diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzatle" "b/data_all_eng_slimpj/shuffled/split2/finalzzatle" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzatle" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nQuantum computing is highly promising for simulating\nchallenging molecules and materials \\cite{bauer2020quantum,mcardle2020quantum}. It is most likely beneficial for systems with the presence of strong correlations where perturbative techniques fail\\cite{cao2019quantum,motta2021emerging}. Unfortunately, current quantum hardware that is called noisy intermediate-scale quantum (NISQ) is limited due to noise and decoherence\\cite{de2021materials}. The variational quantum eigensolver (VQE) has been proposed as a low-depth quantum algorithm \\cite{peruzzo2014variational,mcclean2016theory,kandala2017hardware} to take advantage of NISQ computers. It is a hybrid quantum-classical method that must be carried out on quantum and classical computers. On the quantum computer, quantum states depending on a set of variational parameters are prepared, and the expectation value of the Hamiltonian is then measured. Since most operations on a quantum computer are unitary, unitary coupled-cluster (UCC) wavefunction \\cite{bartlett1989alternative,taube2006new,romero2018strategies,anand2022quantum} has been proposed as a low-circuit-depth state-preparation ansatz for VQE. Next, the set of variational variables is optimized on classical computers, and the loop is repeated until converged. \n\nHowever, the VQE applicability is limited by the dimensionality of many-body systems associated with the number of variational variables and the circuit depths. Recently, many methodologies have been developed to reduce the resources required in VQE. Grimsley and co-workers proposed the adaptive-VQE (ADAPT-VQE) ansatz\\cite{grimsley2019adaptive} that is progressively built by subsequently incorporating into it the operators that contribute most to minimizing the VQE energy towards the ground-state energy. Later on, several groups have extended ADAPT-VQE to make the algorithm more efficient\\cite{tang2021qubit,zhang2021mutual}. While ADAPT-VQE was shown to outperform standard ansatz, its iterative nature makes calculations more costly. Freericks and co-workers, on the other hand, have devised the factorized UCC ansatz for VQE\\cite{fac-UCC,xu2022decomposition} that employs a Taylor expansion in the small amplitudes, trading off circuit depth for additional measurements. Strong correlations were considered by performing the expansion about a small set of UCC factors that are treated exactly. \n\nAlternatively, one can reduce the dimensionality of the many-body Hamiltonian used in VQE by partitioning the whole system into smaller active spaces that can be handled by quantum computing. Karol and co-workers have employed the downfolding framework based on the double UCC (DUCC) to construct effective active-space Hamiltonians \\cite{bauman2019downfolding,kowalski2021dimensionality} that integrate out high-energy Fermionic degrees of freedom while being capable of reproducing exact energy of quantum systems. To this end, one needs to define subsets of excitations either entirely within the subsets considered or involving some external orbitals. The approach can capture the effect of the whole orbital space in small-size active spaces\\cite{metcalf2020resource,chladek2021variational}. Inspired by the divide-and-conquer technique in classical quantum chemistry, Nakagawa and co-workers proposed a method called deep VQE\\cite{fujii2022deep,mizuta2021deep}. In the first step of this method, the whole system is divided into much smaller subsystems, each of which is solved independently using VQE. In the next step, the ground states of subsystems are used as a basis with reduced degrees of freedom to construct an effective Hamiltonian considering the inter-subsystem interactions. The resulting effective Hamiltonian is finally solved using VQE. \n\nQuantum embedding frameworks, such as density matrix embedding theory (DMET)\\cite{mineh2022solving,tilly2021reduced,ralli2022scalable}, dynamical mean-field theory (DMFT)\\cite{keen2020quantum,bauer2016hybrid}, and density functional embedding theory\\cite{ralli2022scalable,gujarati2022quantum}, have been employed to make quantum computation feasible for real molecules and materials. Recently, Rossmannek et al. have demonstrated the performance of the VQE embedding into classical mean-field methods, including Hartree-Fock (HF) and density functional theory (DFT)\\cite{rossmannek2021quantum}. Those authors restricted the quantum computation to a critical subset of molecular orbitals, whereas the remaining electrons provide the embedding potential computed using classical mean-field theories. The proposed embedding schemes obtained significant energy corrections to the HF and DFT reference for several simple molecules in their strongly correlated regime and larger systems of the oxirane size. However, most of those calculations were limited to a minimal basis. It may be interesting to explore the performance of the active-space VQE approach on larger basis sets.\n\nIn the present work, we propose an active-space VQE approximation where VQE is naturally embedded in a correlated mean-field reference. To this end, we start from the UCC ansatz and divide the total excitation operator into internal and external contributions accordingly to the active-inactive partitioning of the orbital space. For the inactive space, instead of HF and DFT, we employ our recently-developed correlated mean-field theory called one-body second-order M{\\o}ller-Plesset perturbation theory (OBMP2)\\cite{OBMP2-JCP2013,tran2021improving}. Unlike standard MP2, OBMP2 is self-consistent, meaning that it can bypass challenges caused by the non-iterative nature of standard MP2. For the active space, we employ VQE to solve the effective Hamiltonian composing of the bare Hamiltonian and a potential caused by the internal-external interaction. Details of the procedure are given in Section~\\ref{sec:theo}. We demonstrate the performance of our approach by considering different systems with singlet and doublet ground states in Section~\\ref{sec:result}. We examine the accuracy in predicting both energy and density matrix (by evaluating dipole moment). We show that the active-space VQE with the correlated reference outperforms the standard active-space VQE. \n\n\\section{Theory}{\\label{sec:theo}}\n\\subsection{Variational quantum eigensolver: UCC ansatz}\n\nVQE relies on the variational principle, which states that the ground-state energy $E_0$ is always less than or equal to the expectation value of Hamiltonian $\\hat{H}$ calculated with the trial wavefunction $\\left|\\psi\\right>$\n\\begin{align}\n E_0 \\leq \\frac{\\left<\\psi\\right| \\hat{H} \\left|\\psi\\right>}{\\left<\\psi\\right|\\left|\\psi\\right>}\n\\end{align}\nwith the molecular Hamiltonian as\n\\begin{align}\n \\hat{H} = \\hat{h} + \\hat{v} = \\sum_{pq}h^{p}_{q} \\hat{a}_{p}^{q} + \\tfrac{1}{2}\\sum_{pqrs}g^{p r}_{q s}\\hat{a}_{p r}^{q s}\\label{eq:h1}\n\\end{align}\nwhere $\\left\\{p, q, r, \\ldots \\right\\}$ indices refer to general ($all$) sin orbitals. The objective of the VQE is to minimize the expectation value of the Hamiltonian with respect to $\\left|\\psi\\right>$. \n\nTo implement this optimization problem on the quantum computer, one has to start by defining a wavefunction ansatz that can be expressed as a series of quantum gates. To this end, we express $\\left|\\psi\\right>$ as the application of a parametrized unitary operator $U(\\boldsymbol \\theta)$ to an initial state $\\left|\\boldsymbol 0 \\right>$ for $N$ qubits, with $\\boldsymbol \\theta$ representing a set of parameters varying values in $\\left(-\\pi, \\pi\\right]$. Given that trial wavefunctions, $\\left|\\psi\\right>$, are necessarily normalized, we can now write the VQE optimization problem as follows.\n\\begin{align}\n E_{\\text{VQE}} = \\min_{\\boldsymbol \\theta} \\left<\\boldsymbol 0\\right| U^\\dagger(\\boldsymbol \\theta) \\hat{H} U(\\boldsymbol \\theta) \\left|\\boldsymbol 0\\right> \\label{eq:E-vqe}\n\\end{align}\n\nThe unitary coupled cluster (UCC) ansatz is perhaps the most widely-used ansatz for VQE and given as.\n\\begin{align}\n \\left|\\psi_\\text{UCC} \\right> = e^{\\hat{A}} \\left|\\boldsymbol 0\\right>,\n\\end{align}\nwhere $\\left|\\boldsymbol 0 \\right>$ is the HF reference and $\\hat{A}$ is an anti-Hermitian combination of particle-hole excitation and de-excitation:\n\\begin{align}\n \\hat{A} &= \\hat{T} - \\hat{T}^\\dagger \\label{eq:A_op}\\\\\n \\hat{T} &= \\sum_{i}^{occ}\\sum_{a}^{vir} T_{i}^{a} \\hat{a}_a^i + \\sum_{ij}^{occ}\\sum_{ab}^{vir} T_{ij}^{ab} \\hat{a}_{ab}^{ij} + ... \\label{eq:T_exc} \n\\end{align}\nwhere $\\left\\{i, j, k, \\ldots \\right\\}$ indices refer to occupied ($occ$) spin orbitals and\n$\\left\\{a, b, c, \\ldots \\right\\}$ indices refer to virtual ($vir$) spin orbitals. \nThe amplitudes $T_{i}^{a}$ and $T_{ij}^{ab}$ are parameterized into rotation angles $\\boldsymbol\\theta$ that are the variationally optimized. Because the computational cost scales exponentially with the system size, the excitation operator is usually truncated at single and double excitations, resulting in UCC singles and doubles (UCCSD).\n\nVQE employs HF wavefunction as a reference, and orbitals are fixed during calculation. However, it is well-known that HF orbitals are not optimal for correlated methods. Recently, several works have proposed the orbital-optimized VQE (OO-VQE) method, in which orbitals are optimized by making the energy stationary with respect to orbital rotation parameters\\cite{sokolov2020-ooVQE,mizukami2020-ooVQE,yalouz2021-SA-ooVQE,ratini2022-WAHTOR}. This approach requires the orbital gradient of VQE energy, demanding additional computational costs. \n\n\\subsection{Correlated mean-field theory: OBMP2}{\\label{sec:obmp2}}\nLet us recap the OBMP2 theory, whose formulation details are presented in Refs.~\\citenum{tran2021improving} and ~\\citenum{OBMP2-JCP2013}. The OBMP2 approach was derived through the canonical transformation \\cite{CT-JCP2006,CT-JCP2007,CT-ACP2007,CT-JCP2009,CT-JCP2010,CT-IRPC2010}, in which an effective Hamiltonian that includes dynamic correlation effects is achieved by a similarity transformation of the molecular Hamiltonian $\\hat{H}$ using a unitary operator $e^{\\hat{A}}$ :\n\\begin{align}\n\\hat{\\bar{H}} = e^{\\hat{A}^\\dagger} \\hat{H} e^{\\hat{A}},\n\\label{Hamiltonian:ct}\n\\end{align}\nwith the anti-Hermitian excited operator $\\hat{A}$ defined as in Eq~\\ref{eq:A_op}. In OBMP2, the cluster operator $\\hat{A}$ is modeled such that including only double excitation. \n\\begin{align}\n \\hat{A} = \\hat{A}_\\text{D} = \\tfrac{1}{2} \\sum_{ij}^{occ} \\sum_{ab}^{vir} T_{ij}^{ab}(\\hat{a}_{ab}^{ij} - \\hat{a}_{ij}^{ab}) \\,, \\label{eq:op1}\n\\end{align}\nwith the MP2 amplitude \n\\begin{align}\n T_{i j}^{a b} = \\frac{g_{i j}^{a b} } { \\epsilon_{i} + \\epsilon_{j} - \\epsilon_{a} - \\epsilon_{b} } \\,, \\label{eq:amp}\n\\end{align}\nwhere $\\epsilon_{i}$ is the orbital energy of the spin-orbital $i$. Using the Baker\u2013Campbell\u2013Hausdorff transformation, the OBMP2 Hamiltonian is defined as\n\\begin{align}\n \\hat{H}_\\text{OBMP2} = \\hat{H}_\\text{HF} + \\left[\\hat{H},\\hat{A}_\\text{D}\\right]_1 + \\tfrac{1}{2}\\left[\\left[\\hat{F},\\hat{A}_\\text{D}\\right],\\hat{A}_\\text{D}\\right]_1.\n \\label{eq:h2}\n\\end{align}\nwith\n\\begin{align}\n \\hat{H}_\\text{HF} &= \\hat{F} + C = \\hat{h} + \\hat{v}_{\\text{HF}} +C \\label{eq:h1hf}\n\\end{align}\nWhere $\\hat{h}$ is the one-electron Hamiltonian defined in Eq~\\ref{eq:h1}. The HF potential $\\hat{v}^{\\text{HF}}$ and the constant $C$ is given as:\n\\begin{align}\n \\hat{v}_{\\text{HF}} &= \\sum_{pq}^{all}\\sum_{i}^{occ}\\left(g^{p i}_{q i} - g^{p i}_{i q} \\right) \\label{eq:vhf}\\\\\n C &= \\,\\, \\sum_{ij}^{occ} \\left(g^{ij}_{ji} - g_{ij}^{ij} \\right) \\,. \n\\end{align}\nIn Eq.\\ref{eq:h2}, commutators with the subscription 1, $[\\ldots]_1$, involve one-body operators and constants that are reduced from many-body operators using the cumulant approximation\\cite{cumulant-JCP1997,cumulant-PRA1998,cumulant-CPL1998,cumulant-JCP1999}. Doing some derivation, we eventually arrive at the OBMP2 Hamiltonian as follows\n\\begin{align}\n \\hat{H}_\\text{OBMP2} = & \\,\\, \\hat{H}_\\text{HF} + \\hat{V}_\\text{OBMP2} \\label{eq:h4}\n\\end{align}\nwhere $\\hat{v}_\\text{OBMP2}$ is a correlated potential composing of one-body operators. The working expression is given as\n\\begin{align}\n\\hat{V}_{\\text{OBMP2}} = & \\overline{T}_{i j}^{a b} \\left[ f_{a}^{i} \\,\\hat{\\Omega}\\left( \\hat{a}_{j}^{b} \\right) \n + g_{a b}^{i p} \\,\\hat{\\Omega} \\left( \\hat{a}_{j}^{p} \\right) - g^{a q}_{i j} \\,\\hat{\\Omega} \\left( \\hat{a}^{b}_{q} \\right) \\right] \\nonumber \\\\ &- 2 \\overline{T}_{i j}^{a b}g^{i j}_{a b} \n + \\,f_{a}^{i}\\overline{T}_{i j}^{a b}\\overline{T}_{j k}^{b c} \\,\\hat{\\Omega} \\left(\\hat{a}_{c}^{k} \\right) \\nonumber \\\\ \n &+ f_{c}^{a}T_{i j}^{a b}\\overline{T}_{i l}^{c b} \\,\\hat{\\Omega} \\left(\\hat{a}^{l}_{j} \\right) + f_{c}^{a}T_{i j}^{a b}\\overline{T}_{k j}^{c b} \\,\\hat{\\Omega} \\left(\\hat{a}^{k}_{i} \\right) \\nonumber \\\\ \n &- f^{k}_{i}T_{i j}^{a b}\\overline{T}_{k l}^{a b} \\,\\hat{\\Omega} \\left(\\hat{a}_{l}^{j} \\right)\n - f^{p}_{i}T_{i j}^{a b}\\overline{T}_{k j}^{a b} \\,\\hat{\\Omega} \\left(\\hat{a}^{p}_{k} \\right) \\nonumber \\\\ \n & + f^{k}_{i} T_{i j}^{a b}\\overline{T}_{k j}^{a d} \\,\\hat{\\Omega}\\left(\\hat{a}_{b}^{d} \\right) + f_{k}^{i}T_{i j}^{a b}\\overline{T}_{k j}^{c b} \\,\\hat{\\Omega} \\left(\\hat{a}_{a}^{c} \\right) \\nonumber \\\\ \n &- f_{c}^{a}T_{i j}^{a b}\\overline{T}_{i j}^{c d} \\,\\hat{\\Omega} \\left(\\hat{a}^{b}_{d} \\right) \\,\n - f_{p}^{a}T_{i j}^{a b}\\overline{T}_{i j}^{c b} \\,\\hat{\\Omega} \\left(\\hat{a}^{p}_{c} \\right) \\nonumber \\\\\n & - 2f_{a}^{c}{T}_{i j}^{a b}\\overline{T}_{i j}^{c b} + 2f_{i}^{k}{T}_{i j}^{a b}\\overline{T}_{k j}^{a b}. \\label{eq:vobmp2} \n\\end{align}\nwith $\\overline{T}_{ij}^{ab} = {T}_{ij}^{ab} - {T}_{ji}^{ab}$, the symmetrization operator $\\hat{\\Omega} \\left( \\hat{a}^{p}_{q} \\right) = \\hat{a}^{p}_{q} + \\hat{a}^{q}_{p}$, and the Fock matrix \n\\begin{align}\n f_p^q = h_p^q + \\sum_{i}^{occ}\\left(g^{p i}_{q i} - g^{p i}_{i q} \\right).\n\\end{align}\nNote that, for convenience, we have used Einstein's convention in Eq.~\\ref{eq:vobmp2} to present the summations over repeated indices. We rewrite $\\hat{H}_\\text{OBMP2}$ (Eqs. \\ref{eq:h2} and \\ref{eq:h4}) in a similar form to Eq. \\ref{eq:h1hf} for $\\hat{H}_\\text{HF}$ as follows:\n\\begin{align}\n \\hat{H}_\\text{OBMP2} = & \\hat{\\bar{F}} + \\bar{C} \\label{eq:h5}\n\\end{align}\nwith $\\hat{\\bar{F}} = \\bar{f}^{p}_{q} \\hat{a}_{p}^{q}$.\n$\\bar{f}^{p}_{q}$ is so-called {\\it correlated} Fock matrix and written as\n\\begin{align}\n\\bar{f}^{p}_{q} &= f^{p}_{q} + v^{p}_{q}. \\label{eq:corr-fock\n\\end{align}\n$v^{p}_{q}$ is the matrix representation of the one-body operator $\\hat{V}_{\\text{OBMP2}}$, serving as the correlation potential altering the uncorrelated HF picture. We update the MO coefficients and energies by diagonalizing the matrix $\\bar{f}^{p}_{q}$, leading to orbital relaxation in the presence of dynamic correlation effects. The OBMP2 method is implemented within a local version of PySCF\\cite{pyscf-2018}. \n\\subsection{VQE with OBMP2 reference}\n\nWe can see that both UCC and OBMP2 are formulated using a unitary exponential operator $e^{\\hat{A}}$ (Eqs~\\ref{eq:A_op} and \\ref{eq:op1}), implying that one can combine these two naturally. Partitioning the whole orbital space into active and inactive spaces, one can decompose excitation operators into internal and external contributions as:\n\\begin{align}\n \\hat{A} = \\hat{A}_{\\text{int}} + \\hat{A}_{\\text{ext}}.\n\\end{align}\nHere, the internal (int) defines excitations within active space, and the external (ext) is the remaining excitations involving at least one inactive space orbital. \n\nThe total energy can be written as:\n\\begin{align}\n E &= \\left< \\boldsymbol 0 \\right| e^{\\hat{A}^\\dagger} \\hat{H} e^{\\hat{A}} \\left| \\boldsymbol 0 \\right> \\\\\n &= \\left< \\boldsymbol 0 \\right| e^{\\hat{A}_{\\text{int}}^\\dagger} e^{\\hat{A}_{\\text{ext}}^\\dagger} \\hat{H} e^{\\hat{A}_{\\text{ext}}} e^{\\hat{A}_{\\text{int}}} \\left| \\boldsymbol 0 \\right>\n\\end{align}\nIn the standard active space approximation, the external contribution is zero. Here, we treat it using OBMP2 described in the subsection~\\ref{sec:obmp2}. We arrive then at \n\\begin{align}\n E &= \\left< \\boldsymbol 0 \\right| e^{\\hat{A}_{\\text{int}}^\\dagger} \\hat{H}^{\\text{act}}_{\\text{eff}} e^{\\hat{A}_{\\text{int}}} \\left| \\boldsymbol 0 \\right> + E_{\\text{OBMP2}}^{\\text{ext}} \\label{eq:ene-part}\n\\end{align}\nwhere $\\hat{H}^{\\text{act}}_{\\text{eff}}$ is the effective Hamiltonian within the active space including the bare Hamiltonian $\\hat{H}^{\\text{act}}$ and an effective one-body potential $v^{\\text{act}}_{\\text{eff}}$ from the remaining electrons outside this space: \n\\begin{align}\n \\hat{H}^{\\text{act}}_{\\text{eff}} &= \n \\hat{H}^{\\text{act}} + \\hat{v}^{\\text{act}}_{\\text{eff}},\\\\\n \\hat{v}^{\\text{act}}_{\\text{eff}} &= \\hat{v}^{\\text{act}}_{\\text{HF}} + \\hat{V}^{\\text{act}}_{\\text{OBMP2}}.\n\\end{align}\nIn the present implementation, we have not included $\\hat{V}_{\\text{OBMP2}}^{\\text{act}}$ in the active-space Hamiltonian. Instead, the active-inactive interaction is buried in the external contribution determined by: \n\\begin{align}\n E_{\\text{OBMP2}}^{\\text{ext}} = E_{\\text{OBMP2}}^{\\text{tot}} - E_{\\text{OBMP2}}^{\\text{act}} \n\\end{align} \nwith $E_{\\text{OBMP2}}^{\\text{tot}}$ the total OBMP2 energy of the whole system and $E_{\\text{OBMP2}}^{\\text{act}}$ the OBMP2 energy of the active space. \n\nCalculations start by running OBMP2 for the whole system and selecting active space using OBMP2 orbitals. VQE is then used for the active space with the effective Hamiltonian. We use UCC with doubles (UCCD) as the ansatz in the present work. The total double-excitation amplitude for the whole system $T_2$ is the summation of internal and external contributions: \n\\begin{align}\n T_2 = T_2^{\\text{int}}+T_2^{\\text{ext}}. \\label{eq:total-amp} \n\\end{align}\nThe density matrix needed for properties is then evaluated using the total amplitude. It is essential to remind that, unlike standard MP2, the OBMP2 amplitude is relaxed through a self-consistency, bypassing issues caused by the non-self-consistent nature of standard MP2. The classical calculation is carried out using PySCF\\cite{pyscf-2018}, and the quantum part is done using the Qiskit package\\cite{Qiskit}.\n\n\\section{Results and discussion}{\\label{sec:result}}\n\\subsection{Full-space VQE with OBMP2 orbitals}\n\n\\begin{figure}[t!]\n \\includegraphics[width=8cm,]{H4chain-sto.png}\n \\caption{Potential energy curves of H$_4$ chain in STO-6G. VQE was performed for the full orbital space.}\n \\label{fig:h4c-sto}\n\\end{figure}\n\n\\begin{figure}[t!]\n \\includegraphics[width=8cm,]{LiH-sto.png}\n \\caption{Potential energy curves of LiH in STO-6G. VQE was performed for the full orbital space.