n& e'_{tj} = {\\bm{v}'}^T\\tanh (\\bm{W'_p} \\bm{s}^p_j + \\bm{W'_a} \\bm{s}^a_t + \\bm{b}'_{attn}) \\label{eq:kattn_5}\n\\end{align}\n}\n\n\\noindent where $\\bm{s}^p_j$ is the hidden state of keyphrase decoder at position $j$, $\\bm{s}^a_t$ is the hidden state of argument decoder at timestep $t$, and $\\bm{c}_t$ is computed in Eq.~\\ref{eq:attn_1}.\n\n\\smallskip\n\\noindent \\textbf{Decoder Sharing.} \nWe also experiment with a shared decoder between keyphrase generation and argument generation: the last hidden state of the keyphrase decoder is used as the initial hidden state for the argument decoder. A special token \\texttt{} is inserted between the two sequences, indicating the start of argument generation. \n\n\\subsection{Hybrid Beam Search Decoding} \nHere we describe our decoding strategy on the argument decoder. We design a hybrid beam expansion method combined with segment-based reranking to promote diversity of beams and informativeness of the generated arguments. \n\n\\smallskip\n\\noindent \\textbf{Hybrid Beam Expansion.} \nIn the standard beam search, the top $k$ words of highest probability are selected deterministically based on the softmax output to expand each hypothesis. However, this may lead to suboptimal output for text generation~\\cite{wiseman-rush:2016:EMNLP2016}, e.g., one beam often dominates and thus inhibits hypothesis diversity. \nHere we only pick the top $n$ words ($n1$ means that the angular momentum decreases as\nthe flux rope expands. Combine Eq.\\ref{eq_rrx}, \\ref{eq_vr}, \\ref{eq_mom_2} and\n\\ref{eq_vphi}, we can rewrite the momentum conservation equation\nwith $\\hat{r}$ as\n\\begin{eqnarray}\n(\\vec{j}\\times\\vec{B})_r=\\rho(a_ex-k_1^2R^{-2k_2-1}x^{2k_2-3})+R^{-1}\\frac{\\partial p}{\\partial x}\n\\label{eq_mom_4}\n\\end{eqnarray}\nFor a thermodynamic process, we relate the thermal\npressure $p$ with the density $\\rho$ by the polytropic equation of state\n\\begin{eqnarray}\n p=k_3\\rho^\\Gamma \\label{eq_prho}\n\\end{eqnarray}\nwhere $k_3$ is a positive constant and $\\Gamma$ is a variable treated as\nthe polytropic index, and Eq.\\ref{eq_mom_4} becomes\n\\begin{eqnarray}\n(\\vec{j}\\times\\vec{B})_r=\\rho(a_ex-k_1^2R^{-2k_2-1}x^{2k_2-3})+k_3R^{-1}\\frac{\\partial \\rho^\\Gamma}{\\partial x} \\label{eq_mom_5}\n\\end{eqnarray}\nDefine a quantity $f_{em}$ to be the average Lorentz force\nover $\\hat{r}$ from the axis to the boundary of the flux rope,\n$f_{em}=\\frac{1}{R}\\int_0^R(\\vec{j}\\times\\vec{B})_rdr$.\nFrom Eq.\\ref{eq_mom_5}, we get\n\\begin{eqnarray}\nf_{em}=a_e\\int_0^1\\rho xdx-k_1^2R^{-2k_2-1}\\int_0^1\\rho x^{2k_2-3}dx+k_3R^{-1}\\int_0^1\\frac{\\partial\\rho^\\Gamma}{\\partial x}dx \\label{eq_pem1}\n\\end{eqnarray}\n$f_{em}>0$ means that the average Lorentz force directs outward from the\naxis of the flux rope, causing expansion. On the other hand, $f_{em}<0$\nprevents the expansion of the flux rope.\n\nWe assume that the mass of a CME is conserved when it propagates in the\nouter corona and interplanetary space, where the CME has fully developed.\nThe mass conservation gives\n\\begin{eqnarray}\n \\int\\rho rdr d\\phi dz=2\\pi lR^2\\int_0^1\\rho xdx=M \\mathrm{\\ (constant)} \\label{eq_rho}\n\\end{eqnarray}\nwhere $l$ is the axial length of the flux rope (Fig.\\ref{fg_coor}).\nSince the flux rope is assumed to be self-similar and it is generally\naccepted that the magnetic field lines are frozen-in with the plasma flows\nin corona\/interplanetary space, the density in the flux rope has a fixed\ndistribution $f_\\rho(x)$, and therefore\n\\begin{eqnarray}\n\\rho(t, x)=f_\\rho(x)\\rho_0(t)\n\\end{eqnarray}\nDefine positive constants\n\\begin{eqnarray}\nk_4=\\int_0^1f_\\rho xdx \\\\\nk_5=\\int_0^1f_\\rho x^{2k_2-3}dx\n\\end{eqnarray}\nand a variable\n\\begin{eqnarray}\nq(\\Gamma)=f_\\rho^\\Gamma(0)-f_\\rho^\\Gamma(1)\n\\end{eqnarray}\nThen it can be inferred from Eq.\\ref{eq_rho} that\n\\begin{eqnarray}\n\\rho_0=\\frac{1}{2\\pi}k_{4}^{-1}MR^{-2}l^{-1}\\label{eq_rho_3}\n\\end{eqnarray}\nand Eq.\\ref{eq_pem1} can be written as\n\\begin{eqnarray}\nf_{em}=\\frac{M}{2\\pi}(a_eR^{-2}l^{-1}-k_1^2k_4^{-1}k_5R^{-2k_2-3}l^{-1})-f_{th} \\label{eq_pem2}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nf_{th}=\\frac{1}{R}\\int_0^R-\\frac{\\partial p}{\\partial r}dr=k_3q\\rho_0^\\Gamma R^{-1} \\label{eq_pth}\n\\end{eqnarray}\nis the average thermal pressure force. Like $f_{em}$, $f_{th}$ points outward\nif it is larger than zero.\n\nOn the other hand, in an axisymmetric cylindrical flux rope,\n\\begin{eqnarray}\n&\\vec{B}=B_\\phi\\hat{\\phi}+B_z\\hat{z}=\\nabla\\times\\vec{A} \\label{eq_bcomp} \\\\\n&B_\\phi=-\\frac{\\partial A_z}{\\partial r} \\label{eq_b_phi} \\\\\n&B_z=\\frac{1}{r}\\frac{\\partial}{\\partial r}(rA_\\phi) \\label{eq_b_z} \\end{eqnarray}\nAs the magnetic flux is conserved in both $\\hat\\phi$ and $\\hat{z}$ directions,\nwe get\n\\begin{eqnarray}\n&\\Phi_\\phi=-l\\int_0^R\\frac{\\partial A_z}{\\partial r} dr=l(A_z(0)-A_z(R)) \\label{eq_Phi_phi} \\\\\n&\\Phi_z=2\\pi\\int_0^R\\frac{\\partial}{\\partial r}(rA_\\phi) dr=2\\pi RA_\\phi(R) \\label{eq_Phi_z}\n\\end{eqnarray}\nIn order to satisfy the self-similar expansion assumption, $A_\\phi$ and $A_z$\nhave to keep their own distributions, respectively. Thus, according to the above two\nequations,\n\\begin{eqnarray}\n A_\\phi(t, x)&=&\\frac{f_\\phi(x)}{R} \\label{eq_A_phi} \\\\\n A_z(t, x)&=&\\frac{f_z(x)}{l} \\label{eq_A_z}\n\\end{eqnarray}\nIt can be proved that the conservation of helicity is satisfied automatically\n\\begin{eqnarray}\nH_m&=&\\int\\vec{B}\\cdot\\vec{A}rdrd\\phi dz \\nonumber \\\\\n&=&2\\pi \\int_0^1 [\\frac{f_z}{x}\\frac{\\partial}{\\partial x}(xf_\\phi)-f_\\phi\\frac{\\partial f_z}{\\partial x}] xdx \\nonumber \\\\\n&=&\\mathrm{constant}\n\\end{eqnarray}\nCombining Eq.\\ref{eq_b_phi}, \\ref{eq_b_z}, \\ref{eq_A_phi}\nand \\ref{eq_A_z}, we can calculate the Lorentz force in the flux\nrope\n\\begin{eqnarray}\n\\vec{j}\\times\\vec{B}&=&\\frac{1}{\\mu_0}(\\nabla\\times\\vec{B})\\times\\vec{B} \\nonumber \\\\\n&=&-\\mu_0^{-1}R^{-5}\\{x^{-2}\\frac{\\partial}{\\partial x}(xf_\\phi)\\frac{\\partial^2}{\\partial x^2}(xf_\\phi)-x^{-3}[\\frac{\\partial}{\\partial x}(xf_\\phi)]^2\\}\\hat{r} \\nonumber\\\\\n&&-\\mu_0^{-1}R^{-3}l^{-2}x^{-1}\\frac{\\partial f_z}{\\partial x}\\frac{\\partial}{\\partial x}(x\\frac{\\partial f_z}{\\partial x})\\hat{r} \\label{eq_jb_3}\n\\end{eqnarray}\nand therefore\n\\begin{eqnarray}\nf_{em}&=&-\\mu_0^{-1}R^{-5}\\int_0^1\\{x^{-2}\\frac{\\partial}{\\partial x}(xf_\\phi)\\frac{\\partial^2}{\\partial x^2}(xf_\\phi)-x^{-3}[\\frac{\\partial}{\\partial x}(xf_\\phi)]^2\\}dx \\nonumber\\\\\n&&-\\mu_0^{-1}R^{-3}l^{-2}\\int_0^1x^{-1}\\frac{\\partial f_z}{\\partial x}\\frac{\\partial}{\\partial x}(x\\frac{\\partial f_z}{\\partial x})dx \\label{eq_jbr_0} \\nonumber \\\\\n&=&-\\mu_0^{-1}k_6R^{-5}-\\mu_0^{-1}k_7R^{-3}l^{-2} \\label{eq_pem3}\n\\end{eqnarray}\nwhere $k_6$ and $k_7$ are both constants. It could be proved that\nthe sign of $k_6$ is determined by $B_z^2(R)-B_z^2(0)$, and $k_7\\geq0$.\n\nThe two forms of $f_{em}$, Eq.\\ref{eq_pem2} and \\ref{eq_pem3}, result in\n\\begin{eqnarray}\n&a_e-k_1^2k_4^{-1}k_5R^{-2k_2-1} \\nonumber\\\\\n&=-2\\pi\\mu_0^{-1}M^{-1}(k_6R^{-3}l+k_7R^{-1}l^{-1})+2\\pi M^{-1}k_3(2\\pi k_4M^{-1}R^2l)^{-\\Gamma}[f_\\rho^\\Gamma(0)-f_\\rho^\\Gamma(1)]Rl \\label{eq_acs}\n\\end{eqnarray}\nin which $f_{th}$ is substituted by Eq.\\ref{eq_pth}.\nAs at present it is impossible to practically detect the axial length of a\nflux rope, here we will relate it with a measurable variable, $L$, the distance\nbetween the flux rope axis and the solar surface (Fig.\\ref{fg_coor}),\nat which altitude the flux rope originates, by the assumption\n\\begin{eqnarray}\nl=k_8L \\label{eq_ll}\n\\end{eqnarray}\nwhere $k_8$ is a positive constant. The topology of\nflux rope as shown in Figure \\ref{fg_coor} implies that this\nassumption is reasonable.\nFinally, Eq.\\ref{eq_acs} can be simplified to\n\\begin{eqnarray}\na_e-c_0R^{-c_1-3}=-c_2R^{-3}L-c_3R^{-1}L^{-1}+c_4(c_5^\\Gamma-c_6^\\Gamma)R^{1-2\\Gamma}L^{1-\\Gamma} \\label{eq_sase_fitting}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n&c_0=k_1^2k_4^{-1}k_5\\geq 0 \\\\\n&c_1=2k_2-2 \\geq 0 \\\\\n&c_2=2\\pi\\mu_0^{-1}M^{-1}k_6k_8 \\\\\n&c_3=2\\pi\\mu_0^{-1}M^{-1}k_7k_8^{-1}\\geq 0 \\\\\n&c_4=2\\pi M^{-1}k_3k_8>0 \\\\\n&c_5=(2\\pi)^{-1}Mk_4^{-1}k_8^{-1}f_\\rho(0) \\geq0 \\label{eq_density_center} \\\\\n&c_6=(2\\pi)^{-1}Mk_4^{-1}k_8^{-1}f_\\rho(1) \\geq0 \\label{eq_density_boundary}\n\\end{eqnarray}\nThe left-hand side of Eq.\\ref{eq_sase_fitting} describes the motion of the fluids\nin the flux rope. Its first item is the acceleration due to the radial motion (i.e., expansion)\nand the second one gives the acceleration due to the poloidal motion. The right-hand\nside reflects the contributions from the Lorentz force (the first two items) and\nthermal pressure force (the last one).\nThe constants $k_{1-8}$ and $c_{0-6}$ appeared above are summarized in Table \\ref{tb_constants}.\n\n\\begin{table*}[t\n\\caption{List of the constants $k_{1-8}$ and $c_{0-6}$}\n\\label{tb_constants}\n\\begin{tabular}{cp{120pt}|cp{120pt}}\n\\hline\nConstant & Interpretation & Constant & Interpretation \\\\\n\\hline\n$k_1$ & Scale the initial magnitude of the poloidal motion &$k_4$ and $k_5$ & Integral constants related to the density distribution \\\\\n$k_2$ & Decrease rate of the angular momentum as the flux rope expands &$k_6$ and $k_7$ & Scale the initial magnitude of the Lorentz force contributed by the axial and poloidal fields \\\\\n$k_3$ & Coefficient in the polytropic equation of state &$k_8$ & Assumed constant to relate the length of flux rope $l$ to distance $L$ \\\\\n\\hline\\hline\n$c_0$ & Scale the initial magnitude of the acceleration due to the poloidal motion &$c_2$ and $c_3$ & Similar to $k_6$ and $k_7$ \\\\\n$c_1$ & Similar to $k_2$ &$c_4(c_5^\\Gamma-c_6^\\Gamma)$ & Scale the initial magnitude of the contribution by thermal pressure force \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\nThe Lorentz force and thermal pressure force can be rewritten in terms of the\nconstants $c_{0-6}$, $k_8$ and the total mass $M$ as follows\n\\begin{eqnarray}\nf_{em}=-\\frac{M}{2\\pi k_8}(c_2R^{-5}+c_3R^{-3}L^{-2})\\label{eq_fem}\\\\\nf_{th}=\\frac{M}{2\\pi k_8}c_4(c_5^\\Gamma-c_6^\\Gamma)R^{-2\\Gamma-1}L^{-\\Gamma}\\label{eq_fth}\n\\end{eqnarray}\nand their ratio is\n\\begin{eqnarray}\n\\frac{f_{em}}{f_{th}}=\\frac{c_2R^{2\\Gamma-4}L^\\Gamma+c_3R^{2\\Gamma-2}L^{\\Gamma-2}}{c_4(c_6^\\Gamma-c_5^\\Gamma)}\\label{eq_inff}\n\\end{eqnarray}\n\nIn summary, starting from MHD equations with the three major assumptions that\n(1) the flux-rope CME has an axisymmetric cylinder configuration, (2) its cross-section is\nself-similarly evolving, and (3) its axial length is proportional to the distance\nfrom the solar surface, we find that the polytropic index, $\\Gamma$, can be related to the\nmeasurable parameters: the distance, $L$, the radius, $R$, and another derived\nquantity, the expansion acceleration ($a_e$),\nas shown in Eq.\\ref{eq_sase_fitting}.\nIf we have enough measurement points, the unknown constants $c_{0-6}$ and variable\n$\\Gamma$ could be obtained through some fitting techniques (e.g., that described in the first\nparagraph of Sec.\\ref{sec_results}), and then the relative strength\nof the Lorentz force and thermal pressure force can also be easily calculated by Eq.\\ref{eq_fem} and \\ref{eq_fth}.\n\n\\begin{comment}\nFurther, we define the ratio of the Lorentz\nforce to thermal pressure gradient force as\n\\begin{eqnarray}\n\\mathrm{Plasma\\ } \\delta=\\frac{|f_{em}|}{|f_{th}|}\n\\end{eqnarray}\nIt reflects the relative significance of the Lorentz\nforce compared to the thermal pressure gradient force. Plasma $\\delta=0$ means $\\vec j\\times\\vec B=0$, implying a strictly force-free\nstate. From Eq.\\ref{eq_rho_3}, \\ref{eq_pth} and \\ref{eq_pem3}, we find that\n$\\delta$ has the following form\n\\begin{eqnarray}\n\\delta=\\left|\\frac{c_2R^{2\\Gamma-4}L^{\\Gamma}+c_3R^{2\\Gamma-2}L^{\\Gamma-2}}{c_4(c_5^\\Gamma-c_6^\\Gamma)}\\right|\n\\label{eq_inff}\n\\end{eqnarray}\nthat can be simply calculated after the parameters $c_{0-6}$\nand $\\Gamma$ in Eq.\\ref{eq_sase_fitting} are obtained.\n\\end{comment}\n\n\\subsection{Asymptotic Value of Polytropic Index $\\Gamma$} \\label{sec_asymptotic}\nHere, we consider the case of a nearly force-free expanding flux rope.\nIt is generally true that most CMEs are almost force-free at least near 1 AU though\nthey may be far away from a froce-free state at initial stage.\nIt can be proved that $R\\propto L$\n(ref. to Appendix), i.e., $R=\\alpha L$ where $\\alpha$ is a positive constant.\nThen Eq.\\ref{eq_inff} becomes\n\\begin{eqnarray}\n\\frac{f_{em}}{f_{th}}=\\frac{(c_2\\alpha^{2\\Gamma-4}+c_3\\alpha^{2\\Gamma-2})L^{3\\Gamma-4}}{c_4(c_6^\\Gamma-c_5^\\Gamma)}\n\\end{eqnarray}\nIt is found that $\\Gamma=\\frac{4}{3}$ is a critical point, above\/below which\nthe absolute value of Lorentz force decreases slower\/faster than that of thermal\npressure force as increasing distance $L$. This value of $\\Gamma$ is the same as\nthat obtained by \\citet{Low_1982} and \\citet{Kumar_Rust_1996} for a self-similar\nexpanding flux rope. This inference is reasonable because smaller $\\Gamma$ implies the\nplasma absorbs more heat for the same expansion and therefore the thermal pressure\nshould decrease slower.\n\nUnder force-free condition, Eq.\\ref{eq_sase_fitting} can also reduce to\n\\begin{eqnarray}\na_e=c_0(\\alpha L)^{-c_1-3}-c_2\\alpha^{-3}L^{-2}-c_3\\alpha^{-1}L^{-2}+c_4(c_5^\\Gamma-c_6^\\Gamma)\\alpha^{1-2\\Gamma}L^{2-3\\Gamma}\n\\end{eqnarray}\nand at infinite distance, $L\\rightarrow+\\infty$, we have\n\\begin{eqnarray}\na_{e\\infty}\\sim c_4(c_5^{\\Gamma_\\infty}-c_6^{\\Gamma_\\infty})\\alpha^{1-2{\\Gamma_\\infty}}L_\\infty^{2-3{\\Gamma_\\infty}}\n\\end{eqnarray}\nThe above equation indicates that $\\Gamma=\\frac{2}{3}$ is another critical point.\nThe polytropic index $\\Gamma$ should be larger than $\\frac{2}{3}$ to make sure that the flux rope will finally approach a\nsteady expansion and propagation state (including the case that the flux rope stop somewhere\nwithout expansion). Otherwise, the flux rope will always accelerated expanding.\n\nBased on the current observations, the expansion behavior of CMEs at large heliocentric\ndistance is not as clear as that in the inner heliosphere. The investigations on\nthe radial widths of CMEs suggest that CMEs at least keep expanding within about 15 AU\n\\citep[e.g.,][]{Wang_Richardson_2004, Wang_etal_2005a}, but the expansion speeds seem\nto be slower and slower. Although the number of CMEs observed near and beyond 15 AU is\nsmall and the uncertainty of statistics is large, it is likely that a CME may not be\nable to keep an accelerated expansion always. Thus, in practice, the polytropic index\nof the CME plasma should be larger than $\\frac{2}{3}$.\n\n\n\\section{The 2007 October 8 CME} \\label{sec_case}\n\\subsection{Observations and Measurements} \\label{sec_measurements}\nThe suite of SECCHI instruments on board STEREO spacecraft provide an\nunprecedented continuous view of CMEs from the surface of the Sun through\nthe inner heliosphere. The instruments, EUVI, COR1, COR2, HI1 and HI2,\nmake the images of the solar corona in the ranges of 0--1.5, 1.4--4.0,\n2.5--15.0, $\\sim$15--90, and $\\sim$90--300 solar radii ($R_S$),\nrespectively \\citep{Howard_etal_2008}. The SECCHI observations present\nus the great opportunity to study the evolution of CMEs over an extended\ndistance. The CME launched on 2007 October 8 is a well observed event,\nwhich is used to study the CME evolution and the applicability of our model.\n\n\\begin{figure*}[p]\n \\centering\n \\includegraphics[width=0.9\\hsize]{f2low.eps}\n \\caption{Images of the 2007 October 8 CME taken by (a) EUVI at 304\\AA,\n(b) COR1, (c) COR2, (d) HI1 and (e) HI2 on board STEREO B. The panel\nat lower-right corner shows a direct image of the CME in the HI1\nFOV. This image has been corrected to the plane perpendicular to the\nline between the Sun and STEREO B, because the CME is assumed to be\na limb event in CORs and the direction that HI1 camera faces to is\ndifferent with CORs'.}\\label{fg_20071008cme}\n\\end{figure*}\n\nThis CME was initiated close to the western limb as seen from STEREO\nB. Hereafter all the observations used are from instruments on board\nthe B spacecraft. Figure \\ref{fg_20071008cme} shows five images of\nthe CME at different distances from the Sun. The CME was accompanied\nby a prominence eruption starting at about 07:00 UT on October 8, as\nseen by EUVI. The CME source region is clearly shown in the EUVI\n304\\AA\\ image on the top-left panel of the figure. The erupting\nprominence was also seen in the COR1 running-difference image (the\ntop-right panel). The CME was first observed in COR1 at about 08:46\nUT on October 8, and continuously ran through COR2 and HI1 fields of\nview (FOV). It even showed in the HI2 FOV after about 12:00 UT on\nOctober 10. Since the CME was launched from the western limb and\nshowed a circular-like structure, we believe that the CME was viewed\nby the instruments through an axial-view angle. Therefore, the\nprojection of the CME on the plane of the sky can be treated as the\ncross-section of the CME.\n\nTo obtain the two quantities, $R$ and $L$, for necessary model inputs,\nhere we simply measure three parameters,\nthe heliocentric distance of the CME leading edge, $h$, and the maximum\nand minimum position angles, $\\theta_{max}$ and $\\theta_{min}$, of the CME\nas shown in Figure \\ref{fg_coor}. Under the assumption of a circular\ncross-section, $R$ and $L$ could be derived by\n\\begin{eqnarray}\nR&=&h-L-R_S \\\\\nL&=&\\frac{h}{1+\\sin \\frac{\\theta_{max}-\\theta_{min}}{2}}-R_S\n\\end{eqnarray}\nIt should be noted that the measurements in HI2 images are not included\nin the following analysis, because the elongation effect is not negligible.\n\n\\begin{figure*}[tbh]\n \\centering\n \\includegraphics[width=0.48\\hsize]{f3a.eps}\n \\includegraphics[width=0.45\\hsize]{f3b.eps}\\\\\n \\hskip -170pt\n \\includegraphics[width=0.45\\hsize]{f3c.eps}\n \\caption{{\\it Left-upper panel:} The measurements of the heliocentric distance,\n$h$, of the flux-rope CME leading edge and its angular width,\n$\\Delta\\theta=\\theta_{max}-\\theta_{min}$. {\\it Right panel:} The\nderived distance, $L$, of the flux rope axis from the solar surface\nand the flux rope radius, $R$. {\\it Left-lower panel:} The\npropagation, $v_c$, and expansion, $v_e$, speeds derived from $L$\nand $R$, respectively.}\\label{fg_cme_motion}\n\\end{figure*}\n\nFigure \\ref{fg_cme_motion} shows the measurements and the derived parameters.\nThe CME is a slow and gradually accelerated event. It took about 46 hours\nfor its leading edge to reach 70 $R_S$. Nevertheless, because of its slowness,\nwe are able to make about one hundred measurement points for this CME. The\nred crosses plotted in the leftupper panel suggests that the CME angular width\nincreased at the early phase (mainly in the COR1 FOV), and then reached to\na near-constant value in the COR2 and HI1 FOVs. The right panel presents\nthe evolution of the derived $R$ and $L$.\nIt is shown that the radius of the flux-rope CME is about 20 $R_S$ when it\npropagated nearly 50 $R_S$ away from the Sun, which put the leading edge at\nabout 70 $R_S$. The left-lower panel exhibits the speeds derived from the $R$ and $L$,\nnamely expansion, $v_e$ and propagation, $v_c$, speeds, respectively. At the early\nphase, the expansion speed was very close to the propagation speed.\nIn the later phase, the propagation speed increased more\nquickly than the expansion speed. The increased difference between $v_c$ and\n$v_e$ is probably because of (1) the enhanced drag force of the ambient solar wind,\nwhich is fully formed in the outer corona and (2) the weakened pressure in the CME.