}\n \\label{fig:lih-sto}\n\\end{figure}\n\nSeveral authors have shown that orbital relaxation is important to reduce VQE errors \\cite{sokolov2020-ooVQE,mizukami2020-ooVQE,yalouz2021-SA-ooVQE,ratini2022-WAHTOR}. In these studies, the energy of VQE is minimized concerning both cluster amplitudes and orbitals, resulting in a self-consistency that demands higher computational costs than standard VQE. It is thus interesting to examine whether correlated orbital reference preoptimized using a lower-level method can improve the accuracy of ``single-shot'' VQE. Here, we performed VQE only once on OBMP2 orbitals that are relaxed in the presence of dynamical correlation at the MP2 level. In fact, since MP2 is the first-order amplitude truncation of coupled-cluster doubles (CCD), one can consider OBMP2 is an approximation to orbital-optimized CCD (OO-CCD) \\cite{mizukami2020-ooVQE,sokolov2020-ooVQE}.\n\nIn Figure~\\ref{fig:h4c-sto}, we plot the potential energy curves of the H$_4$ chain in the STO-6G basis. VQE with the UCCD solver for the entire orbital space was performed using HF and OBMP2 orbitals. While both cases are almost identical and close to the FCI reference when $R < 1.5$\\r{A}, errors become more prominent as stretching the distance. Noticeably, VQE with the OBMP2 orbitals is better than that with the HF orbitals, and its error is twice smaller than the HF counterpart. We consider another example, LiH in the STO-6G basis. As shown in Figure~\\ref{fig:lih-sto}, VQE again benefits from correlated orbital reference. In general, OBMP2 orbital reference can help to reduce VQE errors without any additional costs. Using OBMP2 orbitals as a basis may be a good approximation to OO-CCD. \n\n\\subsection{Active-space VQE with restricted OBMP2}\n\n\\begin{figure}[t!]\n \\includegraphics[width=8cm,]{H4chain-ccpvdz.png}\n \\caption{Potential energy curves of H$_4$ chain in cc-pVDZ. VQE was performed in the active space of four orbitals.}\n \\label{fig:h4c-dz}\n\\end{figure}\n\n\\begin{figure}[h!]\n \\includegraphics[width=8cm,]{H4square-ccpvdz.png}\n \\caption{Potential energy curves of H$_4$ square in cc-pVDZ. VQE was performed in the active space of four orbitals .}\n \\label{fig:h4s-dz}\n\\end{figure}\n\nThis subsection considers the active-space approximation for molecules with the ground state singlet. The restricted OBMP2 is used as the reference for VQE. \n\n\\begin{figure*}[t]\n \\includegraphics[width=14cm,]{LiH-ccpvdz.png}\n \\caption{Potential energy curves of LiH in cc-pVDZ with different active spaces for VQE.}\n \\label{fig:lih-dz}\n\\end{figure*}\n\nFigure~\\ref{fig:h4c-dz} represents potential energy curves of H$_4$ chain in cc-pVDZ. VQE was performed within an active space of four orbitals. For comparison, we also plot HF and OBMP2 curves. Due to the lack of dynamic correlation outside the active space, VQE is far from the FCI reference. While OBMP2 is better than VQE around equilibrium, its errors are significantly larger than VQE errors at long distances. Also, the VQE curve is much more parallel to FCI than OBMP2, resulting from a proper description of static correlation in VQE. Capturing dynamical correlation outside the active space, VQE-in-OBMP2 dramatically outperforms VQE and yields errors relative to FCI much smaller than those of VQE. Interestingly, the non-parallelity error (NPE), defined as the difference between the minimum and maximum errors, is smaller for VQE-in-OBMP2 than VQE. \n\nThe next system we consider is the H$_4$ ring as depicted in the inset of Figure~\\ref{fig:h4s-dz}. We analyze the change of energy when the four atoms move along the circumference with radius $R=1.8$\\r{A} by varying the angle $\\theta$. The square geometry corresponds to $\\theta = 90^{\\circ}$. We perform VQE calculation within the active space of four valence orbitals. For comparison, we also present the CCSD result. As we can see in Figure ~\\ref{fig:h4s-dz}, while CCSD agrees very well with the FCI reference for $\\theta$ far from $90^{\\circ}$, it fails to predict the convexity of the energy around $90^{\\circ}$. In contrast, VQE with UCC ansatz can properly describe the energy profile, which is consistent with the work of Sokolov {\\it et al.} \\cite{sokolov2020-ooVQE}. Harsha {\\it et al.} showed that the variational UCC is superior to standard CC, particularly when strong electron correlation is involved\\cite{harsha2018difference}. Due to the lack of dynamical correlation outside the active space, there is a large up-shift of the VQE curve with a maximum error of 40 mHa. When the correlation outside the active space is taken into account, VQE-in-OBMP2 can significantly improve upon standard VQE with a maximum error four times smaller than VQE. \n\nWe now examine the systematic improvement with respect to the size of the active space. To this end, we consider LiH in cc-pVDZ and gradually enlarge the active space by adding $\\sigma$-type orbitals. Here, the core orbital Li $1s$ is not included in the active space and treated at the (correlated) mean-field level. All results are summarized in Figure~\\ref{fig:lih-dz}. We can see a systematic improvement when the size of the active space increases. However, the largest active space of seven orbitals is still insufficient for standard VQE to achieve satisfying accuracy. While retaining the systematic improvement with respect to the size of active space, VQE-in-OBMP2 can dramatically reduce errors. Its NPEs are also smaller than those of standard VQE. With the largest active space considered here, VQE-in-OBMP2 can approach closer to the FCI reference with a maximum error of 6 mHa. \n\n\\subsection{Active-space VQE with unrestricted OBMP2}\n\n\\begin{figure*}[t!]\n \\includegraphics[width=8cm,valign=t]{CH_pes_dz.png}\n \\includegraphics[width=8cm,valign=t]{CH_dipole_dz.png}\n \\caption{Left: potential energy curves of CH with the ground state doublet. Right: The change of CH dipole moment. The basis set is cc-pVDZ. VQE was performed in the active space of five orbitals.}\n \\label{fig:ch-pes-dip}\n\\end{figure*}\n\nThis subsection considers two systems with the ground state doublet (e.g., having one unpaired electron): CH and H$_2$O$^+$. The unrestricted HF (UHF) and OBMP2 (UOBMP2) are used as the reference for VQE. In addition to potential energy curves, we also calculate dipole moments that are a direct measure of the density matrix. The cc-pVDZ basis set is used for all calculations. \n\nFigure~\\ref{fig:ch-pes-dip} represents the potential energy curves and dipole moments of CH from different methods and their errors relative to the CCSD reference. VQE is performed in the active space of five orbitals, composed of C $2s2p$ and H $1s$. Although unrestricted HF (UHF) can describe the dissociation adequately, a large NPE is observed due to a bump at the unrestricted point 1.5\\r{A}. Standard VQE performed on UHF can reduce errors and parallel the curve to the FCI reference. When VQE is performed with the UOBMP2 reference, the errors in energy dramatically decrease. We plot the change of dipole moments by stretching the C--H bond in the right panel of Figure~\\ref{fig:ch-pes-dip}. All the methods yield curves that behave similarly to the CCSD reference. VQE-in-OBMP2 predicts dipole moment closer to the CCSD reference than standard VQE for $R < 1.5 \\r{A}$, indicating the importance of dynamic correlation in the accurate prediction of density-related properties. However, its errors are still significant in the stretched regime, demanding the enlargement of the active space to get more accurate dipole moments. \n\\begin{figure*}[t!]\n \\includegraphics[width=8cm,valign=t]{H2O_pes_dz.png}\n \\includegraphics[width=8cm,valign=t]{H2O_dipole_dz.png}\n \\caption{Left: potential energy curves of H$_2$O$^+$ with the ground state doublet. Right: The change of H$_2$O$^+$ dipole moment. The basis set is cc-pVDZ. VQE was performed in the active space of six orbitals.}\n \\label{fig:h2o-pes-dip}\n\\end{figure*}\n\nLet us now consider H$_2$O$^+$ for which VQE is performed in the active space of six orbitals composed of O $2s2p$ and H $1s$. Its potential energy curves and dipole moments evaluated using different methods are depicted in Figure~\\ref{fig:h2o-pes-dip}. As for the potential energy curves, although VQE with six orbitals in the active space does not significantly improve UHF, it makes the curve more parallel to the CCSD reference, in particular, at the unrestricted point ($R \\approx 1.3 \\r{A}$). When combined with UOBMP2, VQE yields the curve much closer to the reference and further reduces NPE. As for dipole moments, the VQE-in-OBMP2 curve is overall closest to the CCSD reference. It agrees very well with CCSD for short distances ($R < 1.5\\r{A}$). However, it deviates from the reference at long distances, requiring a larger active space for VQE. \n\n\\section{Conclusion}\nWe have proposed an active space approximation in which VQE is naturally embedded in a correlated mean-field reference derived from the UCC ansatz. By partitioning the whole orbital space into active and inactive spaces, we divided the total excitation operator in the UCC ansatz into internal and external contributions. The inactive space is treated using OBMP2, a correlated mean-field theory recently developed by us. The effective Hamiltonian for the active space is derived as the sum of the bare Hamiltonian in the active space and a potential describing the internal-external interaction. Considering different systems with singlet and doublet ground states in the minimal and larger basis sets, we demonstrated the accuracy of our approach in predicting energies and dipole moments. We show that the VQE with the OBMP2 reference significantly improves upon the standard active-space VQE with the uncorrelated HF reference. \n\nOur proposed approach is generally applicable to different types of UCC ansatz, such as generalized UCC\\cite{lee2018generalized}, paired UCC\\cite{lee2018generalized,stein2014seniority}, and pair-natural orbital-UCC\\cite{kottmann2021reducing}. We believe it is useful to study real chemistry and materials on quantum computers. Further work is planned to develop more sophisticated schemes of active-space selection to treat systems with large active spaces. For example, one can split the active space into smaller subspaces and treat them independently using VQE as has been done in quantum embedding methods \\cite{welborn2016bootstrap,wouters2016practical,seet-jctc2016,seet-jpcl2017}. In the current work, orbitals are only optimized in OBMP2, and VQE is performed as a \"single-shot\" calculation. We are now implementing a fully self-consistent scheme, in which the total amplitude (Eq.~\\ref{eq:total-amp}) is used to construct correlated Fock (Eq.~\\ref{eq:corr-fock}). Molecular orbitals and orbital energies are then relaxed in the presence of both UCC and MP2 correlations. We hope to report these results in near-future publications.\n\n\\section*{Acknowledgments}\nThis work is supported by the Vietnam Academy of Science and Technology (VAST) under the grant number CSCL14.01\/22-23. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIf a random walk is started in a known state and run for several steps, one may examine the probabilities that it is now in each possible state. A likelihood order is a partial order on the state space, so that if one state is larger than another, then the random walk is always more likely to be in the former state than the latter, after any number of steps. The main result of this paper is that for any Coxeter system $(W,S)$, the weak Bruhat order is a likelihood order for the simple random walk on $W$ generated by $S \\cup \\{1_W\\}$. \n\n\\begin{Theorem}\n\\label{the:main}\nFor any Coxeter system $(W,S)$, consider the simple random walk on $W$, starting at the identity, and at each step multiplying on the right by an element of $S$ or by the identity, each with probability $\\frac{1}{|S|+1}$. Then for any $n$, and any two states $w$ and $w'$, if $w \\leq_B w'$ then the probability that the random walk is at $w$ after $n$ steps is at least the probability that it is at $w'$. \n\\end{Theorem}\n\nIn type A, this result describes a partial order for the (appropriately lazy) adjacent transposition walk on the symmetric group. For analysis of the mixing time of this walk, see Section 4 of \\cite{Compgroups}. The adjacent transposition walk is a special case of the interchange process \\cite{Interchange}.\n\nLikelihood orders can describe the most and least likely states of a random walk. In particular, if a random walk has uniform stationary distribution, then the separation distance from the stationary distribution depends only on the probability of being at the least likely state. Thus, knowing which state is the least likely, together with a lower bound on the probability of being at that state, produces an upper bound on the separation distance mixing time. Upper bounds on separation distance give upper bounds on total variation distance, so upper bounds on total variation mixing times follow. \n\nLower bounds on mixing times are obtained by analysing a set of unlikely states. Knowledge of which states are the least likely via a likelihood order can inform the choice of such a set. For instance, see Section 4 of \\cite{Thumb}.\n\nSome results regarding likelihood orders for various random walks on the symmetric group $S_n$ are given in \\cite{MeganLikelihood} and \\cite{MeganInvolutions}. These likelihood orders are shown to hold after enough steps (for example, after $O(n^2)$ steps). Likelihood orders have also been considered by Diaconis and Isaacs in \\cite{DiaconisIsaacs}, where they prove that for any symmetric random walk on a group, after any even number of steps the most likely state is the initial one. They also give likelihood orders for several random walks on the cycle.\n\n\\section{Preliminaries}\n\nIn this section, some necessary background results are recalled. Section \\ref{sec:cayley} defines Cayley graphs and a useful family of their symmetries, and Section \\ref{sec:distances} discusses the sets of vertices in the Cayley graph which are closer to one end of a given edge than to the other. Section \\ref{sec:bruhat} defines the weak Bruhat order.\n\nA Coxeter system $(W,S)$ is a group $W$ together with a presentation of a certain form.\n\n\\begin{Definition}\nA \\emph{Coxeter presentation} is a presentation of the form $$\\pres{s_1,s_2,\\dots,s_n}{\\{s_i^2\\}_{i=1}^n, \\{(s_is_j)^{m_{ij}}\\}_{i \\neq j}}$$ where each $m_{ij}, i \\neq j$ is either a positive integer at least two, or $\\infty$, indicating the lack of that relation. \n\\end{Definition}\n\nA good example of a Coxeter group is the symmetric group $S_n$, which has the Coxeter presentation $$S_n = \\left\\langle s_1, s_2, \\dots, s_{n-1}\\left|\\begin{array}{lcl}\ns_i^2 & \\text{for each} & 1 \\leq i \\leq n-1 \\\\\n(s_is_{i+1})^3 & \\text{for each} & 1 \\leq i \\leq n-2 \\\\\ns_is_js_i^{-1}s_j^{-1} & \\text{if} & |i-j| > 1 \\\\\n\\end{array}\\right.\\right\\rangle.$$ \n\nIn this presentation, the generator $s_i$ represents the transposition $(i \\; i+1)$. This presentation has $m_{ij} = 3$ when $|i-j|=1$ and $m_{ij} = 2$ for $|i-j|>1$.\n\n\\subsection{Cayley graphs}\n\\label{sec:cayley}\n\nGiven a group $W$ and a generating set $S$, the Cayley graph $\\Gamma(W,S)$ is defined as follows\n\n\\begin{Definition}\nThe graph $\\Gamma(W,S)$ has a vertex for each element of $W$, and for each $w \\in W$ and each $s \\in S$, there is an edge from $w$ to $ws$. It will often be convenient to label the edge $(w,ws)$ by the generator $s$. \n\\end{Definition} \n\nIn the present setting, groups will always be generated by elements of order two, so Cayley graphs will be undirected.\n\nIt will be necessary to have the following results regarding certain symmetries of Cayley graphs.\n\n\\begin{Definition}\nConsider a Cayley graph $\\Gamma(W,S)$. For any $x \\in W$, let $L_x$ be the left multiplication map on $\\Gamma(W,S)$ which takes $w$ to $xw$, for each $w \\in W$.\n\\end{Definition}\n\n\\begin{Proposition}\nFor any Cayley graph $\\Gamma(W,S)$ and any $x \\in W$, the map $L_x$ is an automorphism of $\\Gamma(W,S)$. Further, $L_x$ preserves the edge labels of $\\Gamma(W,S)$.\n\\end{Proposition}\n\\begin{proof}\nThe map $L_{x^{-1}}$ is the inverse of $L_x$, so $L_w$ is a bijection. To check that $L_x$ preserves edges of $\\Gamma(W,S)$, observe that for each edge $(w,ws)$ of $\\Gamma(W,S)$, the image under $L_x$, $(xw,xws)$, is also an edge of $\\Gamma(W,S)$, and that these two edges have the same label.\n\\end{proof}\n\nRandom walks on the group $W$ can be understood via the Cayley graph. In particular, if a random walk is defined by at each step multiplying by an element of $S$, then consider the set of paths in $\\Gamma(W,S)$ of length $n$ which start at the identity. The probability $P^n(w)$ that the walk is at $w$ after $n$ steps is equal to the proportion of these paths which end at $w$. Lazy walks can be considered by including $1_W$ in $S$.\n\n\\subsection{Distances in $\\Gamma(W,S)$}\n\\label{sec:distances}\n\nIt will be important to understand relative distances in the Cayley graphs of Coxeter groups. Consider the Cayley graph with the usual graph metric --- that is, each edge has length $1$, and the distance $d(w,x)$ between two vertices $w$ and $x$ is the number of edges in the shortest path connecting them. As usual in the theory of Coxeter groups, $l(w)$ will denote the distance from the identity $d(1_W,w)$. Equivalently, $l(w)$ is the fewest number of generators which can be multiplied to produce $w$.\n\nFor this section, fix $w \\in W$ and $s \\in S$, with $l(w) < l(ws)$. That $l(w) < l(ws)$ is not used in this section, but is consistent with how these results will be used in Section \\ref{sec:main}.\n\n\\begin{Definition}\n\\label{def:colours}\nLet $\\Gamma(W,S)$ be the Cayley graph of a Coxeter system $(W,S)$. For the fixed adjacent vertices $w$ and $ws$ of $\\Gamma(W,S)$, colour each vertex of $\\Gamma(W,S)$ white if it is closer to $w$ than to $ws$ and black if it is closer to $ws$ than to $w$. \n\\end{Definition}\n\n\\begin{Proposition}\nEach vertex of $\\Gamma(W,S)$ is coloured white or black, but not both.\n\\end{Proposition}\n\\begin{proof}\nEach vertex of $\\Gamma(W,S)$ has at most one colour, because it cannot be both closer to $w$ than to $ws$ and the reverse. To show that each vertex is coloured, it is sufficient to show that no vertex can be equidistant from $w$ and $ws$.\n\nThe Coxeter relations of $(W,S)$ each have even length, so $\\Gamma(W,S)$ is a bipartite graph, and hence the distances from any vertex to $w$ and $ws$ have opposite parities. Thus each vertex is coloured, completing the proof.\n\\end{proof}\n\n\\begin{Definition}\nContinuing from Definition \\ref{def:colours}, colour grey each edge which connects a white vertex to a black vertex. \n\\end{Definition}\n\n\\begin{Lemma}\n\\label{lem:greydist}\nIf $(x,xt)$ is a grey edge, with $x$ white and $xt$ black, then $d(x,w) = d(xt,ws)$. (The generator $t$ may be equal to $s$, but need not be.)\n\\end{Lemma}\n\\begin{proof}\nThe two vertices $w$ and $ws$ are adjacent, as are the vertices $x$ and $xt$. The vertex $x$ is white, and $xt$ is black. Thus, the following relations between distances hold \n\\begin{align*}\nd(x,w) + 1 &= d(x,ws) \\\\ \nd(xt,ws) + 1 &= d(xt,w) \\\\\nd(x,ws) &= d(xt,ws) \\pm 1\\\\\nd(xt,w) &= d(x,w) \\pm 1\\\\\n\\end{align*} \nAdding these four equations, each of the two $\\pm$ signs must be a plus. Thus $d(x,w) = d(xt,ws)$, as required.\n\\end{proof}\n\n\\begin{Lemma}\n\\label{lem:greypath}\nUnder the conditions of Lemma \\ref{lem:greydist}, $w^{-1}x = sw^{-1}xt$.