\nThe issue of CME acceleration, which is as important as CME expansion, is not\naddressed in our model presented in this paper.\n\nIn the measurements, the CME radius obtained is the one along the\nlatitudinal direction on the meridional plane. This radius would be\nthe same as the radius along the radial direction if the\ncross-section is a perfect circle. However, the true cross-section\ndeviates from the perfect circle, and the deviation becomes larger\nas the CME is further from the Sun\n\\citep[e.g.,][]{Riley_Crooker_2004}. The distortional stretching of\nthe cross-section is caused by the divergent radial expansion of the\nbackground solar wind, which causes kinematic expansion of CMEs\nalong both the meridional and azimuthal directions, but not at all\nalong the radial direction. The CME expansion along the radial\ndirection is mostly driven by the dynamic effect, such as pressure\ngradient forces, while the expansion along the other two directions\nthat lie on the spherical surface is caused by the combination of\nthe dynamic and kinematic effects. As a result, the overall\ncross-section is a convex-outward ``pancake\" shape\n\\citep{Riley_Crooker_2004}. Figure \\ref{fg_20071008cme}f shows such\na distortion of the 2008 November 8 CME as observed in HI1 FOV; the\naspect ratio, defined by the ratio of the radius along the\nmeridional direction and that along the radial direction, is about\n1.4 when the CME leading edge is at $\\sim70 R_S$.\n\nDue to this stretching effect, our measurements assuming a circular cross-section lead to the inaccuracy of the\nmeasured parameters and the inferred parameters as well. In order to study the internal state of a CME, the radius\nof the CME, $R$, should be the one along the radial direction, and it is apparently overestimated when the radius\nalong the meridional direction is adopted. The derived expansion speed of CME is thus larger than the true value.\nSuch simplified measurements would infer unrealistic parameters of CME at 1 AU. For instance, the observed radius\nof 20 $R_S$ of the CME at a distance of 50 $R_S$ from the Sun would imply a CME cross-section of 0.8 AU at 1 AU, which is\ntoo larger to be true. The observed speeds of $v_e$ and $v_c$ would imply a speed of about 150 km\/s at the trailing\nedge of the CME, which is much smaller than the observed solar wind speed, i.e., about 300 km\/s. Therefore, one\nshould be cautious when our method is applied to CMEs at a large distance from the Sun (e.g, $> 70 R_S$). The\nheliospheric region we investigated in this paper is within about 70 $R_S$, and the stretching effect is\nrelatively small. Nevertheless, we will carefully estimate the errors on CME parameters in the second\nparagraph of Sec.\\ref{sec_discussion}.\nWe point out here that there is an observational difficulty in measuring the radius of CMEs along the radial\ndirection in a consistent way, mainly because of the low brightness contrast of the CME trailing edge in coronagraph\nimages. This difficulty might be overcome if the CME of interest is particularly bright.\n\n\nBefore modeling the CME, we fit the measurement points with a certain function\nto retrieve the smooth evolution process of the CME, which is required for the\nmodel. We use the modified function of log normal distribution to fit the speeds.\nWe did not fit the expansion acceleration directly, because any small\nerror in measurements of $R$ will be dramatically amplified in its second derivative\n$a_e$.\nThe fitting function of velocity has the form\n\\begin{eqnarray}\nv(t)=\\frac{v_\\infty}{2}\\left[1+\\mathrm{erf}\\left(\\frac{\\ln(t-t_0)-M}{S\\sqrt{2}}\\right)\\right]\n\\end{eqnarray}\nwhere $\\mathrm{erf}(z)$ is the erf function or error function, defined by\n\\begin{eqnarray}\n\\mathrm{erf}(z)\\equiv\\frac{2}{\\sqrt{\\pi}}\\int_0^ze^{-t^2}dt\n\\end{eqnarray}\nThis function has a value range from 0 to $v_\\infty$. It is chosen\nbecause the measurements show a trend that, at least within the FOVs of SECCHI,\nboth the speeds will not increase forever, but instead\nasymptotically approach a constant speed, $v_\\infty$. The acceleration\ncan be derived by\n\\begin{eqnarray}\na_e(t)=\\frac{v_\\infty}{S\\sqrt{2\\pi}(t-t_0)}e^{-\\frac{[\\ln(t-t_0)-M]^2}{2S^2}}\n\\end{eqnarray}\nThe solid lines in the left-lower panel of Figure\n\\ref{fg_cme_motion} show the fitting results. The fitted parameter\n$v_\\infty$ is 118 km\/s for expansion and 246 km\/s for propagation.\nAs a comparison with the measurements, the integrals of the fitting\ncurves of the speeds are also plotted in the right panel. It has\nbeen mentioned before that these estimated speeds suffer the solar\nwind stretching effect. Particularly, the estimated expansion speed\nis larger than that it should be. The error will be discussed in\nSec.\\ref{sec_discussion}.\n\n\\subsection{Results}\\label{sec_results}\nTo fit the above curves with the model, Eq.\\ref{eq_sase_fitting}, we use\nan iterative method. Generally speaking, first we solve this equation\nin every 8 neighboring measurement points to obtain a set of parameters $c_{0-6}$\nand $\\Gamma$. The segment of the 8 points is a running box through the entire\nevolution process of the CME. Secondly, input the obtained variable $\\Gamma$\ninto the model as guess values to fit the global constants $c_{0-6}$. Thirdly,\nuse the fitted $c_{0-6}$ to update the variable $\\Gamma$ by solving\nEq.\\ref{eq_sase_fitting} again. Then iterate the above 2nd and 3rd steps to\nmake constants $c_{0-6}$ and $\\Gamma$ converging to a steady solution. For the sake\nof simplicity, we ignore the poloidal motion of the fluid by setting $c_0$ zero. It\nis also because there seems no strong observational evidence showing a ring flow\ninside a CME.\n\nThe model results are shown in Figure \\ref{fg_cme_state}.\nThe uncertainty of the model results is estimated from the relative error\nof $a_e$, which is given by\n\\begin{eqnarray}\nE=\\left|\\frac{a_{em}-a_{ei}}{a_{ei}}\\right|\n\\end{eqnarray}\nwhere $a_{ei}$ is the value calculated by the input data, and $a_{em}$ is\nthe model value. The error curve is plotted in the left-upper panel of Figure\n\\ref{fg_cme_state}. It is found that the error is\nsmaller than 1\\%, except during 12:00 -- 18:00 UT.\nA possible explanation of the large uncertainty during that time has been\ngiven in the last second paragraph of Sec.\\ref{sec_discussion}.\n\n\\subsubsection{Polytropic Index}\nFrom the right-upper panel of Figure \\ref{fg_cme_state}, it is found\nthat $\\Gamma$ was less than 1.4 throughout the interplanetary space.\nIn the inner corona, say $L\\lsim2R_S$, it was about 1.24. After\nentering the outer corona, it quickly increased to above 1.35 at\n$L\\approx5 R_S$, and then slowly approached down to about 1.336,\nwhich is very close to the first critical point $\\frac{4}{3}$. This\nvalue of $\\Gamma$ is consistent with the observational value\nobtained from \\citet{Liu_etal_2006} statistics for protons. As the\nCME kept expanding during its propagation in the FOVs, the\npolytropic index less than $\\frac{5}{3}$ means that there must be\nsome mechanisms to inject heat from somewhere into the CME plasma.\nAlthough the CME plasma continuously got thermal energy, the proton\ntemperature may be still much lower than that in the ambient solar\nwind, as revealed by many in-situ observations of MCs\n\\citep[e.g.,][]{Burlaga_etal_1981}.\n\n\\begin{figure*}[tbh]\n \\centering\n\\hskip -20pt \\includegraphics[width=0.45\\hsize]{f4a.eps}\n\\hskip 10pt \\includegraphics[width=0.41\\hsize]{f4b.eps}\\\\\n\\hskip 0pt \\includegraphics[width=0.415\\hsize]{f4c.eps} \\hskip\n10pt\n\\includegraphics[width=0.495\\hsize]{f4d.eps}\n \\caption{{\\it Left-upper panel:} The profile of $a_e$ (black), the modeled result (dashed green),\nand the relative error (see text for details). {\\it Right-upper panel:} The variation of the polytropic index, $\\Gamma$,\nof the CME plasma. {\\it Left-lower panel:} The variations of the average Lorentz force, $f_{em}$, and the average\nthermal pressure force, $f_{th}$. Their signs have been marked on the upper right corner. {\\it Right-lower panel:}\nThe sum and ratio of the two forces.}\\label{fg_cme_state}\n\\end{figure*}\n\nWe believe that the hot plasmas in the lower solar atmosphere is\nprobably a major heat source of CMEs in the interplanetary space. As shown\nin Figure \\ref{fg_coor}, a CME is believed to be a looped structure\nwith two ends rooted on the solar surface in a global scale.\nBidirectional electron streams are one of the evidence of it\n\\citep[e.g.][]{Farrugia_etal_1993b, Larson_etal_1997}. Thus it is\npossible that heat is conducted from the bottom to CMEs. The ambient\nsolar wind with higher temperature might be an additional source\nbecause the temperature difference between the two mediums is\nsignificant. However, the cross-field diffusion of particles are\nmuch more difficult than the motion parallel to magnetic field\nlines, especially in a nearly force-free flux rope; the coefficient\nratio $\\kappa_\\perp\/\\kappa_\\parallel$ of perpendicular to parallel\ndiffusion roughly locates in the range of $0.005 - 0.05$\n\\citep[e.g.,][]{Jokipii_etal_1995, Giacalone_Jokipii_1999,\nZank_etal_2004, Bieber_etal_2004}. Thus the contribution of the\nambient high-temperature solar wind should be very limited.\n\nIt is well known that the magnetic energy decreases as CMEs\npropagate away from the Sun. According to our model, the total\nmagnetic energy is given by\n\\begin{eqnarray}\nE_m=\\frac{1}{2\\mu_0}\\int B^2rdrd\\phi dz=\\pi\\mu_0^{-1}(k_9l^{-1}+k_{10}R^{-2}l)\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n&k_9=\\int_0^1(\\frac{\\partial f_z}{\\partial x})^2xdx \\\\\n&k_{10}=\\int_0^1[\\frac{\\partial}{x\\partial x}(xf_\\phi)]^2xdx \\label{eq_menergy}\n\\end{eqnarray}\nare both positive integral constants. The magnetic energy generally\ndissipates at the rate of $\\sim l^{-1}$, which is a significant\ndissipation as CMEs move outward. However, such magnetic energy\ndissipation does not necessarily mean to be a major source of the\nheat. According to MHD theory, magnetic energy partially goes into\nkinetic energy and partially converts to thermal energy. The former\nis due to the work done by Lorentz force ($\\vec j\\times\\vec\nB\\cdot\\vec u$), and the latter is through the Joule heating\n($\\frac{j^2}{\\sigma}$) process, where $j$ is the current density and\n$\\sigma$ is the electrical conductivity. Since $\\sigma$ usually has\na large value in interplanetary medium, without anomalous\nresistivity, the magnetic energy does not have an efficient way to\nbe converted to thermal energy. However, there are possibly many\nnon-ideal processes, such as turbulence, but not accounted by MHD\ntheory. Thus we do not know whether the dissipated magnetic energy\nis a major source of heating or not.\n\n\\subsubsection{Internal Forces}\nThe averaged Lorentz force, $f_{em}$, and thermal pressure force,\n$f_{th}$, have been presented in the left-lower panel of Figure\n\\ref{fg_cme_state}. Their absolute values are very close to each\nother, and both of them decreased continuously throughout the\ninterplanetary space. The signs of the two forces are opposite.\n$f_{em}$ is negative indicating a centripetal force, whereas\n$f_{th}$ is positive, centrifugal. This result suggests that the\nthermal pressure force contributed to the CME expansion, but the\nLorentz force prevented the CME from expanding.\n\nThe difference between the two forces can be seen more clear from\nthe right-lower panel of Figure \\ref{fg_cme_state}. The black line\nexhibits the net force, $f_{em}+f_{th}$, inside the CME. It directed\noutward and reached the maximum at about 10:30 UT. The profile is\nconsistent with the expansion acceleration presented in the\nleft-upper panel (the black line). Thus the net force just shows us\nthe internal cause of the CME expansion. The red line is the ratio\nof their absolute values. Its value changed in a very small range\nfrom about 1.0 to 0.98. It suggests that such a small difference\nbetween the two forces is able to drive the CME expanding with the\nacceleration at the order of 1 $m\/s^2$. Moreover, the ratio decrease\nmeans that the Lorentz force decreased slightly faster than the thermal\npressure force. One may notice that, since $\\Gamma$ was larger than\nthe first critical point $\\frac{4}{3}$ at $L\\gsim5R_S$, according to the analysis in\nSec.\\ref{sec_asymptotic}, the Lorentz force should drop slower than\nthermal pressure force. Actually it may not be an inconsistency, because\nthe inference derived in Sec.\\ref{sec_asymptotic} is established on\nthe force-free assumption, the CME we studied may not be\nperfectly force-free, and therefore the first critical point of $\\Gamma$ probably shifts a little bit.\n\nUsually, CMEs are a flux rope with two ends rooted on the Sun. The\naxial curvature of the flux rope may cause the magnetic strength at\nthe Sun-side of the flux rope larger than that at the opposite side,\nwhich leads the Lorentz force having an additional component to\ndrive the flux rope moving outward away from the Sun\n\\citep[e.g.,][]{Garren_Chen_1994, Lin_etal_1998, Lin_etal_2002,\nKliem_Torok_2006, Fan_Gibson_2007}. Thus, as the flux rope we\napplied here is assumed to be a straight cylinder, the Lorentz force\n$f_{em}$ we derived does not include the component caused by the\naxial curvature of the flux rope. This component is important in\nstudying the propagation properties of a CME. However, our model is\nto study the CME internal state (specifically the thermodynamic process\nand expansion behavior), and its propagation behavior is obtained\ndirectly from coronagraph observations, thus the neglect of this\ncomponent should be acceptable although it does bring on some error,\nwhich has been briefly mentioned in the\nsecond paragraph of Sec.\\ref{sec_discussion}.\n\n\n\\section{Summary} \\label{sec_summary}\nIn this paper, we developed an analytical flux rope model for the\npurpose of probing the internal state of CMEs and understanding its\nexpansion behavior. The model suggests that, if the flux rope\nis force free, there are two critical\nvalues for the polytropic index $\\Gamma$. One is $\\frac{4}{3}$, above\/below\nwhich the absolute value of the Lorentz force decreases slower\/faster than\nthat of the thermal pressure force as the flux-rope CME propagates away\nfrom the Sun. The other is $\\frac{2}{3}$, above which the flux-rope CME\nwill essentially approach a steady expansion and propagation state.\n\nBy applying this model to the 2007 October 8 CME event, we find that\n(1) the polytropic index of the CME plasma increased from initially 1.24\nto more than 1.35 quickly, and then slowly decreased to about 1.336; it\nsuggests that there be continuously heat injected\/converted into\nthe CME plasma and the value of $\\Gamma$ tends to be the first critical value $\\frac{4}{3}$;\n(2) the Lorentz force directed inward while the thermal pressure force outward, both\nof them decreased rapidly as the CME moved out, and the small difference between\nthem is consistent with the expansion acceleration of the CME; the direction of\nthe two forces reveal that the thermal pressure force is the internal driver of the CME expansion,\nwhereas the Lorentz force prevented the CME from expanding.\n\n\\section{Discussion}\\label{sec_discussion}\nIn our model, the interaction between CMEs and the solar wind has\nbeen implicitly included to certain extent, though we do not\nexplicitly address these effects. The consequences of the\ninteraction, in terms of the effects on the CME dynamic evolution\ncan be roughly classified into the following three types: (1) the\nsolar wind dragging effect, which is is due to the momentum exchange\nbetween the CME plasma and the ambient solar wind and mainly affects\nthe CME's propagation speed or the bulk motion speed, (2) the solar\nwind constraint effect on expansion, which is caused by the presence\nof the external magnetic and thermal pressures and mainly prevents a\nfree expansion of the CME (i.e., in all directions), and (3) the\nsolar wind stretching effect on expansion, which is caused by the\ndivergent radial expansion of solar wind flow, and causes flattening\nor ``pancaking\" of CMEs. The first two effects are indirectly\nincluded in the model through the measurements of $L$ and $R$.\nDifferent dragging and\/or constraint force(s) may result in\ndifferent variation of $L$ and\/or $R$ with time (or heliocentric\ndistance). Particularly, we do not need to explicitly put the solar\nwind dragging term in the model, because we are addressing the\ninternal state of CMEs, not the bulk acceleration. The stretching\neffect, which is of a kinematic effect, is not included in our\nmodel. As discussed in the sixth paragraph of\nSec.\\ref{sec_measurements}, this is largely due to the limitation of\nthe measurements. The possible errors caused by such effect are\nexplicitly addressed in the next paragraph.\n\nThe main uncertainty of this model, we believe, comes from the\nassumption of an axisymmetric cylinder, in which the curvature of\nthe axis of the flux rope and the distortion of the circular\ncross-section are not taken into account. As to the first one, the\nneglect of the axial curvature generally results in the Lorentz\nforce underestimated. As to the second one, as discussed earlier,\nthe distortion of the CME cross-section is due to the kinematic\nstretching effect of a spherically divergent solar wind flow\n\\citep[e.g.,][]{Crooker_Intriligator_1996, Russell_Mulligan_2002,\nRiley_etal_2003, Riley_Crooker_2004, Liu_etal_2006}. In the case of\nthe particular CME studied in this paper, the aspect ratio is about\n1.4 when the CME leading edge is at $\\sim70 R_S$ (or the flux rope\naxis is at $\\sim56 R_S$). The overall shape of the CME looks like an\nellipse. To estimate the errors caused by the circular assumption,\nwe approximate the ellipse to be a circle of the same area. With\nthis treatment, we estimate that $R$ is overestimated by 19\\%, and\n$L$ is underestimated by 11\\%. Therefore, the expansion speed is\noverestimated by 19\\%, and the propagation speed is underestimated\nby 11\\%. Further, we find that the density is underestimated by\n21\\% (ref to Eq.\\ref{eq_rho_3}), $f_{th}$ is underestimated by 39\\%\n(ref to Eq.\\ref{eq_pth} and assume $\\Gamma=\\frac{4}{3}$), $f_{em}$\nis underestimated by 25--58\\% (ref to Eq.\\ref{eq_pem3}), and the\nerror of the polytropic index is probably neglected (ref. to\nEq.\\ref{eq_fth}). These errors are evaluated for the CME at $\\sim70\nR_S$. At a smaller distance, we expect that the errors be smaller,\nsince the distortion is less severe.\n\nThe self-similar assumption made in our model may be another error\nsource, in which we assume that the distributions of the quantities\nalong $\\hat{\\vec r}$ in the flux rope remain unchanged during the\nCME propagates away from the Sun. Self-similar evolution is a frequently\nused assumption in modeling \\citep[e.g.,][]{Low_1982, Kumar_Rust_1996,\nGibson_Low_1998, Krall_StCyr_2006}. The recent research by\n\\citet{Demoulin_Dasso_2009} suggested that, when $l$, the length of flux\nrope, is proportional to $p_t^{-1\/4}$, the total pressure in the ambient\nsolar wind, a force-free flux rope will evolve self-similarly. The\ntotal pressure of solar wind consists of\nthermal pressure $p_{th}=nkT$ and magnetic pressure\n$p_{m}=\\frac{B^2}{2\\mu}$. Near the Sun, we can assume that the magnetic\npressure is dominant, thus it is approximated that $p_t\\approx p_m\\propto L^{-4}$,\ni.e., $p_t^{-1\/4}\\propto L$. Since the length of a flux rope is\nusually proportional to the distance $L$, we have\n$l\\propto p_t^{-1\/4}$. It means that self-similar assumption should\nbe a good approximation when the CME is nearly force-free and not too\nfar away from the Sun. Other previous studies also showed that the\nself-similar evolution of CMEs is probably true within tens solar radii\n\\citep[e.g.,][]{Chen_etal_1997, Krall_etal_2001, Maricic_etal_2004}.