\n\\end{Lemma}\n\\begin{proof}\nLet $\\boldsymbol{\\omega}$ be a reduced word for $w^{-1}x$. From Lemma \\ref{lem:greydist}, $l(sw^{-1}xt) = l(w^{-1}x)$. Thus, $s\\boldsymbol{\\omega} t$ is a word of length two greater than the minimum length of any equivalent word. By the deletion condition (Section 1.7 of \\cite{Humphreys}), there is a reduced word for $sw^{-1}xt$ which can be obtained by deleting two letters from $s\\boldsymbol{\\omega} t$. However, Lemma \\ref{lem:greydist} also implies that the words $s\\boldsymbol{\\omega}$ and $\\boldsymbol{\\omega} t$ are reduced, so the two letters removed from the word $s\\boldsymbol{\\omega} t$ must be the initial $s$ and the final $t$. Therefore $s\\boldsymbol{\\omega} t$ and $\\boldsymbol{\\omega}$ are equivalent words, so $w^{-1}x = sw^{-1}xt$.\n\\end{proof}\n\n\\begin{Proposition}\n\\label{prop:greyflip}\nThe map $L_{wsw^{-1}}$ interchanges the endpoints of any grey edge.\n\\end{Proposition}\n\\begin{proof}\nLet $(x,xt)$ be a grey edge, with $x$ white and $xt$ black. The image $L_{wsw^{-1}}(x)$ is\n\\begin{align*}\nL_{wsw^{-1}}(x) &= wsw^{-1}x \\\\\n& = wssw^{-1}xt \\text{ (by Lemma \\ref{lem:greypath})} \\\\\n& = xt \\\\\n\\end{align*}\nThe map $L_{wsw^{-1}}$ is an involution, so $L_{wsw^{-1}}(xt) = x$.\n\\end{proof}\n\n\\begin{Corollary}\nAny vertex of $\\Gamma(W,S)$ is adjacent to at most one grey edge.\n\\end{Corollary}\n\\begin{proof}\nThe function $L_{wsw^{-1}}$ is well defined. If any vertex were adjacent to more than one grey edge, then Proposition \\ref{prop:greyflip} implies that $L_{wsw^{-1}}$ is multivalued, a contradiction.\n\\end{proof}\n\nThe results in this section are not new --- they are standard facts about the geometry of $\\Gamma(W,S)$, proven again here to draw out the key pieces. The set of grey edges is commonly referred to as a wall, and could be defined as the set of edges preserved by the reflection $L_{wsw^{-1}}$.\n\n\\subsection{The weak Bruhat order}\n\\label{sec:bruhat}\n\nLet $(W,S)$ be a Coxeter system. The (right) weak Bruhat order on $(W,S)$ is defined as follows (Chapter 3 of \\cite{Bjorner}).\n\n\\begin{Definition}\nLet $w$ and $w'$ be elements of $W$. Then $w \\leq_B w'$ if there is a reduced word for $w$ which can be multiplied on the right by elements of $S$ to produce a reduced word for $w'$.\t\n\\end{Definition}\n\nAn equivalent formulation in terms of the Cayley graph of $(W,S)$ is\n\n\\begin{Definition}\nLet $w$ and $w'$ be elements of $W$. Then $w \\leq_B w'$ if there is a minimal length path in $\\Gamma(W,S)$ from the identity element $1_W$ to $w'$ which passes through $w$.\t\n\\end{Definition}\n\nThese two definitions are equivalent because edges in the Cayley graph $\\Gamma(W,S)$ correspond to multiplication on the right by an element of $S$.\n\n\\section{Proof of main result}\n\\label{sec:main}\n\nThe main result of this paper is that the weak Bruhat order arises as a likelihood order for random walks on $(W,S)$.\n\n\\begin{repTheorem}{the:main}\nFor any Coxeter system $(W,S)$, consider the simple random walk on $W$, starting at the identity, and at each step multiplying on the right by an element of $S$ or by the identity, each with probability $\\frac{1}{|S|+1}$. Then for any $n$, and any two states $w$ and $w'$, if $w \\leq_B w'$ then the probability that the random walk is at $w$ after $n$ steps is at least the probability that it is at $w'$. \n\\end{repTheorem}\n\nTo prove Theorem \\ref{the:main}, it suffices to consider $w$ and $w'$ which are adjacent --- that is, when $w' = ws$ for some $s \\in S$, with $l(w) < l(ws)$. If the result is true for adjacent vertices, then the general case follows by induction. Thus, the theorem reduces to the following proposition.\n\n\\begin{Proposition}\n\\label{prop:likelihood}\nLet $w \\in W$ and $s \\in S$, with $l(ws) > l(w)$. Then for any $n$, $P^n(ws) < P^n(w)$.\n\\end{Proposition}\n\\begin{proof}\nLet $\\cP$ be the set of paths of length $t$ from the identity $1_W$ to $w$, with each step being either the identity or an element of $S$. Let $\\cP'$ be the set of paths of length $t$ from $1_W$ to $ws$. To prove this proposition, it suffices to construct an injection from $\\cP'$ to $\\cP$. Consider an element $\\alpha$ of $\\cP'$, and write it as $$\\alpha = (1_W=a_0,a_1,a_2,\\dots,a_n=ws).$$\n\nThe path $\\alpha$ starts at a point closer to $w$ than to $ws$, because $l(ws) > l(w)$, and $\\alpha$ ends at $ws$, which is closer to $ws$ than to $w$. Thus at some point, $\\alpha$ crosses from a vertex closer to $w$ to a vertex closer to $ws$ --- that is, this path crosses a grey edge. Given $\\alpha$, let $i$ be the last time at which $\\alpha$ either just crossed a grey edge or just stayed in place on an endpoint of a grey edge. Using $\\oplus$ to denote concatenation, define the sequence of vertices \n\\begin{align*}\nf(\\alpha) &= (a_j)_{j=0}^{i-1} \\oplus (L_{wsw^{-1}}(a_j))_{j=i}^{n} \\\\\n &= (1_W=a_0,a_1,\\dots,a_{i-1},L_{wsw^{-1}}(a_i),L_{wsw^{-1}}(a_{i+1}),\\dots,L_{wsw^{-1}}(a_n)) = w.\\\\\n\\end{align*}\n\nThat is, the sequence $f(\\alpha)$ is identical to $\\alpha$ up until time $i-1$, and from time $i$ onwards, it is reflected by $L_{wsw^{-1}}$. \n\n\\begin{Proposition}\nThe sequence $f(\\alpha)$ is a path. That is, each two consecutive entries are either adjacent in the graph $\\Gamma(W,S)$, or equal.\n\\end{Proposition}\n\\begin{proof}\nIt must be checked that each two consecutive entries in $f(\\alpha)$ are either adjacent in the graph $\\Gamma(W,S)$, or equal. This is immediate for each consecutive pair except for $a_{i-1}$ and $L_{wsw^{-1}}(a_i)$.\n\nBy the definition of $i$, the vertex $a_i$ is a black vertex adjacent to exactly one grey edge, and $a_{i-1}$ is one of the two endpoints of that edge. From Proposition \\ref{prop:greyflip}, $L_{wsw^{-1}}(a_i)$ is either equal to $a_{i-1}$ or connected to $a_{i-1}$ by this grey edge.\n\\end{proof}\n\nThe map $L_{wsw^{-1}}$ interchanges $ws$ and $w$, so $f$ is a function from $\\cP'$ to $\\cP$. All that remains is to show that $f$ is an injection. The function $f$ is an involution, because it applies $L_{wsw^{-1}}$ to the part of the path from time $i$ onwards, the map $L_{wsw^{-1}}$ is an involution, and moving from $\\alpha$ to $f(\\alpha)$ does not change the definition of $i$ (but rather interchanges the two cases in the definition of $i$).\n\nThus $f$ is an involution from $\\cP'$ to $\\cP$, so there are at least as many paths of length $n$ from $1_W$ to $w$ as from $1_W$ to $ws$, completing the proof of Proposition \\ref{prop:likelihood}.\n\\end{proof}\n\n\\begin{Corollary}\nFor any Coxeter system $(W,S)$ and the corresponding random walk described by Theorem \\ref{the:main}, the most likely element after $n$ steps is the identity. If $W$ is finite, then the least likely element is the longest element. Here, most and least likely are not necessarily strict.\n\\end{Corollary}\n\n\\begin{Example}\nConsider the random walk on $S_4$ generated by the adjacent transpositions $(1 \\; 2)$, $(2 \\; 3)$, and $(3 \\; 4)$, as well as the identity. For any $n$, the most likely element after $n$ steps is the identity and the least likely is the reversal $(1 \\; 4)(2 \\; 3)$. Theorem \\ref{the:main} does not address the relative likelihoods of the transpositions $s = (1 \\; 2)$ and $t = (2 \\; 3)$, but both are more likely than $(1 \\; 3) = sts = tst$.\n\\end{Example}\n\n\\begin{Example}\nTake the simple random walk on the square grid $\\Z \\times \\Z$ generated by $(\\pm 1,0)$, $(0,\\pm 1)$ and $(0,0)$. This is a relabelling of the Cayley graph of the Coxeter group $$D_\\infty \\times D_\\infty = \\pres{s,t,u,v}{s^2,t^2,u^2,v^2,susu,svsv,tutu,tvtv}.$$ For any $n$, the most likely element after $n$ steps is the identity $(0,0)$, and the next most likely are the four adjacent vertices (which are equally likely, by symmetry). The vertex $(1,3)$ is always more likely than $(2,3)$, but Theorem \\ref{the:main} doesn't address the relative likelihoods of $(1,3)$ and $(2,2)$.\n\\end{Example}\n\n\\begin{Example}\nFinally, consider the simple random walk on a $d$--regular tree, with laziness $\\frac{1}{d+1}$. This is the Cayley graph of the free product of several copies of the group $\\Z \/ 2\\Z$. For any $n$, the most likely vertex after $n$ steps is the initial vertex, the next most likely are the adjacent vertices, then the vertices at distance two, and so on.\n\\end{Example}\n\nTheorem \\ref{the:main} may be strengthened to not require that all generators have equal probability.\n\n\\begin{Theorem}\nFor any Coxeter system $(W,S)$, consider a random walk on $W$, starting at the identity, and at each step multiplying on the right by an element $s \\in S$ or by the identity, with probabilities $p_s$ or $p_{\\text{id}}$. As long as each $p_s$ is less than $p_{\\text{id}}$, the conclusion of Theorem \\ref{the:main} holds.\n\\end{Theorem}\n\\begin{proof}\nThe only part of the proof that must be changed is the proof of Proposition \\ref{prop:likelihood}, comparing the probabilities of the states $ws$ and $w$, for $w \\in W$ and $s \\in S$ with $l(ws) > l(w)$. With this choice of $s$, divide the probability of multiplying by $1_W$ into two parts, of probabilities $p_s$ and $p_{\\text{id}} - p_s$. Where the proof of Proposition \\ref{prop:likelihood} pairs up the events of multiplying by $s$ or by $1_W$, use only the first of these parts, which is an event of equal probability to that of multiplication by $s$.\n\\end{proof}\n\n\\bibliographystyle{hplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{I}\nHydrologists have studied air-water flow in soils, mainly using the so-called Richards approximation. At least two\nhypotheses are physically required for this model to be applicable: the water pressure in the saturated region must\nbe larger than the atmospheric pressure and all the unsaturated regions must have a boundary connected to the\nsurface. However, in many situations, these hypotheses are not satisfied and a more general two-phase flow model\nmust be considered. This work explores the limit of this general model as the viscosity of the air tends to zero,\nwhich is one of the hypotheses required in the Richards model. To that purpose we prove the existence of a weak\nsolution of the two-phase flow problem and prove estimates which are uniform in the air viscosity. In this paper, we\nassume that the air and water phases are incompressible and immiscible. The geometric domain is supposed to be\nhorizontal, homogeneous and isotropic. Our starting point is the following two-phase flow model, which one can\ndeduce from Darcy's law\n$$(\\cal{TP})\\left\\{\\begin{array}{ll}\n&u_t -div(k_w(u)\\nabla(p)) = s_w\\nonumber\\\\\n&(1-u)_t - div(\\Frac{1}{\\mu} k_a(u)\\nabla(p + p_c(u))) = s_a, \\nonumber\n\\end{array}\\right.$$\nwhere $u$ and $p$ are respectively the saturation and the pressure of the water phase, $k_w$ and $k_a$ are\nrespectively the relative permeabilities of the water and the air phase, $\\mu$ is the ratio between the viscosity of\nthe air phase and that of the water phase, $p_c$ is the capillary pressure, $s_w$ is an internal source term for the\nwater phase and $s_a$ is an internal source term for the air phase; these source terms are used to represent\nexchanges with the outside. We suppose in particular that the physical functions $k_w$, $k_a$ and $p_c$ only depend\non the saturation $u$ of the water phase, and that $ k_w(1) = k_a(0) = 1$. The aim of this paper is the study of the\nlimit of the two-phase flow problem as $\\mu \\downarrow 0$.\\\\\n$\\\\ $\nThe classical Richards model as formulated by the\nengineers is given by\n$$({\\cal{R}})\\left\\{\\begin{array}{ll}\n&u_t -div(k_w(u)\\nabla p) = s_w\\nonumber\\\\\n&u=p_c^{-1}(p_{atm}-p). \\nonumber\n\\end{array}\\right.$$\nwhere the properties of capillary pressure $p_c=p_c(u)$ are describes in hypothesis $(H_8)$ below. For the existence\nand uniqueness of the solution of Richards model together with suitable initial and boundary conditions as well as\nqualitative properties of the solution and methods for numerical approximations we refer to \\cite{AL}, \\cite{HW},\n\\cite{P}, \\cite{RPK}. In this article, we will show that the singular limit as $\\mu\\downarrow 0$ of the two phase\nflow problem $(\\cal{TP})$ has the form\n$$({\\cal{FBP}})\\left\\{\\begin{array}{ll}\n&u_t -div(k_w(u)\\nabla p) = s_w\\nonumber\\\\\n&u=1\\mbox{ or }\\nabla(p+p_c(u))=0\\mbox{ a.e. in }\n\\Omega\\times(0,T). \\nonumber\n\\end{array}\\right.$$\nWe remark that a solution of $({\\cal{R}})$ with $u > 0$ satisfies $({\\cal{FBP}})$. \\\\\n$\\\\ $ This paper is organized as follows. In Section 2 we present a complete mathematical formulation of the\nproblem, and state the main mathematical results, which include a precise formulation of the singular limit problem.\nWe give a sequence of regularized problems in Section 3, and prove the existence of a classical solution. In Section\n4 we present a priori estimates, which are uniform in an extra regularization parameter $\\delta$ and in the air\nviscosity $\\mu$. In Section 5, we let $\\delta\\downarrow 0$ and prove that the solution converges to a solution of\nthe two phase flow problem. We study its limiting behavior as the air viscosity $\\mu$ tends to zero in Section 6.\nFinally in Section 7 we propose a finite volume algorithm in a one\ndimensional context and present a variety of numerical solutions.\\\\\n\n\\section{Mathematical formulation and main results}\\label{I}\nWe consider the two-phase flow problem\n$$\n\\vgl=0\\vglb=0 \\mbox{$(S^{\\mu})$} \\left\\{ \\eqalijna{ &u_t =div\\bigg(k_w(u)\\nabla p\\bigg)+f^{\\mu}(c)\\overline s-f^{\\mu}(u)\\underline s,\n~~~~~&\\mbox{ in }Q_T,\\cr &(1-u)_t = div\\bigg(\\Frac{1}{\\mu}k_a(u)\\nabla(p+p_c(u))\\bigg)\\cr &~~~~~~~~~~~~~~+\n(1-f^{\\mu}(c))\\overline s-(1-f^{\\mu}(u))\\underline s,~~~~&\\mbox{ in }Q_T,\\cr &\\Int_{\\Omega} p(x,t)dx=0,&\\mbox{ for }t \\in (0,T),\\cr &\\nabla\np.n=0, &\\mbox{ on }\\partial \\Omega \\times (0,T),\\cr &\\nabla (p+p_c(u)).n=0, &\\mbox{ on }\\partial \\Omega \\times\n(0,T),\\cr &u(x,0)=u_0(x),&\\mbox{ for }x \\in \\Omega, } \\right. \\eqno\\eqalijnb{\\latexeqno{n1}\\cr\\nonumber\\cr\n \\latexeqno{n2} \\cr\n \\latexeqno{n4} \\cr\\latexeqno{n3} \\cr \\latexeqno{n5} \\cr \\latexeqno{CImu} \\cr\n }\n$$\nwhere $T$ is a positive constant, $Q_T:=\\Omega \\times (0,T)$ and where we suppose that\n$$\\begin{array}{lll}\n&(H_1) &\\Omega \\mbox{ is a smooth bounded domain of ${\\mathrm {I\\mkern-5.5mu R\\mkern1mu}}^N$ where the space dimension $N$ is arbitrary},\\\\\n&(H_2) &u_m\\in(0,1),\\\\\n&(H_3) &c\\in L^\\infty({\\Omega}\\times(0,T)) \\mbox{ and }u_m\\leq c \\leq 1,\\\\\n&(H_4) &u_0\\in L^\\infty({\\Omega}) \\mbox{ and }u_m\\leq u_0 \\leq 1,\\\\\n&(H_5) &\\overline s\\in L^2(\\Omega),~~\\overline s\\geq 0,~~\\underline s\\in L^2(\\Omega),~~\\underline s\\geq 0\\mbox{ and }\\Int_{\\Omega}(\\overline s(x)-\\underline s(x))dx=0,\\\\\n&(H_6) &k_w\\in C^2([0,1]),~~k_w'\\geq 0,~~k_w(0)=0,~~k_w(1)=1\\mbox{ and }k_w(u_m)>0,\\\\\n&(H_7) &k_a\\in C^2([0,1]),~~k_a'\\leq 0,~~k_a(1)=0,~~k_a(0)=1\\mbox{ and }k_a(s)>0\\mbox{ for all }s\\in[0,1),\\\\\n&(H_8) &p_c\\in C^0([0,1])\\cup C^3([0,1)),~~p_c'<0\\mbox{ and }\\sup_{s\\in[0,1)}(-k_a(s)p_c'(s))<+\\infty,\\\\\n&(H_9) &\\mu\\in(0,1].\n\\end{array}$$\nIn this model, $u$ and $p$ are respectively the saturation and the pressure of the water phase, $k_w$ and $k_a$ are\nrespectively the mobilities of the water phase and the mobility of the non-water phase and $p_c$ is the capillary\npressure. We assume in particular that the permeability functions $k_w$, $k_a$ and the capillary pressure $p_c$ only\ndepend on the saturation $u$ of the water phase. Here, we suppose that the flow of the water phase in the reservoir\nis driven by an injection term $f^\\mu(c)\\overline s$ and an extraction term $f^\\mu(u) \\underline s$ where $\\overline s$ and $\\underline s$ are given\nspace dependent functions, $c$ is the saturation of the injected fluid; if $c=1$, only water will be injected, if\n$c=0$, only air will be injected, whereas a mixture of water and air will be injected if $0 < c < 1$. The function\n$f^\\mu$ is the fractional flow of the water phase, namely\n\\begin{equation}\\label{deffmu}\nf^{\\mu}(s)=\\Frac{k_w(s)}{M^\\mu(s)}, \\mbox{ with\n}M^\\mu(s)=k_w(s)+\\Frac{1}{\\mu}k_a(s).\n\\end{equation}\nIn particular, we remark that\n\\begin{equation}\\label{fmucroissante}\nf^{\\mu}(s) \\mbox{ is non decreasing. }\n\\end{equation}\nNext we introduce a set of notations, which will be useful in the sequel.\n\\begin{equation}\\label{defg}\ng(s)=-\\Int_0^sk_a(\\tau)p_c'(\\tau)d\\tau,\n\\end{equation}\n\\begin{equation}\\label{defzeta}\n\\zeta(s)=\\Int_0^s\\sqrt{k_a(\\tau)}p_c'(\\tau)d\\tau,\n\\end{equation}\n\\begin{equation}\\label{Q}\n{\\cal{Q}}^\\mu(s)=\\Int_0^s f^{\\mu}(\\tau)p_c'(\\tau)d\\tau,\n\\end{equation}\nand\n\\begin{equation}\\label{R}\n{\\cal{R}}^\\mu(s)=\\Int_0^s\\Frac{k_a(\\tau)}{k_a(\\tau)+\\muk_w(\\tau)}p_c'(\\tau)d\\tau,\n\\end{equation}\nfor all $s\\in[0,1]$. This implies in particular that\n\\begin{equation}\\label{P+Q} {\\cal{R}}^\\mu(s)+{\\cal{Q}}^\\mu(s)=p_c(s)-p_c(0), \\mbox{ for all }s\\in[0,1].\n\\end{equation}\n\\begin{definition} The pair $(u^{\\mu},p^{\\mu})$ is a weak solution of Problem $(S^\\mu)$ if\n$$\\begin{array}{lll}\n&u^{\\mu}\\in L^\\infty({\\Omega}\\times(0,T)),~~~~&\\mbox{with~}0\\leq u^{\\mu}\\leq 1 \\mbox{ in }Q_T,\\\\\n&&\\\\\n&p^{\\mu}\\in L^2(0,T;H^1({\\Omega})),~~~~&\\Int_\\Op^{\\mu}(x,t)dx=0 \\mbox{~for~almost~every~} t \\in (0,T),\\\\\n&&\\\\\n&g(u^{\\mu})\\in L^2(0,T;H^1({\\Omega})),&\\\\\n\\end{array}$$\nwith\n\\begin{eqnarray}\n&\\Int_0^T\\displaystyle \\Int _{\\Omega}u^{\\mu} \\varphi_t dxdt=&\n\\Int_0^T\\displaystyle \\Int _{\\Omega}k_w(u^{\\mu})\\nablap^{\\mu}.\\nabla \\varphi dx dt\n-\\Int_0^T\\displaystyle \\Int _{\\Omega}\\bigg(f^{\\mu}(c)\\overline s-f^{\\mu}(u^{\\mu})\\underline s \\bigg) \\varphi dxdt\\nonumber\\\\\n&&-\\displaystyle \\Int _{\\Omega} u_0(x)\\varphi(x,0)dx,\\label{defsol1}\n\\end{eqnarray}and\n\\begin{eqnarray}\n&&\\Int_0^T\\displaystyle \\Int _{\\Omega}\\bigg(1-u^{\\mu}\\bigg)\\varphi_tdxdt=\\Int_0^T\\displaystyle \\Int _{\\Omega}\\Frac{1}{\\mu}k_a(u^{\\mu})\\bigg(\\nablap^{\\mu}+\\nabla p_c(u^{\\mu})\\bigg).\\nabla \\varphi dx dt\\nonumber\\\\\n&&-\\Int_0^T\\displaystyle \\Int _{\\Omega}\\bigg((1-f^{\\mu}(c))\\overline s-(1-f^{\\mu}(u^{\\mu}))\\underline s\\bigg)\n\\varphi dx dt-\\displaystyle \\Int _{\\Omega}\n\\bigg(1-u_0(x)\\bigg)\\varphi(x,0)dx,~~~~\\label{defsol2}\n\\end{eqnarray}\nfor all $\\varphi$ in ${\\cal{C}}:=\\{w\\in W_2^{2,1}(Q_T), w(.,T)=0 \\mbox{ in }\\Omega \\}.$\n\\end{definition}\n$\\\\ $ Our first result, which we prove in Section \\ref{Existencesolution}, is the following\n\\begin{theorem}\\label{thexistence}\nSuppose that the hypotheses $(H_1)-(H_9)$ are satisfied, then there exists a weak solution $(u^{\\mu},p^{\\mu})$ of Problem\n$(S^\\mu)$.\n\\end{theorem}\nNext we define the discontinuous function $\\chi$ by $$\\chi(s):=\\left\\{\\begin{array}{ll}0&\\mbox{ if }s\\in[0,1)\\\\\n1&\\mbox{ if }s=1,\\end{array}\\right.$$ as well as the graph\n$$H(s):=\\left\\{\\begin{array}{lll}&0&\\mbox{ if }s\\in[0,1)\\\\ &[0,1]&\\mbox{ if }s=1.\\end{array}\\right.$$\nThe main goal of this paper is to prove the following convergence result,\n\\begin{theorem}\\label{thlim}\nSuppose that the hypotheses $(H_1)-(H_9)$ are satisfied, then there\nexists a subsequence $((u^{\\mu_n},p^{\\mu_n}))_{n\\in N}$ of\nweak solutions of Problem $(S^{\\mu_n})$ and functions $u$, $p$, $\\hat f$ such that\n$$\\begin{array}{ll}\n&u\\in L^\\infty(Q_T),~~~0\\leq u\\leq 1\\mbox { in }Q_T,\\\\\n&\\\\\n&\\hat f\\in L^\\infty(Q_T),~~~0\\leq \\hat f\\leq 1\\mbox { in }Q_T,\\\\\n&\\\\\n&p\\in L^2(0,T;H^1({\\Omega})),\\\\\n&\\\\\n&k_a(u)\\nablap_c(u)\\in L^2({\\Omega}\\times(0,T)),\\\\\n\\end{array}$$\nand\n$$\\begin{array}{ll}\n&(u^{\\mu_n})_{n\\in N}\\mbox{ tends to }u\\mbox{ strongly in }L^2(Q_T),\\\\\n&(p^{\\mu_n})_{n\\in N}\\mbox{ tends to }p\\mbox{ weakly in }L^2(0,T;H^1({\\Omega})),\\\\\n\\end{array}$$\nas $\\mu_n$ tends to zero and\n\\begin{eqnarray}\n&\\Int_0^T\\displaystyle \\Int _{\\Omega} u \\varphi_tdxdt =&\\Int_0^T\\displaystyle \\Int _{\\Omega}k_w(u)\\nabla p.\\nabla \\varphi dxdt-\\Int_0^T\\displaystyle \\Int _{\\Omega}\\bigg(\\chi(c)\\overline s-\\hat f\\underline s \\bigg) \\varphi dxdt\\nonumber\\\\\n&&-\\displaystyle \\Int _{\\Omega} u_0(x)\\varphi(x,0)dx,\\label{sollim1}\n\\end{eqnarray}\nfor all $\\varphi \\in{\\cal{C}}$, where $\\hat f(x,t)\\in H(u(x,t))$\nfor $(x,t)\\in Q_T$. Moreover we also have that\n\\begin{equation}\\label{lim1}\n\\Int_0^T\\displaystyle \\Int _{\\Omega} \\bigg[k_a(u)\\bigg]^2\\bigg[\\nabla p+\\nabla p_c(u)\\bigg]^2dxdt=0\n\\end{equation}\nand\n\\begin{equation}\\label{lim2}\n\\Int_{\\Omega} p(x,t) dx=0, \\mbox{ for almost every }t\\in(0,T).\n\\end{equation}\n\\end{theorem}\nFormally, $u$ satisfies the following limit problem\n$$\n\\left\\{\n\\begin{array}{ll}\nu_t =div\\bigg(k_w(u)\\nabla p \\bigg)+ \\chi(c)\\bar s - \\hat\nf\\underline{s},\n~~&\\mbox{ in }Q_T, \\\\\n\\nabla u.n=0, &\\mbox{ on }\\partial \\Omega \\times (0,T),\\\\\nu(x,0)=u_0(x),~~&\\mbox{ for }x \\in \\Omega.\n\\end {array}\n\\right.\n$$\nMore precisely the following corollary holds\n\\begin{corollary}\\label{thcoro} Suppose that $u<1$ in ${\\cal O}=\\cup_{t\\in[\\tau,T]}\\Omega_t$, where $\\tau >0$\nand $\\Omega_t$, for $t\\in[\\tau,T]$, are smooth subdomains of ${\\Omega}$\nand ${\\cal O}$ is a smooth domain of $\\Omega\\times[\\tau,T]$ and that $u=1$ in\n$Q_T\\setminus\\overline{\\cal{O}}$ then\n$$p(x,t)=-p_c(u(x,t))+constant(t), \\mbox{ for all }(x,t)\\in {\\cal O}$$ and $u$ satisfies\n$$\n\\left\\{\n\\begin{array}{ll}\nu_t =-div\\bigg(k_w(u)\\nabla p_c(u)\\bigg)+ \\chi(c)\\bar s, ~~&\\mbox{ in } {\\cal O}, \\\\\n\\Frac{\\partial u}{\\partial n}=0,&\\mbox{ on }\\partial {\\cal O}\\cap\n\\bigg(\\partial\\Omega\\times(0,T)\\bigg),\\\\\nu=1, &\\mbox{ elsewhere on }\\partial {\\cal O},\\\\\nu(x,0)=u_0(x),&\\mbox{ for }x \\in \\Omega.\n\\end {array}\n\\right.\n$$\n\\end{corollary}\n$\\\\ $ Finally we remark that another form of the limit problem involves a parabolic equation, which is close to the\nstandard Richards equation. Indeed if we set $\\phi(s):=p_c(0)-p_c(s)$ and denote by $\\beta$ the inverse function of\n$\\phi$, the function $v:=\\phi(u)$ is a weak solution of the problem\n$$\n\\left\\{\n\\begin{array}{ll}\n\\beta(v)_t =div\\bigg(k_w(\\beta(v))\\nabla v \\bigg)+ \\chi(c)\\bar s -\n\\hat f\\underline{s},\n~~&\\mbox{ in }Q_T, \\\\\n\\nabla \\beta(v).n=0, &\\mbox{ on }\\partial \\Omega \\times (0,T),\\\\\n\\beta(v)(x,0)=u_0(x),~~&\\mbox{ for }x \\in \\Omega,\n\\end {array}\n\\right.\n$$\nwith $\\hat f\\in H(\\beta(v))$.\n\\setcounter{equation}{0}\n\\section{Existence of a solution of an approximate problem $(S^{\\mu}_{\\delta})$ of Problem\n$(S^{\\mu})$}\\label{Existencesolution}\nLet $\\delta$ be an arbitrary positive constant. In order to prove the\nexistence of a solution of Problem $(S^{\\mu})$ we introduce a\nsequence of regularized problems $(S^{\\mu}_{\\delta})$, namely\n$$\n\\vgl=0\\vglb=0 \\mbox{$(S^{\\mu}_{\\delta})$} \\left\\{ \\eqalijna{ &u_t~~ =\ndiv\\bigg(k_w(u)\\nabla p\\bigg)+f^{\\mu}(c_\\d)\\overline s_\\delta\\cr\n&~~~~~~~~-f^{\\mu}(u)\\bigg(\\underline s_\\delta+\\displaystyle {\\Int \\kern -0.961em -}_{\\Omega}(\\overline s_\\delta-\\underline s_\\delta )dx\\bigg),\n~~&\\mbox{ in }\\Omega \\times (0,T),\\cr &(1-u)_t\n=div\\bigg(\\Frac{1}{\\mu}k_a(u)\\nabla(p+p_c(u))\\bigg)+\\bigg(1-f^{\\mu}(c_\\d)\\bigg)\\overline s_\\delta~~\\cr\n&~~~~~~~~~~~~~~-\\bigg(1-f^{\\mu}(u)\\bigg)\\bigg(\\underline s_\\delta+\\displaystyle {\\Int \\kern -0.961em -}_{\\Omega}(\\overline s_\\delta-\\underline s_\\delta\n)dx\\bigg), &\\mbox{ in }\\Omega \\times (0,T),\\cr &\\Int_{\\Omega}\np(x,t)dx=0,&\\mbox{ for }t\\in(0,T),\\cr\n &\\nabla p.n=0, &\\mbox{ on }\\partial \\Omega\n\\times (0,T),\\cr &\\nabla (p +p_c(u)).n=0, &\\mbox{ on }\\partial\n\\Omega \\times (0,T),\\cr &u(x,0)=u_{0}^\\delta(x),& \\mbox{ for }x \\in\n\\Omega,\\cr } \\right. \\eqno\\eqalijnb{\\nonumber\\cr\\latexeqno{1d}\\cr\n\\nonumber\\cr\n \\latexeqno{2d} \\cr\n \\latexeqno{3d} \\cr\n \\latexeqno{4d} \\cr \\latexeqno{5d} \\cr\\latexeqno{CId} \\cr\n }\n$$\nwhere $u_{0}^\\delta$, $c_\\delta$, $\\overline s_\\delta$ and $\\underline s_\\delta$ are smooth functions such that $u_{0}^\\delta$ tends to $u_0$ in\n$L^2(\\Omega)$ and $c_\\delta$, $\\overline s_\\delta$ and $\\underline s_\\delta$ tend respectively to $c$, $\\overline s$ and $\\underline s$ in $L^2(Q_T)$, as\n$\\delta\\downarrow 0$. In particular we suppose that there exists a positive constant $C$ such that\n\\begin{equation}\\label{sdeltaborne}\n\\underline s_\\delta\\geq 0,~~\\overline s_\\delta\\geq 0 \\mbox{ and }\\Int_{\\Omega} \\underline s_\\delta^2+\\Int_{\\Omega} \\overline s_\\delta^2 \\leq C.\n\\end{equation}\nMoreover we suppose that $u_{0}^\\delta$, $c_\\delta$ satisfy\n\\begin{eqnarray}\n&0From the standard theory of parabolic equations, we have that\n\\begin{equation}\\label{n8f}\n|\\hat V|^{2+\\alpha,\\frac{2+\\alpha}{2}}_{Q_T}\\leq\nD_3\\bigg(|W|^{1+\\alpha,\\frac{1+\\alpha}{2}}_{Q_T}+|\\nabla W|^{1+\\alpha,\\frac{1+\\alpha}{2}}_{Q_T}\\bigg) + D_4.\\label{n8ebis}\\\\\n\\end{equation}\nMoreover defining by $\\cal L$ the parabolic operator arising in\n$(Q^2_W)$, namely\n$${\\cal {L}}(\\hat V)(x,t):=\\hat V_t-\\Delta\\psi^\\mu_\\varepsilon(\\hat V)\n-M^\\mu(\\hat V)\\nablaf^{\\mu}(\\hat V).\\nabla W\n-[f^{\\mu}(c_\\delta)-f^{\\mu}(\\hat V)]\\overline s_\\delta,$$ we remark that\n(\\ref{fmucroissante}), the property (\\ref{Cdeltaborne}) of\n$c_\\delta$ and the fact that, by (\\ref{sdeltaborne}), $\\overline s_\\delta$ is positive imply that\n\\begin{equation}\\label{calO}\n{\\cal {L}}(u_m)\\leq 0\\mbox{ and }{\\cal {L}}(1-\\delta)\\geq 0.\n\\end{equation}\nSetting $T:=T^2\\circ T^1$, the inequalities (\\ref{calO}) ensure that $T$ maps the convex set ${\\cal{K}}$ into\nitself. Moreover we deduce from (\\ref{n8f}) that $T({\\cal{K}})$ is relatively compact in ${\\cal{K}}$.\n\\\\Next, we check that $T$ is continuous. Suppose that a sequence $(V_m)_{m\\inN}$ converges to a limit $V\\in\n{\\cal{K}}$ in $C^{1+\\alpha,\\frac{1+\\alpha}{2}}(\\overline Q_T)$, as $m\\rightarrow\\infty$. Since $(V_m)_{m\\inN}$ is bounded\nin $C^{1+\\alpha,\\frac{1+\\alpha}{2}}(\\overline Q_T)$, it follows from (\\ref{n8e}) that the sequence\n$(W_m:=T^1(V_m))_{m\\inN}$, where $W_m$ is the solution of $(Q^1_{V_m})$, is bounded in\n$C^{1+\\alpha,\\frac{1+\\alpha}{2}}(\\overline Q_T)$, so that as $m\\rightarrow\\infty$, $W_m$ converges to the unique solution\n$W$ of Problem $(Q^1_V)$ in $C^{1+\\beta,\\frac{1+\\beta}{2}}(\\overline Q_T)$ for all $\\beta\\in(0,\\alpha)$. Moreover $W\\in\nC^{1+\\alpha,\\frac{1+\\alpha}{2}}(\\overline Q_T)$. Further it also follows from (\\ref{n8e}) that $(\\nabla\nW_m)_{m\\inN}$ is bounded in $C^{1+\\alpha,\\frac{1+\\alpha}{2}}(\\overline Q_T)$, so that the solution $\\hat V_m=T^2(W_m)$ of\nProblem $(Q^2_{W_m})$ is bounded in $C^{2+\\alpha,\\frac{2+\\alpha}{2}}(\\overline Q_T)$. Since $\\hat V_m=T^2(W_m)=T(V_m)$,\n$(T(V_m))_{m\\inN}$ converges to the unique solution $\\hat V$ of Problem $(Q^2_{W})$ in\n$C^{2+\\beta,\\frac{2+\\beta}{2}}(\\overline Q_T)$ for all $\\beta\\in(0,\\alpha)$, as $m\\rightarrow\\infty$, so that $\\hat\nV=T^2(W)=T^2\\circ T^1(V)$. Therefore we have just proved that $(T^2\\circ T^1(V_m))_{m\\inN}$ converges to $T^2\\circ\nT^1(V)$ in $C^{2+\\beta,\\frac{2+\\beta}{2}}(\\overline Q_T)$ for all $\\beta\\in(0,\\alpha)$, as $m\\rightarrow\\infty$, which\nensures the continuity of the map $T$. It follows from the Schauder fixed point theorem that there exists a solution\n$(u^{\\mu}_\\delta,{\\cal{P}}^\\mu_\\delta)$ of $(\\tilde{S}^{\\mu}_{\\delta})$ such that\n$$u^{\\mu}_{\\delta}\\in K\\cap C^{2+\\alpha,\\frac{2+\\alpha}{2}}(\\overline Q_T) \\mbox{ and ${\\cal{P}}^\\mu_{\\delta}$, $\\nabla {\\cal{P}}^\\mu_{\\delta}$, $\\in C^{1+\\alpha,\\frac{1+\\alpha}{2}}(Q_T)$,\n$\\Delta {\\cal{P}}^\\mu_{\\delta} \\in C^{\\alpha,\\frac{\\alpha}{2}}(Q_T)$.}$$ $\\\\\n$ This concludes the proof of Lemma \\ref{leexistencetildeeps}.\nMoreover we deduce from Lemma \\ref{leexistencetildeeps} the\nexistence of a solution of $(S^{\\mu}_{\\delta})$, namely\n\n\\begin{corollary}\\label{leexistenceeps}\nAssume the hypotheses $(H_1)-(H_9)$ then there exists $(u^{\\mu}_{\\delta},p^{\\mu}_{\\delta})$ solution of $(S^{\\mu}_{\\delta})$ such that\n$u^{\\mu}_{\\delta}\\in C^{2+\\alpha,\\frac{2+\\alpha}{2}}(\\overline Q_T)$,\n\\begin{equation}\\label{borneu}\nu_m\\leq u^{\\mu}_{\\delta}(x,t)\\leq 1-\\delta\n\\end{equation}\nand $p^{\\mu}_{\\delta}$, $\\nabla p^{\\mu}_{\\delta}$ $\\in C^{1+\\alpha,\\frac{1+\\alpha}{2}}(Q_T)$, $\\Delta p^{\\mu}_{\\delta} \\in\nC^{\\alpha,\\frac{\\alpha}{2}}(Q_T)$.\n\\end{corollary}\n\n\\setcounter{equation}{0}\n\\section{A priori Estimates}\nIn view of (\\ref{fmucroissante}) and (\\ref{borneu}) we deduce the following bounds\n\\begin{eqnarray}\n&&0=f^{\\mu}(0)\\leq f^{\\mu}(u^{\\mu}_{\\delta}(x,t))\\leq 1=f^{\\mu}(1),\\label{Eb1}\\\\\n&&0=f^{\\mu}(0)\\leq f^{\\mu}(c_{\\delta}(x,t))\\leq 1=f^{\\mu}(1),\\label{Eb1bis}\\\\\n&&00$. Moreover we have by Poincar\\'e-Wirtinger inequality\nthat\n$$\\Int_{Q_T}(p^{\\mu}_{\\delta}+{\\cal{R}}^\\mu(u^{\\mu}_{\\delta}))^2\\leq C_1\\bigg[\\Int_{Q_T}|\\nabla(p^{\\mu}_{\\delta}+{\\cal{R}}^\\mu(u^{\\mu}_{\\delta}))|^2+\n\\bigg(\\Int_{Q_T}p^{\\mu}_{\\delta}+{\\cal{R}}^\\mu(u^{\\mu}_{\\delta})\\bigg)^2 \\bigg].$$ Using\n(\\ref{3d}) and (\\ref{Eb5}), it follows that\n$$\\Int_{Q_T}(p^{\\mu}_{\\delta}+{\\cal{R}}^\\mu(u^{\\mu}_{\\delta}))^2\\leq C_1\\Int_{Q_T}|\\nabla(p^{\\mu}_{\\delta}+{\\cal{R}}^\\mu(u^{\\mu}_{\\delta}))|^2+ C_2,$$\nwhich we substitute into (\\ref{E1}) with $h=\\Frac{k_w(u_m)}{2C_1}$ to deduce, also in view of (\\ref{sdeltaborne})\nand (\\ref{Eb3}), that\n\\begin{equation}\\label{E2}\n\\Int_{Q_T}|\\nabla(p^{\\mu}_{\\delta}+{\\cal{R}}^\\mu(u^{\\mu}_{\\delta}))|^2\\leq C_3\\mbox{ and\n}\\Int_{Q_T}|p^{\\mu}_{\\delta}+{\\cal{R}}^\\mu(u^{\\mu}_{\\delta})|^2\\leq C_3.\n\\end{equation}\nFurthermore multiplying (\\ref{1d}) by $p^{\\mu}_{\\delta}$ and (\\ref{2d}) by $p^{\\mu}_{\\delta}+p_c(u^{\\mu}_{\\delta})$, adding up both\nresults and integrating on $Q_T$ we obtain\n\\begin{eqnarray}\n&-\\Int_{Q_T}(u^{\\mu}_{\\delta})_tp_c(u^{\\mu}_{\\delta})+\\Int_{Q_T}\nk_w(u^{\\mu}_{\\delta})|\\nablap^{\\mu}_{\\delta}|^2+\\Frac{1}{\\mu}k_a(u^{\\mu}_{\\delta})\\big|\\nablap^{\\mu}_{\\delta}+\\nabla\np_c(u^{\\mu}_{\\delta})\\big|^2=I,~~~~\\label{E3}\n\\end{eqnarray}\nwhere\n$$\\begin{array}{ll}\nI:=&\\Int_{Q_T}\\bigg[f^{\\mu}(c_\\delta)\\overline s_\\delta-f^{\\mu}(u^{\\mu}_{\\delta})\\bigg(\\underline s_\\delta +\\displaystyle {\\Int \\kern -0.961em -}_{\\Omega}(\\overline s_\\delta-\\underline s_\\delta )\\bigg)\\bigg]p^{\\mu}_{\\delta} dxdt\\\\\n&+\\Int_{Q_T}\\bigg[(1-f^{\\mu}(c_\\delta))\\overline s_\\delta-\\bigg(1-f^{\\mu}(u^{\\mu}_{\\delta})\\bigg)\\bigg(\\underline s_\\delta+\\displaystyle {\\Int \\kern -0.961em -}_{\\Omega}(\\overline s_\\delta-\\underline s_\\delta )\n\\bigg)\\bigg](p^{\\mu}_{\\delta}+p_c(u^{\\mu}_{\\delta}))dxdt.\n\\end{array}\n$$\nWe check below that first term on the left-hand-side of (\\ref{E3}) and $I$ are bounded. Denoting by ${\\cal{P}}_c$ a\nprimitive of $p_c$ we have that\n$$\\Int_{Q_T}p_c(u^{\\mu}_{\\delta})(u^{\\mu}_{\\delta})_t=\\Int_{\\Omega}\\Int_0^T\\Frac{\\partial}{\\partial\nt}\\big[{\\cal{P}}_c(u^{\\mu}_{\\delta})\\big].$$ Since ${\\cal{P}}_c$ is continuous and $u^{\\mu}_{\\delta}$ is bounded this gives\n\\begin{equation}\\label{E4}\n\\bigg|\\Int_{Q_T}p_c(u^{\\mu}_{\\delta})(u^{\\mu}_{\\delta})_tdxdt\\bigg|\\leq C_4.\n\\end{equation}\nMoreover we have using (\\ref{3d}) and (\\ref{P+Q}) that\n\\begin{eqnarray}\n&I=&\\Int_{Q_T}\\bigg(p^{\\mu}_{\\delta}+{\\cal{R}}^\\mu(u^{\\mu}_{\\delta})\\bigg)(\\overline s_\\delta-\\underline s_\\delta)\ndxdt\\\\\n&&-\\Int_{Q_T}{\\cal{R}}^\\mu(u^{\\mu}_{\\delta})\\bigg[f^{\\mu}(c_\\delta)\\overline s_\\delta-f^{\\mu}(u^{\\mu}_{\\delta})\\underline s_\\delta+\\bigg(1-f^{\\mu}(u^{\\mu}_\\delta)\\bigg)\\bigg( \\displaystyle {\\Int \\kern -0.961em -}_{\\Omega}(\\overline s_\\delta-\\underline s_\\delta) \\bigg)\\bigg] dxdt\\nonumber\\\\\n&&+\\Int_{Q_T}\\bigg[(1-f^{\\mu}(c_\\delta))\\overline s_\\delta-\\bigg(1-f^{\\mu}(u^{\\mu}_{\\delta})\\bigg)\\bigg(\\underline s_\\delta\n+\\displaystyle {\\Int \\kern -0.961em -}_{\\Omega}(\\overline s_\\delta-\\underline s_\\delta) \\bigg)\n\\bigg]\\bigg[{\\cal{Q}}^\\mu(u^{\\mu}_{\\delta})+p_c(0)\\bigg] dxdt.\\nonumber\n\\end{eqnarray}\nIn view of $(H_5)$, (\\ref{sdeltaborne}), (\\ref{Eb1}),\n(\\ref{Eb1bis}), (\\ref{Eb5}) and (\\ref{Eb6}) we obtain\n$$\nI\\leq C_5\\Int_{Q_T}|p^{\\mu}_{\\delta}+{\\cal{R}}^\\mu(u^{\\mu}_{\\delta})|^2+ C_6.\n$$\nThis together with (\\ref{E2}) yields $I\\leq C_5C_3+C_6$. Substituting this into (\\ref{E3}) and also using (\\ref{E4})\nwe obtain that\n\\begin{equation}\\label{E5}\n\\Int_{Q_T} k_w(u^{\\mu}_{\\delta})|\\nablap^{\\mu}_{\\delta}|^2+\\Frac{1}{\\mu}k_a(u^{\\mu}_{\\delta})\\big|\\nablap^{\\mu}_{\\delta}+\\nabla\np_c(u^{\\mu}_{\\delta})\\big|^2dxdt\\leq C_7,\n\\end{equation}\nwhich implies (\\ref{Est1}).\nIn view of (\\ref{Eb2}), we also deduce from (\\ref{E5}) the estimate (\\ref{Est1.1}). \\\\\nNext we prove (\\ref{Est1.2}). By the definition (\\ref{defg}) of $g$, we obtain from (\\ref{7d}) that\n\\begin{equation}\\label{E5b}\n-div\\bigg(M^\\mu(u^{\\mu}_{\\delta})\\nablap^{\\mu}_{\\delta}\\bigg)\n+\\Frac{1}{\\mu}\\Delta g(u^{\\mu}_{\\delta})=\\overline s_\\delta-\\underline s_\\delta-\\displaystyle {\\Int \\kern -0.961em -}_{\\Omega}(\\overline s_\\delta-\\underline s_\\delta)dx.\n\\end{equation}\nMultiplying (\\ref{E5b}) by $f^{\\mu}(u^{\\mu}_{\\delta})$ and subtracting the result from $(\\ref{1d})$ we deduce that\n\\begin{equation}\\label{E5t}\n(u^{\\mu}_{\\delta})_t=\\Frac{1}{\\mu}f^{\\mu}(u^{\\mu}_{\\delta})\\Delta\ng(u^{\\mu}_{\\delta})+div\\bigg(k_w(u^{\\mu}_{\\delta})\\nabla p^{\\mu}_{\\delta}\\bigg)\n-f^{\\mu}(u^{\\mu}_{\\delta})div\\bigg(M^\\mu(u^{\\mu}_{\\delta})\\nabla(p^{\\mu})\\bigg)+ \\overline s_\\delta\\big[f^{\\mu}(c_\\delta)-f^{\\mu}(u^{\\mu}_{\\delta})\\big].\n\\end{equation}\nMoreover using the definition (\\ref{deffmu}) of $f^{\\mu}$ and $M^\\mu$ we note that\n$$\n\\begin{array}{lll}\n&div\\bigg(M^\\mu(u^{\\mu}_{\\delta})f^{\\mu}(u^{\\mu}_{\\delta})\\nabla p^{\\mu}_{\\delta}\\bigg)&=\ndiv\\bigg(k_w(u^{\\mu}_{\\delta})\\nabla p^{\\mu}_{\\delta}\\bigg)\\\\\n&&=M^\\mu(u^{\\mu}_{\\delta})\\nabla(f^{\\mu}(u^{\\mu}_{\\delta})).\\nablap^{\\mu}_{\\delta}\n+f^{\\mu}(u^{\\mu}_{\\delta})div\\bigg(M^\\mu(u^{\\mu}_{\\delta})\\nabla(p^{\\mu})\\bigg),\n\\end{array}\n$$\nwhich we substitute into (\\ref{E5t}) to obtain\n\\begin{eqnarray}\n(u^{\\mu}_{\\delta})_t-\\Frac{1}{\\mu}f^{\\mu}(u^{\\mu}_{\\delta})\\Delta\ng(u^{\\mu}_{\\delta})- M^\\mu(u^{\\mu}_{\\delta})\\nabla(f^{\\mu}(u^{\\mu}_{\\delta})).\\nablap^{\\mu}_{\\delta}\n=\\overline s_\\delta\\big[f^{\\mu}(c_\\delta)-f^{\\mu}(u^{\\mu}_{\\delta})\\big].\\label{E6}\n\\end{eqnarray}\nWe set\n\\begin{equation}\\label{defD}\nD^\\mu(a):=p_c(a)f^{\\mu}(a)-{\\cal{Q}}^\\mu(a),\n\\end{equation}\nfor all $a\\in[0,1]$, so that by the definition (\\ref{Q}) of ${\\cal{Q}}^\\mu$ we have $\\nabla\nD^\\mu(u^{\\mu}_{\\delta})=p_c(u^{\\mu}_{\\delta})\\nabla(f^{\\mu}(u^{\\mu}_{\\delta}))$. Substituting this into (\\ref{E6}), which we have multiplied\nby $p_c(u^{\\mu}_{\\delta})$, we deduce that\n\\begin{equation}\\label{E7}\np_c(u^{\\mu}_{\\delta})(u^{\\mu}_{\\delta})_t -\\Frac{1}{\\mu}f^{\\mu}(u^{\\mu}_{\\delta})p_c(u^{\\mu}_{\\delta})\\Delta\ng(u^{\\mu}_{\\delta})-M^\\mu(u^{\\mu}_{\\delta})\\nabla\nD^\\mu(u^{\\mu}_{\\delta}).\\nablap^{\\mu}_{\\delta}=p_c(u^{\\mu}_{\\delta})\\overline s_\\delta\\big[f^{\\mu}(c_\\delta)-f^{\\mu}(u^{\\mu}_{\\delta})\\big].\n\\end{equation}\nMultiplying (\\ref{E5b}) by $D^\\mu(u^{\\mu}_{\\delta})$, adding the result\nto (\\ref{E7}) and also using the fact that\n$$div\\bigg(M^\\mu(u^{\\mu}_{\\delta})D^\\mu(u^{\\mu}_{\\delta})\\nablap^{\\mu}_{\\delta}\\bigg)=\nM^\\mu(u^{\\mu}_{\\delta})\\nabla D^\\mu(u^{\\mu}_{\\delta}).\\nablap^{\\mu}_{\\delta}\n+D^\\mu(u^{\\mu}_{\\delta})div\\bigg(M^\\mu(u^{\\mu}_{\\delta})\\nablap^{\\mu}_{\\delta}\\bigg),$$\nwe deduce that\n\\begin{eqnarray}\n&p_c(u^{\\mu}_{\\delta})(u^{\\mu}_{\\delta})_t-\\Frac{1}{\\mu}\\bigg(f^{\\mu}(u^{\\mu}_{\\delta})p_c(u^{\\mu}_{\\delta})-D^\\mu(u^{\\mu}_{\\delta})\\bigg)\\Delta\ng(u^{\\mu}_{\\delta})\n-div\\bigg(M^\\mu(u^{\\mu}_{\\delta})D^\\mu(u^{\\mu}_{\\delta})\\nablap^{\\mu}_{\\delta}\\bigg)\n\\nonumber\\\\\n&=p_c(u^{\\mu}_{\\delta})\\overline s_\\delta\\bigg[f^{\\mu}(c_\\delta)-f^{\\mu}(u^{\\mu}_{\\delta})\\bigg]+\nD^\\mu(u^{\\mu}_{\\delta})\\bigg(\\overline s_\\delta-\\underline s_\\delta-\\displaystyle {\\Int \\kern -0.961em -}_{\\Omega}(\\overline s_\\delta-\\underline s_\\delta)\\bigg). \\label{E8}\\end{eqnarray} Integrating (\\ref{E8})\non $Q_T$ and using the fact that the definition (\\ref{defD}) of $D^\\mu$ implies\n$$p_c(u^{\\mu}_{\\delta})f^{\\mu}(u^{\\mu}_{\\delta})-D^\\mu(u^{\\mu}_{\\delta})={\\cal{Q}}^\\mu(u^{\\mu}_{\\delta}),$$\nwe obtain\n\\begin{eqnarray}\n\\Int_{Q_T}p_c(u^{\\mu}_{\\delta})(u^{\\mu}_{\\delta})_tdxdt\n-\\Frac{1}{\\mu}\\Int_{Q_T}{\\cal{Q}}^\\mu(u^{\\mu}_{\\delta})\\Delta\ng(u^{\\mu}_{\\delta})dxdt=J,\\label{E9}\n\\end{eqnarray}\nwhere\n$$\n\\begin{array}{ll}\nJ:=&\\Int_{Q_T}p_c(u^{\\mu}_{\\delta})\\overline s_\\delta\\big[f^{\\mu}(c_\\delta)-f^{\\mu}(u^{\\mu}_{\\delta})\\big]dxdt\\\\\n&+\\Int_{Q_T}\\bigg(p_c(u^{\\mu}_{\\delta})f^{\\mu}(u^{\\mu}_{\\delta})-{\\cal{Q}}^\\mu(u^{\\mu}_{\\delta})\\bigg)\n\\bigg(\\overline s_\\delta-\\underline s_\\delta-\\displaystyle {\\Int \\kern -0.961em -}_{\\Omega}(\\overline s_\\delta-\\underline s_\\delta)\\bigg)dxdt.\n\\end{array}\n$$\nIt follows from (\\ref{Eb1}), (\\ref{Eb1bis}), (\\ref{Eb4}), (\\ref{Eb6}) and\n(\\ref{sdeltaborne}) that $|J|\\leq C_8$. Substituting this\ninto (\\ref{E9}) and also using (\\ref{E4}) we obtain that\n\\begin{equation}\\label{E11}\n0\\leq -\\Frac{1}{\\mu}\\Int_{Q_T}\\nabla{\\cal{Q}}^\\mu(u^{\\mu}_{\\delta}).\\nabla(\ng(u^{\\mu}_{\\delta}))dxdt\\leq C_9.\n\\end{equation}\nFurthermore we remark that\n$$\n\\Frac{1}{\\mu}f^{\\mu}(u^{\\mu}_{\\delta})\\geq \\Frac{k_w(u_m)}{2},\n$$\nwhich together with (\\ref{E11}) and the fact that $\\nabla {\\cal{Q}}^\\mu(u^{\\mu}_{\\delta})=f^{\\mu}(u^{\\mu}_{\\delta})\\nablap_c(u^{\\mu}_{\\delta})$\nyields\n\\begin{equation}\\label{E12}\n0\\leq -\\Int_{Q_T}\\nablap_c(u^{\\mu}_{\\delta})\\nabla(g(u^{\\mu}_{\\delta}))dxdt\\leq C_{10}.\n\\end{equation}\nBy the definition (\\ref{defzeta}) of $\\zeta$, we have $-\\nablap_c(u^{\\mu}_{\\delta})\\nabla\ng(u^{\\mu}_{\\delta})=|\\nabla\\zeta(u^{\\mu}_{\\delta})|^2$. This together with (\\ref{E12}) implies (\\ref{Est1.2}) and\n(\\ref{Est1.2bis}), which in view of (\\ref{Eb2bis}) gives (\\ref{Est1.2ter}). This completes the proof of Lemma\n\\ref{leest1}. $\\\\ $$\\\\ $ In what follows we give estimates of differences of space translates of $p^{\\mu}_{\\delta}$ and\n$g(u^{\\mu}_{\\delta})$. We set for $r\\in{\\mathrm {I\\mkern-5.5mu R\\mkern1mu}}^+$ sufficiently small:\n$$\\Omega_r=\\{x\\in\\Omega,~~B(x,2r)\\subset\\Omega\\}.$$\n\\begin{lemma}\\label{leest2} Let $(u^{\\mu}_{\\delta},p^{\\mu}_{\\delta})$ be a solution of Problem $(S^\\mu_\\delta)$;\nthere exists a positive constant $C$ such that\n\\begin{equation}\\label{Est2}\n\\Int_0^T\\Int_{{\\Omega}_r}\\bigg|p^{\\mu}_{\\delta}(x+\\xi,t)-p^{\\mu}_{\\delta}(x,t)\\bigg|^2(x,t)dxdt\\leq\nC\\xi^2\n\\end{equation}\nand\n\\begin{equation}\\label{Est2.2}\n\\Int_0^T\\Int_{{\\Omega}_r}\\bigg|g(u^{\\mu}_{\\delta})(x+\\xi,t)-g(u^{\\mu}_{\\delta})(x,t)\\bigg|^2dxdt\\leq\nC\\xi^2,\n\\end{equation}\nwhere $\\xi\\in{\\mathrm {I\\mkern-5.5mu R\\mkern1mu}}^N$ and $|\\xi|\\leq 2r.