\nOn the other hand, however, the self-similar assumption must be broken\ngradually. An obvious evidence is from the solar wind stretching effect\nas have been addressed before. Another evidence is that a CME may relax\nfrom a complex structure to a nearly force-free flux rope structure, for\nexample the simulation by \\citet{Lynch_etal_2004}.\n\n\\begin{comment}\nIn addition, we have treated CMEs as a one-fluid plasma, i.e., we do not distinguish between\nelectrons and protons. The model results mainly address the behavior\nof protons, since protons are much heavier than electrons, and thus form\nthe most mass of CMEs. Further, protons obey the\nMaxwellian distribution, while electrons seem not\n\\citep[e.g.,][]{Scudder_1992, Liu_etal_2005}. The non-Maxwellian\ndistribution means that electrons may not be treated as an ideal gas,\nand therefore the equations used in the\nmodel may not be appropriate as long as electrons are concerned.\n\\end{comment}\n\n\nFor the CME plasma, neglecting the viscous stress tensor\nin Eq.\\ref{eq_mom_1} might be appropriate. The viscous stress tensor\nof protons can be approximately given by\n\\begin{eqnarray}\nS_{ij}\\approx3\\eta_0\\left(\\frac{\\delta_{ij}}{3}-\\frac{B_iB_j}{B^2}\\right)\\left(\\frac{\\vec{B}\\cdot\\vec{B}\\cdot\\nabla\\vec{v}}{B^2}-\\frac{\\nabla\\cdot\\vec{v}}{3}\\right)\n\\end{eqnarray}\nand $\\eta_0$ is the coefficient of viscosity that could be estimated\nby $\\eta_0\\approx10^{-17}T_p^{\\frac{5}{2}}$ kg$\\cdot$m$^{-1}\\cdot$s$^{-1}$\n\\citep{Braginskii_1965, Hollweg_1985}.\nHere $\\delta_{ij}$ is the unit tensor, $\\vec{v}$ is flow velocity, and\n$T_p$ is the proton temperature.\nSince the proton temperature in CMEs is low, $\\eta_0$ and therefore\nthe viscous stress tensor is very small. Thus, we guess that the\nviscosity in the momentum equation might be ignored.\n\nBoth forces ignored in Eq.\\ref{eq_mom_1}, the gravity $\\vec{F}_g$ and the equivalent fictitious force $\\vec{F}_a$\ndue to the use of a non-inertial reference frame, are in the radial direction in the solar frame.\nTheir effects can be evaluated by comparing them with the acceleration of\nthe expansion of the fluids in flux rope CMEs. The solar\ngravity acceleration is about 270 m\/s$^2$ at the surface, and decreases\nat the rate of $r^{-2}$, which makes it as low as $\\sim$2.7 m\/s$^2$\nat 10 $R_S$. And $\\vec{F}_a$ should be also very small for most CMEs beyond\n10 $R_S$. Thus both forces would significantly distort the model results\nonly on CMEs with slow expansion acceleration in the lower corona, but not on\nthose with large expansion acceleration or in the outer corona.\nThis may be the reason why a large error of $a_e$ appears during 12:00 -- 18:00 UT\nin modeling this CME (left-upper panel of Fig.\\ref{fg_cme_state}).\n\nThe flux-rope model presented in this paper might be the first of its kind\nto provide a way to infer the inter state of CMEs directly based on coronagraph\nobservations. It is different from other CME dynamic models, such as those\nby \\citet{Chen_1989} and \\citet{Gibson_Low_1998}, which were designed to study\nthe interaction of CMEs with the ambient solar wind and other dynamic processes\nby adjusting the initial conditions of CMEs and the global parameters of the\nambient solar wind. Besides, \\citet{Kumar_Rust_1996} proposed a current-core\nflux rope model with self-similar evolution (ref. to KR model thereafter).\nAlthough a self-similar flux rope\nis also employed in their model, our model is largely different from theirs.\nFirst, the flux rope in KR model is assumed force-free and the Lundquist\nsolution \\citep{Lundquist_1950} is applied to describe the internal\nmagnetic structure, but our model does not specify the magnetic field\ndistribution and it may be non-force-free. Secondly, the self-similar assumption\nin KR model limits the radius of the flux rope to be proportional to the\ndistance, whereas our self-similar condition is held only in the cross-section\nof the flux rope; the $R$ and $L$ in our model are two independent measurements\n(see Fig.\\ref{fg_cme_motion}). Thirdly, KR model did not consider the solar wind\neffects on the flux rope, while two of three solar wind effects are implicitly\nincluded in our model. Thus one can treat our model a more generic one.\nUndoubtedly, KR model is an excellent model for force-free\nflux ropes, and got many interesting results. For example, it is suggested that the\npolytropic index is $\\frac{4}{3}$ for a CME far from the Sun. It is an inference\nfrom their self-similar assumption, and it seems to be true for the 2007 October\nCME we studied here. In our model, the $\\Gamma$ value of $\\frac{4}{3}$ implies a\nspecial case (Sec.\\ref{sec_asymptotic}) in which the two internal forces $f_{em}$\nand $f_{th}$ vary at the same rate. Further work will be performed to test whether\nit holds for all CME events.\n\n\n\n\\paragraph{Acknowledgments.}\nWe acknowledge the use of the data from\nSTEREO\/SECCHI. We are grateful to James Chen and Yong C.-M. Liu for\ndiscussions. We also thank the referees for valuable comments.\nY. Wang and J. Zhang are supported by grants from\nNASA NNG05GG19G, NNG07AO72G, and NSF ATM-0748003. Y. Wang and C. Shen\nalso acknowledge the support of China grants from NSF 40525014, 973\nkey project 2006CB806304, and Ministry of Education 200530.\n\n\\section*{Appendix}\nIn cylindrical coordinate system, the magnetic field of a force-free\nflux rope has the \\citet{Lundquist_1950} solution\n\\begin{eqnarray}\nB_r&=&0 \\nonumber\\\\\nB_\\phi&=&HB_0J_1(2.41x) \\\\\nB_z&=&B_0J_0(2.41x) \\nonumber\n\\end{eqnarray}\nwhere $x=\\frac{r}{R}$ is the normalized radial distance as defined in Sec.\\ref{sec_model},\n$J_0$ and $J_1$ are the zero and first order Bessel functions, $H=\\pm1$\nindicates the sign of the handedness and $B_0$ is the magnetic field\nmagnitude at the axis of the flux rope. According to the properties\nof Bessel function, we have the magnetic vector potential\n\\begin{eqnarray}\nA_\\phi&=&\\frac{RB_0}{2.41}J_1(2.41x) \\label{eq_ffaphi}\\\\\nA_z&=&\\frac{HRB_0}{2.41}J_0(2.41x) \\label{eq_ffaz}\n\\end{eqnarray}\nThe conservation of $\\Phi_z$\n\\begin{eqnarray}\n\\Phi_z=2\\pi\\int_0^R\\frac{\\partial}{\\partial r}(rA_\\phi)dr=\\frac{2\\pi R^2B_0}{2.41}J_1(2.41)=\\mathrm{constant}\n\\end{eqnarray}\nrequires that\n\\begin{eqnarray}\nB_0=2.41a_1R^{-2} \\label{eq_ffb0}\n\\end{eqnarray}\nwhere $a_1$ is a constant. The magnetic vector potential can be\nrewritten as\n\\begin{eqnarray}\nA_\\phi&=&\\frac{a_1}{R}J_1(2.41x) \\label{eq_ffA_phi}\\\\\nA_z&=&\\frac{a_1H}{R}J_0(2.41x) \\label{eq_ffA_z}\n\\end{eqnarray}\nMeanwhile, the magnetic helicity is\n\\begin{eqnarray}\nH_m=\\int\\vec{B}\\cdot\\vec{A}drrd\\phi dz=4.82\\pi a_1^2a_2HR^{-1}l\n\\end{eqnarray}\nwhere $a_2=\\int_0^1x(J_0^2+J_1^2)dx$ is a constant.\nThe conservation of $H_m$ results in\n\\begin{eqnarray}\n&R\\propto l\n\\end{eqnarray}\nCombined it with the assumption Eq.\\ref{eq_ll}, it is inferred that\n\\begin{eqnarray}\n&R\\propto L\n\\end{eqnarray}\nwhich means that the force-free flux rope expands radially.\n\n\\bibliographystyle{agufull08}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nSymbiotic stars are interacting binaries. They consist of an evolved red giant \nor Mira-type variable, and a hot component --- white dwarf, subdwarf, neutron \nstar or main-sequence star. The hot component accretes material from the stellar \nwind of the donor (Sokoloski 2003). The wind is ionized by the hot component, \ncausing the rise of a nebula. Orbital periods of the symbiotic stars range from \nyears to decades.\n\nCH~Cyg is an eclipsing symbiotic star composed of a M6-7~III star and an \naccreting white dwarf, so the system belongs to the S-type symbiotics. The \nbinary separation is $8.7^{+1.1}_{-0.7}$~AU (Miko{\\l}ajewska et al., 2010). The \nmasses of the components are $M_{\\rm rg}=2^{+1}_{-0.5}\\, \\rm M_{\\sun}$ and \n$M_{\\rm wd}=0.70^{+0.22}_{-0.09}\\, \\rm M_{\\sun}$ (Miko{\\l}ajewska et al., 2010). \nBased on {\\it Hipparcos} satellite measurements, the distance to CH~Cyg is \nestimated to $d=244^{+49}_{-35}\\, \\rm pc$ (van Leeuven 2007). CH~Cyg ejects \ncollimated bipolar outflows with velocity of $\\sim$700~\\kms, detectable in the \nradio (Taylor, Seaquist \\& Mattei 1986; Crocker et al., 2001). The system is \ndetectable also in X-rays (Galloway \\& Sokoloski 2004). The orbital period of \nCH~Cyg is $\\sim$15.6~yr (Hinkle, Fekel \\& Joyce 2009). The light curve of CH~Cyg \nis very complex. The detected variability varies from dozens of years caused by \nthe orbital motion and dust obscuration events (Bogdanov \\& Taranova 2001), \nthrough periodicity with periods of several hundred days caused by pulsations of \nthe giant (Mikolajewski, Mikolajewska \\& Khudyakova 1992), to flickering \nactivity with time-scales of a few minutes (Dobrzycka, Kenyon \\& Milone 1996).\n\nFlickering is broad-band stochastic light variations on time-scales of a few \nminutes with amplitude from a few$\\times$0.01~mag to more than one magnitude. \nFlickering activity is detected in only 10 symbiotic stars --- RS~Oph, T~CrB, \nMWC~560, V2116~Oph, CH~Cyg, RT~Cru, $o$~Cet, V407~Cyg, V648~Car and EF~Aql \n(Dobrzycka, Kenyon \\& Milone 1996; Sokoloski, Bildsten \\& Ho 2001; Gromadzki et \nal. 2006; Angeloni et al. 2012; \nZamanov et al., 2017).\n\nThe flickering of CH~Cyg was first detected by Wallerstein (1968) and Cester \n(1968) and it was studied in detail later (Skopal 1988; Mikolajewski et al. \n1990; Mikolajewski et al. 1992; Panov \\& Ivanova 1992; Kuczawska, Mikolajewski \n\\& Kirejczyk 1992; Hric et al. 1993; Dobrzycka, Kenyon \\& Milone 1996; Sokoloski \n\\& Kenyon 2003). CH~Cyg is an eclipsing system (Mikolajewski, Mikolajewska \\& \nTomov 1987), and the flickering activity disappears during the eclipses \n(Sokoloski \\& Kenyon 2003). In 2010, the flickering from CH~Cyg became \nnon-detectable (Sokoloski et al., 2010) until it renewed its activity in 2014 \n(Stoyanov et al., 2014). CH~Cyg probably enters a new active stage in 2017 \n(Iijima 2017).\n\nHere we present photometric observations of the flickering of CH Cyg and \ncalculations of the flickering source parameters.\n\n\\section{Observations}\n\nThe observations are performed with the following three telescopes equipped with \nCCD cameras:\n\\begin{itemize}\n \\item the 60 cm Cassegrain telescope of Rozhen NAO\n \n \\item the 50\/70 cm Schmidt telescope of Rozhen NAO \n \n \\item the automated 41 cm telescope of the University of Ja\\'en, Spain \n(Mart{\\'{\\i}}, Luque-Escamilla, \\& Garc\\'{\\i}a-Hern\\'andez 2017)\n\\end{itemize}\n\nIn Fig.~\\ref{fig.obs} are plotted the light curves from a few nights. The \nobservations consisted of repeated exposures in U, B and V bands, or in B and V \nbands. On 20110609 the total duration of the run is 86 minutes; 20141001 --- 90 \nminutes; 20170724 --- 66 minutes; 20170809 --- 147 minutes; 20170811 --- 284 \nminutes.\n\nThe data reduction was done using IRAF (Tody 1993) following standard procedures \nfor aperture photometry. A few comparison stars from the list of Henden \\& \nMunari (2006) have been used, bearing in mind that SAO~31628 is an eclipsing \nbinary (Sokoloski \\& Stone 2000).\n\nThe journal of observations is given in Table~\\ref{table1}. In the table are \ngiven the telescope, band, number of exposures, exposure time, average \nmagnitude, minimum and maximum magnitude during the run, and typical \nobservational error.\n\n\\section{Flickering source parameters}\n\nIn our observations obtained during the period 2010--2013 the flickering of \nCH~Cyg was not detectable (see Table~\\ref{flick} and Fig.~\\ref{fig.obs}). It \nre-appeared in August 2014. After that CH~Cyg exhibited variability on a time \nscale of 1--30 minutes with amplitude $0.2-0.3$~mag in $V$. The amplitude \nincreases in B and U bands.\n\nBruch (1992) proposed that the light curve of CVs can be separated into two \nparts --- constant light, and variable (flickering) source. We assume that all \nthe variability in each night is due to flickering. In these suppositions the \nflickering light source is considered 100\\% modulated. Following these \nassumptions, we calculate the flux of the flickering light source as $F_{\\rm \nfl}=F_{\\rm av}-F_{\\rm min}$, where $F_{\\rm av}$ is the average flux during the \nrun and $F_{\\rm min}$ is the minimum flux during the run (corrected for the \ntypical error of the observations). $F_{\\rm fl}$ has been calculated for each \nband, using the values given in Table~1 and the Bessel (1979) calibration for \nthe fluxes of a zero magnitude star.\n\nA modification of this method is given in Nelson et al. (2011), which proposed \nto use the $F_{\\rm fl}=F_{\\rm max}-F_{\\rm min}$, where $F_{\\rm max}$ is the \nmaximum flux during the run. Adopting these, we find that the flickering light \nsource contributes about 4\\% in $V$, 6\\% in $B$, and 8\\% in $U$ (October 2014).\n\nThe calculated colours of the flickering light source are given in \nTable~\\ref{table2}, where $T_{(B-V)_1}$ is calculated using $F_{av}$ and \n$T_{(B-V)_2}$ is calculated using $F_{max}$.\n\nAssuming that the flickering source radiates as a black body, for the colours of \nthe black body we use the calibration given in Strai\\v zis (1977). The use of \nother formulae (e.g. Ballesteros 2012) could introduce a difference of about \n$\\pm 500$~K.\n\nAn independent estimate of the black body temperatures from the B-V colour using \nan analytic approximation (Ballesteros 2012) provides similar results with a \ndifference not exceeding 1000 K.\n\nThe radius is calculated for B-V colour and B band flux, assuming effective \nwavelength of B band $\\lambda = 4400$~\\AA, the temperature calculated from the \nB-V colour, assuming black body, spherical form of the flickering source, and a \ndistance $d=244 pc$.\n\nIn a comparison of the temperatures, calculated using the method of Bruch (1992) \nand Nelson et al. (2011), we find that the temperatures are in agreement.\n\nAs expected, the method of Nelson et al. (2011) gives higher values for the size \nof the flickering source, because it uses greater flux.\n\nIn Table~\\ref{table2} are summarized the calculated flickering source \nparameters: \n (1) date of observations; \n (2) U-B colour of the flickering source calculated following Bruch (1992); \n (3) temperature corresponding to the U-B colour;\n (4) B-V colour of the flickering source calculated following Bruch (1992); \n (5) temperature corresponding to B-V colour;\n (6) radius of the flickering source calculated following Bruch (1992);\n (7) U-B colour of the flickering source calculated following Nelson et al. \n(2011); \n (8) temperature corresponding to the U-B colour;\n (9) B-V colour of the flickering source calculated following Nelson et al. \n(2011); \n (10) temperature corresponding to B-V colour;\n (11) radius of the flickering source calculated following Nelson et al. \n(2011).\n\n\\begin{figure*} \n \\vspace{16.0cm} \n \\special{psfile=20110609.eps hoffset=-10 voffset=200 hscale=38 \nvscale=38 angle=0} \n \\special{psfile=20141001.eps hoffset=90 voffset=200 hscale=38 \nvscale=38 angle=0} \n \\special{psfile=20170724.J.eps hoffset=210 voffset=200 hscale=38 \nvscale=38 angle=0} \n \\special{psfile=20170809.J.eps hoffset=-20 voffset=-40 hscale=38 \nvscale=38 angle=0} \n \\special{psfile=20170811.eps hoffset=140 voffset=-40 hscale=38 \nvscale=38 angle=0} \n \\caption[]{Flickering of CH~Cyg in U, B and V bands.}\n \\label{fig.obs} \n\\end{figure*}\t \n\n\\begin{table}\n \\begin{center}\n \\caption{Journal of observations.}\n\\begin{tabular}{l cccc ccccc cc}\ndate-obs & telescope & band & exposures & average & min & max & merr & \\\\\n\\\\\n20110609 & 60cm Roz & B & 191 $\\times$ 10s & 10.317 & 10.301 & 10.336 & \n0.005 & \\\\\n20110609 & 60cm Roz & V & 190 $\\times$ 5s & 8.796 & 8.784 & 8.812 & \n0.002 & \\\\\n\\\\\n20141001 & 60cm Roz & U & 69 $\\times$ 15s & 7.578 & 7.344 & 7.714 & \n0.009 & \\\\\n20141001 & 60cm Roz & B & 71 $\\times$ 10s & 7.916 & 7.735 & 8.017 & \n0.003 & \\\\\n20141001 & 60cm Roz & V & 71 $\\times$ 4s & 7.197 & 7.074 & 7.294 & \n0.004 & \\\\\n\\\\\n20170724 & 41cm Jaen & B & 200 $\\times$ 3s & 8.264 & 8.040 & 8.497 & \n0.009 & \\\\\n20170724 & 41cm Jaen & V & 196 $\\times$ 2s & 7.533 & 7.399 & 7.686 & \n0.005 & \\\\\n\\\\\n20170809 & 41cm Jaen & B & 417 $\\times$ 6s & 8.410 & 8.248 & 8.600 & \n0.003 & \\\\ \n20170809 & 41cm Jaen & V & 389 $\\times$ 2s & 7.666 & 7.547 & 7.827 & \n0.004 & \\\\ \n\\\\\n20170811 &50\/70cm Roz& U & 318 $\\times$ 20s & 7.748 & 7.476 & 8.041 & \n0.007 & \\\\\n20170811 &50\/70cm Roz& B & 318 $\\times$ 4s & 8.298 & 8.068 & 8.545 & \n0.005 & \\\\ \n20170811 &50\/70cm Roz& V & 318 $\\times$ 2s & 7.649 & 7.453 & 7.826 & \n0.005 & \\\\\n\\\\\n\\end{tabular}\n\\label{table1}\n\\end{center}\n\\end{table}\n\n\\begin{table}\n\\caption{Flickering source parameters.}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{c | cc | ccc | cc | ccccc }\ndate-obs & $U-B$ & $T_{(U-B)_1}$ & $B-V$ & $T_{(B-V)_1}$ & $R\/R_\\odot$ & $U-B$ &\n$T_{(U-B)_2}$ & $B-V$ & $T_{(B-V)_2}$ & $R\/R_\\odot$ & \\\\\n& & & & & & & & & & & \\\\\n1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & \\\\\n& & & & & & & & & & & \\\\\n20141001 & -0.6439 & 9292 & 0.6782 & 5554 & 1.94 & -0.6441 & 9284 & 0.4212 & \n6992 &\n2.07 & \\\\\n20170724 & --- & --- & 0.3126 & 7895 & 1.20 & --- & --- & 0.2161 & 9049 & 1.42 & \n\\\\\n20170809 & --- & --- & 0.5430 & 6231 & 1.62 & --- & --- & 0.5322 & 6299 & 2.24 & \n\\\\\n20170811 & -0.7134 & 9834 & 0.3211 & 7824 & 1.23 & -0.7353 & 10723 & 0.3966 & \n7195 &\n2.10 & \\\\\n\\end{tabular}\n}\n\\label{table2}\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\caption{List of observations of CH~Cyg. The last column indicates if the \nflickering\nis present or not.}\n\\begin{tabular}{c | c c | c | c | c | c | c | c c c c}\ndate-obs & bands & flickering & date-obs & bands & flickering \\\\\n& & & & & \\\\\n20100430 & BV & no & 20110918 & B & no \\\\ \n20100501 & UB & no & 20111006 & B & no \\\\\n20100502 & B & no & 20111118 & B & no \\\\\n20100506 & B & no & 20120618 & B & no \\\\\n20100507 & UB & no & 20120820 & B & no \\\\\n20100509 & BV & no & 20130514 & BV & no \\\\\n20100816 & UB & no & 20130703 & B & no \\\\\n20100817 & BV & no & 20130803 & B & no \\\\\n20100818 & UB & no & 20140814 & B & yes \\\\\n20100909 & BV & no & 20141001 & UB & yes \\\\\n20101029 & UB & no & 20170724 & BV & yes \\\\\n20101030 & BV & no & 20170809 & BV & yes \\\\\n20110529 & B & no & 20170811 & UBV & yes \\\\\n20110609 & BV & no & & &\\\\\n\\end{tabular}\n\\label{flick}\n\\end{center}\n\\end{table}\n\n\\section{Discussion}\n\n\\subsection*{Parameters of the flickering source}\n\nThe temperature of the flickering source in similar to the temperature of the \nbright spot in cataclysmic variable stars (e. g. Marsh 1988;\nWood et al., 1989; Zhang \\& Robinson 1987) and it is significantly lower than \nthe temperature of the boundary layer. This is a hint that the flickering \nprobably originates from the bright spot, but its nature is still unknown. For \ncomparison, the temperatures and radii of the flickering source in other \nsymbiotic stars are comparable with those that we estimate for CH~Cyg. For \nRS~Oph, Zamanov et al. (2010) give $T_{fl}=9500 \\pm 500$~K and $R_{fl} = 3.5 \\pm \n0.5 R_\\odot$. For MWC~560, Zamanov et al. (2011) give $T_{fl}=13550 \\pm 500$~K \nand $R_{fl} = 1.68 \\pm 0.16 R_\\odot$.\n\n\\subsection*{Why was the flickering missing for 4 years?