$\n\\end{lemma}\n{\\underline{Proof}}: The inequalities (\\ref{Est2}) and (\\ref{Est2.2}) follow from (\\ref{Est1.1}) and (\\ref{Est1.2ter})\nrespectively. $\\\\ $$\\\\ $ Next we estimate differences of time translates of $g(u^{\\mu}_{\\delta})$.\n\\begin{lemma}\\label{leest3} Let $(u^{\\mu}_{\\delta},p^{\\mu}_{\\delta})$ be a solution of Problem $(S^\\mu_\\delta)$ then\nthere exists a positive constant $C$ such that\n\\begin{equation}\\label{Est3}\n\\Int_0^{T-\\tau}\\Int_{{\\Omega}}\\big[g(u^{\\mu}_{\\delta})(x,t+\\tau)-g(u^{\\mu}_{\\delta})(x,t)\\big]^2dxdt\\leq\nC\\tau,\n\\end{equation}\nfor all $\\tau\\in(0,T)$.\n\\end{lemma}\n{\\underline{Proof}}: We set\n$$A(t):=\\Int_{\\Omega}[g(u^{\\mu}_{\\delta})(x,t+\\tau)-g(u^{\\mu}_{\\delta})(x,t)]^2dx.$$\nSince $g$ is a non decreasing Lipschitz continuous function with the Lipschitz constant $C_g$ we have that\n\\begin{eqnarray}\nA(t)&\\leq& C_g\\Int_{\\Omega}[g(u^{\\mu}_{\\delta}(x,t+\\tau))-g(u^{\\mu}_{\\delta}(x,t))][u^{\\mu}_{\\delta}(x,t+\\tau)-u^{\\mu}_{\\delta}(x,t)]dx\\nonumber\\\\\n&\\leq & C_g\\Int_{\\Omega}[g(u^{\\mu}_{\\delta}(x,t+\\tau))-g(u^{\\mu}_{\\delta}(x,t))]\\left[\\Int_t^{t+\\tau}(u^{\\mu}_\\delta)_t(x,\\theta)d\\theta\\right]dx\\nonumber\\\\\n&\\leq& C_g\\Int_{\\Omega}\\Int_t^{t+\\tau}\\bigg[g(u^{\\mu}_{\\delta}(x,t+\\tau))-g(u^{\\mu}_{\\delta}(x,t))\\bigg]\\nonumber\\\\\n&~~~&~\\bigg[div(k_w(u^{\\mu}_{\\delta})\\nablap^{\\mu}_{\\delta})+f^{\\mu}(c_\\delta)\\overline s_\\delta-f^{\\mu}(u^{\\mu}_\\delta)\n\\bigg(\\underline s_\\delta+\\displaystyle {\\Int \\kern -0.961em -}_{\\Omega}(\\overline s_\\delta(y)-\\underline s_\\delta(y))dy\\bigg)\\bigg](x,\\theta) d\\theta dx,\\nonumber\n\\end{eqnarray}\nwhere we have used (\\ref{1d}). Integrating by parts this gives\n\\begin{eqnarray}\n&A(t)\\leq\nC_g\\bigg\\{\\Int_t^{t+\\tau}\\Int_{\\Omega}\\bigg|k_w(u^{\\mu}_{\\delta})(x,\\theta)\\nablap^{\\mu}_{\\delta}(x,\\theta)\n\\nabla g(u^{\\mu}_{\\delta})(x,t+\\tau)\\bigg|dxd\\theta \\nonumber\\\\\n&+\\Int_t^{t+\\tau}\\Int_{\\Omega}\\bigg|k_w(u^{\\mu}_{\\delta})(x,\\theta)\\nablap^{\\mu}_{\\delta}(x,\\theta)\\nabla\ng(u^{\\mu}_{\\delta})(x,t)\\bigg|\ndxd\\theta \\nonumber\\\\\n&+\\bigg|\\Int_{\\Omega}\\bigg[g(u^{\\mu}_{\\delta})(x,t+\\tau)-g(u^{\\mu}_{\\delta})(x,t)\\bigg]K(x,t,\\tau)dx\n\\bigg|\\bigg\\},\\label{Est20}\n\\end{eqnarray}\nwhere\n\\begin{equation}\\label{defK}\nK(x,t,\\tau):=\\Int_t^{t+\\tau}\\bigg(f^{\\mu}(c_\\delta(x,\\theta))\\overline s_\\delta(x)-\nf^{\\mu}(u_\\delta(x,\\theta))\\bigg[\\underline s_\\delta(x)+\\displaystyle {\\Int \\kern -0.961em -}_{\\Omega}(\\overline s_\\delta(y)-\\underline s_\\delta(y))dy\\bigg]\\bigg)d\\theta.\n\\end{equation}\nNext we estimate the right hand side of (\\ref{Est20}). Using (\\ref{Eb2}) we have that\n\\begin{eqnarray}\n&&\\Int_t^{t+\\tau}\\Int_{\\Omega}\\bigg|k_w(u^{\\mu}_{\\delta})(x,\\theta)\\nablap^{\\mu}_{\\delta}(x,\\theta)\n\\nabla g(u^{\\mu}_{\\delta})(x,t+\\tau)\\bigg|dxd\\theta\\nonumber\\\\\n&&\\leq\n\\Frac{1}{2}\\bigg(\\Int_t^{t+\\tau}\\Int_{\\Omega}|\\nablap^{\\mu}_{\\delta}(x,\\theta)|^2dxd\\theta+\n\\Int_t^{t+\\tau}\\Int_{\\Omega}|\\nabla\ng(u^{\\mu}_{\\delta})(x,t+\\tau)|^2dxd\\theta\\bigg)\\nonumber\\\\\n&&\\leq\n\\Frac{1}{2}\\bigg(\\Int_t^{t+\\tau}\\Int_{\\Omega}|\\nablap^{\\mu}_{\\delta}(x,\\theta)|^2dxd\\theta+\n\\tau\\Int_{\\Omega}|\\nabla g(u^{\\mu}_{\\delta})(x,t+\\tau)|^2dx\\bigg).\n\\label{Est21}\n\\end{eqnarray}\nSimilarly we have that\n\\begin{eqnarray}\n&&\\Int_t^{t+\\tau}\\Int_{\\Omega}\\bigg|k_w(u^{\\mu}_{\\delta})(x,\\theta)\\nablap^{\\mu}_{\\delta}(x,\\theta)\\nabla\ng(u^{\\mu}_{\\delta})(x,t)\\bigg|dxd\\theta\\nonumber\\\\\n&&\\leq \\Frac{1}{2}\\bigg(\\Int_t^{t+\\tau}\\Int_{\\Omega}|\\nablap^{\\mu}_{\\delta}(x,\\theta)|^2dxd\\theta+ \\tau\\Int_{\\Omega}|\\nabla\ng(u^{\\mu}_{\\delta})(x,t)|^2dx\\bigg). \\label{Est22}\n\\end{eqnarray}\nMoreover using (\\ref{Eb1}) and (\\ref{Eb1bis}) we obtain from the definition (\\ref{defK}) of $K$ that\n$$|K(x,t,\\tau)|\\leq\n\\Int_t^{t+\\tau}\\bigg[|\\overline s_\\delta|+|\\underline s_\\delta|+\\displaystyle {\\Int \\kern -0.961em -}_{\\Omega}|\\overline s_\\delta-\\underline s_\\delta|dx\\bigg]d\\theta\\leq\n\\bigg[|\\overline s_\\delta|+|\\underline s_\\delta|+\\displaystyle {\\Int \\kern -0.961em -}_{\\Omega}|\\overline s_\\delta-\\underline s_\\delta|dx\\bigg]\\tau.$$ This together with $(\\ref{sdeltaborne})$ and the fact\nthat the function $g(u^{\\mu}_{\\delta})$ is bounded uniformly on $\\mu$ and $\\delta$ yields\n\\begin{equation}\\label{Est22.2}\n\\bigg|\\Int_{\\Omega}\\bigg[g(u^{\\mu}_{\\delta})(x,t+\\tau)-g(u^{\\mu}_{\\delta})(x,t)\\bigg]K(x,t,\\tau)\\bigg|dx\\leq \\tilde C\\tau.\n\\end{equation}\nSubstituting (\\ref{Est21}), (\\ref{Est22}) and (\\ref{Est22.2}) into (\\ref{Est20}) we deduce that\n\\begin{eqnarray}\nA(t)&\\leq\n&C_g\\bigg(\\Int_t^{t+\\tau}\\Int_{\\Omega}|\\nablap^{\\mu}_{\\delta}(x,\\theta)|^2dxd\\theta+\n\\Frac{\\tau}{2}\\Int_{\\Omega}|\\nabla\ng(u^{\\mu}_{\\delta})(x,t+\\tau)|^2dx\\nonumber\\\\\n&&+ \\Frac{\\tau}{2}\\Int_{\\Omega}|\\nabla g(u^{\\mu}_{\\delta})(x,t)|^2dx+\\tilde C\\tau\\bigg), \\nonumber\n\\end{eqnarray}\nwhich we integrate on $[0,T-\\tau]$ to obtain\n\\begin{eqnarray}\n\\Int_0^{T-\\tau}A(t)dt&\\leq &C_g \\bigg(\n\\Int_0^{T-\\tau}\\Int_t^{t+\\tau}\\Int_{\\Omega}|\\nablap^{\\mu}_{\\delta}(x,\\theta)|^2dxd\\theta\ndt+\n\\tau \\Int_0^{T}\\Int_{\\Omega}|\\nabla g(u^{\\mu}_{\\delta})|^2dxdt+\\tilde C\\tau T\\bigg)\\nonumber\\\\\n&\\leq &C_g \\bigg( \\tau\\Int_0^{T}\\Int_{\\Omega}|\\nablap^{\\mu}_{\\delta}(x,\\theta)|^2dxd\\theta + \\tau \\Int_0^{T}\\Int_{\\Omega}|\\nabla\ng(u^{\\mu}_{\\delta})|^2dxdt+\\tilde C\\tau T\\bigg).\\nonumber\n\\end{eqnarray}\nIn view of (\\ref{Est1.1}) and (\\ref{Est1.2ter}) we deduce (\\ref{Est3}), which completes the proof of Lemma\n\\ref{leest3}.\n\\setcounter{equation}{0}\n\\section{Convergence as $\\delta\\downarrow 0$.}\\label{deltavers0}\nLetting $\\delta$ tend to 0, we deduce from the estimates given in Lemmas \\ref{leest1} and \\ref{leest2} the existence of\na weak solution of Problem $(S^\\mu)$. More precisely, we have the following result,\n\\begin{lemma}\\label{ledelta=0}\nThere exists a weak solution $(u^{\\mu},p^{\\mu})$ of Problem $(S^\\mu)$, which satisfies\n\\begin{equation}\\label{Est1delta=0} \\Int_0^T\\displaystyle \\Int _{\\Omega} \\bigg[k_a(u^{\\mu})\\bigg]^2\\bigg[\\nablap^{\\mu}+\\nabla\np_c(u^{\\mu})\\bigg]^2dxdt\\leq C\\mu,\n\\end{equation}\n\\begin{equation}\\label{Est1.1delta=0}\n\\Int_0^T\\displaystyle \\Int _{\\Omega} |\\nablap^{\\mu}|^2dxdt\\leq C,\n\\end{equation}\n\\begin{equation}\\label{Est1.2delta=0}\n\\Int_0^T\\displaystyle \\Int _{\\Omega} |\\nabla g(u^{\\mu})|^2dxdt\\leq C,\n\\end{equation}\n\\begin{equation}\\label{Est2.2delta=0}\n\\Int_0^T\\Int_{{\\Omega}_r}\\big[g(u^{\\mu})(x+\\xi,t)-g(u^{\\mu})(x,t)\\big]^2dxdt\\leq\nC\\xi^2,\n\\end{equation}\nwhere $\\xi\\in{\\mathrm {I\\mkern-5.5mu R\\mkern1mu}}^N$ and $|\\xi|\\leq 2r$. Moreover the following estimate of differences of time translates holds\n\\begin{equation}\\label{Est3delta=0}\n\\Int_0^{T-\\tau}\\Int_{{\\Omega}}\\big[g(u^{\\mu})(x,t+\\tau)-g(u^{\\mu})(x,t)\\big]^2dxdt\\leq\nC\\tau,\n\\end{equation}\nfor all $\\tau\\in(0,T)$.\n\\end{lemma}\n{\\underline{Proof}}: We deduce from (\\ref{Est1.1}), (\\ref{Est2.2}) and (\\ref{Est3}) that there exist functions $\\hat g^\\mu$ and\n$p^\\mu$ and a subsequence $((u^{\\mu}_{\\delta_n},p^{\\mu}_{\\delta_n}))_{n\\in N}$ of weak solutions of Problem $(S^{\\mu}_{{\\delta_n}})$ such\nthat\n\\begin{eqnarray}\n&&(g(u^{\\mu}_{\\delta_n}))_{n\\in N}\\mbox{ tends to }\\hat g^\\mu\\mbox{ strongly in }L^2(Q_T), ~~~~~~\\label{borneudelta=0.a}\n\\\\\n&&(p^{\\mu}_{\\delta_n})_{n\\in N}\\mbox{ tends to }p^\\mu\\mbox{ weakly in }L^2(0,T;H^1({\\Omega})),\\nonumber\n\\end{eqnarray}\nas $\\delta_n$ tends to zero. Thus for a subsequence, which we denote again by $\\delta_n$, we have that\n\\begin{equation}\\label{borneudelta=0.b}\n(g(u^{\\mu}_{\\delta_n}))_{n\\in N}\\mbox{ tends to }\\hat g^\\mu\\mbox{ for almost }(x,t)\\in Q_T.\n\\end{equation}\nUsing the fact that g is bijective we deduce that\n\\begin{equation}\\label{borneudelta=0.c} (u^{\\mu}_{\\delta_n})_{n\\in N} \\mbox{ tends to $u^\\mu:=g^{-1}(\\hat g^\\mu)$ strongly in $L^2(Q_T)$ and almost everywhere in }Q_T,\n\\end{equation}\nas $\\delta_n$ tends to zero. Moreover we have in view of (\\ref{Est1.2ter}) and (\\ref{borneudelta=0.a}) that $\\nabla\ng(u^{\\mu}_{\\delta_n})$ tends to $\\nabla g(u^{\\mu})$ weakly in $L^2(Q_T)$ as $\\delta_n\\downarrow 0$, so that by the\ndefinition (\\ref{defg}) of $g$\n\\begin{equation}\\label{cvfaible1}\nk_a(u^{\\mu}_{\\delta_n})\\nabla p_c(u^{\\mu}_{\\delta_n}) \\mbox{ tends to $k_a(u^{\\mu})\\nabla p_c(u^{\\mu})$ weakly in $L^2(Q_T)$ as\n$\\delta_n\\downarrow 0$.}\n\\end{equation}\nLetting $\\delta_n$ tend to 0 in (\\ref{borneu}) we deduce that\n\\begin{equation}\\label{borneudelta=0.1}\nu_m\\leq u^{\\mu}(x,t)\\leq 1. \\end{equation} Moreover we deduce from\n(\\ref{3d}) that\n\\begin{equation}\\label{delta=0.2}\n\\Int_\\Op^{\\mu}(x,t)dx=0, \\mbox{ for almost every } t \\in (0,T).\n\\end{equation} Multiplying (\\ref{1d}) by $\\varphi\\in{\\cal{C}}$,\nintegrating by parts and letting $\\delta_n$ tend to 0 we obtain\n\\begin{eqnarray}\n&\\Int_0^T\\displaystyle \\Int _{\\Omega} u^\\mu \\varphi_tdxdt =&\\Int_0^T\\displaystyle \\Int _{\\Omega}k_w(u^\\mu)\\nabla p^\\mu.\\nabla \\varphi dxdt-\\Int_0^T\\displaystyle \\Int _{\\Omega}\\bigg( f^\\mu(c)\\overline s- f^\\mu(u^\\mu)\\underline s \\bigg) \\varphi dxdt\\nonumber\\\\\n&&-\\displaystyle \\Int _{\\Omega} u_0(x)\\varphi(x,0)dx,\\label{delta=0.3}\n\\end{eqnarray}\nwhere we have used that $u_{0}^\\delta$ tends to $u_0$ in $L^2(\\Omega)$ and that $c_\\delta$, $\\overline s_\\delta$ and $\\underline s_\\delta$ tend\nrespectively to $c$, $\\overline s$ and $\\underline s$ in $L^2(Q_T)$ as $\\delta\\downarrow 0$. Similarly, multiplying (\\ref{2d}) by\n$\\varphi\\in{\\cal{C}}$, integrating by parts and letting $\\delta_n$ tend to 0 we deduce that\n\\begin{eqnarray}\n&&\\Int_0^T\\displaystyle \\Int _{\\Omega}\\bigg(1-u^{\\mu}\\bigg)\\varphi_tdxdt=\\Frac{1}{\\mu}\\Int_0^T\\displaystyle \\Int _{\\Omega}\\bigg(k_a(u^{\\mu})\\nablap^{\\mu}+\\nabla g(u^{\\mu})\\bigg).\\nabla \\varphi dx dt\\nonumber\\\\\n&&-\\Int_0^T\\displaystyle \\Int _{\\Omega}\\bigg((1-f^{\\mu}(c))\\overline s-(1-f^{\\mu}(u^{\\mu}))\\underline s\\bigg) \\varphi dx dt-\\displaystyle \\Int _{\\Omega}\n\\bigg(1-u_0(x)\\bigg)\\varphi(x,0)dx,~~\\label{delta=0.4}\n\\end{eqnarray}\nwhich since $\\nabla g(u^{\\mu})=k_a(u^{\\mu})\\nabla p_c(u^{\\mu})$ coincides with (\\ref{defsol2}). Next we prove\n(\\ref{Est1delta=0}). We first check that\n\\begin{equation}\\label{cvfaible2}\nk_a(u^{\\mu}_{\\delta_n})\\nabla p^{\\mu}_{\\delta_n}\\mbox{ tends to }k_a(u^{\\mu})\\nabla p^{\\mu} \\mbox{ weakly in }L^2(Q_T),\n\\end{equation}\nas $\\delta_n$ tends to 0. Let $\\varphi\\in L^2(Q_T)$, we have that\n\\begin{eqnarray}\n\\bigg|\\Int_{Q_T} \\bigg(k_a(u^{\\mu}_{\\delta_n})\\nabla p^{\\mu}_{\\delta_n}-k_a(u^{\\mu})\\nabla p^{\\mu}\\bigg)\\varphi ~dxdt\\bigg| \\leq\n|I^1_{\\delta_n}|+|I^2_{\\delta_n}|,\\label{delta=0.4bis}\n\\end{eqnarray}\nwhere\n$$\nI^1_{\\delta_n}:= \\Int_{Q_T} \\bigg(k_a(u^{\\mu}_{\\delta_n})-k_a(u^{\\mu})\\bigg) \\nabla p^{\\mu}_{\\delta_n}\\varphi~dxdt\n$$and\n$$I^2_{\\delta_n}=\\Int_{Q_T}\nk_a(u^{\\mu})\\varphi\\bigg(\\nabla p^{\\mu}_{\\delta_n} -\\nabla p^{\\mu}\\bigg)~dxdt.\n$$\nUsing the fact that $\\nabla p^{\\mu}_{\\delta_n}$ converges to $\\nabla p^{\\mu}$ weakly in $L^2(Q_T)$ as $\\delta_n\\downarrow\n0$, we deduce, since $k_a(u^{\\mu})\\varphi\\in L^2(Q_T)$, that\n\\begin{equation}\\label{delta=0.4ter}\n\\mbox{ $|I^2_{\\delta_n}|$ tends to $0$ as $\\delta_n\\downarrow 0$.}\n\\end{equation}\nMoreover we have by (\\ref{Est1.1}) that\n\\begin{eqnarray}\n|I^1_{\\delta_n}|&&\\leq \\bigg(\\Int_{Q_T} \\bigg|k_a(u^{\\mu}_{\\delta_n})-k_a(u^{\\mu})\\bigg|^2\\varphi^2dxdt\\bigg)^{1\/2}\n\\bigg(\\Int_{Q_T}|\\nabla p^{\\mu}_{\\delta_n}|^2dxdt\\bigg)^{1\/2} \\nonumber\\\\&&\\leq C\\bigg(\\Int_{Q_T}\n\\bigg|k_a(u^{\\mu}_{\\delta_n})-k_a(u^{\\mu})\\bigg|^2\\varphi^2dxdt\\bigg)^{1\/2}.\\nonumber\n\\end{eqnarray}\nSince $\\bigg|k_a(u^{\\mu}_{\\delta_n})-k_a(u^{\\mu})\\bigg|^2\\varphi^2\\leq 4\\varphi^2$ and since $k_a(u^{\\mu}_{\\delta_n})$ tends\nto $k_a(u^{\\mu})$ almost everywhere, we deduce from the Dominated Convergence Theorem that $I^1_{\\delta_n}$ tends to\n$0$ as $\\delta_n\\downarrow 0$. This with (\\ref{delta=0.4ter}) implies (\\ref{cvfaible2}), which with\n(\\ref{cvfaible1}) gives that\n\\begin{equation}\\label{cvfaible3}\nk_a(u^{\\mu}_{\\delta_n})\\bigg[\\nabla p^{\\mu}_{\\delta_n}+\\nabla p_c(u^{\\mu}_{\\delta_n})\\bigg]\\mbox{ tends to\n}k_a(u^{\\mu})\\bigg[\\nabla p^{\\mu}+\\nabla p_c(u^{\\mu})\\bigg] \\mbox{ weakly in }L^2(Q_T).\n\\end{equation}\nThe functional $v\\mapsto \\Int_{Q_T}v^2dxdt$ is convex and lower semi continuous from $L^2(Q_T)$ to $\\overline R$\ntherefore it is also weakly l.s.c. (see \\cite{B} Corollary III.8) and thus we deduce from (\\ref{Eb2bis}),\n(\\ref{Est1}) and (\\ref{cvfaible3}) that\n$$\n\\begin{array}{lll}\n\\Int_{Q_T}\\bigg[k_a(u^{\\mu})\\bigg]^2\\bigg[\\nabla p^{\\mu} +\\nabla p_c(u^{\\mu})\\bigg]^2dxdt\n&\\leq& \\liminf_{\\delta_n\\downarrow 0\n}\\Int_{Q_T}\\bigg[k_a(u^{\\mu}_{\\delta_n})\\bigg]^2\\bigg[\\nabla p^{\\mu}_{\\delta_n} +\\nabla\np_c(u^{\\mu}_{\\delta_n})\\bigg]^2dxdt\\\\\n&\\leq& \\liminf_{\\delta_n\\downarrow 0}\\Int_{Q_T}k_a(u^{\\mu}_{\\delta_n})\\bigg[\\nabla p^{\\mu}_{\\delta_n} +\\nabla\np_c(u^{\\mu}_{\\delta_n})\\bigg]^2dxdt \\\\\n&\\leq &C\\mu,\n\\end{array}\n$$ which coincides with (\\ref{Est1delta=0}). Finally, we deduce\nrespectively from (\\ref{Est1.1}), (\\ref{Est1.2ter}), (\\ref{Est2.2}) and (\\ref{Est3}) the estimates\n(\\ref{Est1.1delta=0}), (\\ref{Est1.2delta=0}), (\\ref{Est2.2delta=0}) and (\\ref{Est3delta=0}). This concludes the\nproof of Lemma \\ref{ledelta=0}.\n\n\\setcounter{equation}{0}\n\\section{Convergence as $\\mu\\downarrow 0$.}\\label{muvers0}\nThe goal of this section is to prove Theorem \\ref{thlim}. We first deduce from the estimates (\\ref{Est1.1delta=0}),\n(\\ref{Est2.2delta=0}) and (\\ref{Est3delta=0}) that there exists a couple of functions $(u,p)$ and a subsequence\n$((u^{\\mu_n},p^{\\mu_n}))_{n\\in N}$ such that $$\\begin{array}{ll}\n&(u^{\\mu_n})_{n\\in N}\\mbox{ tends to }u\\mbox{ strongly in }L^2(Q_T)\\mbox{ and almost everywhere in }Q_T ,\\\\\n&(p^{\\mu_n})_{n\\in N}\\mbox{ tends to }p\\mbox{ weakly in }L^2(0,T;H^1({\\Omega})),\\\\\n\\end{array}$$\nas $\\mu_n$ tends to zero. Moreover since\n$$0\\leq f^{\\mu_n}(u^{\\mu_n})\\leq 1,$$\nthere exists a function $\\hat f\\in L^2(Q_T)$ with $0\\leq \\hat f\\leq 1$ and a subsequence\n$(f^{\\mu_{n_m}}(u^{\\mu_{n_m}}))_{n_m\\in N}$ of $(f^{\\mu_{n}}(u^{\\mu_{n}}))_{n\\in N}$ such that\n$(f^{\\mu_{n_m}}(u^{\\mu_{n_m}}))_{n_m\\in N}$ tends to $\\hat f$ weakly in $L^2(Q_T)$ as $\\mu_{n_m}$ tends to zero.\nMoreover we deduce respectively from (\\ref{borneudelta=0.1}) and (\\ref{delta=0.2}) that $0\\leq u\\leq 1$ and that\n$$\\Int_\\Omega p(x,t)dx=0,\\mbox{ for almost every }t\\in(0,T),$$\nwhich gives (\\ref{lim2}). As it is done in Section \\ref{muvers0} in the proof of (\\ref{Est1delta=0}), one can first\ncheck that\n$$\nk_a(u^{\\mu_{n_m}})(\\nabla p^{\\mu_{n_m}} +\\nabla p_c(u^{\\mu_{n_m}}))\\mbox{ tends to }k_a(u)(\\nabla p +\\nabla\np_c(u))\\mbox{ weakly in }L^2(Q_T),\n$$\nas $\\mu_{n_m}\\downarrow 0$ and then deduce from (\\ref{Est1delta=0}) the estimate (\\ref{lim1}). Furthermore letting $\\mu_{n_m}$ tends to zero into\n(\\ref{delta=0.3}) we obtain, since $\\lim_{\\mu_{n_m}\\downarrow 0}f^{\\mu_{n_m}}(s)=\\chi(s)$ for all $s\\in[0,1]$, that\n$$\n\\begin{array}{ll}\n\\Int_0^T\\displaystyle \\Int _{\\Omega} u \\varphi_tdxdt =&\\Int_0^T\\displaystyle \\Int _{\\Omega}k_w(u)\\nabla p.\\nabla \\varphi dxdt-\\Int_0^T\\displaystyle \\Int _{\\Omega}\\bigg(\\chi(c)\\overline s-\\hat f\\underline s \\bigg) \\varphi dxdt\\nonumber\\\\\n&-\\displaystyle \\Int _{\\Omega} u_0(x)\\varphi(x,0)dx,\n\\end{array}\n$$\nwhich coincides with (\\ref{sollim1}) and concludes the proof of Theorem \\ref{thlim}.\n\n\\setcounter{equation}{0}\n\\section{Numerical simulations}\n\\subsection{The saturation equation and the numerical algorithm}\nIn this section we present numerical simulations in one space dimension. To that purpose we apply the finite volume\nmethod, which we present below. To begin with, we rewrite the equations (\\ref{n1}) and (\\ref{n2}) in the case that\n$\\Omega=(0,1)$; this gives for $(x,t)\\in(0,1)\\times(0,T)$\n\\begin{eqnarray}\n&&u^\\mu_t =\\partial_x\\bigg(k_w(u^\\mu)\\partial_x\np^\\mu\\bigg)+f^{\\mu}(c)\\overline s-f^{\\mu}(u^\\mu)\\underline s, ~~\\label{D1} \\\\\n&&(1-u^\\mu)_t =\n\\partial_x\\bigg(\\Frac{1}{\\mu}k_a(u^\\mu)\\partial_x(p^\\mu+p_c(u^\\mu))\\bigg)+ (1-f^{\\mu}(c))\\overline s-(1-f^{\\mu}(u^\\mu))\\underline s.~~\\label{D2}\n\\end{eqnarray}\nAdding up both equations and using the boundary conditions (\\ref{n3}) and (\\ref{n5}) we obtain\n\\begin{equation}\\label{D3}\n\\partial_x p^\\mu=-\\Frac{k_a(u^\\mu)}{k_a(u^\\mu)+\\muk_w(u^\\mu)}\\partial_x(p_c(u^\\mu)).\n\\end{equation}\nSubstituting (\\ref{D3}) into (\\ref{D1}) yields\n\\begin{equation}\\label{D4}\nu^\\mu_t\n=-\\partial_x\\bigg[f^{\\mu}(u^\\mu)\\Frac{k_a(u^\\mu)}{\\mu}\\partial_x(p_c(u^\\mu))\\bigg]+f^{\\mu}(c)\\overline s-f^{\\mu}(u^\\mu)\\underline s.\n\\end{equation}\nMoreover we deduce from (\\ref{D3}) and the definition (\\ref{R}) of ${\\cal{R}}^\\mu$ that $\\partial_x\np^\\mu=-\\partial_x({\\cal{R}}^\\mu(u^\\mu))$, so that in view of (\\ref{n4}) we have\n\\begin{equation}\\label{D5}\np^\\mu(x,t)=-{\\cal{R}}^\\mu(u^\\mu)(x,t)+\\Int_0^1{\\cal{R}}^\\mu(u^\\mu)(y,t)dy.\n\\end{equation}\nIn the sequel, we compare numerically the solution $u^\\mu$ of (\\ref{D4}) with the solution $u$ of the limit equation\nin the case that $u<1$, namely\n\\begin{equation}\\label{D5.b}\nu_t =-\\partial_x\\bigg(k_w(u)\\partial_xp_c(u)\\bigg)+ \\chi(c)\\bar s.\n\\end{equation}\nWe discretize the time evolution equation (\\ref{D4}) together with the initial condition and the homogeneous Neumann\nboundary condition. The time explicit finite volume scheme is defined by the following equations in which ${\\cal{K}}>0$ and\n${\\cal{J}}>0$ denote respectively the time and the space step. \\\\\n(i) The discrete initial condition is given for $i\\in\\{0,...,[1\/{\\cal{J}}]\\}$ by\n\\begin{equation}\\label{Condinitialdiscre}\n[U^\\mu]_i^{0}=u^\\mu(i{\\cal{J}},0).\n\\end{equation}\n(ii) For $i\\in\\{0,...,[1\/{\\cal{J}}]\\}$ and for $n\\in\\{0,...,[T\/{\\cal{K}}]\\}$ the discrete equation is given by\n\\begin{eqnarray}\n\\Frac{1}{{\\cal{K}}}\\bigg([U^\\mu]_i^{n+1}-[U^\\mu]_i^{n}\\bigg)&=&\n[F^\\mu]_{i+1}^n-[F^\\mu]_i^n\n+f^{\\mu}(C_i^n)\\overline{S}_i^n-f^{\\mu}([U^\\mu]_i^n)\\underline{S}_i^n,\\label{D6}\n\\end{eqnarray}\nwhere\n$$[F^\\mu]_i^n=-\\Frac{1}{{\\cal{J}}}\\bigg(p_c([U^\\mu]_{i+1}^n)-p_c([U^\\mu]_{i}^n\\bigg)\n\\Frac{k_w([U^\\mu]_{i+1}^n)k_a([U^\\mu]_{i}^n)}{\\muk_w([U^\\mu]_{i+1}^n)+k_a([U^\\mu]_{i}^n)}.$$\n\n\\noindent(iii) For $n\\in\\{0,...,[T\/{\\cal{K}}]\\}$ the discrete Neumann condition is defined by\n\\begin{equation}\\label{Condborddiscre}\n[F^\\mu]_0^{n}=0 \\mbox{ and } [F^\\mu]_{[1\/{\\cal{J}}]}^{n}=0.\n\\end{equation}\nThe numerical scheme (\\ref{Condinitialdiscre})-(\\ref{Condborddiscre}) allows to build an approximate solution,\n$u_{{\\cal{J}},{\\cal{K}}}:[0,1]\\times[0,T]\\rightarrow {\\mathrm {I\\mkern-5.5mu R\\mkern1mu}}$ for all $i\\in\\{0,...,[1\/{\\cal{J}}]\\}$ and all $n\\in\\{0,...,[T\/{\\cal{K}}]\\}$, which is\ngiven by\n\\begin{equation}\\label{schema}\nu_{{\\cal{J}},{\\cal{K}}}(x,t)=u_i^n,\\mbox{ for all }x\\in(i{\\cal{J}},(i+1){\\cal{J}}]\\mbox{ and\nfor all }t\\in (n{\\cal{K}},(n+1){\\cal{K}}].