}\n\nFlickering is expected to arise in the vicinity of the accretion disc around the \nwhite dwarf companion in CH~Cyg. \nIts absence during a nearly four year time interval is most naturally \ninterpreted as a major disruption of the inner disc structure. \nThis is probably due to reduced supply of mass flow from the M-type giant across \nthe L$_1$ Lagrangian point of the system. \nThis situation, lasting in the period 2010--2013 according to Table~\\ref{flick}, \nrenders now difficult other alternative interpretations \nfor the lack of flickering based on an eclipse configuration (Stoyanov et al. \n2014). The required eclipse duration, at \nleast four years, appears to be too long unless a highly eccentric system is \ninvoked. Interestingly, in the CH~Cyg case there \nis observational evidence based on infrared observations for episodic creation \nand dissipation of a dust envelope around it \n(Taranova \\& Shenavrin 2004). Such a behavior has been observed in the past. \nThe creation and dissipation of a dust envelope is connected with a change of the mass-loss of the giant, which \nunderwent a significant reduction during the \n2010--2013 time interval. The time scale reported by \nTaranova \\& Shenavrin (2004) for previous dust envelope creation and dissipation \n events is $\\sim$ 10 yr. The cessation of the flickering is consistent with \nthe giant undergoing a reduced mass-loss episode with a shorter, few year \nduration. Unfortunately, no contemporaneous infrared \nobservations are available to us to confirm this hypothesis.\n\n\n\n\n\n\n\n\n\\section*{Conclusions}\n\nWe performed quasi-simultaneous multicolour observations of the flickering of \nthe symbiotic star CH Cyg in Johnson U, B and V bands. We calculated the \nflickering source parameters --- the temperature is in the range $5000 < T < \n11~000$~K and the radius is in the range 1.42 $ 10$~Gyr) populations in these\nfields must be predominantly metal-poor ([Fe\/H]$\\leq -1$), and that\nthe metal-rich populations ([Fe\/H]$> -1$) must be of intermediate age\n($\\sim 6$--8~Gyr). An old metal-rich population would have a MSTO\nmuch redder and fainter than observed, while an intermediate-age\nmetal-poor population would have a MSTO much bluer and brighter than\nobserved. That said, there is a minority population of young stars\nspanning a wide range in metallicity, with the brightest and bluest\nstars in the plume matched by the 3~Gyr isochrone at [Fe\/H]~=~$-2$.\n\n\\begin{figure*}[t]\n\\epsscale{1.1}\n\\plotone{f4.eps}\n\\epsscale{1.0}\n\\caption{{\\it Top panels:} The CMDs of the spheroid ($a$), stream\n($b$) and disk ($c$), compared to the ridge lines of the Galactic globular\nclusters in Table~\\ref{tabclus} ({\\it colored curves}). \nThe Andromeda data are shown as Hess diagrams with the same binning\nused in Figure~\\ref{cmdsvels}, but over a narrower range of \ncolor and luminosity. The ridge\nlines shift redward with increasing cluster metallicity. {\\it Bottom\npanels:} The CMDs of the spheroid ($d$), stream ($e$) and disk ($f$),\ncompared to isochrones at [Fe\/H]~=~$-2$ ({\\it yellow curves}), $-1$\n({\\it pink curves}), and 0 ({\\it light blue curves}), and ages of 3,\n8, and 13~Gyr (running from left to right for each color). It is\nclear that the old ($> 10$~Gyr) populations in these fields must be\npredominantly metal-poor ([Fe\/H]$\\leq -1$), and that most of the\nmetal-rich populations ([Fe\/H]$> -1$) must be of intermediate age\n($\\sim 6$--8~Gyr).}\n\\label{comparegrid}\n\\end{figure*}\n\nThe implications of the SGB distribution warrant additional\ndiscussion. The isochrones in Figure~\\ref{comparegrid} show that the\nluminosity of the SGB decreases with either increasing age or\nincreasing metallicity. Thus, different age-metallicity relations for\nthe stars in our CMDs would be expected to produce different\nluminosity distributions across the SGB. To evaluate the implications\nof this constraint, we show in Figure~\\ref{sgbwidth} hypothetical\npopulations of stars in the vicinity of the SGB as they would appear\nif observed under the same conditions as in our spheroid field. The\nupper panels present the age-metallicity relations of the isochrones\nemployed to construct each model population, with the stars divided\nequally among the isochrones. The lower panels show the corresponding\nCMDs resulting from these hypothetical populations. Even with a very\nwide range in age, a single metallicity does not reproduce the width\nof the RGB (panels $a$ and $e$); this is because the RGB is far more\nsensitive to metallicity than to age. Moreover, the SGB luminosity\ndistribution is much wider than observed. If one has old metal-rich\nstars and young metal-poor stars (panels $b$ and $f$), the RGB becomes\nmuch wider, but the SGB luminosity distribution is still much wider\nthan observed. If all of the stars are at a single age (panels $c$\nand $g$), the SGB narrows, but it is still wider than the SGB observed\nin our fields. It is only when one has young metal-rich stars and old\nmetal-poor stars (panels $d$ and $h$) that the SGB locus becomes very\ntight and horizontal, as observed for the dominant populations in our\nthree CMDs, while at the same time reproducing a wide RGB. Because\nthe RGB is more sensitive to metallicity than to age, while the MSTO\nis very sensitive to both, one is able to break the age-metallicity\ndegeneracy in studies employing this region of the CMD. Note\nthat relatively young and metal-poor stars (panels $a$, $b$,\n$e$, and $f$) are needed to explain the brightest and bluest stars in\nthe blue plume of our observed CMDs.\n\n\\begin{figure*}[t]\n\\epsscale{1.1}\n\\plotone{f5.eps}\n\\epsscale{1.0}\n\\caption{{\\it Top panels:} Four hypothetical populations of stars. \nIn each population, the stars are equally distributed among 20 isochrones\nwith distinct distributions in age and metallicity.\n{\\it Bottom panels:} Model CMDs for these hypothetical\npopulations, with the observational errors and completeness of our\nspheroid data, shown at a logarithmic stretch.}\n\\label{sgbwidth}\n\\end{figure*}\n\nThe similarities at the HB and RGB between the stream and spheroid\nimply that these populations have very similar metallicity distributions,\nat least at the positions of our fields (Brown et al.\\ 2006). Much\nfarther out in the galaxy (31 kpc from the center), Guhathakurta et\nal.\\ (2006) found that the stream was more metal-rich than the\nsurrounding spheroid, but this finding is not inconsistent with our\nresults. Kalirai et al.\\ (2006a) have shown that the spheroid of\nAndromeda has a metallicity gradient, such that it is significantly\nmore metal poor at 30 kpc than it is close to the galaxy's center.\nOur finding of similar metallicities between the stream and\nspheroid in our interior fields, when combined with the Guhathakurta\net al. (2006) results, reaffirms the existence of this metallicity\ngradient.\n\nAlthough the CMDs for each field have many similarities, closer\ninspection reveals significant distinctions, especially between the\ndisk and the other two fields. We highlight these distinctions in\nFigure~\\ref{compare}, which shows the differences between the stream\nand spheroid and also those between the disk and spheroid. The\nspheroid data used in each comparison are a subset that reaches\napproximately the same depth as the stream and disk data. The\nspheroid CMD was also scaled to the number of stars in each of the\nother two CMDs before subtracting; note that it makes little\ndifference if this normalization is done based on the total number of\nstars in each field or just those well above the detection limits\n(e.g., $m_{F814} < 28$~mag). Relative to the spheroid main sequence,\nthe stream main sequence extends somewhat farther to the blue, even\nthough the RGB and HB distributions are nearly identical. Thus, the\nage distribution in the stream must extend to slightly younger ages\nthan those in the spheroid (as also noted by Brown et al.\\ 2006). In\ncontrast, the distributions of age and metallicity in the disk extend\nto significantly younger ages and higher metallicities than those in\nthe spheroid and stream, and the old metal-poor population is not as\nprominent. The RGB stars in the disk are skewed toward redder colors,\nwhile the HB population is largely restricted to the red clump; both\nof these features indicate a higher metallicity in the disk. In the\ndisk population, the red clump HB is also somewhat extended in\nluminosity, indicating a younger age distribution (an excellent\nexample of the variation in clump luminosity with age can be seen in\nthe Monelli et al.\\ [2003] study of the Carina dwarf spheroidal).\nThere does not appear to be a significant population on the blue HB,\nalthough a trace population might be hidden in the blue plume of stars\nrising above the dominant MSTO; Figure~\\ref{compare}f shows an\noversubtraction of the blue HB from the spheroid ({\\it dark boxes})\nappearing within the cloud of undersubtracted blue plume stars from\nthe disk ({\\it light boxes}). The stronger plume in the disk\npopulation indicates an extension to significantly younger ages. The\nplume in the disk population includes $\\approx$40 stars that are\nbrighter than the region where the blue end of the HB would nominally\nfall, implying that these bright blue stars have masses of\n$\\sim$2--5~$M_\\odot$ and ages of $\\sim$0.2--1 Gyr. Note that\nCuillandre et al.\\ (2001) have also seen evidence for trace\npopulations of young stars in the outer disk of M31. However, the\ndisk does not look quite as young as one might expect if there were a\nsignificant thin disk population -- a point we will return to later.\n\n\\begin{figure*}[t]\n\\epsscale{1.1}\n\\plotone{f6.eps}\n\\epsscale{1.0}\n\\caption{Comparisons of the CMDs for our three fields. The ridge line\nfor NGC~104 ({\\it curve}) is shown for reference. {\\it a)} The CMD of the\nspheroid field, shown at its full depth, at a logarithmic stretch.\n{\\it b)} The CMD of the stream field. {\\it c)} The CMD of the disk\nfield. {\\it d)} The CMD of the spheroid field, shown at a depth that\nmatches that in the stream and disk. {\\it e)} The difference between\nthe stream and spheroid CMDs (with the latter scaled to match the\nnumber of stars in the former). The RGB and HB distributions are very\nsimilar, but the locus of stars at the MS extends slightly brighter and\nbluer than that in the spheroid. {\\it f)} The difference between\nthe disk and spheroid CMDs (with the latter scaled to match the number\nof stars in the former). The RGB of the disk is considerably redder\nthan that of the spheroid, indicating higher metallicities in the\ndisk. The HB of the disk is almost entirely in the red clump, with a\nspread to brighter luminosities, indicating higher metallicities and\nyounger ages in the disk. The blue plume of stars above the MSTO is\nmuch stronger in the disk, indicating younger ages than in the\nspheroid.}\n\\label{compare}\n\\end{figure*}\n\nIn a field population, it is difficult to distinguish between young\nmetal-poor stars and old blue stragglers (see Carney, Latham, \\& Laird\n2005 and references therein). Thus, some of the apparently young\nstars in our CMDs ($\\lesssim 6$~Gyr) might instead be blue stragglers.\nHowever, whether blue stragglers form via merger or mass transfer, in\nan old population they will be limited to $M \\lesssim $2~$M_\\odot$.\nAll three of our fields show blue plume stars as bright as the HB over\na wide range of color, and in the disk these stars continue to\nluminosities significantly brighter than the HB. The high masses\nrequired to explain the brightest stars in the blue plume population\nimply that truly young stars are present, and these stars appear to be\na smooth extension of the fainter blue plume population. This argues\nagainst a significant contribution from blue stragglers in the blue plume.\n\nIf we fit Gaussian distributions to the velocity data in our fields\n(Figure~\\ref{cmdsvels}), we find that the spheroid is a $\\sim$25\\%\ncontamination in our stream field and a $\\sim$33\\% contamination in\nour disk field. Given the wide separation between our fields\n(Figure~\\ref{mapfig}), we cannot necessarily assume that the\npopulation in our spheroid field is representative of the spheroid\ncontamination in our stream and disk fields. However, it is natural\nto ask how the stream and disk CMDs would look if the spheroid\ncontamination were subtracted under the assumption that the population\nin our spheroid field is in fact representative of this contamination.\nTo show this, we used that subset of the spheroid data that is matched\nto the depth of the stream and disk data. We randomly drew a star\nfrom these spheroid data, found the star in the stream data that most\nclosely agreed in its photometry, and then subtracted that star from\nthe stream data. These subtractions were repeated until 25\\% of the\nstream stars were removed. In 99\\% of the subtractions, the star\nsubtracted from the stream data was within 0.02~mag of the spheroid\nstar, and in 99.9\\% of the subtractions, the star subtracted from the\nstream data was within 0.1~mag of the spheroid star; the handful of\nstars that could not be matched at this level fell very far from the\ndominant stellar locus (in the negligible cloud of sparse stars at\nrandom colors and magnitudes), and these were not subtracted. We\nrepeated this process on the disk data, but there subtracted 33\\% of\nthe disk stars; again, 99\\% of the subtractions matched disk to\nspheroid stars within 0.02~mag, while 99.9\\% of the subtractions\nmatched disk to spheroid stars within 0.1~mag. The resulting CMDs are\nshown in Figure~\\ref{subtract}. Because of the many similarities\nbetween the original three CMDs (Figure~\\ref{cmdsvels}), the changes\ndue to the subtraction of the spheroid contamination are subtle. To\nhelp highlight the differences between the three fields, we also show\nluminosity and color cuts across the CMDs ({\\it colored boxes});\npanels $d$ and $e$ show the color distributions on the lower RGB and\nHB, respectively, while panels $f$ and $g$ show the luminosity\ndistributions at the red clump and SGB, respectively. The color and\nluminosity cuts help quantify the similarities and differences between\nthe populations discussed above and shown in Figure~\\ref{compare}.\nCompared to the spheroid population, the stream population exhibits\nsimilar RGB and HB morphologies, but its main sequence extends\nsomewhat brighter and bluer. In contrast, the disk population\nexhibits RGB and HB morphologies that are skewed toward redder colors,\nwith the main sequence showing a strong extension to brighter and\nbluer colors.\n\n\\begin{figure*}[t]\n\\epsscale{1.1}\n\\plotone{f7.eps}\n\\epsscale{1.0}\n\\caption{The spheroid CMD compared to the stream and disk CMDs, where\nwe have attempted to subtract the spheroid contamination \nfrom the stream and disk.\n{\\it a)} The CMD of the spheroid field, shown at a depth that\nmatches that in the stream and disk fields. Cuts across the CMD ({\\it\nblue boxes}) are used to make comparisons with the stream and disk;\nthe histograms in each cut ({\\it panels d-g}) are normalized to the\nnumber of stars in the spheroid. Labels refer to subsequent panels in\nthis figure. {\\it b)} The CMD of the stream field, with a subtraction\nof spheroid stars assumed to contaminate at the 25\\% level, and with\nthe same cuts indicated ({\\it green boxes}). {\\it c)} The CMD of the\ndisk field, with a subtraction of spheroid stars assumed to\ncontaminate at the 33\\% level, and with the same cuts indicated ({\\it\nred boxes}). {\\it d)} Histograms for stars along the RGB color cut\nfor the spheroid ({\\it blue}), stream ({\\it green}) and disk ({\\it\nred}). {\\it e)} Histograms along the HB color cut. {\\it f)}\nHistograms along the HB luminosity cut. {\\it g)} Histograms along the\nSGB luminosity cut. Compared to the spheroid and stream, the disk\npopulation has a redder RGB (indicating higher metallicities), an HB\nthat falls mostly in a red clump that extends to brighter luminosities\n(indicating younger ages and higher metallicities), and a stronger\nblue plume above the MSTO (indicating younger ages).}\n\\label{subtract}\n\\end{figure*}\n\n\\subsection{Maximum Likelihood Fitting of Isochrones}\n\nWe turn now to the quantitative fitting of our CMDs.\nOur characterization of the star formation history in each field\nprimarily uses the StarFish code of Harris \\& Zaritsky (2001). This\ncode takes a grid of isochrones, populates them according to the \ninitial mass function (IMF),\nthen applies the photometric scatter and incompleteness (as a function\nof magnitude and color) determined in the artificial star tests. The\ncode then fits the observed CMD by employing linear combinations of the\nscattered isochrones. The fitting can be done via minimization of\neither a $\\chi^2$ statistic or the Maximum Likelihood statistic of\nDolphin (2002). We found little difference between fits done with\neither statistic, and ultimately used the Maximum Likelihood statistic\nin our analysis.\n\nIn the StarFish fitting, each isochrone at a given age and metallicity\nis varied independently, resulting in a large number of free\nparameters in the fit. This method is similar to most of the star\nformation history methods used in the literature (e.g., Dolphin 2002;\nSkillman et al.\\ 2003). Although the term ``star formation history''\nmight imply a physical connection between the subpopulations, this\nmethod is really a fit to the age and metallicity distributions. In\naddition to StarFish, we wrote our own codes that fit the isochrones to\nthe data according to mathematical and physical restrictions that\ngreatly reduce the number of free parameters; these models will be \nthe subject of a future paper.\n\nWe do not fit the entire range of stars observed in the CMD. Instead,\nwe restrict our fits to the lower RGB (below the level of the HB),\nSGB, and upper main sequence. Specifically, we fit over $-0.9 \\leq\nm_{F606W}-m_{F814W} \\leq -0.1$~mag in color, and $26.5 \\leq m_{F814W}\n\\leq 30.5$~mag in magnitude for the spheroid data and $26.5 \\leq\nm_{F814W} \\leq 30.0$~mag in magnitude for the stream and disk data\n(which are $\\approx$0.5~mag shallower). This region of the CMD\noffers excellent sensitivity to age and metallicity while avoiding\nthose regions of the CMD that have low signal-to-noise ratio or that are poorly\nconstrained by the models (such as the HB, the upper RGB, the RGB\nluminosity function bump, and the faint end of the CMD). The HB is a\nqualitative indicator of age and metallicity, becoming\nredder at younger ages and higher metallicities, and eventually forming a\nred clump with a significant spread in luminosity. However,\ndisentangling the effects of age and metallicity is highly uncertain;\nindeed, the ``second parameter debate'' in the study of HB morphology\nrefers to the dependence of the HB morphology on parameters other than\nmetallicity, such as age and helium abundance. Although Galactic\nforeground stars comprise much less than 1\\% of the stars in our\nfield, they tend to fall near the upper RGB in M31, which is sparsely\npopulated in our data; the upper RGB is thus the one region of our\nCMDs with significant foreground dwarf contamination. In addition,\nthe upper RGB is contaminated by asymptotic giant branch (AGB) stars,\nwhich in turn have a distribution depending on the age and [Fe\/H] of\ntheir progenitor HB stars. The RGB luminosity function bump is a\nqualitative metallicity indicator, and it is most prominent in CMDs of\nmetal-rich populations, where it appears as an overdensity on the RGB\nimmediately below the luminosity of the HB; theoretical models\nreproduce the general trend for the bump luminosity to brighten with\ndecreasing metallicity, but the zeropoint of the relationship is\nuncertain, and the mix of age and metallicity in our populations makes\nit difficult to interpret this feature in the data. The faintest main\nsequence stars in the CMD suffer from large photometric scatter and\nlow completeness.\n\nWe use the Victoria-Regina Isochrones (VandenBerg et al.\\ 2006) in all\nof our fitting. These isochrones do not include core He diffusion,\nwhich would decrease their ages at a given turnoff luminosity by $\\sim\n10$\\% (VandenBerg et al.\\ 2002). Although the ages of isochrones with\ncore He diffusion are likely more accurate, models in which diffusion\nis allowed to act efficiently on other elements in the surface layers\nshow significant discrepancies when compared to observed CMDs,\nindicating that there must be some other mechanism at work, such as\nturbulence in the surface layers (see Brown et al.\\ 2005 and\nreferences therein). Helium diffusion can still occur in the core,\nand thus the ages discussed herein should be reduced by $\\sim$10\\% to\nobtain absolute ages.\n\nThe Victoria-Regina Isochrones are distributed with a ground-based\nmagnitude system. Sirianni et al.\\ (2005) provide an iterative\ntransformation to put ACS data in a ground-based system, but warn\nagainst its use, given the systematic errors intrinsic to such a\nprocess. The biggest problem is that the F606W bandpass is very\ndifferent from Johnson $V$, although the difference between F814W and\nCousins $I$ is nonnegligible, too. To properly make the\ntransformation from one system to the other, one must know the\nintrinsic spectral energy distribution of the source, and this is\ndifficult to estimate based on photometry in two broad bandpasses.\nIt is much more straightforward to use the physical parameters along\neach model isochrone (effective temperature and surface gravity) to\ntransform the models into the observational system using synthetic\nspectra of the appropriate metallicity. We use the transformation of\nBrown et al.\\ (2005), which produces good agreement between these\nisochrones and the ACS observations of Galactic clusters spanning a\nwide range in metallicity (Table~\\ref{tabclus}). Over most of the CMD\n(including the region we use here for fitting), the agreement is\nbetter than $\\sim$0.02~mag. In this sense, we are using the\nisochrones to provide relative changes in age and metallicity, once\nthey have been anchored to observations of Galactic clusters. We are\nthus providing star formation histories in a reference frame based on\nthe ages and metallicities of the clusters listed in\nTable~\\ref{tabclus}.\n\n\\subsubsection{The Isochrone Grid}\n\nWe fit a large grid of isochrones spanning $1 \\leq$~age~$\\leq 14$~Gyr\n(with 0.5 Gyr steps) and $-2.3 \\leq $~[Fe\/H]~$\\leq +0.5$ (with\n$\\approx$0.1 dex steps) using the StarFish code. The fine spacing in\nage and metallicity avoids artificial lumpiness in the synthetic\nCMDs but means that neighboring isochrones in the grid are nearly\ndegenerate. Such degeneracies, plus the large number of free\nparameters, do not allow a fit to converge in a reasonable time.\nFortunately, the StarFish code allows groups of neighboring isochrones\nto be locked such that their amplitudes vary together; one of these\nisochrone groups is treated as a single isochrone as far as the\nfitting is concerned, even if its stars span a small range in age and\nmetallicity (see Harris \\& Zaritsky 2001 for details). We locked our\nfull grid of isochrones into 117 independent isochrone groups, with\nthe sampling chosen to match the nonlinear changes in the CMD with age\nand metallicity (the CMD changes more rapidly at higher metallicities\nand younger ages). The grid of isochrones and the locked isochrone\ngroups are shown in Figure~\\ref{gridfig}.\n\n\\begin{figure}[h]\n\\epsscale{1.2}\n\\plotone{f8.eps}\n\\epsscale{1.0}\n\\vskip -0.1in\n\\caption{The isochrones used in StarFish fitting. A fine grid of \nisochrones ({\\it crosses}) was used to avoid artificial lumpiness\nin the synthetic CMD, but these isochrones were locked together in\ngroups ({\\it boxes}) to reduce the number of free parameters and\nto avoid degeneracies in the fit.}\n\\label{gridfig}\n\\end{figure}\n\n\\subsubsection{Fixed Parameters}\n\nBesides distance and reddening, there are several other parameters\nthat must be fixed before proceeding with a fit. The binary fraction\nis highly uncertain, and not even well-constrained in the field or\ncluster populations of our own Galaxy; the value appears to be in the\nrange of 10--30\\% in the field population of the Galactic halo (Ryan\n1992 and references therein). The fits to our data are best when the\nbinary fraction is near 10\\%, whereas fits with the binary fraction\nsignificantly deviating from 10\\% show noticeable residuals. Thus,\nunless specified otherwise, the binary fraction is assumed to be 10\\%\nthroughout this paper. The binary fraction is set in the StarFish\ncode (Harris \\& Zaritsky 2001) at the stage where it scatters the\nisochrones; specifically, for a given fraction of stars, it draws a\nsecond star randomly from the IMF and produces a single unresolved\nobject with the combined color and magnitude of the two stars. For\nthe IMF index, we chose the Salpeter (1955)\nvalue of $-1.35$. For the isochrone abundances, we did not assume a\nscaled-solar abundance pattern. Instead, we assumed that the alpha\nelements are enhanced at low metallicity and unenhanced (scaled-solar)\nat high metallicity; specifically, we assumed [$\\alpha$\/Fe]~=~0.3 at\n[Fe\/H]~$\\leq -0.7$ and [$\\alpha$\/Fe]~=~0.0 at [Fe\/H]~$> -0.7$. At the\n[$\\alpha$\/Fe] resolution available in our isochrone grid, this trend\nroughly reproduces that seen in the Galaxy (Pritzl, Venn, \\& Irwin\n2005 and references therein), although bulge populations appear to be\nenhanced in alpha elements even at high metallicity (McWilliam \\& Rich\n1994; Rich \\& Origlia 2005). As it turns out, the IMF and\nalpha-enhancement assumptions make little difference in our results.\nAll of these assumptions (distance, reddening, binary fraction, IMF,\nand alpha enhancement) are varied in our exploration of systematic\nerrors (see \\S\\ref{syssec}).\n\n\\subsubsection{Uncertainties}\n \nIn the fits below, we do not plot error bars for the weights of the\nindividual isochrones. This is because the uncertainty associated\nwith the normalization of any individual isochrone is very large, and\ncorrelated with the normalization of neighboring isochrones. If any\none isochrone in the best-fit model is deleted from the fit,\ncompensating changes can be made in neighboring\nisochrones that restore the quality of the fit. The result is that\nthe uncertainty on any individual isochrone weight is largely\nmeaningless. These difficulties are a continuing plague for studies of\nstar formation histories in complex populations (e.g., Skillman et\nal.\\ 2003; Harris \\& Zaritsky 2004). If one is fitting a simple\nstellar population (single age and single metallicity), one can trace\nout confidence contours in the age-metallicity plane according to the\nchange in fit quality, but with a complex star formation history, it\nis the distribution of ages and metallicities that matters. What one\nreally wants is a set of isochrones that are truly eigenfunctions of\nan orthogonal basis set. However, there is not an obvious basis\nfunction that relates in a simple way back to physical parameters.\nThe sampling in our isochrone grid is fine enough to avoid artificial\nstructure in the synthetic CMDs, yet coarse enough to avoid isochrones\nthat are completely degenerate within the photometric errors.\n\nEven though some of the isochrone weights in the final fits are very\nsmall, the ensemble of these small weights is necessary for a good\nfit. One way of demonstrating this assertion is by repeating the fits\nafter deleting isochrones with low weights. Starting with the best\nfit, we first sorted the isochrones by their fitted weights, and then\nretained only those whose weight exceeded a specified cutoff;\nspecifically, the cutoff in weight was chosen so that this subset of\nisochrones accounted for 90\\% of the stars in the best fit. Refitting\nwith this reduced set of isochrones produced terrible fits (fit score\n$\\sim$50\\% larger). The fit was also poor when we retained those\nisochrones responsible for 95\\% of the stars in the best fit. The fit\ndid not become acceptable until we had retained those isochrones\nresponsible for 99\\% of the stars in the best fit ($\\sim$50 of the\noriginal 117 isochrone groups).\n\n\\subsection{Results for the Spheroid}\n\\label{secsph}\n\nThe distribution of age and metallicity in our best fit to the\nspheroid data is shown in Figure~\\ref{halofit}. In this figure, the\narea of the symbols ({\\it filled circles}) is proportional to the\nnumber of stars in each isochrone group. Note that the spacing of the\nisochrone groups is irregular, so that if one were to plot a star\nformation rate in units of $M_\\odot$ per unit time per unit\nlogarithmic metallicity, the relative sizes of the symbols would be\nsomewhat increased at younger ages and higher metallicities (where the\nspacing is finer). As noted by Brown et al.\\ (2003), the\nspheroid CMD is best fitted by a wide range of age and metallicity, and\nis strikingly different from the old, metal-poor halo of the Milky\nWay. Approximately 40\\% of the stars are less than 10~Gyr old, and\napproximately 50\\% of the stars are more metal-rich than 47~Tuc\n([Fe\/H]$\\approx-0.7$). The mean metallicity, $<$[Fe\/H]$>$=$-0.6$, is\nidentical to that found by Durrell et al.\\ (1994) at 9~kpc on the\nminor axis, and slighter higher than the $<$[m\/H]$>$=$-0.6$ found by\nHolland et al.\\ (1996) from earlier WFPC2 photometry of our field,\nwith similar spreads to both higher and lower metallicities. Although\nour mean metallicity is much higher than that in the Milky Way halo,\nthe metallicity distribution definitely has a tail extending to\nmetal-poor stars. These include the RR Lyrae stars in our field,\nwhich have a mean metallicity of [Fe\/H]~=~$-1.7$ (Brown et al.\\\n2004a), and the minority population of blue HB stars.\n\n\\begin{figure}[h]\n\\epsscale{1.2}\n\\plotone{f9.eps}\n\\epsscale{1.0}\n\\vskip -0.1in\n\\caption{The distribution of age and metallicity in the best-fit model\nof the spheroid data. The area of the filled circles is proportional\nto the number of stars in each isochrone group.\n}\n\\label{halofit}\n\\end{figure}\n\nAlthough we used the Dolphin (2002) Maximum Likelihood statistic to\nperform our fits, we also compared the results with those obtained\nfrom a traditional $\\chi^2$ statistic, and the fits were\nsimilar. Dolphin (2002) also provides a goodness of fit statistic,\n$\\chi^2_{\\rm eff}$, for those more familiar\nwith $\\chi^2$ fitting (with values close to unity indicating a good\nfit). The best fit model (Figure~\\ref{halofit}) has $\\chi^2_{\\rm eff}\n=1.11$ per degree of freedom (8000 CMD bins minus 117 freely varying\nisochrone weights). This score clearly implies an imperfect fit. To\ndemonstrate this, we ran Monte Carlo simulations of the idealized\ncase. We created random realizations of the data drawn from the\nbest-fit model to obtain the distribution of the Maximum Likelihood\nstatistic, and found that the Maximum Likelihood statistic obtained in\nour best-fit model exceeds the mean score by 6$\\sigma$ (where $\\sigma$ is\none standard deviation in the distribution of the Maximum Likelihood\nstatistic from the Monte Carlo runs).\n\nThere are many reasons why the model should not\nexactly reproduce the data. These include imperfections in the\nisochrones (they are calibrated at the $\\sim$0.02~mag level against\nGalactic globular clusters observed in the same filters), deviations\nfrom a Salpeter (1955) IMF, deviation from our assumed binary fraction\nof 10\\% (e.g., one might imagine that the binary fraction varies with\nage and metallicity depending on the variations in the formation\nenvironment), and the limitations of the artificial star tests used to\nscatter the isochrones (artificial stars are created, with noise, from\nthe same PSF model used in the PSF fitting, while real stars will\ndeviate from the PSF model due to noise and true intrinsic\ninaccuracies in the PSF model). Although the model does not exactly\nreproduce the data distribution over 8000 CMD bins, the deviations are\nremarkably small, as we show in Figure~\\ref{halomod}. In the top row\nof panels, we show the data in the fitting region ({\\it yellow}), the\nbest-fit model ({\\it blue}), and the differences between the two ({\\it\nyellow} and {\\it blue}) shown at the same linear stretch; i.e., \nthe CMD bins in panel $c$ are shaded blue where the model exceeds the\ndata, and shaded yellow where the data exceeds the model, with the\nshading on the same linear scale employed in panels $a$ and $b$. The\ndifferences between the data and model appear almost completely\nrandom, with minimal systematic residuals; in fact, the upper panels look\nmuch like the idealized case shown in the bottom row of panels, where\nthe residuals are completely random.\nThere, we show a random realization of the best-fit model ({\\it\nyellow}), a repeat of the best-fit model ({\\it blue}), and the\ndifferences between the two ({\\it yellow} and {\\it blue}). The\nrealization ({\\it bottom left}) is nearly indistinguishable from the\nactual data ({\\it top left}). The difference between the realization\nand the model ({\\it bottom right}) demonstrates the noise residuals one can\nexpect when comparing a smooth model to discrete data in the idealized\ncase ($\\chi^2_{\\rm eff} = 1$). \n\n\\begin{figure}[h]\n\\epsscale{1.2}\n\\plotone{f10.eps}\n\\epsscale{1.0}\n\\vskip -0.1in\n\\caption{{\\it Top panels:} The CMD of the spheroid data ({\\it\nyellow}), the best-fit model to those data ({\\it blue}), and the\ndifferences between the data and model ({\\it yellow} and {\\it blue}),\nall shown at the same linear stretch. {\\it Bottom panels:} An\nartificial CMD drawn from the best-fit model ({\\it yellow}), the same\nbest-fit model ({\\it blue}), and the differences between the\nartificial data and model ({\\it yellow} and {\\it blue}), all shown at\nthe same linear stretch employed in the top panels. }\n\\label{halomod}\n\\end{figure}\n\nGiven the large number of free parameters, one might also wonder if\nthe ``best-fit'' model has truly converged on the best fit. One way\nto test this is through repeated fitting with distinct initial\nconditions. We show in Figure~\\ref{converge} the results of three\n``best-fit'' models to the spheroid data, each of which started from a\ndistinct random set of isochrone weights. Although there are small\nvariations in the final individual isochrone weights, it is clear that\nthe overarching result is the same in each case. As stated earlier,\nthe degeneracies in the isochrone set mean that any individual\nisochrone can be varied significantly without changing the fit\nquality. For example, in Figure~\\ref{converge}, the relatively low\nweight at [Fe\/H]=$-1.7$, compared to the weights at [Fe\/H]=$-1.4$\nand [Fe\/H]=$-2.1$, is not meaningful; for the isochrones at 13~Gyr,\nwe can redistribute the weights at [Fe\/H]=$-1.4$, $-1.7$, and $-2.1$\nso that they are the same in each of these bins, and the fit quality\ndoes not suffer.\n\n\\begin{figure}[h]\n\\epsscale{1.2}\n\\plotone{f11.eps}\n\\epsscale{1.0}\n\\vskip -0.1in\n\\caption{Three different attempts at fitting the spheroid data, using\nthe same isochrones in each case, but where the initial guess in each\npanel was a distinct random distribution of isochrones.\nThe area of the filled circles is proportional to the\nnumber of stars in each isochrone group. Although there are small\nvariations in the individual amplitude weights from panel to panel, it\nis clear that the best-fit model is well converged.}\n\\label{converge}\n\\end{figure}\n\nAlthough the uncertainties on the individual isochrone weights in the\nbest-fit model are large, one can ask what classes of models, in a\nbroad sense, produce fits that are as good as the best-fit model. If\none restricts the fit to isochrones of ages$<10$~Gyr, the quality of\nthe fit is noticeably reduced, with $\\chi^2_{\\rm eff} = 1.18$ (a fit that\nis an additional 5$\\sigma$ worse than the best-fit model). Much of\nthe weight in this fit falls at the top end of the allowed age range,\nand the difference between the model CMD and the data CMD shows\nsignificant residuals (Figure~\\ref{haloyngold}). Alternatively, if\none restricts the fit to isochrones with ages$\\ge 10$~Gyr, the quality\nof the fit is grossly reduced, with $\\chi^2_{\\rm eff} = 3.09$ and\nvery obvious differences between the model CMD and the data CMD\n(Figure~\\ref{haloyngold}). This is consistent with the results of\nBrown et al.\\ (2003), who showed that the spheroid CMD is inconsistent\nwith a purely old population of stars.\n\n\\begin{table*}[t]\n\\begin{center}\n\\caption{Summary of Spheroid Fitting}\n\\begin{tabular}{lcccl}\n\\tableline\nModel & $<$[Fe\/H]$>$ & $<$age$>$ & $\\chi^2_{\\rm eff}$ & Comment \\\\\n\\tableline\nStandard model & $-0.6$ & 9.7 & 1.11 & Minimal residuals in fit\\\\\nAge $< 10$~Gyr & $-0.5$ & 8.4 & 1.18 & Significant residuals in fit\\\\\nAge $\\ge 10$~Gyr & $-0.8$ & 10.9 & 3.09 & Gross residuals in fit\\\\\nNo old metal-rich stars & $-0.6$ & 9.6 & 1.11 & Minimal residuals in fit\\\\\nNo young metal-poor stars & $-0.6$ & 9.7 & 1.18 & Misses part of plume\\\\\n\\tableline\n\\end{tabular}\n\\label{tabspheroid}\n\\end{center}\n\\end{table*}\n\n\\begin{figure}[h]\n\\epsscale{1.2}\n\\plotone{f12.eps}\n\\epsscale{1.0}\n\\vskip -0.1in\n\\caption{{\\it Top panels:} The CMD of the spheroid data ({\\it\nyellow}), the best-fit model to those data using a set of isochrones\nrestricted to ages less than 10~Gyr ({\\it blue}), and the\ndifferences between the data and model ({\\it yellow} and {\\it blue}),\nall shown at the same linear stretch. {\\it Bottom panels:} The same\nCMD of the spheroid data ({\\it yellow}), the best-fit model to those\ndata using a set of isochrones restricted to ages $\\ge 10$~Gyr\n({\\it blue}), and the differences between the data and\nmodel ({\\it yellow} and {\\it blue}), all shown at the same linear\nstretch employed in the top panels. It is clear that neither model\nis acceptable, given the residuals ({\\it right hand panels}). }\n\\label{haloyngold}\n\\end{figure}\n\nThe best-fit model has minority populations in the isochrones\nrepresenting old metal-rich stars and young metal-poor stars. If\ntruly present, these populations are extremely interesting, because\nthe former imply that at least some of the stars were formed in\nsomething like a bulge environment (with rapid early enrichment),\nwhile the latter imply the accretion of metal-poor stars from dwarf\ngalaxies or star formation following the infall of\nrelatively pristine material. To test this, we repeated the fit while\nexcluding two regions from the input grid of isochrones: age$\\ge\n10$~Gyr at [Fe\/H]$\\ge 0$, and age$< 5$~Gyr at [Fe\/H]$< -0.5$; each of\nthese regions contains 3\\% of the stellar mass in the best-fit model.\nIf the old metal-rich isochrones are excluded from the fit, the fit\nquality in the resulting model does not suffer at all; thus, the CMD\nis consistent with either a small population of such old metal-rich\nstars or none at all. In contrast, if the young metal-poor isochrones\nare excluded from the fit, the fit quality is somewhat reduced, with\n$\\chi^2_{\\rm eff} = 1.18$, due to the model missing the brightest and\nbluest stars in the blue plume above the dominant main sequence. \nThis is not surprising, given our visual inspection of the\nCMD and comparison to young isochrones of various metallicities\n(Figure~\\ref{comparegrid}). Note that the scattered model isochrones\ninclude the effects of blends (determined by the artificial star tests)\nbut not any contribution from blue stragglers; thus, some (but not all) of the\nyoung stars in the fit ($\\lesssim 6$~Gyr) could be an attempt to\naccount for blue stragglers (see \\S\\ref{secinspec}).