\n\\end{equation}\nIn order to also compute the pressures, we propose the following\ndiscrete equation corresponding to (\\ref{D5})\n\\begin{equation}\\label{D7}\n[P^\\mu]_i^{n}=-{\\cal{R}}([U^\\mu]_i^{n})+{\\cal{J}}\\Sigma_{j=1}^{[1\/{\\cal{J}}]}{\\cal{R}}([U^\\mu]_j^{n}).\n\\end{equation}\nFinally, setting $p^\\mu_g(x,t)=p^\\mu(x,t)+p_c(u^\\mu)(x,t)$ we\ndeduce that\n\\begin{equation}\\label{D8}\n([P_g^\\mu])_i^{n}=-{\\cal{R}}([U^\\mu]_i^{n}+p_c([U^\\mu]_i^{n})\n+{\\cal{J}}\\Sigma_{j=1}^{[1\/{\\cal{J}}]}{\\cal{R}}([U^\\mu]_j^{n}),\n\\end{equation}\nfor all $i\\in\\{0,...,[1\/{\\cal{J}}]\\}$ and all $n\\in\\{0,...,[T\/{\\cal{K}}]\\}$.\nSimilarly we propose a finite volume scheme corresponding to the\nequation (\\ref{D5.b}), namely\n\\begin{eqnarray}\n\\Frac{1}{{\\cal{K}}}\\bigg(U_i^{n+1}-U_i^{n}\\bigg)= F_{i+1}^n-F_i^n\n+\\chi(C_i^n)\\overline{S}_i^n\\label{D6},\n\\end{eqnarray}\nwhere\n$$F_i^n=-\\Frac{1}{{\\cal{J}}}\\bigg(p_c(U_{i+1}^n)-p_c(U_{i}^n\\bigg)\nk_w(U_{i+1}^n),$$ for all\n$(i,n)\\in\\{0,...,[1\/{\\cal{J}}]\\}\\times\\{0,...,[T\/{\\cal{K}}]\\}$.\n\n\\subsection{Numerical tests}\nFor the numerical computation we take $\\mu=10^{-8}$, $p_c(z)=0,1\\sqrt{1-z}$, $k_a(z)=(1-z)^2$, $k_w(z)=\\sqrt z$ and\n$\\overline s(z)=\\delta_0(z)$, $\\underline s(z)=\\delta_1(z)$, where $\\delta_a$ is the Dirac function at the point $a$. Furthermore\n$u^\\mu$ is\ngiven by the line with crosses, $p^\\mu_g$ is given by the lines with diam and the limit function $u$ corresponds to the continuous line.\\\\\n$\\\\ $ \\underline{\\bf{First test case}:} The case that $c=0,7$ and\n$u_0=1$ on $[0,1]$. We obtain at $t=0,01$ the following pictures\n\\begin{center}\n\\includegraphics[width=0.5\\hsize]{test1articlet_0_01}\\\\\n\\it{Figure 1 : t=0,01}\n\\end{center}\nWe note that, for $\\mu$ small, the functions $u$ and $u^\\mu$ are very close. Here we only start with water and\ninject a mixture of water and air. The air immediately invades the whole domain. Figure 1 illustrates the result\nwhich we proved in this paper, namely that $u^\\mu$ tends to the solution $u$ of the limit equation (\\ref{D5.b}) as\n$\\mu$ tends to 0 and moreover that the pressure\n$p^\\mu_a$ is constant. This is indeed the case since $u<1$.\\\\\n$\\\\ $\n\\underline{\\bf{Second test case}:} The case that $c=0,7$ and\n$u_0(x)=\\left\\{\\begin{array}{ll} 0,1\\mbox{ on }[0,1\/3]\\\\0,7\\mbox{\non }(1\/3,1]\\end{array}\\right.$. We obtain the following pictures\nfor $t=0,01$ and for $t=0,1$ respectively\n\\begin{center}\n\\includegraphics[scale=0.4]{test2articlet_0_01}\\\\\n\\it{Figure 2 : t=0,01}\n\\end{center}\n$\\\\ $\n\\begin{center}\n\\includegraphics[scale=0.4]{test2articlet_0_1}\\\\\n\\it{Figure 3 : t=0,1}\n\\end{center}\nThe injection of a mixture of water and air $(c=0.7)$ takes place in a region of low water saturation. We first\nremark that both functions $u^\\mu$ and $u$ evolve very slowly. Here again we have that $u(x,t)<1$ for all\n$(x,t)\\in(0,1)\\times(0,T)$ and we remark that the graphs of the two functions $u^\\mu$ and $u$ nearly\ncoincide.\\\\\n$\\\\ $ \\underline{\\bf{Third test case}:} The case that $c=1$ and\n$u_0(x)=\\left\\{\\begin{array}{ll} 0,1\\mbox{ on }[0,1\/3]\\\\0,7\\mbox{\non }(1\/3,1]\\end{array}\\right.$. We obtain the following pictures\nfor $t=0,01$ and for $t=0,1$ respectively\n\\begin{center}\n\\includegraphics[scale=0.4]{test2articlet_0_01_c_1}\\\\\n\\it{Figure 4 : t=0,01}\n\\end{center}\n$\\\\ $\n\\begin{center}\n\\includegraphics[scale=0.4]{test2articlet_0_1_c_1}\\\\\n\\it{Figure 5 : t=0,1}\n\\end{center}\nHere only water is injected; note that the saturation $u^{\\mu}$ evolves rather fast.\n\n\\newpage \\noindent\n{\\Large\\bf Acknowledgement} \\\\\n\\vspace{0.05 cm} \\\\\nWe would like to thank Professor Rapha\\`ele Herbin from the University of Provence for her interest in this work.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec-intro}\n\nThe rotational period of an asteroid is a physical property that is important in a wide range of planetary science and space exploration contexts. A rotational period measurement is essential to the characterization of the rotational state~\\citep[e.g.,][]{prav02}, which informs our understanding of an asteroid's interior and morphology~\\citep[e.g.,][]{Sche15AIV}, dynamical evolution through the Yarkovsky and YORP effects~\\citep[e.g.,][]{Vokr15AIV,gree20}, and the formation and evolution of binaries, triples, and pairs~\\citep[e.g.,][]{Marg15AIV,Wals15AIV}. Spin rate distributions place bounds on the dynamical and collisional evolutions of the main belt of asteroids~\\citep[e.g.,][]{bott15AIV}, and therefore the characteristics of the near-Earth asteroid population, which governs the history of impact cratering in the inner solar system and affects planetary defense efforts. Spin periods also provide useful initial conditions when modeling the shape~\\citep[e.g.,][]{Ostr02,Benn15AIV,Dure15AIV} and thermophysical properties~\\citep[e.g.,][]{Delb15AIV} of asteroids.\nIn these contexts, the availability of a small number of candidate spin periods is extremely valuable. One can test the model with each trial period and promptly identify the correct period. This fact motivated in part our inclusion of secondary spin period solutions in our results, in addition to our primary, best-fit period solutions.\n\n\n\nSeveral approaches have been used to measure the rotational periods of asteroids with high precision, including Earth-based~\\citep[e.g.,][]{prav02} or space-based~\\citep[e.g.,][]{hora18} photometric observations or a combination of the two~\\citep[e.g.,][]{dure18}, as well as Earth-based radar observations~\\citep[e.g.,][]{ostr06,naid15dp}. Optical lightcurve photometry based on wideband measurements of the sunlight reflected by the asteroid is the most common approach.\nHere we use infrared lightcurves to determine asteroid spin periods.\n\n\n\nDuring its six-month primary mission, the WISE spacecraft~\\citep{wrig10} conducted a whole-sky infrared survey at four infrared bands (W1--4) centered at 3.4, 4.6, 12, and 22 $\\mu$m. All four detectors were simultaneously exposed, producing up to four independent photometric measurements. The high-quality, multi-band IR observations of $\\sim$100,000 asteroids have been used to estimate asteroid diameters and albedos~\\citep[e.g.,][]{main15AIV}. Improved algorithms applied to a curated set of thousands of asteroids yielded refined estimates as well as estimates for asteroids not previously analyzed~\\citep{myhr22}.\n\n\n\n\nBoth the cadence and length of an asteroid lightcurve observing campaign determine the parameters that can be reliably recovered from the observations. Durations on the order of days\/months\/years, such as survey data from the Palomar Transient Factory (PTF) \\citep{wasz15}, provide an opportunity to sample the object at different phase angles and to recover parameters of the phase function \\citep{muin10}. Densely sampled observations that span at least a complete rotational cycle provide the best opportunity to determine the spin period. WISE observations of asteroids present a challenge for lightcurve analyses because they take place over short intervals with sparse cadence, typically yielding only $\\sim$16 observations over a $\\sim$36-hour interval~\\citep{wrig10}. Nevertheless, most asteroids experience a few rotations in 36 hr, such that WISE data can in principle be used to estimate the spin periods of thousands of asteroids.\n\n\\citet{dure15} combined sparse photometry, including WISE data, to derive the spin period of\none asteroid. \\citet{hanu15} combined WISE thermal infrared data and other data to obtain shape models and spin periods of\nsix asteroids. Their work has since been expanded to derive 1451 spin periods for asteroids observed by WISE \\citep{dure18}. Here, we use the well-curated data set of ~\\citet{myhr22} from the fully cryogenic phase of the WISE mission to estimate spin periods for hundreds of asteroids.\n\n\n\n\n\n\n\n\n\nThe Lightcurve Database (LCDB) \\citep{lcdb} is a compilation of most known lightcurve measurements from various sources. Each lightcurve is assigned a quality code\nbetween 0 (incorrect) and 3 (best) by the database curators to convey the confidence level in the uniqueness and accuracy of the rotational period estimate.\nWe used the LCDB to evaluate the reliability of our solutions and to train a machine learning reliability classifier.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Methods}\n\\label{sec-methods}\n\n\n\\subsection{Overview}\n\\label{sec-overview} \n\n\n\n\n\n\nOur methods closely follow those of \\citet{wasz15}, who used sparse photometry from the Palomar Transient Factory (PTF) to determine $\\sim$9,000 reliable asteroid spin periods. Their method is conceptually straightforward. For each trial period, one fits a Fourier series model \\citep[][Equation 1]{harr89} to the observed flux values and computes the sum of squares of the flux residuals. The Fourier series is truncated after the second harmonic, a simplification that rests on the assumption that the object is approximately ellipsoidal in shape. It is also consistent with the fact that the second harmonic dominates asteroid lightcurves with amplitudes greater than 0.4 magnitudes \\citep[][]{harr14}. Although this model is insufficient to capture the full details of the lightcurve, it is adequate to recover the spin period in most instances, as can be verified by comparing the solutions to high-quality (quality code 3- or above) solutions published in the lightcurve database (LCDB) \\citep{lcdb}. A similar method was also used by \\citet{chan17} to determine 2780 reliable asteroid rotation periods from the PTF.\n\n\n\n\n\n\n\\citet{wasz15}'s method is directly applicable to WISE photometry, which typically contains\nat least $\\sim$16 observations of each asteroid\nwith a nominal 1.59 hr cadence \nover a $\\sim$36 hr period. \nAlthough this observational\nmode prevents the determination of spin periods for fast ($P<$ 3.17 hr) and\nslow ($P>$ 72 hr) rotators, \nmost asteroids have spin periods that are amenable to characterization with this technique.\nBased on reliable LCDB statistics (quality code 3- or higher) and the range of diameters (0.28--72.2 km) in the \\citet{myhr22} sample, we estimate that fewer than 17\\% of asteroids in our sample have a spin period smaller than 3.17 hr.\n\n\\citet{wasz15} were able to fit for a photometric phase function because their observations were obtained over a wide range of phase angles. In contrast, WISE observations typically span a narrow range of phase angles and we did not attempt to evaluate the phase function. Phase angles remained nearly constant during the short observation intervals, and phase angle effects were absorbed by the zeroth-order coefficient of the Fourier series, i.e., mean magnitude.\n\nWe compared our method to results obtained with the more traditional Lomb-Scargle periodogram \\citep{press92} (Sections~\\ref{methods-ls} and \\ref{res-ls}).\n\n\\subsection{Data Set}\n\\label{sec-data} \n\n\n\\subsubsection{Initial Data Set}\n\\label{sec-data-gold4}\n\nWe used the carefully curated data set of \\citet{myhr22}, who eliminated measurements with artifacts, low signal-to-noise ratio (S\/N), poor photometric quality, saturation, questionable PSF fits, background confusion, problematic near-conjunction conditions, or large discrepancies between ephemeris predictions and reported position. Their data set provides high-quality flux measurements in all four infrared bands for 4420 asteroids. For a small fraction (6\\%, 265 asteroids),\nWISE observations were obtained in distinct (almost always two) epoch clusters separated by more than 30 days.\nWe determined independent solutions for each cluster of observations because they were obtained at different phase angles.\nAlthough our analysis is done on individual clusters, we may refer to the clusters as asteroids for ease of presentation.\n\n\n\\subsubsection{Assignment of Flux Uncertainties}\n\\label{sec-sigma}\n\nBoth \\citet{hanu15} and \\citet{myhr18empirical} have shown that uncertainties reported by the WISE pipeline underestimate actual flux uncertainties. \\citet{hanu15} used $\\sim$400 pairs of asteroid detections observed in quick succession ($\\sim$11 s) to quantify actual flux uncertainties in W3 and W4. They found that uncertainties reported in the WISE database underestimate actual uncertainties by factors 1.4 and 1.3 in W3 and W4, respectively. \\citet{myhr18empirical} expanded this analysis to bands W1--W4 and included a much larger number of pairs (7834, 11202, 125318, and 59049 in W1--W4, respectively). Here, we further expanded the number of pairs and fit Gaussians to the distributions of Z values $(Z=(f_1 - f_2)\/\\sqrt{\\sigma_1^2+\\sigma_2^2})$ \\citep[][Equation 3]{myhr18empirical} after removing $\\sim$1\\% of pairs with $|Z|>5$. The elimination is required because the tails of the distributions are non-Gaussian even though the cores of the distributions are well approximated by Gaussians. We list the correction factors in all four bands for completeness (Table \\ref{tab-factors}). We assigned uncertainties to the flux measurements by multiplying the uncertainties reported in the WISE database with the relevant correction factor.\n\n\\begin{table}[h]\n \\begin{center}\n \\begin{tabular}{lrrr}\n %\n\n Band & Number of pairs & Correction factor & Median $\\sigma$ (corrected) \\\\\n \\hline\n W1 & 29936 & 1.224 & 0.195 \\\\\n W2 & 34462 & 1.120 & 0.132 \\\\\n W3 & 170232 & 1.479 & 0.035 \\\\\n W4 & 120225 & 1.218 & 0.058 \\\\\n \\end{tabular}\n \\caption{Correction factors that are required to convert flux uncertainties reported in the WISE pipeline to actual uncertainties. The second column shows the number of remaining pairs after elimination of $\\sim$1\\% of pairs with uncharacteristically large flux differences.\n %\n %\n %\n %\n The last column shows the median flux uncertainties in magnitude units after correction.\n }\n \\label{tab-factors}\n\\end{center} \n \\end{table}\n\n\n\\subsubsection{Band Selection}\n\\label{sec-band-selection} \n\n\nWe focus on observations in W4 for two reasons. First, thermal emission from asteroids is stronger, S\/N is higher, and the number of observations in the curated data set is higher in W3 and W4 than in W1 and W2. \nSecond, the observation cadence of the original and curated WISE data \nis generally non-uniform, with time intervals as small as 11~s, a most common time interval of 1.59 hr, a frequent time interval of 2 $\\times$ 1.59 hr = 3.17 hr, and occasional time intervals at higher multiples of 1.59 hr.\nW4 observations provide the best observational cadence, whereas W3 observations have a\nhigher fraction\nof asteroids with a longer cadence (Figure \\ref{fig-w3-cadence}), which increases the susceptibility to aliasing difficulties (Section {\\ref{sec-alias}}). \n\n\n\n\n\n\n\\begin{figure}[hbt]\n\\begin{center}\n \\begin{tabular}{cc}\n \\includegraphics[width=3in]{plots\/WISE_Cadance_W3.pdf} & \\includegraphics[width=3in]{plots\/WISE_Cadance_W4.pdf}\n \\end{tabular}\n %\n \\caption{Histograms of W3 (left) and W4 (right) sampling intervals in the \\citet{myhr22} data set.}\n %\n\\label{fig-w3-cadence}\n\\end{center}\n\\end{figure}\n\n\nWe are not concerned with thermal lags or differences in lightcurve amplitudes compared to optical lightcurves, as they are inconsequential to the determination of spin periods. In addition, \\citet{dure18} demonstrated that thermal lightcurves in W3 and W4 are qualitatively consistent with optical lightcurves. \n\nWhile multiband lightcurve fits could in principle be envisioned, their utility in the context of spin period determinations with WISE data is limited because the exposures in the four bands are simultaneous, such that they sample roughly the same rotational phase. The determination of phase lags among observations in the four WISE bands is potentially informative but beyond the scope of the current work.\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{Data Selection Filters}\n\\label{sec-data-prefilters}\n\nWe applied several preprocessing filters in order to identify lightcurves most suitable for rotational period determination. Each observation cluster must pass all of the following filters to qualify for analysis, otherwise it is discarded.\nFirst, we focus on\nlightcurves that have at least 12 data points in W4.\nSecond, we eliminated\nlightcurves where the peak-to-peak variation in flux magnitude W4$_{{\\rm red}} < 0.3$ mag,\nwhere we used magnitudes reduced to the values that would be observed at an asteroid-sun distance $r_{\\rm as}$ = 1 au and asteroid-observer distance $r_{\\rm ao}$ = 1 au.\nThis filter eliminates low-amplitude lightcurves, which may be ambiguous with respect to spin period determination \\citep{harr14}.\nThis filter effectively eliminates very slow rotators, which are not considered in our analysis anyway (Section \\ref{sec-fit}).\nThird, we eliminated any\nlightcurve that does not have an adequate observation cadence. Specifically, we required at least one time interval between consecutive observations to fall in the range 1.55--1.58 h, which corresponds to the WISE spacecraft orbital period \\citep{wrig10}. Observation clusters that have a longer (3.10--3.16 h)\nminimum sampling interval are more susceptible to severe aliasing (Section \\ref{sec-alias}) and are discarded.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nThe filtered data set contains a total of 3061 asteroids, with 3225 observation epochs and 57,532 W4 photometric measurements. In this data set, 164 (5.3\\%) out of the 3061 asteroids contain two observation clusters, and none have more than 2 observation clusters.\n\nOn average, an observation cluster contains 16 data points and spans 36 hr.\n\n\n\n\n\n\n\n\n\\begin{table}[h]\n \\begin{center}\n \\begin{tabular}{lr}\n W4 flux observations & 57,532 \\\\ %\n %\n %\n Median flux uncertainty & 0.067 \\\\ %\n Median number of data points per cluster & 16 \\\\\n Median observation span of clusters & 1.522 days\\\\\n\n \\end{tabular}\n %\n \\caption{Characteristics of filtered data set}\n \\label{tab-preproc-stats}\n\\end{center} \n \\end{table}\n\n\n\n\n\n\n\n\n\\subsection{Fitting Procedure}\n\\label{sec-fit}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nAn effective method for identifying periodicities in sparse data is a ``period scan'', where a model lightcurve is fitted to the data for a range of trial spin periods and a measure of the misfit (i.e., fit dispersion) is evaluated at each trial period \\citep{harr12}. The misfit metric is the usual sum of squares of residuals:\n\\begin{equation}\n\\chi^2 = \\sum_{i=1}^{N} \\frac{(O_i - C_i)^2}{\\sigma_i^2},\n\\end{equation}\nwhere $O_i$ is the $i$-th observation, $C_i$ is the $i$-th computed (modeled) value, $\\sigma_i$ is the uncertainty associated with the $i$-th observation, and the index $i$ ranges from 1 to $N$, the total number of observations. The observations are the reduced flux magnitudes (Section \\ref{sec-data-prefilters}), the uncertainties are the magnitude uncertainties from the WISE database multiplied by the relevant correction factors (Section \\ref{sec-sigma}), and the computed values are obtained by fitting a second-order Fourier model to the data similar to \\citet{wasz15}'s formulation. Specifically, \n\\begin{equation}\n C_i = A_{0,0}\n + A_{1,1} \\sin{\\left(\\frac{2\\pi}{P} t_i\\right)}\n + A_{2,1} \\cos{\\left(\\frac{2\\pi}{P} t_i\\right)}\n + A_{1,2} \\sin{\\left(\\frac{4\\pi}{P} t_i\\right)}\n + A_{2,2} \\cos{\\left(\\frac{4\\pi}{P} t_i\\right)},\n \\label{eq-fourier}\n\\end{equation}\nwhere $t_i$ is the light-time corrected epoch of the $i$-th observation, $P$ is the trial spin period, and the $A_{i,j}$ are the five adjustable Fourier coefficients.