\n\nWe summarize the fits to the spheroid data in Table~\\ref{tabspheroid}.\nOur standard model is that which simply allows the full grid\n(Figure~\\ref{gridfig}) to vary freely, while the other models are\nself-explanatory. Mean values of [Fe\/H] and age are not as useful as\nthe full age and metallicity distributions, given the complicated star\nformation history present in the field, but these mean values do serve\nas a yardstick to gauge differences between the fits. \\\\\n\n\\subsection{Results for the Stream}\n\nThe distribution of age and metallicity in our best fit to the\nstream data is shown in Figure~\\ref{strmfit}. Given the qualitative\nsimilarities between the stream and spheroid CMDs, it is not\nsurprising that the best-fit distribution of age and metallicity in\nthe stream resembles that in the spheroid. However, as noted above,\nthere are some distinctions. The mean age in the stream (8.8~Gyr) is\n$\\sim$1~Gyr younger than that in the spheroid (9.7~Gyr), while the\nmean metallicities are nearly the same ($-0.6$ in the spheroid and\n$-0.7$ in the stream). The fit quality for the best-fit stream model\nis similar to that for the spheroid, with $\\chi^2_{\\rm eff} = 1.08$. \nIn Figure~\\ref{strmmod}, we show the comparison of the best-fit\nmodel to the data, as well as the residuals.\n\nGiven that the stream and spheroid are so similar, we also explored to\nwhat extent both populations might be consistent with a single star\nformation history. First, we simply used the spheroid star formation\nhistory (Figure~\\ref{halofit}) to normalize a set of isochrones\nscattered according to the stream artificial star tests, and then\nscaled the result to match the number of stars in the stream. This\ncreated a model with the spheroid star formation history but the\nobservational properties of the stream data, enabling a fair\ncomparison of the two. The result is shown in\nFigure~\\ref{strmhalomod}. It is obvious that there are gross\nresiduals in the model. Although this was not a fit (given that we\nsimply applied the star formation history of the spheroid), if this\nmodel had resulted from our standard isochrone fitting, it would have\nproduced a $\\chi^2_{\\rm eff}$ of 1.32. The comparison of the spheroid\ndata with this model population yielded a $\\chi^2_{\\rm eff}$ of 1.11\n(\\S\\ref{secsph}); the much larger discrepancy of the stream data\nwith this model population strongly implies that the spheroid and\nstream data were drawn from distinct populations, at a confidence\nlevel exceeding 99\\%.\n\n\\begin{figure}[h]\n\\epsscale{1.2}\n\\plotone{f13.eps}\n\\epsscale{1.0}\n\\vskip -0.1in\n\\caption{The distribution of age and metallicity in the best-fit model\nof the stream data. The area of the filled circles is proportional to\nthe number of stars in each isochrone group. The total area within\nthe filled symbols has been normalized to that in\nFigure~\\ref{halofit}, to ease comparison (but in reality the surface\nbrightness in the stream is $\\sim$0.5~mag fainter). }\n\\label{strmfit}\n\\end{figure}\n\n\\begin{figure}[h]\n\\epsscale{1.2}\n\\plotone{f14.eps}\n\\epsscale{1.0}\n\\vskip -0.1in\n\\caption{{\\it Top panels:} The CMD of the stream data ({\\it yellow}),\nthe best-fit model to those data ({\\it blue}), and the differences\nbetween the data and model ({\\it yellow} and {\\it blue}), all shown at\nthe same linear stretch. {\\it Bottom panels:} An artificial CMD drawn\nfrom the best-fit model ({\\it yellow}), the same best-fit model ({\\it\nblue}), and the differences between the artificial data and model\n({\\it yellow} and {\\it blue}), all shown at the same linear stretch\nemployed in the top panels. Note that the magnitude range of the\nstream fit is smaller than that in the spheroid fit, because the\nspheroid data are $\\sim$0.5 mag deeper than the stream data.}\n\\label{strmmod}\n\\end{figure}\n\nNext, we tried fitting both the spheroid and stream simultaneously\nwith the same star formation history. Specifically, a model for the\nstream was constructed from isochrones appropriately matching the\nstream observations (utilizing the stream artificial star tests), and\na model for the spheroid was constructed from isochrones appropriately\nmatching the spheroid observations (utilizing the spheroid artificial\nstar tests), but the relative weights of the isochrones used to\nconstruct these stream and spheroid models came from a single\ndistribution of age and metallicity. This distribution was varied\nuntil the best fit to both the stream and spheroid data was achieved.\nThe resulting age and metallicity distribution is shown in\nFigure~\\ref{strmhalofit}. Curiously, this compromise solution to both\nCMDs is a bit older and more metal-poor than that found for either CMD\nindividually; this is likely due to the fact that the spheroid and\nstream are distinct, resulting in a poor fit when fitting both at the\nsame time. The poor quality of the fit can be seen when this\ncompromise model is compared to the stream data, as shown in\nFigure~\\ref{strmhalomod}. The value for $\\chi^2_{\\rm eff}$ is not\nterrible (1.14), but there are approximately twice the number of\ndegrees of freedom in this fit, given that we are fitting two CMDs of\ndata simultaneously, so the deviation from unity is more significant.\nBoth of these tests imply that while the stream and spheroid CMDs are very\nsimilar, they are not drawn from exactly the same population.\n\n\\begin{figure}[h]\n\\epsscale{1.2}\n\\plotone{f15.eps}\n\\epsscale{1.0}\n\\vskip -0.1in\n\\caption{{\\it a)} The CMD of the stream data ({\\it yellow}). {\\it b)}\nA model for the stream data ({\\it blue}), but constructed from the\nspheroid star formation history, scattered with the observational\nerrors of the stream data and normalized to the stream star counts. {\\it c)}\nThe differences between the data and model ({\\it\nyellow} and {\\it blue}), all shown at the same linear stretch. {\\it\nd)} The same CMD of the stream data ({\\it yellow}). {\\it e)} The\nbest-fit compromise model fit simultaneously to the spheroid and\nstream datasets ({\\it blue}). {\\it f)} The differences between the data and\nmodel ({\\it yellow} and {\\it blue}), all shown at the same linear\nstretch employed in the top panels. Significant residuals can be seen\nin either case ({\\it right hand panels}) implying that the stream and\nspheroid CMDs are not drawn from the same population.}\n\\label{strmhalomod}\n\\end{figure}\n\n\\begin{figure}[h]\n\\epsscale{1.2}\n\\plotone{f16.eps}\n\\epsscale{1.0}\n\\vskip -0.1in\n\\caption{\nThe distribution of age and metallicity in the best-fit model\nsimultaneously fit to the spheroid and stream data. The area of the\nfilled circles is proportional to the number of stars in each\nisochrone group. The distribution shown here is clearly a compromise\nbetween those shown in Figures~\\ref{halofit} and \\ref{strmfit}.\n}\n\\label{strmhalofit}\n\\end{figure}\n\nAs done with the spheroid data, we also explored to what extent stream models\nwith more restricted age ranges are consistent with the data. When\nthe isochrones are restricted to ages$< 10$~Gyr, the quality of the\nfit is nearly unchanged (with $\\chi^2_{\\rm eff}=1.10$), although the\nresulting distribution of age and metallicity looks somewhat skewed,\nwith much of the weight falling at the top end of the age range. When\nthe isochrones are restricted to ages$\\ge 10$~Gyr, the quality of the\nfit is very poor, with $\\chi^2_{\\rm eff}=2.80$. If old metal-rich\nstars are removed from the input isochrone grid, the quality of the\nfit is unchanged from the best fit model, while if young metal-poor\nstars are removed, the quality of the fit is noticeably affected, with\n$\\chi^2_{\\rm eff} = 1.14$, but the model is only\nmissing the brightest and bluest stars in the blue plume.\n\n\\begin{table*}[t]\n\\begin{center}\n\\caption{Summary of Stream Fitting}\n\\begin{tabular}{lcccl}\n\\tableline\nModel & $<$[Fe\/H]$>$ & $<$age$>$ & $\\chi^2_{\\rm eff}$ & Comment \\\\\n\\tableline\nStandard model & $-0.7$ & 8.8 & 1.08 & Minimal residuals in fit\\\\\nAge $< 10$~Gyr & $-0.6$ & 8.1 & 1.10 & Minimal residuals in fit\\\\\nAge $\\ge 10$~Gyr & $-1.0$ & 11.0 & 2.80 & Gross residuals in fit\\\\\nNo old metal-rich stars & $-0.7$ & 8.8 & 1.09 & Minimal residuals in fit\\\\\nNo young metal-poor stars & $-0.7$ & 8.8 & 1.14 & Misses part of plume\\\\\nBest-fit spheroid model & $-0.6$ & 9.7 & 1.32\\tablenotemark{a} & Gross residuals \\\\\nSimultaneous fit to spheroid stream & $-0.8$ & 10.1 & 1.14\\tablenotemark{b} & Significant residuals\\\\\nFixed 25\\% spheroid contamination & $-0.8$ & 8.8 & 1.10 & Similar to standard model\\\\\n\\tableline\n\\multicolumn{5}{l}{$^a$Not actually a fit. See text for details.}\\\\\n\\multicolumn{5}{l}{$^b$Twice the degrees of freedom. See text for details.}\n\\end{tabular}\n\\label{tabstream}\n\\end{center}\n\\end{table*}\n\nThe Keck data for our stream field imply that 75\\% of its stars fall\nin two kinematically cold stream components (Kalirai et al.\\ 2006b),\nand that 25\\% of its stars are in the underlying spheroid. Although\nthe population in our spheroid field might not be representative of\nthe underlying spheroid in the stream field, it is reasonable to\nwonder how the fitting of the stream star formation history is\naffected if this spheroid contamination is taken into account. To\nexplore this, we fitted the stream with the same set of isochrones, but\nadded an additional component to the model, fixed at 25\\% of the\npopulation, representing the spheroid contamination. This\ncontamination component was constructed from the best-fit model to the\nspheroid but using the isochrones scattered with the stream artificial\nstar tests; thus the contamination component appropriately represents\nthe spheroid population as it would appear in the stream data. The\nresults are shown in Figure~\\ref{strmfit_cont}. The quality of the\nfit is good; $\\chi^2_{\\rm eff} = 1.10$. In the top panel, we show the\ntotal star formation history (combining the fixed spheroid\ncontamination and the fit to the stream). In the bottom panel, we\nhave subtracted the spheroid contamination component from the star\nformation history, to show the star formation history of the stream in\nisolation. The isolated star formation history of the stream\n(Figure~\\ref{strmfit_cont}b) is very similar to the best-fit model to\nthe stream that did not try to account for the spheroid contamination\n(Figure~\\ref{strmfit}). Given the similarity between the stream and\nspheroid CMDs, and the fact that the spheroid contamination is only\n25\\%, this is not that surprising. We summarize the results of the\nstream fitting in Table~\\ref{tabstream}.\n\n\\begin{figure}[h]\n\\epsscale{1.2}\n\\plotone{f17.eps}\n\\epsscale{1.0}\n\\vskip -0.1in\n\\caption{ The best-fit model to the stream, assuming a fixed 25\\%\ncontamination from the underlying spheroid that matches the population\nin Figure~\\ref{halofit}. The area of the filled circles is\nproportional to the number of stars in each isochrone group. {\\it Top\npanel:} The complete star formation history, including the fixed\nspheroid contamination. {\\it Bottom panel:} The star formation\nhistory for the stream population in isolation, excluding that part of\nthe fit representing the spheroid contamination. The population has\nbeen normalized such that the total area in the symbols is the same in\nboth panels.}\n\\label{strmfit_cont}\n\\end{figure}\n\n\\subsection{Results for the Disk}\n\nThe distribution of age and metallicity in our best fit to the disk\ndata is shown in Figure~\\ref{diskfit}. As expected from our earlier\ninspection of the CMDs, the star formation history in the disk is\nmarkedly distinct from that in the spheroid or stream, in the sense\nthat the population is younger and significantly more metal-rich, with\na mean age of 7.5~Gyr and a mean metallicity of [Fe\/H]~=~$-0.2$. The\nfit quality for the best-fit disk model is excellent, with\n$\\chi^2_{\\rm eff} = 1.05$. In Figure~\\ref{diskmod}, we show the\ncomparison of the best-fit model to the data, as well as the residuals.\n\n\\begin{figure}[h]\n\\epsscale{1.2}\n\\plotone{f18.eps}\n\\epsscale{1.0}\n\\vskip -0.1in\n\\caption{The distribution of age and metallicity in the best-fit model\nof the disk data. The area of the filled circles is proportional to\nthe number of stars in that isochrone group. The distribution shown\nhere is clearly distinct from those shown in Figures~\\ref{halofit} and\n\\ref{strmfit}. The total area within the filled symbols has been\nnormalized to that in Figures~\\ref{halofit} and \\ref{strmfit}, to ease\ncomparisons.}\n\\label{diskfit}\n\\end{figure}\n\n\\begin{figure}[h]\n\\epsscale{1.2}\n\\plotone{f19.eps}\n\\epsscale{1.0}\n\\vskip -0.1in\n\\caption{{\\it Top panels:} The CMD of the disk data ({\\it\nyellow}), the best-fit model to those data ({\\it blue}), and the\ndifferences between the data and model ({\\it yellow} and {\\it blue}),\nall shown at the same linear stretch. {\\it Bottom panels:} An\nartificial CMD drawn from the best-fit model ({\\it yellow}), the same\nbest-fit model ({\\it blue}), and the differences between the data and\nmodel ({\\it yellow} and {\\it blue}), all shown at the same linear\nstretch employed in the top panels. Note that the magnitude range\nof the disk fit is smaller than that in the spheroid fit, because\nthe spheroid data are $\\sim$0.5 mag deeper than the disk data.}\n\\label{diskmod}\n\\end{figure}\n\nThe best-fit model to the disk is dominated by stars at ages of less\nthan 10~Gyr. In fact, if we fit the data with a subset of the\nisochrones restricted to ages$<$10~Gyr (i.e., remove all old\nisochrones from the input grid, not just the metal-rich ones), the fit\nis only negligibly worse than that achieved with the full set of\nisochrones ($\\chi^2_{\\rm eff} = 1.06$), and the resulting distribution\nof age and metallicity looks very similar to that in the best-fit\nmodel. In contrast, a fit restricted to ages$\\ge 10$~Gyr is grossly\ninadequate, with $\\chi^2_{\\rm eff}=5.07$. If young metal-poor stars\nare removed from the fit (as done with the stream and spheroid\nfitting), the fit quality drops, with $\\chi^2_{\\rm eff} = 1.14$, and\nthe model misses the bright blue stars in the plume.\n\n\\begin{table*}[t]\n\\begin{center}\n\\caption{Summary of Disk Fitting}\n\\begin{tabular}{lcccl}\n\\tableline\nModel & $<$[Fe\/H]$>$ & $<$age$>$ & $\\chi^2_{\\rm eff}$ & Comment \\\\\n\\tableline\nStandard model & $-0.2$ & 7.5 & 1.05 & Minimal residuals in fit\\\\\nAge $< 10$~Gyr & $-0.1$ & 6.9 & 1.06 & Minimal residuals in fit\\\\\nAge $\\ge 10$~Gyr & $-0.9$ & 11.0 & 5.07 & Gross residuals in fit\\\\\nNo young metal-poor stars & $-0.2$ & 7.6 & 1.14 & Misses part of plume\\\\\nFixed 33\\% spheroid contamination & $+0.1$ & 6.6 & 1.05 & Younger. Minimal residuals\\\\\n\\tableline\n\\end{tabular}\n\\label{tabdisk}\n\\end{center}\n\\end{table*}\n\nThe metallicity distribution in our best fit to the disk CMD is\nsomewhat more metal-rich than that typically found in the outer disk\nof M31 (e.g., Worthey et al.\\ 2005). There are several possible\nreasons for this. First, the greatest color dependence upon [Fe\/H] is\nat the tip of the RGB, which in our data is both sparsely populated\nand seriously contaminated by foreground dwarf stars. Instead, we are\nusing the lower RGB, which offers the advantage of large numbers of\nM31 stars and little contamination, but the penalty is a reduced color\nsensitivity to [Fe\/H]. Second, the use of distinct isochrone sets and\ndistinct observing bands results in significant scatter for abundance\ndeterminations even when the population is a simple one, such as a\nglobular cluster. The metallicities we derive are calibrated to the\nglobular cluster metallicities given in Table~\\ref{tabclus}.\nPublished abundances for globular clusters of intermediate metallicity\nvary by $\\sim$0.2 dex in the recent literature, while abundances for\nhigh metallicity clusters vary by even more (see Brown et al. 2005 and\nreferences therein). Moreover, isochrones at high metallicity are\ndifficult to calibrate, given that appropriate clusters tend to be in\nheavily reddened regions, such as the Galactic bulge. Finally,\nprevious [Fe\/H] distributions for M31 fields invariably employed old\nisochrones or the ridge lines of old globular clusters as reference\npoints. This will bias the results toward lower metallicity if the\nmetal-rich population is in fact significantly younger than Galactic\nglobular clusters. For example, the upper RGB for a 13 Gyr population at\n[Fe\/H]~=~0.0 is very similar to that for a 6 Gyr population at\n[Fe\/H]~=~+0.230.\n\nThe Keck kinematics of our disk field imply that $\\sim$67\\% of its\nstars are moving in the disk (Kalirai et al.\\ 2006b; Reitzel et al.\\ in\nprep.), and that $\\sim$33\\% of its stars are in the underlying\nspheroid. As with our analysis of the stream, the population in our\nspheroid field might not be representative of the underlying spheroid\nin the disk field, but it is reasonable to explore a fit to the disk\nwith a fixed contamination component from the spheroid. We repeated\nthe disk fitting with an additional model component held fixed at 33\\%\nof the population, representing spheroid contamination. This\ncontamination component was constructed from the best-fit model to the\nspheroid but using the isochrones scattered with the disk artificial\nstar tests; thus the contamination component appropriately represents\nthe spheroid population as it would appear in the disk data. The\nresults are shown in Figure~\\ref{diskfit_cont}. The quality of the\nfit is excellent, with $\\chi^2_{\\rm eff} = 1.05$. In the top panel, we\nshow the total star formation history (which includes the fixed\nspheroid component in the fit to the disk field). In the bottom\npanel, we show the same fit to the star formation history, but\nsubtract that fixed component representing spheroid contamination, in\norder to show the star formation history of the disk population in\nisolation. The isolated disk population (Figure~\\ref{diskfit_cont}b)\nis significantly younger and more metal-rich than that found in our\ninitial model (Figure~\\ref{diskfit}), where we did not try to account\nfor the spheroid contamination. The isolated point at 13 Gyr and\n[Fe\/H]~=~0.5 is not significant (repeating the fit with no isochrones\nolder than 10~Gyr yields $\\chi^2_{\\rm eff} = 1.06$). The similarities\nbetween Figures~\\ref{diskfit} and Figure~\\ref{diskfit_cont}a are\nreassuring; the fit in Figure~\\ref{diskfit} did not employ any\nknowledge of the spheroid contamination, yet it is clear that this fit\ntried to reproduce the old metal-poor component that is present in\nFigure~\\ref{diskfit_cont}a, where we explicitly specified a spheroid\ncontamination component to the model. Because the spheroid\ncontamination can completely account for the old and metal-poor stars\nin the disk field, the dearth of metal-poor stars is another example\nof the ``G dwarf problem'' -- that a simple closed box model of\nchemical evolution predicts a longer tail of metal-poor stars than\nseen in all massive galaxies (see Worthey et al.\\ 2005 and references\ntherein). We summarize the results of the disk fitting in\nTable~\\ref{tabdisk}.\n\n\\begin{figure}[h]\n\\epsscale{1.2}\n\\plotone{f20.eps}\n\\epsscale{1.0}\n\\vskip -0.1in\n\\caption{ The best-fit model to the disk, assuming a fixed 33\\%\ncontamination from the underlying spheroid that matches the population\nin Figure~\\ref{halofit}. {\\it Top panel:} The complete star formation\nhistory, including the fixed spheroid contamination. {\\it Bottom\npanel:} The star formation history for the disk population in\nisolation, excluding that part of the fit representing the spheroid\ncontamination. The population has been normalized such that the total\narea in the symbols is the same in both panels.}\n\\label{diskfit_cont}\n\\end{figure}\n\n\\subsection{Systematic Effects of Binaries, Alpha-enhancement, \nIMF, Distance, and Reddening}\n\\label{syssec}\n\nThe fits above make assumptions about the binary fraction,\nalpha-element enhancement, IMF, distance, and reddening. Of these\nthree parameters, the binary fraction and reddening uncertainties\ntranslate into the largest uncertainties in the resulting fits, but do\nnot change the gross interpretation of the CMDs. We assumed an IMF\nindex of $-1.35$ (Salpeter 1955), and assumed that [$\\alpha$\/Fe]~=~0.3\nat [Fe\/H]~$\\leq -0.7$ and [$\\alpha$\/Fe]~=~0.0 at [Fe\/H]~$> -0.7$. We\nassumed 770~kpc for the M31 distance (Freedman \\& Madore 1990), which\nis based on Cepheids and falls in the middle of the range generally\nquoted in the literature (e.g., Pritchet \\& van den Bergh 1987; Stanek\n\\& Garnavic 1998; Holland 1998; Durrell et al.\\ 2001; McConnachie et\nal.\\ 2005; Ribas et al.\\ 2005). \nWe assumed $E(B-V)=0.08$~mag in each field, but as noted\nearlier, the Schlegel et al.\\ (1998) map is uncertain at the\n$\\sim$0.02~mag level in random fields, with somewhat higher\nuncertainties near Local Group galaxies.\n\nWe chose a binary fraction of 10\\%, because grossly changing this\nvalue produced lower quality fits with obvious residuals in the\ncomparison of the models and data. Given that we chose a binary\nfraction that minimized fit residuals, in a sense we ``fit'' the\nbinary fraction, but did so on a very coarse scale. Fortunately all\nthree fields can be reasonably fit with the same binary fraction,\nbecause this avoids complications in the interpretation of the fits.\nIf we assumed distinct binary fractions in the fitting to each field,\none could attribute some of the age variations to this varying binary\nfraction. At larger binary fractions, the features in the synthetic\nCMD become brighter, and the age distribution must shift to older ages\nto compensate, while lower binary fractions result in younger age\ndistributions.\n\nTo demonstrate the sensitivity of our fits to these parameters, we\nrepeated our fits while varying our assumptions. The results are\nshown in Table~\\ref{systab} for all three fields, and in\nFigure~\\ref{sysfig} for the spheroid field. Reducing the binary\nfraction to 0\\% would decrease our ages by 0--0.4~Gyr, while\nincreasing the binary fraction to 40\\% would increase our ages by\n$\\sim$1~Gyr. Changing the alpha enhancement has almost no effect,\nother than a slight shift in the metallicity distribution. The\ninsensitivity to alpha enhancement makes sense, because in these\nbandpasses, isochrones with enhanced alpha elements look much like\nscaled-solar isochrones at slightly higher metallicity (note that the\nisochrones are always transformed to the ACS bandpasses using\nsynthetic spectra of a consistent alpha enhancement; see Brown et al.\\\n2005). Changing the IMF index from $-1.35$ to $-1.15$ also has little\neffect on the metallicity and age distributions; this is\nbecause our CMDs are sampling a fairly small range in stellar mass\n(the bulk of the stars brighter than the faint limit in our fitting\nregion fall in the mass range 0.7$\\lesssim M \\lesssim 1.2$~$M_\\odot$).\nChanging the extinction by 0.03~mag in either direction (assuming the\naverage Galactic extinction curve of Fitzpatrick 1999) primarily\naffects the metallicity distribution; an increase in the assumed\nextinction (redder stars) is compensated by a lower metallicity (bluer\nstars), and vice versa. Changing the distance modulus by 0.03~mag in\neither direction primarily affects the age distribution; an increase\nin the assumed distance (fainter apparent magnitudes) is compensated by\na younger age (brighter absolute and apparent magnitudes), and vice\nversa. Note that no change in assumptions for the spheroid\n(Figure~\\ref{sysfig}) can make the spheroid population look like that\nof the disk (Figure~\\ref{diskfit}).\n\n\\begin{table*}[t]\n\\begin{center}\n\\caption{Systematic Effects of Assumptions}\n\\begin{tabular}{lccccccccc}\n\\tableline\n & \\multicolumn{3}{c}{spheroid} & \\multicolumn{3}{c}{stream} & \\multicolumn{3}{c}{disk} \\\\\nFit & $<$age$>$ & $<$[Fe\/H]$>$ & $\\chi^2_{\\rm eff}$ & $<$age$>$ & $<$[Fe\/H]$>$ & $\\chi^2_{\\rm eff}$ & $<$age$>$ & $<$[Fe\/H]$>$ & $\\chi^2_{\\rm eff}$ \\\\\n\\tableline\nStandard model & 9.7 & $-$0.6 & 1.11 & 8.8 & $-$0.7 & 1.08 & 7.5 & $-$0.2 & 1.05 \\\\\nBinary fraction = 0.0 & 9.3 & $-$0.5 & 1.14 & 8.7 & $-$0.7 & 1.09 & 7.5 & $-$0.1 & 1.11 \\\\\nBinary fraction = 0.2 & 10.1 & $-$0.8 & 1.17 & 9.2 & $-$0.8 & 1.13 & 7.7 & $-$0.2 & 1.05 \\\\\nBinary fraction = 0.4 & 10.9 & $-$0.9 & 1.42 & 9.9 & $-$1.0 & 1.25 & 8.5 & $-$0.4 & 1.13 \\\\\n$[\\alpha$\/Fe$]=0.3$ at [Fe\/H]$\\leq$0 & 9.7 & $-$0.7 & 1.11 & 8.9 & $-$0.8 & 1.08 & 7.7 & $-$0.3 & 1.05 \\\\\n$[\\alpha$\/Fe$]=0.0$ at all [Fe\/H] & 9.7 & $-$0.6 & 1.11 & 8.9 & $-$0.7 & 1.08 & 7.6 & $-$0.1 & 1.05 \\\\\nIMF index $-$1.15 & 9.9 & $-$0.7 & 1.11 & 9.0 & $-$0.7 & 1.08 & 7.6 & $-$0.2 & 1.04 \\\\\nDistance = 760 kpc & 9.8 & $-$0.7 & 1.12 & 9.1 & $-$0.7 & 1.09 & 7.6 & $-$0.2 & 1.04 \\\\ \nDistance = 780 kpc & 9.5 & $-$0.6 & 1.11 & 8.7 & $-$0.7 & 1.08 & 7.4 & $-$0.1 & 1.05 \\\\\n$E(B-V)=0.05$~mag & 9.8 & $-$0.2 & 1.15 & 9.1 & $-$0.3 & 1.08 & 7.7 & $+$0.2 & 1.14 \\\\\n$E(B-V)=0.11$~mag & 9.4 & $-$1.0 & 1.17 & 8.6 & $-$1.1 & 1.16 & 7.3 & $-$0.5 & 1.05 \\\\\n\\tableline\n\\end{tabular}\n\\label{systab}\n\\end{center}\n\\end{table*}\n\n\\begin{figure*}[t]\n\\epsscale{1.1}\n\\plotone{f21.eps}\n\\epsscale{1.0}\n\\caption{The distribution of age and metallicity in the best-fit model\nto the spheroid data, making different assumptions about the binary\nfraction, IMF, alpha-enhancement, distance, and reddening. The area\nof the filled circles is proportional to the number of stars in each\nisochrone group. {\\it a)} Our standard model: binary fraction 10\\%,\nSalpeter (1955) IMF, alpha-element enhancement at [Fe\/H]$<-0.7$,\ndistance of 770 kpc, $E(B-V)=0.08$~mag. {\\it b)} A binary fraction\nof 0\\%. {\\it c)} A binary fraction of 20\\%. {\\it d)} A binary\nfraction of 40\\%. {\\it e)} Isochrones at [Fe\/H]$\\leq 0$ are\nalpha-enhanced. {\\it f)} None of the isochrones are alpha-enhanced.\n{\\it g)} An IMF index of $-1.15$. {\\it h)} Distance is 760~kpc. {\\it\ni)} Distance is 780~kpc. {\\it j)} $E(B-V)=0.05$~mag. {\\it k)}\n$E(B-V)=0.11$~mag. }\n\\label{sysfig}\n\\end{figure*}\n\n\\section{Discussion}\n\\label{secdisc}\n\nThe quantitative fitting to the CMDs of the spheroid, stream, and\nouter disk reaffirmed our general impressions from the qualitative\ninspection of the CMDs. In Figure~\\ref{compfits}, we compare the star\nformation histories for the three fields. The star formation history\nin the spheroid is simply our standard model (Figure~\\ref{halofit}),\nwhile the star formation histories in the stream and disk are those\nthat have had an assumed spheroid contamination subtracted\n(Figures~\\ref{strmfit_cont}b and \\ref{diskfit_cont}b). All three\nfields show an extended star formation history. The star formation\nhistory in the stream is similar to that in the spheroid, but is\nshifted somewhat younger. The disk population is dominated by\nintermediate-age stars, with little evidence for the old metal-poor\npopulation present in the spheroid and stream. All three fields have\na trace population of young metal-poor stars, presumably due to the\naccretion of metal-poor stars from dwarf galaxies or due to stars\nforming from the infall of relatively pristine material. The fact\nthat such material continues to fall into Andromeda is evidenced by the\nextensive population of \\ion{H}{1} clouds recently found in the\noutskirts of the galaxy (Thilker et al.\\ 2004).\n\n\\begin{figure}[h]\n\\epsscale{1.1}\n\\plotone{f22.eps}\n\\epsscale{1.0}\n\\caption{The best-fit star formation histories for the\nspheroid ($a$), stream ($b$), and disk ($c$). The area\nof the filled circles ({\\it grey}) is proportional to the\nnumber of stars falling in the given isochrone. For comparison,\nthe star formation history of the spheroid is overplotted in\neach panel ({\\it black open circles}) The stream and disk fits\neach assumed a fixed contamination from the spheroid, which has\nbeen subtracted.\n}\n\\label{compfits}\n\\end{figure}\n\n\\subsection{Disk}\n\nMost hierarchical CDM models predict that a spiral disk forms\ninside-out, generally leading to a disk that becomes progressively\nyounger at increasing radius. For example, the simulated disk of\nAbadi et al.\\ (2003a, 2003b) has a mean age of $\\sim 8$--10~Gyr within\n2 kpc of the center and $\\sim$6--8~Gyr near 20 kpc. However, the\nliterature does include counter-examples with more complex age\ngradients. The simulated galaxy of Robertson et al.\\ (2004) exhibits\na mean stellar age of $\\sim$7.5~Gyr in the center (within 2 kpc) and\n$\\sim$10~Gyr in the disk outskirts (beyond 14 kpc). The CDM models of\nSommer-Larson, G$\\ddot{\\rm o}$tz, \\& Portinari (2003) result in disk\ngalaxies that sometimes form inside-out and sometimes form outside-in.\nBoth classes predict mean ages of 6--8~Gyr in the outer disk (6 scale\nlengths from the center), but the age distributions differ, with the\ninside-out galaxy hosting a significantly larger fraction of young\nstars ($\\lesssim 3$~Gyr) in the outskirts. In a sophisticated model\nof the chemical evolution in the Milky Way disk, Chiappini et al.\\\n(2001) demonstrate an inside-out formation scenario where the stellar\nage is not a monotonically varying function of distance from the\nGalactic center; in the inner disk (4--10 kpc), the stellar ages are\ndecreasing with increasing radius, as expected, but beyond this radius,\nthe stellar ages increase with radius, because the thick disk and halo\nbegin to dominate over the thin disk. All of these models can be\ncompared to the solar neighborhood (e.g., Ibukiyama \\& Arimoto 2002;\nSandage, Lubin, \\& VandenBerg 2003; Fontaine, Brassard, \\& Bergeron\n2001), but we know little of the detailed star formation histories for\nother giant spiral galaxies. As far as the structures are concerned,\nobservations of high-redshift disk galaxies (e.g. Ferguson, Dickinson\n\\& Williams 2000; Ravindranath et al.\\ 2004) suggest that disks were\nlargely in place 8 Gyr ago. Since then, they have increased their\nstellar masses and increased their sizes consistent with an inside-out\nsequence of star formation (Trujillo et al.\\ 2005), with the average\nstellar surface mass density staying roughly constant from $z=1$ to\nthe present (Barden et al.\\ 2005).\n\nOur mean age in the outer disk (6.6~Gyr) is in good agreement with the\nmodels of Abadi et al.\\ (2003a, 2003b), and significantly younger than\nthe models of Robertson et al.\\ (2004); these comparisons\nsuggest a consistency with an inside-out formation scenario. Our mean\nage also falls in the range found in both the inside-out and\noutside-in models of Sommer-Larson et al.\\ (2003), but our age\ndistribution, with a significant dearth of stars younger than 3~Gyr,\nis in somewhat better agreement with their outside-in model. However,\nthese are all hydrodynamical models that track the birth of\nparticles but largely ignore the details of chemical evolution. It\nwould be interesting to compare our age-metallicity distribution with\nsuch a distribution in a true chemical evolution model under an\ninside-out formation scenario (e.g., Chiappini et al.\\ 2001). \n\nOur star formation history in the disk is probably saying less about\nthe validity of the inside-out formation scenario and more about the\nrelative scales of the thin and thick disk; because our disk field is\n25~kpc from the galactic center, it is well into the regime where one\nmight expect the thick disk to dominate (Chiappini et al.\\ 2001).\nIndeed, there is evidence that the thick disk begins to dominate well\ninside this radius; in their WFPC2 images of an off-axis field 5~kpc\nfrom the nucleus, Sarajedini \\& Van Duyne (2001) found a population\napparently dominated by thick disk stars.\nNote that Morrison et al.\\ (2004) apparently found a subsystem of the\nM31 globular cluster system with thin disk kinematics, but this\nsubsystem is largely restricted to that part of the disk plane\ninterior to our own disk field. In Figure~\\ref{diskfit_comp}, we\ncompare the age and metallicity distribution in our disk field to\nthose distributions in the solar neighborhood. The outer disk of\nAndromeda is clearly similar to the thick disk population of the solar\nneighborhood (dominated by intermediate-age stars at relatively high\nmetallicities), but looks nothing like the thin disk of the solar\nneighborhood (dominated by stars younger than 5~Gyr). The hydrogen\ncolumn density in our disk field (Table~\\ref{fieldtab}; Braun et al.\\\nin prep.) is below the threshold typically assumed for star formation\nin disk galaxies ($N_{HI} \\sim 10^{21}$ cm$^{-2}$; Kennicutt 1989),\nand so the dearth of very young stars should not be surprising.\n\n\\begin{figure}[ht]\n\\epsscale{1.1}\n\\plotone{f23.eps}\n\\epsscale{1.0}\n\\caption{{\\it a)} The distribution of age and metallicity in the\nbest-fit model of the disk data (assuming a 33\\% contamination from\nthe spheroid, which has been subtracted). The area of the filled circles is\nproportional to the number of stars in that isochrone group. The\ndistribution shown here is clearly distinct from those shown in\nFigures~\\ref{halofit} and \\ref{strmfit}. {\\it b)} The distribution of\nage and metallicity for individual thick disk stars in the solar\nneighborhood, from the photometric ({\\it open boxes}) and\nspectroscopic ({\\it filled boxes}) measurements of Ibukiyama \\&\nArimoto (2002; their Figure~8). {\\it b)} The distribution of age and\nmetallicity for individual thin disk stars in the solar neighborhood,\nfrom the photometric ({\\it open circles}) and spectroscopic ({\\it\nfilled circles}) measurements of Ibukiyama \\& Arimoto (2002; their\nFigure~5).}\n\\label{diskfit_comp}\n\\end{figure}\n\nOur star formation history in the disk is in rough agreement with that\nfound by other groups studying the outskirts of the disk with\nshallower {\\it HST} data. Looking at a field $\\sim$15$^{\\prime}$\nfurther away from the galaxy center than our own field, Ferguson \\&\nJohnson (2001) found a somewhat older and more metal-poor population;\nthey quoted a mean age $\\gtrsim 8$~Gyr and a metallicity of\n$<$[Fe\/H]$>$~$\\sim -0.7$. They reported trace populations of young\nstars ($\\sim 1.5$--3 $M_\\odot$) and ancient metal-poor stars ($\\gtrsim\n10$~Gyr and [Fe\/H]$\\sim -1.7$), which we also find in our field.\nFerguson \\& Johnson (2001) assumed that disk stars comprised\n$\\sim$95\\% of their field population, based on an extrapolation of\nthe Walterbos \\& Kennicutt (1988) decomposition. During our\nobservation planning, we also used the work of Walterbos \\& Kennicutt\n(1988) as a guide, and estimated that the disk contribution in our own\nfield was similarly high. We were subsequently surprised to find that\nthe kinematic data in our field imply that the disk in fact comprises only\n67\\% of the population (Figure~\\ref{cmdsvels}); it must be even lower\nin the Ferguson \\& Johnson (2001) field. The disk is clearly falling\noff more rapidly than an extrapolation of the Walterbos \\& Kennicutt\n(1988) data from the interior. Note that on the other side of the\ngalaxy, looking in the outer disk near the massive cluster G1, Rich et\nal.\\ (2004) also found a population dominated by intermediate-age\nstars (6--8~Gyr). The dominance of intermediate-age stars in the\nouter disk of Andromeda appears to be ubiquitous.\n\nLooking at fields sampling a wide range of radial distance and\nazimuthal angle in Andromeda, Ibata et al.\\ (2005) found significant\nnumbers of stars moving with velocities close to the expected mean\nvelocity for circular orbits. They found these stars primarily at\ndistances of 15--40~kpc from the center, with possible detections out\nto 70~kpc. Their extended disk has an exponential scale length of\n5.1~kpc, similar to that of the bright inner disk, but its irregular\nmorphology and substructure strongly suggest that it is dominated by tidal\ndebris. They estimate that the luminosity of this ``disk-like\nstructure'' accounts for $\\sim$10\\% of the total luminosity in the M31\ndisk. For reference, their ``F13'' field is near our outer disk field\n($\\sim$10$^{\\prime}$ away), and shows kinematic structures very\nsimilar to those in Figure~\\ref{cmdsvels}c; their data show a narrow\npeak near the velocity expected for stars orbiting in the disk, and a\nmuch broader peak for spheroid stars that show little rotation with\nthe disk. Ibata et al.\\ (2005) argue that their extended disk is more\nlikely associated with the thin disk than the thick disk of Andromeda.\nHowever, given the kinematic and population data in our outer disk field, it\nwould seem more likely that \ntheir disk-like structure is an extension of the thick\ndisk. This would also be consistent with its irregular morphology,\ngiven that thick disks are thought to form via mergers that disrupt\nthe thin disk (see Wyse et al.\\ 2006 and references therein). \\\\\n\n\\subsection{Spheroid and Stream}\n\nAs found by Brown et al.\\ (2003), the Andromeda spheroid population\nspans a surprisingly wide range of age and metallicity, especially\ncompared to the halo of the Milky Way. Given the substructure in\nAndromeda (Ferguson et al.\\ 2002; Figure~\\ref{mapfig}) and the success\nof $\\Lambda$CDM models, we have strong observational and theoretical\nreasons for turning to merger scenarios as possible explanations for\nthe observed distribution of age and metallicity. One can imagine\nthat, compared to the Milky Way, Andromeda has experienced many more\nsmall mergers or a few more large ones. These mergers may have\npolluted the inner spheroid with their own material and material\nfrom the Andromeda disk and bulge; in this scenario, the declining\npresence of this pollution at increasing radius would account\nfor the appearance of the spheroid beyond 30~kpc, which looks\nmore like a canonical metal-poor halo (Guhathakurta et al.\\ 2005; \nIrwin et al.\\ 2005).\n\nIf the Andromeda spheroid is the result of many smaller mergers that\ndid not occur in the Milky Way, one must ask why there is such a\nstatistically significant distinction between the merger histories of\ntwo similarly-sized spirals in the same galaxy group. Is Andromeda\nthe ``normal'' massive spiral, having cannibalized 10 small galaxies\nin its history, while the Milky Way is a 3$\\sigma$ outlier, having\ncannibalized only 1 small galaxy? Alternatively, if the Andromeda\nspheroid was polluted by one large merger that did not occur in the\nMilky Way, one may ask if such a merger is consistent with the\ndisturbed, but not destroyed, Andromeda disk. Plausible merger\nscenarios must balance both of these concerns.\n\nRecent models by Font et al.\\ (2006a) show promise in this regard. In\ntheir various realizations of a spiral galaxy halo, two models stand\nout. One halo underwent a large accretion event ($10^{8-9}$ $M_\\odot$ stellar\nmass) 11 Gyr ago, and the other underwent two accretion events ($10^9$\n$M_\\odot$ stellar mass) $\\sim$8.5~Gyr ago; in the former case, the resulting\nhalo had a lower mean metallicity, with\n$<$[Fe\/H]$>$=$-1.3$, while in the latter case, the resulting\nhalo had a significantly higher mean metallicity, with $<$[Fe\/H]$>$=$-0.9$.\nVel$\\acute{\\rm a}$zquez \\& White (1999) find that, depending upon the\norbit of the infalling satellite, satellites with up to 20\\% of the\ndisk mass can be accreted without destroying the disk. Clearly the\namount of disk disruption spans a continuum of outcomes depending upon\nthe mass of the infalling satellite and its orbit. Given a mass of\n$\\approx 7 \\times 10^{10}$ $M_\\odot$ in Andromeda's disk (Geehan et\nal.