\n\n\n\nWe considered a range of evenly spaced rotational frequencies between $f$ = 0 and 7.57 rotations per day, where $f_{\\rm max}$~=~7.57 rot\/day ($P \\simeq 3.17$ h) represents the fastest rotation that can be\nNyquist sampled with the WISE observational cadence (Section \\ref{sec-alias}).\nHowever, we showed that it is possible to recover the periods of fast rotators by taking advantage of the mirroring properties of aliased signals.\nWe did not attempt to recover rotational frequencies that exceed 11 rotations per day, which correspond approximately to the rotational frequency at which centrifugal acceleration at the equator exceeds the acceleration due to self-gravity for typical asteroid densities, the so-called spin barrier at $P\\simeq 2.2$ hr \\citep{prav02}. The overwhelming majority of asteroids detected by WISE and included in our data set are large and experience fewer than 11 rotations per day.\n\n\n\nThe least-squares minimizer is an ordinary linear least-squares solver that follows \\citet[][tar.gz file]{wasz15}, except that we do not fit for phase curve parameters.\nWe also computed the reduced chi-squared metric $\\chi^2_\\nu = \\chi^2 \/ (N-5)$, where the number of degrees of freedom $\\nu = N-5$ represents the number of data points minus the number of free model parameters. %\n\nThe discrete Fourier transform of a time series with duration $T$ yields a frequency resolution of $1\/T$. In this work, we chose to increase the frequency resolution by an oversampling factor of 10 in order to better resolve peaks in the %\nperiod scan. The number of trial frequencies was therefore set to $10 \\times T \\times f_{\\rm max}$.\n\n\nWe used an iterative procedure similar to \\citet{wasz15}\nwhere an increasingly large ``cosmic error'' is added to the observation uncertainties.\nThe cosmic error is initialized as 0.002 mag in the first iteration and is multiplied by 1.5 in each subsequent iteration. %\nThe purpose of the cosmic error is to inflate the measurement uncertainties in order to reflect the model's inability to accurately represent asteroid lightcurves with a Fourier series truncated at the second harmonic. The cosmic error does not\naffect the periodicities identified in the lightcurve, but it does affect the confidence intervals assigned to the period estimates. \n\n\nWe followed \\citet{wasz15} and \\citet{harr14} in preferring double-peaked folded lightcurves. \nTo identify the number of peaks in the lightcurve, we generated a synthetic folded lightcurve with the fitted Fourier coefficients and candidate period, sampled it with 10,000 points, and analyzed the samples with Matlab's built-in function {\\tt find\\_peaks}\\footnote{https:\/\/www.mathworks.com\/help\/signal\/ref\/findpeaks.html}.\nFor each double-peaked solution, we computed the heights of each peak relative to the lightcurve's global minimum.\n\n\n\n\n\n\n\n\n\n\nThere are two possible paths to convergence. At the end of each iteration, the solution with the lowest $\\chi^2_\\nu$ is selected if and only if it satisfies three conditions: (1) the folded lightcurve is double-peaked;\n(2) the height of the highest peak is at least twice that of the lowest peak; and (3) $\\chi^2_\\nu < 3$. If conditions (1) or (2) are not satisfied, the solution at half-frequency is considered and is adopted if it satisfies the same three conditions. Otherwise, the cosmic error is increased\nand the next iteration begins.\nIf the cosmic error reaches 0.1 mag, the fit is deemed unsuccessful.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Aliasing}\n\\label{sec-alias}\n\n\n\n\n\n\n\n\nThe cadence of observations determines the sampling intervals between consecutive photometric measurements.\nThe Nyquist sampling criterion requires that at least two samples of a periodic signal be obtained per cycle in order to identify the periodicity unambiguously.\nApart from the fortuitous double detections obtained $\\sim$11 s apart (Section \\ref{sec-sigma}), the smallest sampling interval in WISE data is approximately 1.59 hr, which is dictated by WISE's $\\sim$15 daily orbital revolutions \\citep{wrig10}.\nAsteroids sampled with the 1.59 hr cadence and rotation periods larger than 3.17 hr ($f <$7.57 rot\/day) are usually Nyquist sampled, i.e., suffer no aliasing.\n\nAssuming uniform sampling, the signatures of asteroids with rotation periods between 1.59 hr and 3.17 hr appear aliased in the period scan in a predictable manner, specifically:\n\\begin{equation}\n %\n f_{\\rm alias} = 2 f_{\\rm Nyq} - f_{\\rm spin},\n \\label{eq-nyquist}\n\\end{equation}\nwhere $f_{\\rm Nyq}$ = 7.57 rot\/day is the critical Nyquist frequency or folding frequency, $f_{\\rm spin}$ is the underlying true rotational frequency, and $f_{\\rm alias}$ is the aliased frequency.\nFor example, an asteroid rotating at the spin barrier of 2.2 h (10.9 rot\/day) exhibits signatures at 2.2 h (10.9 rot\/day) and 6.97 h (3.4 rot\/day). \nAbsent additional information, a period scan may yield inconclusive results with respect to these two solutions. \nHowever, folding about the 7.57 rot\/day axis remains limited because only about 17\\% of asteroids in our sample experience rotation rates that exceed 7.57 rot\/day (Section \\ref{sec-overview}).\nThe overwhelming majority of asteroids detected by WISE and included in our data set are large and experience fewer than 11 rotations per day.\n\n\nIt is frequent\nfor the interval between consecutive W4 measurements to be 3.17 hr instead of the nominal cadence of 1.59 hr, e.g., when poor-quality flux measurements are eliminated. As a result, we also expected and observed\n(Section~\\ref{sec-results})\nfolding of the periodogram about $f_{\\rm Nyq'}$ = 3.78 rot\/day.\nThe folded frequency happens to be correct in a substantial fraction of cases, which we used to our advantage as it is calculable and therefore recoverable with no loss of precision.\n\n\n\n\n\n\n\n\n\nWe found that the aliasing behavior is complicated by non-uniform sampling in the WISE data. \nAdditional aliasing considerations are described in Appendix~\\ref{app-aliasing}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{S\/N Calculations}\n\\label{sec-snr}\nA period scan may return multiple peaks with low misfit values. S\/N metrics are useful in determining whether a peak is likely to represent a genuine rotational signature as opposed to a noise artifact. We adopted two S\/N metrics. One metric follows \\citet{wasz15} and quantifies the height of the peak with respect to the median misfit in terms of an estimation to the standard deviations of the misfit variations:\n\\begin{equation}\n S\/N_{\\rm W} = \\frac{|\\chi^2_{\\rm min} - \\chi^2_{\\rm median}|}{(\\chi^2_{84\\%} - \\chi^2_{16\\%})\/2}, \n %\n \\label{eq-snrw}\n\\end{equation}\nwhere the denominator includes percentiles of the misfit distribution corresponding to $\\pm$1 standard deviations from the median.\nThe second metric follows \\citet{harr12} and associates the misfit outside of minima, which we approximated by $\\chi^2_{95\\%}$, to the quadratic sum of the amplitude of lightcurve variation ($a$) and the noise in the data ($n$). It also associates the minimum misfit to the square of the single-point data scatter after removal of the signal ($n^2$), and assigns an overall noise level to the solution equal to $n\/\\sqrt{\\nu} = n\/\\sqrt{(N-5)}$. We have\n\\begin{equation}\n (a^2 + n^2) = \\chi^2_{95\\%} \/ N,\n\\end{equation}\n \\begin{equation}\n n^2 = \\chi^2_{\\rm min} \/ N,\n\\end{equation}\n \\begin{equation}\n {\\rm signal} = a = \\sqrt{(\\chi^2_{95\\%} - \\chi^2_{\\rm min}) \/ N}\n \\end{equation}\n \\begin{equation}\n {\\rm noise} = n' = \\sqrt{\\chi^2_{\\rm min} \/ N} \/ \\sqrt{(N - 5)}\n \\end{equation}\n \\begin{equation}\n S\/N_{\\rm H} = a\/n' = \\sqrt{\\frac{\\chi^2_{95\\%} - \\chi^2_{\\rm min}}{\\chi^2_{\\rm min} }} \\sqrt{(N - 5)}\n \\label{eq-snrh}\n \\end{equation}\n\n\\subsection{Assignment of Spin Period Uncertainties}\n\\label{sec-uncertainties}\n\nOnce the best-fit peak was identified, we assigned a 1$\\sigma$ uncertainty to the fitted period by computing the periods corresponding to a constant chi-square boundary \\citep[][Section 15.6]{press92}. Specifically, we computed the periods at which\n$ \\chi^2 = \\chi^2_{\\rm min} + \\Delta\\chi^2(68.3\\%, \\nu=5) = \\chi^2_{\\rm min} + 5.86$, where $\\chi^2_{\\rm min}$ is the minimum misfit.\n\n\n\n\\subsection{Post-processing Filters}\n\\label{sec-post-fit-filters}\nBecause the duration and cadence of WISE observations are not optimal for the unambiguous determination of spin periods, it was important to\nremove solutions that are\nlikely unreliable.\nWe applied the following filters:\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n %\n %\n \n %\n %\n %\n %\n \n %\n \n %\n %\n \n %\n \n %\n\n %\n\n %\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n(1) Reject slow rotators. \nWe required observations over at least 180 degrees of rotational phase and rejected any solution with a best-fit period that is two or more times longer than the data span.\n\n\n\n\n(2) Reject anomalously high amplitude lightcurves.\nWe eliminated solutions where the peak-to-peak amplitude of the fitted solution was three or more times larger than the peak-to-peak amplitude of the observations.\nThe peak-to-peak amplitude of the lightcurve was determined numerically while evaluating the lightcurve with the best-fit Fourier coefficients. \n\n\n\n %\n(3) Reject low S\/N solutions. \n Low S\/N solutions are likely spurious and were eliminated. In practice, we found that the S\/N formulation of \\citet{harr12} was more effective than that of \\citet{wasz15}, perhaps due in part to the relatively small number of (noisy) observations. \n %\n Solutions were rejected when $(S\/N)_{\\rm H} < 5$.\n %\n %\n %\n \n %\n The solutions that passed all of the above filters are reported below. \n %\n %\n We have evidence that most of these solutions are accurate\n %\n (Section~\\ref{sec-results}),\n but we also expect a fraction of aliased or incorrect solutions in this set.\n\n\n %\n %\n\n\n\n\n\n\\subsection{Machine Learning Reliability Classifier} \n\\label{sec-ml}\n\n\n\n\n\n\n\\citet{wasz15} pioneered the usage of a machine learning (ML) classifier to improve the reliability of asteroid lightcurve fits.\nThey applied a random forest (RF) algorithm, which is \na supervised machine learning algorithm that utilizes an ensemble of weak decision tree predictors to increase prediction power. \nThe hypothesis underpinning a machine learning classifier is that certain appropriately chosen features associated with a lightcurve solution jointly carry non-trivial information regarding the reliability of the solution. Given a labeled training set that includes both the values of the features and a reliability indicator, an ML algorithm can be trained to detect relations within the feature space and predict a reliability indicator for lightcurve solutions that do not have a reliably known period (i.e., solutions that are not in the training set).\nA Random Forest classifier makes predictions via a majority voting process by its ensemble of decision trees. For each sample, the classifier generates a probability derived from the voting process, then makes a binary prediction (i.e., reliable or unreliable) on the basis of a user-defined probability threshold. \n\\citet{wasz15}'s classifier was trained with about 1000 lightcurves with known reference periods and improved the overall\nsuccess rate \nfrom $\\sim$66\\% to $\\sim$80\\% for 19,000 lightcurves. %\n\n\n\n\n\n\n\nBecause our work also involved the analysis of thousands of sparse lightcurve, we initially applied an RF algorithm in an attempt to identify the most reliable lightcurve solutions. The RF classifier was able to provide a modest improvement to the success rate of our primary solutions (from 55\\% to 70\\%), but it also marked correct solutions as incorrect. Because the recovery of spin periods among our three solutions was so high (88\\%) and the performance of the RF classifier was limited, we ultimately decided against providing a potentially flawed reliability indicator.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Lomb-Scargle Periodogram}\n\\label{methods-ls}\n\n\n\n\nThe Lomb-Scargle (LS) periodogram \\citep{Lomb76,Scar82} is a standard algorithm that enables the analysis of periodicities in unevenly sampled time series, \nsuch as asteroid lightcurves. \nWe explored the performance of the LS periodogram\nas an alternative to our default pipeline. Our LS pipeline follows the implementation in standard libraries and does not include iterative adjustment of uncertainties, requirement for double-peaked solutions, and post-fit filters implemented in our default pipeline.\n\n\n\n\nWe used the generalized LS algorithm of \\citet{zech09} as implemented in the astropy package\\footnote{https:\/\/docs.astropy.org\/en\/stable\/timeseries\/lombscargle.html}. This implementation takes observational uncertainties into account and enables the estimation of a floating mean.\nWe deployed both first-order and second-order LS periodograms.\n\nIn the first-order LS implementation, we followed \\citet{McNeill19}\nand set the estimated rotational period at twice the best-fit LS period to conform to the double-peak nature of asteroid lightcurves.\nWe admitted only solutions with a false-alarm probability (FAP) less than 10\\%. %\n\nThe second-order LS implementation is similar to the periodogram calculation used in our default pipeline but does not include any post-fit filter. \n\n\n\n\n\n\\section{Results}\n\\label{sec-results}\n\n\\subsection{Default pipeline}\n\nWe present the period solutions that successfully converged in the Fourier fitting algorithm and passed the post-fit filters (Section~\\ref{sec-post-fit-filters}). \nThe period solutions of 2008 ($\\sim$62\\%) out of the initial 3225 lightcurves fulfilled both inclusion criteria.\n\n\n\n\n\n\n\n\n\n\n\n\n\nTo test the reliability of our results, we compared our spin period estimates to high-quality (quality code 3- or higher) rotational periods published in the LCDB.\nWe quantified the fraction of solutions that were within 5\\% of the LCDB solution, which are deemed to be accurate solutions.\nAt the time of writing, there were 752 solutions (representing 702 unique asteroids) among our 2008\nsolutions with a suitable LCDB estimate.\nWe refer to this set of solutions as the `LCDB reference group'.\n\n\n\n\n\n %\n\n\nIn the LCDB reference group, the fitted period was found to be accurate (within 5\\%) in\n55\\% of the cases. Notably, the relative errors of the fitted periods exhibit a bimodal distribution (Figure \\ref{fig-accuracy-histogram}).\nThe two\nmodes bifurcate at a fractional error of approximately 5\\%,\nwhich provides a posteriori justification for selecting a 5\\% threshold for accuracy.\n\n\n\\begin{figure}[hbt]\n \\begin{center}\n \\includegraphics[width=4in]{plots\/period_relative_error.pdf}\n %\n \\caption{Histogram of fractional accuracy in spin period obtained by comparing our period solutions to the corresponding LCDB reference periods. The left and right clusters correspond to the accurate and inaccurate period estimates, respectively. The red line denotes the 5\\% accuracy threshold.}\n %\n\\label{fig-accuracy-histogram}\n\\end{center}\n\\end{figure}\n\nOur spin period solutions are listed in Table~\\ref{tab-results}. Figure~\\ref{fig-suc-mode-1} illustrates an example of a favorable situation with a short (1.59 hr) cadence and relative long (6 days) duration, which yields a solution with high S\/N and no aliasing. We found that it is possible to successfully recover the spin period even when the observation span is comparable to the spin period (Figure~\\ref{fig-suc-mode-3}). Our analysis also identified correct spin periods in situations where the minimum $\\chi^2$ value is not markedly different from other competing solutions (Figure~\\ref{fig-suc-mode-2}). When multiple periodogram peaks have comparable $\\chi^2$ values, the potential for an incorrect solution exists.\n\n\n\n\n\n\\begin{figure}[p]\n \\begin{center}\n \\includegraphics[width=7in]{plots\/mode_s_1_296.pdf}\n \\caption{\n %\n Spin period solution for asteroid 296 obtained from 35 W4 observations spanning 6 days. \n The period solution at 4.543 $\\pm$ 0.014 hr is in good agreement with the LCDB value of 4.5385 hr.\n The data exhibit short (1.59 h) sampling intervals over a long observation span, and the periodogram is free of aliases.\n }\n\\label{fig-suc-mode-1}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[p]\n \\begin{center}\n \\includegraphics[width=7in]{plots\/mode_s_3_2715.pdf}\n \\caption{\n %\n %\n %\n %\n Spin period solution for asteroid 2715 obtained from 16 W4 observations spanning 1.5 days.\n The period solution at 33.19 $\\pm$ 2.19 hr is in good agreement with the LCDB value of 33.62 hr.\n The correct solution was identified despite a data observation span that is only slightly longer than the spin period.\n }\n\\label{fig-suc-mode-3}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[p]\n \\begin{center}\n \\includegraphics[width=7in]{plots\/mode_s_2_2812.pdf}\n \\caption{\n %\n Spin period solution for asteroid 2812 obtained from 14 W4 observations spanning 1.2 days.\n The period solution at 7.74 $\\pm$ 0.30 hr is in good agreement with the LCDB value of 7.7 hr.\n The correct solution was identified despite a relatively low S\/N and the presence of competing solutions with similar but larger $\\chi^2$ values.\n }\n\\label{fig-suc-mode-2}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[p]\n \\begin{center}\n \\includegraphics[width=3in]{plots\/pp_x_plot.pdf}\n \\caption{\n Best-fit rotational frequency (this work) vs.\\ LCDB\n rotational frequency for the LCDB reference group. The\n dashed grey line is at f = 3.783 rot\/day, which is the\n expected folding frequency for a 3.17 hr cadence and the\n predominant aliasing mode. Most of the best-fit\n solutions (55\\%) are accurate (green dots). The\n solutions mirrored about 3.78 rot\/day (dark blue dots)\n are also accurate in 27\\% of the cases, and the\n solutions mirrored about 7.57 rot\/day (light blue dots)\n are accurate in 6\\% of the cases. The inaccurate\n solutions (red dots) represent 12\\% of the cases.\n The combination of best-fit and\n mirrored solutions yields an aggregate success rate of\n %\n 88\\%.\n}\n\\label{fig-pp-x-plot}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\n\nWe validated our\nresults by plotting our best-fit results against LCDB values for the LCDB reference group (Figure \\ref{fig-pp-x-plot}).\nIn frequency space, which reveals folding behavior, the structure of the solutions is\nstriking. Most of the inaccurate solutions are in fact aliases of the\ncorrect frequencies folded about the f = 3.783 rot\/day axis or, less frequently, the f~=~7.57 rot\/day axis.\nFigures~\\ref{fig-fail-mode-1} and \\ref{fig-fail-mode-2} illustrate two examples.\n\\begin{figure}[p]\n \\begin{center}\n \\includegraphics[width=7in]{plots\/mode_f_1_4715.pdf}\n \\caption{\n %\n %\n %\n %\n Spin period solution for asteroid 4715 obtained from 23 W4 observations spanning 3.6 days.\n The best-fit, primary period solution at 4.962 $\\pm$ 0.01 hr (4.84 rot\/day) is an alias (mirrored around f = 3.78 rot\/day) of the presumed correct LCDB value of 8.8129 hr (2.72 rot\/day). The secondary period solution at 8.7930 hr is accurate.