\\ 2006), the disk could survive the accretion of one or two\n$\\sim$10$^{9-10}$ $M_\\odot$ satellites that would in turn\nsignificantly increase the spheroid metallicity. It is worth noting\nthat in the Font et al.\\ (2006a) models, when metal-rich stars are\npresent in the spheroid, they are still predominantly old, whereas the\nmetal-rich stars are very clearly of intermediate age in our own data.\nWith only 11 of these computationally-intensive realizations, it\nappears that the Font et al.\\ (2006a) simulations do not sufficiently\npopulate the possible parameter space to demonstrate if these old\nmetal-rich stars are a fluke or a general tendency in the models.\nIn contrast, recent simulations by Renda et al.\\ (2005) show that spiral\ngalaxies with more extended merging histories can have halos that are\nboth younger and metal-rich.\nCould the distinction between the spheroids of the Milky Way and\nAndromeda be due to the ingestion of something like the LMC? There is\nalso evidence that the globular cluster system of Andromeda includes\nclusters much younger than those in our own Galaxy, although it is\ndebatable if these clusters could have originated in the accretion of\nsomething like the LMC (e.g., Puzia et al.\\ 2005; Burstein et al.\\\n2004; Beasley et al.\\ 2005), which hosts a large globular cluster\nsystem spanning a wide range of ages.\n\nAndromeda is not alone in having a metal-rich spheroid with an age\ndispersion. The halo of NGC5128 (Cen A) is metal rich, with\n$<$[Fe\/H]$> = -0.41$ (Harris, Harris, \\& Poole 1999). The presence of\nlong period variables with extremely long periods (Rejkuba et al.\\\n2003) implies the presence of young stars, while the analysis of the\nHB, RGB, and AGB populations found in deep {\\it HST} photometry of the\ngalaxy imply an average age of $\\sim$8~Gyr in its halo (Rejkuba et\nal.\\ 2005). The galaxy also shows evidence for mergers in its shells\nand dust lane (Malin, Quinn, \\& Graham 1983).\n\nThe relatively high metallicity of the stream implies its progenitor\nwas at least as massive as $10^9$ $M_\\odot$ (see Dekel \\& Woo 2003);\nas such, most numerical simulations of the stream assume it is a dwarf\ngalaxy that was only recently disrupted by close passage to Andromeda,\nwithin the last $\\sim$0.5~Gyr (Font et al.\\ 2006b; Fardal et al.\\\n2006). The star formation history in the stream is plausible for such\na progenitor, given the wide range of star formation histories seen in\nLocal Group dwarfs (Mateo 1998). As noted by Brown et al.\\ (2006), it\nwould be worth exploring whether or not the progenitor is a disk\ngalaxy, given that the stream combines a relatively high metallicity\nwith a low velocity dispersion; however, models by Font et al.\\\n(2006b) and Fardal et al.\\ (2006) imply this discrepancy in velocity\ncan perhaps be explained by dynamical cooling.\n\nThe strong similarities between the spheroid and stream populations\noffer another clue, but it is a puzzling one. The field population of\nthe Milky Way halo does not look to be comprised of populations like\nthose of present-day dSphs (Shetrone et al.\\ 2003), but the field\npopulation of the Andromeda spheroid looks nearly identical to that of\none of its infalling satellites. A natural question is whether the\n$10^{9-10}$ $M_\\odot$ merger needed to explain the spheroid data is\nsitting in plain sight: the stream. However, if the progenitor of the\nstream really is on its first or second orbit around the galaxy, with\nmuch of its debris coherent on the sky, it is unlikely to comprise a\nsignificant fraction of the population in the relatively smooth\nregions of the spheroid, such as our field. As noted by Brown et al.\\\n(2006), the star count map of Andromeda (Figure~\\ref{mapfig}) and the\nkinematic data (Figure~\\ref{cmdsvels}) imply that the stream dominates\nover the spheroid by a 3:1 ratio in our stream field, but these same\ndata show no evidence for a single dominant stream in our spheroid\nfield. Current orbit models for the stream span a wide range of\npossibilities (e.g., Font et al.\\ 2006b; Fardal et al.\\ 2006); even if\nthe stream wraps around the Andromeda nucleus and then passes through\nour spheroid field (e.g., Ibata et al.\\ 2004), it is implausible that\nit would spread out enough to hide in the star count maps and\nkinematic data, yet still comprise $\\sim$75\\% of the population in our\nspheroid field. Furthermore, the metallicity distribution in our\nspheroid field is clearly very similar to the metallicity distribution\nin other fields throughout the inner spheroid of Andromeda (Ferguson\net al.\\ 2002; Durrell et al.\\ 1994, 2001, 2004). \nThus, arguments (e.g., Ibata et al.\\ 2004) that the intermediate-age \nmetal-rich stars in our spheroid field simply represent contamination \nby the stream would seem to imply\nthat the inner spheroid is metal-rich and ancient everywhere except\nfor our spheroid field, where the $\\sim$40\\% of the population is\nmetal-rich and of intermediate age. Instead of invoking such a\nconspiracy, it is much more plausible that the high metallicities seen\nthroughout the inner spheroid are associated with intermediate-age\npopulations, as in our particular spheroid field.\n\nThe modeling of the stream's progenitor and its possible orbits is\nstill in the early stages. Can a model be constructed where the\ndebris of the stream progenitor dominates the relatively smooth inner\nspheroid everywhere, while maintaining a coherent tidal tail on the\nsky? At the moment, models for the stream progenitor are focused on a\n$\\sim$10$^9$ $M_\\odot$ dwarf galaxy progenitor that only recently\nmerged with Andromeda (within the last few hundred Myr). How far can\nthe models be pushed away from this scenario? At what point does the\ndisruption of the Andromeda disk exceed the level of substructure seen\nby Ferguson et al.\\ (2002)? Depending upon the orbit, the progenitor\ncould be as massive as a few $10^{10}$ $M_\\odot$ without destroying\nthe Andromeda disk. If the progenitor was significantly more massive\nthan the $10^9$ $M_\\odot$ typically assumed now, and perhaps an\ninfalling disk galaxy, could the start of the merger be pushed\nbackward in time, such that its debris could more fully pollute the\ninner spheroid while still leaving a coherent debris stream on the\nsky? Alternatively, the pollution of the inner spheroid might be due\nto a merger event unrelated to that which produced the stream. The\nrecent models of Penarrubia, McConnachie, \\& Babul (2006) are\ninteresting in this regard; they find that an ancient merger with a\nmassive dwarf (10$^{9-10}$ $M_\\odot$) could produce the extended\ndisk-like population found by Ibata et al.\\ (2005).\n\nBrown et al.\\ (2006) offered two other possible explanations for the\nstream and spheroid similarities, but noted that they were\nproblematic. One possibility is that the spheroid is comprised of\nmany disrupted satellites similar to the stream progenitor. However, it is\ndifficult to see how the ensemble average of these disrupted\nsatellites (the spheroid) would so closely resemble the population\nin a single disrupted satellite (the stream). Although the star\nformation history for the stream is plausible for a dwarf galaxy,\nit is not plausible that it is representative for all dwarf\ngalaxies already cannibalized by Andromeda. Another possibility is\nthat the stream is comprised of material disrupted from the Andromeda\ndisk and that the same event polluted the spheroid, but it is unclear\nif the dynamics and energetics of such a scenario can actually work,\nand the stellar populations in our three fields offer evidence against this\nscenario (Figure~\\ref{compfits}). The isolated disk population\n(removing the spheroid contamination) is dominated by metal-rich\n($-0.5 < $ [Fe\/H] $< +0.5$) intermediate-age (4--8 Gyr) stars. The\nisolated stream population (removing the spheroid contamination), on\nthe other hand, also contains stars that are both older and more\nmetal-poor. If our disk population is representative of the outer\ndisk in general, creating the stream from a disruption of disk material\nwould not result in a stream hosting so many old and metal-poor stars.\nThis does not preclude significant contamination of the spheroid by\ndisrupted disk stars -- the population mix in our spheroid field might\nbe an older metal-poor halo with some contribution of disrupted disk\nstars -- but we are still left with coincidence to explain the\nsimilarity between the stream and spheroid populations.\n\n\\subsection{Does the Disk Contribute to our Spheroid Field?}\n\nRecently, Worthey et al.\\ (2005) put forth a provocative hypothesis,\nbased on chemical evolution arguments and the high metallicity of the\nAndromeda spheroid: that all fields in the spheroid observed to date\nare actually dominated by the disk. They suggested that this\nhypothesis could explain the surprisingly broad range of ages found in\nour spheroid field (Brown et al.\\ 2003). More recently, Ibata et al.\\\n(2005) found stars 40 kpc from the center of Andromeda (in all\ndirections) that appear to be moving in the disk. With the kinematic\nand population information available, we can show that the disk\ncontribution in our spheroid field must be very small ($\\lesssim\n1$\\%), as originally claimed by Brown et al.\\ (2003).\n\nThe relevant data are in Figure~\\ref{cmdsvels} and\nTable~\\ref{fieldtab}. Given the disk inclination of 12.5$^{\\rm o}$,\nour spheroid field is 11 kpc from the galactic center in the plane of\nthe sky and 51 kpc from the center in the plane of the disk. The disk\nfield is 25 kpc from the galactic center in both the plane of the sky\nand the plane of the disk. \n\nFigure~\\ref{cmdsvels}c shows the distribution of velocities in our\ndisk field. There are clearly two components. The broader component\n(comprising $\\sim$1\/3 of the population) is at the systemic velocity\nof Andromeda, while the narrower component is redshifted with respect\nto Andromeda due to the rotation of the Andromeda disk. In the\nWorthey et al.\\ (2005) scenario, one would associate the broad\ncomponent with the thick disk and the narrow component with the thin\ndisk, with only the latter component significantly rotating. However,\nwe know from the disk CMD that there is no evidence for a thin disk\npopulation in this field; instead, the population appears to be\ndominated by a thick disk and spheroid. Thus, it is much more\nplausible that the narrow velocity structure is the thick disk and the\nbroad velocity structure is the spheroid. These designations would\nalso explain why the narrow component is significantly rotating but\nthe broad component is not.\n\nCompared to the disk field, the spheroid field is twice as far from\nthe galactic center in the plane of the disk, but half the distance\nfrom the galactic center in the plane of the sky. So, moving our\nattention from the disk field to the spheroid field, we expect the\ncontribution from the disk to decline and the contribution from the\nspheroid to increase. With an exponential disk scale length of\n$\\approx$5~kpc (Walterbos \\& Kennicutt 1988), the disk contribution\nmust drop from the $\\sim$2\/3 in the disk field to $<1$\\% in the\nspheroid field. Indeed, Figure~\\ref{cmdsvels}a shows no indication of\na single narrow component at the Andromeda systemic velocity, as one\nwould expect if the disk were dominating this position, 51 kpc on the\nminor axis. Furthermore, it is worth noting that the hydrogen column\ndensity in the spheroid field is nearly 25 times smaller than that in the disk\nfield (Table~\\ref{fieldtab}).\n\nIbata et al.\\ (2005) found stars moving with disk velocities at\ndistances of 15--40 kpc from the galactic center, but they note that\nour spheroid field lies beyond the break in the density profile of\ntheir ``disk-like structure.'' They show no evidence that this\nstructure should comprise a significant population in our field. The\nvelocity dispersion in their extended disk is 30 km s$^{-1}$, which is\nmuch narrower than the 80 km s$^{-1}$ we see in our spheroid field.\nThe velocity dispersion in our spheroid field is in agreement with the\nkinematics of the planetary nebulae (Halliday et al.\\ 2006;\nHurley-Keller et al.\\ 2004), which show a distribution of similar\nbreadth and evidence for some rotational support.\n\nAn additional piece of evidence comes from the similarity of the\nstream and spheroid populations, given that the Worthey et al.\\ (2005)\nhypothesis rests largely on metallicity. If metallicity alone were\nenough to prove that a field in Andromeda is dominated by disk stars,\none could try to argue that our stream field was dominated by disk\nstars, too. However, it is clear from the morphology, HB luminosity,\nand kinematics in our stream field that $\\sim$75\\% of the population\nin this field is comprised of two kinematically-cold components\nfalling toward Andromeda (Kalirai et al.\\ 2006b). There is no way that\nthe stream is composed of stars residing in the Andromeda disk.\n\nOn all of these grounds, one can see that the spheroid field must have\na negligible contribution from stars currently moving in the Andromeda\ndisk. It is also clear that the spheroid velocity distribution is not\nas hot as one would expect for a hot halo, nor does it reflect the\nkinematics of the halo globular cluster system ($\\sigma \\sim 150$ km\ns$^{-1}$; Perrett et al.\\ 2002). The high metallicity and wide age\ndistribution of the spheroid is likely due to the merger history of\nAndromeda, with the spheroid polluted by a combination of disrupted\nsatellites, stars born in the merger(s), and stars disrupted from the\nAndromeda disk.\n\n\\section{Summary}\n\\label{secsumm}\n\nUsing deep {\\it HST} observations of Andromeda, we have reconstructed\nthe complete star formation history in three fields: the spheroid,\ntidal stream, and outer disk.\n\nIn the best-fit model to the spheroid, 40\\% of the stars are\nmetal-rich and younger than 10~Gyr, in stark contrast to our own\nGalactic halo. The data cannot be reproduced by a population of old\nstars alone (age~$>$~10~Gyr). Although the fit is dominated by old\nmetal-poor stars and young metal-rich stars, a non-negligible\npopulation of young metal-poor stars is also present, implying that at\nleast some stars in the spheroid were accreted from dwarf galaxies or\nformed from relatively pristine infalling material. Since the\ndiscovery of a metal-rich intermediate-age population in our spheroid\nfield (Brown et al.\\ 2003), various explanations have been put forth\nin the literature, including the hypothesis that the disk dominates\nall inner spheroid fields (Worthey et al.\\ 2005), and the idea that\nour spheroid field is contaminated by the tidal stream and not\nrepresentative of the inner spheroid in general (Ibata et al.\\ 2004).\nIn the former scenario, the spheroid field is not special, but it is\nactually the disk instead of the spheroid, whereas in the latter\nscenario, the field is special, because it is the stream and not the\nspheroid. The constraints provided by the population and kinematic\ndata argue that the spheroid field does not have a significant\ncontribution from stars currently residing in Andromeda's disk, but\nthe young metal-rich population may be the result of stars disrupted\nfrom Andromeda's disk by an earlier merger event. The star count maps\nand kinematic data show no evidence for a dominant stream passing\nthrough the spheroid field, as required to explain the similarity\nbetween the spheroid and stream populations by some chance\nintersection of the spheroid field with the stream's orbit.\nFurthermore, the metallicity distribution in the spheroid field looks\nmuch like that observed in various other fields throughout the inner\nspheroid (Ferguson et al.\\ 2002; Durrell et al.\\ 1994, 2001, 2004).\nIt is much more likely that the metal-rich populations throughout the\ninner spheroid are of intermediate age, as found in our spheroid\nfield, instead of invoking the pathological situation where these\nmetal-rich populations are ancient everywhere except in our spheroid\nfield.\n\nIn the best-fit model to the stream, 70\\% of the stars are younger\nthan 10~Gyr. A detailed comparison of the age and metallicity\ndistributions in the stream and spheroid shows them to be remarkably\nsimilar but distinct. It is unclear if the similarity implies that\nthe stream's progenitor is representative of the objects that formed\nthe inner spheroid or if the entire inner spheroid is polluted by\nstars stripped from the stream's progenitor during its particular\ndisruption. The distinction between the disk and stream populations\n-- with the stream including old metal-poor stars that are lacking in\nthe disk -- suggests that the stream is not comprised of stars\ndisrupted from the Andromeda disk.\n\nThe outer disk of Andromeda more closely resembles the thick disk of\nthe solar neighborhood than either the spheroid or the stream.\nAlthough a trace population of 0.2--1.0~Gyr stars is present, there\nare few stars younger than 4~Gyr, and thus the outer disk does not\nappear to host a significant thin disk component. In the best-fit\nmodel to the disk data, 80\\% of the stars are younger than 10~Gyr;\nindeed, we also showed that these data are consistent with a\npopulation that is completely devoid of stars older than 10~Gyr. The\nminority population of old metal-poor stars in the disk field is\nconsistent with the field's kinematics, which show a $\\sim$33\\%\ncontribution from the spheroid. If the population in this spheroid\ncontribution is assumed to be the same as that in our spheroid field,\nthe resulting model reproduces the data extremely well, and implies\nthat $\\sim$70\\% of the stars in the outer disk are 4--8~Gyr old. The\ndisk of Andromeda clearly shares the ``G dwarf problem'' seen in the\nsolar neighborhood.\n\nIn the upcoming {\\it HST} observing cycle, we will be observing four\nmore deep fields in the Andromeda spheroid. One will be at\n$\\sim$22~kpc on the minor axis, and the other three will be in the\nvicinity of $\\sim$35~kpc on the minor axis, thus bracketing that point\nin the spheroid where there is a transition from a bulge-like\npopulation to one that more closely resembles a canonical halo. The\nstar formation history in these additional fields should help to further\ndisentangle the complex formation history of the Andromeda system\nand its various substructures.\n\n\\acknowledgements\n\nSupport for proposals 9453 and 10265 was provided by NASA through a\ngrant from STScI, which is operated by AURA, Inc., under NASA contract\nNAS 5-26555. P.G. would like to acknowledge partial support from NSF\ngrants AST-0307966 and AST-0507483 and NASA\/STScI grants GO-10265\nand GO-10134. R.M.R. also acknowledges support from NSF grant AST-0307931\nand from NASA\/STScI grants GO-9453 and GO-10265. We are\ngrateful to P.\\ Stetson for providing his DAOPHOT code, and to J.\\\nHarris for providing his StarFish code. During our observation\nplanning, A. Ferguson kindly provided ground images of our fields;\nwe also thank her for providing the star count\nmap used in Figure~\\ref{mapfig}. We wish to acknowledge the\nassistance D. VandenBerg provided in determining the transformation of\nhis isochrones to the ACS bandpasses. D. Taylor, P. Royle, and\nD. Soderblom were enormously helpful during the scheduling and\nexecution of these large {\\it HST} programs. D. Thilker kindly\nprovided $N_{HI}$ values at our field locations using his published\nand unpublished maps of M31. We thank J. Kalirai and D. Reitzel for\nproviding the velocity histograms in Figure~\\ref{cmdsvels}, and\nF. Hammer, A. Font, and M. Fardal for enlightening discussions.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}