\n }\n \\label{fig-fail-mode-1}\n\\end{center}\n\\end{figure}\n\\begin{figure}[p]\n \\begin{center}\n \\includegraphics[width=7in]{plots\/mode_f_2_6485.pdf}\n \\caption{\n %\n %\n %\n %\n Spin period solution for asteroid 6485 obtained from 22 W4 observations spanning 5.2 days.\n The best-fit, primary period solution at 3.31 $\\pm$ 0.035 hr (7.25 rot\/day) is an alias (mirrored around f = 3.78 rot\/day) of the presumed correct LCDB value of 75.56 hr (0.318 rot\/day). The secondary period solution at 75.4453 hr is accurate.\n}\n\\label{fig-fail-mode-2}\n\\end{center}\n\\end{figure}\n\\begin{figure}[hbt]\n \\begin{center}\n \\includegraphics[width=7in]{plots\/mode_f_3_10721.pdf}\n \\caption{\n %\n %\n %\n %\n Spin period solution for asteroid 10721 obtained from 18 W4 observations spanning 1.8 days.\n The period solution at 3.2721 $\\pm$ 0.031 hr (7.33 rot\/day) is an alias (mirrored around f = 7.57 rot\/day) of the presumed correct LCDB value of 3.0675 hr (7.82 rot\/day). The tertiary period solution at 3.0514 hr is accurate.\n}\n\\label{fig-fail-mode-3}\n\\end{center}\n\\end{figure}\n\nFor this reason, we list both the best-fit frequency and its mirror values in Table~\\ref{tab-results}. One of these solutions is correct (within 5\\%) in\n88\\% (659\/752)\nof the cases in the LCDB reference group. We posit that the accuracy rate is similar for asteroids that are not in the LCDB reference group.\nThe best-fit, secondary (mirrored about 3.78 rot\/day), and tertiary (mirrored about 7.57 rot\/day) solutions are accurate in 55\\%, 27\\%, and 6\\% of the cases, respectively.\n\nInaccurate solutions that are not mirrors of correct values are visible outside of the X pattern on Figure~\\ref{fig-pp-x-plot}.\n\nA number of solutions deemed to be inaccurate cluster near the accurate solutions along the blue diagonal, especially at low frequencies. This behavior indicates that our 5\\% criterion is a conservative metric of accuracy, and that additional solutions are in fact close to the correct value.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{deluxetable}{rrrrrrrrrrrrrrr}\n %\n \\tablecaption{Spin period solutions for 1929 asteroids. \\label{tab-results}}\n\n\\tablehead{\n %\n\\colhead{Object} & \\colhead{$D$} & \\colhead{span} & \\colhead{idx} & \\colhead{$A$} & \\colhead{$P$} & \\colhead{$\\sigma_P$} & \\colhead{$f$} & \\colhead{$f_{\\xleftrightarrow{1}}$} & \\colhead{$P_{\\xleftrightarrow{1}}$} & \\colhead{$f_{\\xleftrightarrow{2}}$} & \\colhead{$P_{\\xleftrightarrow{2}}$} & \\colhead{$P_{\\rm LCDB}$} & \\colhead{$U$} & \\colhead{flag} \\\\ \n\\colhead{ } & \\colhead{(km)} & \\colhead{(hr)} & \\colhead{ } & \\colhead{(mag)} & \\colhead{(hr)} & \\colhead{(hr)} & \\colhead{(rot\/d)} & \\colhead{(rot\/d)} & \\colhead{(hr)} & \\colhead{(rot\/d)} & \\colhead{(hr)} & \\colhead{(hr)} & \\colhead{} & \\colhead{} \n }\n\n\n\\startdata\n131 & 30.6 & 109.5 & 1 & 0.358 & 5.1919 & 0.019 & 4.6226 & 2.9436 & 8.1532 & 10.5774 & 2.2690 & 5.1812 & 3 & 1 \\\\\n155 & 44.6 & 93.7 & 1 & 0.234 & 5.2919 & 0.016 & 4.5352 & 3.0310 & 7.9183 & 10.6648 & 2.2504 & 7.9597 & 3 & 2 \\\\\n170 & 35.4 & 181.0 & 1 & 0.360 & 13.0199 & 0.250 & 1.8433 & 5.7229 & 4.1937 & 9.4433 & 2.5415 & 13.1200 & 3 & 1 \\\\\n180 & 24.8 & 96.9 & 1 & 0.493 & 23.6223 & 0.477 & 1.0160 & 6.5502 & 3.6640 & 8.6160 & 2.7855 & 23.8660 & 3 & 1 \\\\\n183 & 30.7 & 90.5 & 1 & 0.395 & 11.7553 & 0.095 & 2.0416 & 5.5246 & 4.3442 & 9.6416 & 2.4892 & 11.7700 & 3 & 1 \\\\\n183 & 30.8 & 106.4 & 2 & 0.431 & 11.8183 & 0.115 & 2.0307 & 5.5354 & 4.3357 & 9.6308 & 2.4920 & 11.7700 & 3 & 1 \\\\\n239 & 42.9 & 90.5 & 1 & 0.392 & 18.4676 & 0.263 & 1.2996 & 6.2666 & 3.8298 & 8.8996 & 2.6968 & 18.4707 & 3 & 1 \\\\\n244 & 10.9 & 20.6 & 1 & 1.151 & 3.2773 & 0.083 & 7.3230 & 0.2432 & 98.6986 & 7.8770 & 3.0469 & 129.5100 & 3- & 0 \\\\\n251 & 33.8 & 100.0 & 1 & 0.486 & 20.0040 & 0.301 & 1.1998 & 6.3664 & 3.7698 & 8.7998 & 2.7273 & 20.2160 & 3 & 1 \\\\\n254 & 12.4 & 39.7 & 1 & 0.765 & 6.9642 & 0.188 & 3.4462 & 4.1200 & 5.8252 & 11.0462 & 2.1727 & 5.8949 & 3 & 2 \\\\\n... & ... & ... &...& ... & ... & ... & ... & ... & ... & ... & ...\\\\\n223431 & 5.8 & 119.1 & 1 & 0.214 & 30.5368 & 0.975 & 0.7859 & 6.7803 & 3.5397 & 8.3859 & 2.8619 & & & \\\\\n230118 & 0.9 & 581.1 & 1 & 0.635 & 29.9524 & 0.113 & 0.8013 & 6.7649 & 3.5477 & 8.4013 & 2.8567 & & & \\\\\n232382 & 1.0 & 535.0 & 2 & 0.192 & 67.7217 & 2.077 & 0.3544 & 7.2118 & 3.3279 & 7.9544 & 3.0172 & & & \\\\\n249958 & 5.3 & 134.9 & 1 & 0.231 & 6.8153 & 0.031 & 3.5215 & 4.0447 & 5.9337 & 11.1215 & 2.1580 & & & \\\\\n256155 & 3.1 & 134.9 & 1 & 0.549 & 15.6910 & 0.117 & 1.5295 & 6.0367 & 3.9757 & 9.1295 & 2.6288 & & & \\\\\n307840 & 6.8 & 335.1 & 1 & 0.367 & 16.0317 & 0.150 & 1.4970 & 6.0692 & 3.9544 & 9.0970 & 2.6382 & & & \\\\\n318081 & 5.9 & 42.9 & 1 & 0.410 & 3.6023 & 0.029 & 6.6623 & 0.9039 & 26.5523 & 8.5377 & 2.8111 & & & \\\\\n366774 & 0.9 & 58.8 & 1 & 0.327 & 3.4972 & 0.037 & 6.8625 & 0.7037 & 34.1075 & 8.3375 & 2.8786 & & & \\\\\n386720 & 1.0 & 141.3 & 1 & 0.806 & 4.6034 & 0.015 & 5.2136 & 2.3526 & 10.2013 & 9.9864 & 2.4033 & & & \\\\\n\\enddata \n\\tablecomments{For each object, we show the diameter $D$ (km), observation span (hr), index of the cluster of observations analyzed, the best-fit period $P$ (hr), the peak-to-peak magnitude variation of the fitted lightcurve $A$\n %\n for the primary (best-fit) period solution, the primary period uncertainty $\\sigma_P$ (hr), the best-fit rotational frequency $f$ (rot\/d), the first alternate rotational frequency\n %\n $f_{\\xleftrightarrow{1}}$ (rot\/d) found by folding the best-fit frequency about 3.78 rot\/day, \n the first alternate period\n %\n $P_{\\xleftrightarrow{1}}$ (hr), the second alternate rotational frequency\n %\n $f_{\\xleftrightarrow{2}}$ (rot\/d) found by folding the best-fit frequency about 7.56 rot\/day, \n the second alternate period\n %\n $P_{\\xleftrightarrow{2}}$\n (hr), the LCDB reference period $P_{\\rm LCDB}$ (hr), if known, the corresponding quality code $U$, and a flag indicating agreement for objects in the LCDB Reference Group.\nThe flag is 1 if the best-fit period matches $P_{\\rm LCDB}$ within 5\\%, 2 if the first mirror period matches, 3 if the second mirror period matches, and 0 if none of the three periods match $P_{\\rm LCDB}$. \n(This table is available in its entirety in \\href{https:\/\/ucla.box.com\/s\/49z4fr6pdaalfssg7qhngb30vrwjl9f0}{machine-readable} and \\href{https:\/\/ucla.box.com\/s\/5muceha0gyj7ynsnysfwjt4uxgln7c8t}{CSV} forms in the online journal. A portion is shown here for guidance regarding its form and content.)}\n\\end{deluxetable}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Lomb-Scargle Pipeline}\n\\label{res-ls}\n\n\n\nAs in our default pipeline, we generated three candidate period solutions for each fitted lightcurve (primary: LS solution, secondary: mirror across 3.78 rot\/day, tertiary: mirror across 7.57 rot\/day). We compared the accuracy of the LS pipeline to our default pipeline by computing the number of solutions that are within 5\\% of the high-quality (3\/3-) LCDB solutions (Table \\ref{tab-LS_results}).\n\n\n\n\\begin{table}[h]\n \\begin{center}\n \\begin{tabular}{lrrr}\n %\n\n Accuracy flag & LS first order & LS second order & Default pipeline \\\\\n \\hline\n 0 (inaccurate) & 91 (12\\%) & 518 (52\\%) & 93 (12\\%) \\\\\n 1 (primary) & 320 (43\\%) & 316 (32\\%) & 414 (55\\%) \\\\\n 2 (secondary) & 286 (39\\%) & 105 (11\\%) & 203 (27\\%) \\\\\n 3 (tertiary) & 33 (5\\%) & 56 (6\\%) & 42 (6\\%) \\\\\n Aggregate accuracy & 88\\% & 48\\% & 88\\% \\\\\n Number of reference solutions & 730 & 995 & 752 \\\\\n \n \n \\end{tabular}\n \\caption{ Number of accurate solutions with the Lomb-Scargle and default pipelines.\n }\n \\label{tab-LS_results}\n\\end{center} \n \\end{table}\n\n\nThe first-order LS solutions have an aggregate accuracy comparable to the default pipeline solutions. However, the primary LS solutions were accurate only in 43\\% of the cases, compared to 55\\% in the default pipeline. \nThe second-order LS solutions have a lower accuracy rate than the default pipeline solutions, both for primary solutions and in aggregate. The lower performance of the LS algorithm demonstrates the importance of the\niterative algorithm and post-fit filters in our default, \\citet{wasz15}-inspired pipeline.\n\n\n\n\n\n\n\n\n\n\n\n\n\\FloatBarrier\n\\section{Discussion}\n\\label{sec-discussion}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\\citet{prav02} reviewed the rotation periods of asteroids as a function of diameter and found that the spin distribution\nis Maxwellian for large asteroids ($>$40 km) and strongly non-Maxwellian for smaller asteroids, with an excess of both slowly rotating and rapidly rotating asteroids.\nThe WISE data set can potentially inform these studies because it can yield estimates of both diameter and spin period (Figure~\\ref{fig-period_diam_accurate_fit_period}).\n\n\\begin{figure}[hbt]\n \\begin{center}\n \\includegraphics[width=4.5in]{plots\/fig10.pdf}\n %\n \\caption{Period vs.\\ diameter diagram for the 659 light curves in the LCDB reference group where one of our spin period solutions matched the LCDB to 5\\%. Only the spin period solutions that match the LCDB value are plotted, with color-coding indicating our primary, secondary, or tertiary solutions. The red dashed line at $P=$ 3.17 hr corresponds to the upper range of trial frequencies explored in the fitting process. The grey dashed line at $P=$ 2.2 hr illustrates the spin barrier. Diameters are from \\citet{myhr22}.}\n\\label{fig-period_diam_accurate_fit_period}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\nWe can evaluate the accuracy of our method as function of asteroid diameter and LCDB period (Figure \\ref{fig-period_diam_lcdb_ref_period}).\nDiameter does not have an apparent influence on the accuracy of our method.\nHowever, spin period does affect our ability to recover a correct solution with the WISE data. We found that the spin periods of both slowly rotating ($P > \\sim$100 hr) and rapidly rotating ($P < \\sim$2.5 hr) asteroids are generally not recoverable.\n\n\\begin{figure}[hbt]\n \\begin{center}\n %\n %\n \\includegraphics[width=4.5in]{plots\/fig11.pdf}\n %\n \\caption{LCDB Period vs.\\ diameter \\citep{myhr22} for the 752 lightcurves in the LCDB Reference Group. Our spin period solutions are color-coded according to four possible outcomes: primary solution is accurate, secondary solution is accurate, tertiary solution is accurate, or none are accurate. The red dashed line at $P=$ 3.17 hr corresponds to the upper range of trial frequencies explored in the fitting process. The grey dashed line at $P=$ 2.2 hr illustrates the spin barrier. Diameters are from \\citet{myhr22}.}\n\\label{fig-period_diam_lcdb_ref_period}\n\\end{center}\n\\end{figure}\n\n\n\n\n\nThe distribution of lightcurve amplitudes in our data set is instructive (Figure~\\ref{fig-amp-histogram}). Our distribution underestimates the proportion of low-amplitude lightcurves, because our sample selection enforced a pre-fit filter, which required a magnitude variation of at least 0.3 mag in the observed fluxes. Likewise, one of our post-fit filters eliminated some high-amplitude lightcurve solutions, but it did so only when the data themselves did not exhibit a large magnitude variation.\nWe found that 50\\%, 8.8\\%, 2.2\\%, 1\\%, and 0.25\\% of solutions have amplitudes larger than 0.5 mag, 1 mag, 1.5 mag, 2 mag, and 2.5 mag, respectively.\n\\begin{figure}[hbt]\n \\begin{center}\n \\includegraphics[width=4.5in]{plots\/amp_histogram.pdf}\n \\caption{Histogram of peak-to-peak lightcurve amplitudes.}\n\\label{fig-amp-histogram}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\n\\label{sec-conclusions}\nWe devised a procedure similar to that of \\citet{wasz15} that enables the determination of thousands of asteroid spin periods from WISE data. Despite WISE's suboptimal observation and cadence for asteroid spin measurements,\none of our\nsolutions is accurate 88\\% of the time when compared to a high-quality control group of 752 spin periods. We obtained primary, secondary, and tertiary spin period estimates for 2008 observation clusters representing 1929 unique asteroids. Among those, 1205 asteroids do not currently have a high-quality spin period estimate. Our\nprimary, secondary, or tertiary \nsolution for\nover a thousand asteroids is expected to be accurate at the 5\\% level or better and can greatly facilitate shape or thermal modeling work.\n\n\n\\acknowledgments\n\nWe thank Alan Harris for useful discussions.\n\n\nAL thanks Breann Sitarski for her helpful feedback on this work and the related research presentations, as well as her mentorship throughout his undergraduate studies.\n\nAL was funded in part by the Joe and Andrea Straus Endowment for Undergrad Opportunity\nand the Donald Carlisle Undergrad Research Endowed Fund.\nEW was funded in part by the Nathan P. Myhrvold Graduate Fellowship.\n\nThis publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory\/California Institute of Technology, funded by the National Aeronautics and Space Administration.\n\n\\software{\nNumPy \\citep{numpy},\nSciPy \\citep{scipy},\npandas \\citep{pandas},\nMatplotlib \\citep{mpl}\n}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nIt is widely believed that very few galaxies exist today that have not been formed or shaped in some way by an interaction with another galaxy. \\citet{Toomre_77} identified 11 galaxies that exhibit characteristics of on-going mergers, and arranged them in a chronological order, illustrating how spiral galaxies can merge to produce ellipticals. It is now the view of many astronomers that this process, whilst not the only mechanism to create these systems, plays a vital role in the production of elliptical galaxies. The 11 systems within the `Toomre' sequence, alongside other examples of on-going mergers, have been studied in great detail over a range of wavelengths to try and characterise exactly how these systems evolve. When studying these merging galaxies, X-ray observations are of particular importance, as they are able to probe the dusty nucleus of the system, which can be obscured at other wavelengths, allowing the nature of the point source population to be established. Also imaging of the soft X-ray emission permits the diffuse hot gas to be mapped out. This gives important information about the hot gaseous component associated with the strong starburst, and allows galactic-winds outflowing from the system to be observed, enabling constraints to be placed on the energetics of these outflows.\n\nX-ray observations were initially carried out with the {\\em Einstein Observatory}, providing limited spatial resolution. This instrument was followed by \\emph{ROSAT}, which greatly improved the sensitivity of the observations, allowing the X-ray properties of these merging galaxies to be probed. A study of a sample of interacting systems was carried out by \\citet{Read_98} (from here on RP98), where the X-ray luminosity and properties of the diffuse gas were investigated with \\emph{ROSAT}, alongside the point source population. In this study it was found that the normalised X-ray luminosity of these systems, broadly speaking, followed the normalised \\ensuremath{L_{\\mathrm{FIR}}}\\ luminosity, peaking at the time of coalescence. It was also found that the young merger-remnant systems within the sample exhibited low X-ray luminosities, with no indication from the later stage systems in their sample that these systems would increase in X-ray luminosity. However, it was noted that the $\\sim$1 Gyr system, NGC 7252, still hosts a large amount of molecular gas, in the form of tails and loops, and given more time to evolve, the X-ray properties could resemble those of a mature elliptical galaxy at a greater dynamical time. The main limitation of this study, due to the \\emph{ROSAT}\\ observations, was the inability to disentangle the point sources from the diffuse gas, particularly at larger distances. \n\nWith the next generation of X-ray observatories this issue was addressed with an increase in spatial resolution. This improvement, provided by \\emph{Chandra}, has allowed the investigation of both the diffuse gas and the point source population of galaxies to be carried out in greater detail. The ability to disentangle these two components is vital when investigating interacting and merging galaxies, as studies have shown that both of these components have different evolutionary timescales \\citep{Read_01}. It is therefore important to be able to study both of these separately, to fully understand how they evolve.\n\nFrom observing a selection of interacting galaxy systems, at different stages of evolution, the processes involved in the merging of galaxies pairs can be characterised. In the following sections we will describe the sample we have selected, and investigate the behaviour of the X-ray emission as the galaxies evolve from two spiral galaxies, through to relaxed merger-remnants. Section \\ref{sec:sample} outlines the selection criteria we have used and gives a brief description of each system. In section \\ref{sec_data_red} we present the results from new \\emph{Chandra}\\ observations of Mkn 266 and Arp 222. Correlations and evolution of X-ray emission across the whole sample is presented in section \\ref{sec_evol}, and discussed in section \\ref{sec:dis}. Conclusions are given in section \\ref{sec:con}.\n\n\\section{The Sample}\n\\label {sec:sample}\n\nTo gain a better understanding of galaxy evolution it is important to compare similar systems that are undergoing the same transformation. By compiling a sample of these galaxies, the evolution of the systems' X-ray properties can be investigated. In this paper we have a sample of nine interacting and post-merger systems, carefully selected to ensure that they are representative of galaxies undergoing a major merger. The nine systems were selected using the following criteria;\n\n\\begin{enumerate}\n\\item \nAll systems have been observed with {\\em Chandra}.\n\\item \nSystems comprise of, or, originate from, two similar mass, gas rich, spiral galaxies.\n\\item\nMulti-wavelength information is available for each system.\n\\item\nThe absorbing column is low, maximising the sensitivity to soft X-ray emission.\n\\item\nA wide chronological sequence is covered; from detached pairs to merger-remnants.\n\\end{enumerate}\n\nOnce the nine systems to study had been selected, the issue of merger age had to be addressed. This is one of the main problems when working with a chronological study such as this, and a number of different methods are required to solve this problem. Firstly the point of nuclear coalescence was assigned to be at time 0. From this, the time taken until nuclear coalescence for each pre-merger system can then be estimated. These timescales were derived with a combination of N-body simulations, such as \\citet{Mihos_96}, and dynamical age estimates, where the length and faintness of tidal tails, as well as nuclei separation, were used \\citep{Toomre_72}. For post-merger systems, assigning an age estimate was done by making the assumption that the last widespread episode of star formation within the system took place at the time of nuclear coalescence. Stellar population synthesis models were then used to calculate these timescales, therefore giving a good merger age estimate \\citep{Bruzual_93}. In the following sub-sections a brief description of the nine merger examples, and their X-ray properties, are presented in chronological order.\n\n\\subsection{Arp 270}\n\nThe earliest example of an interacting system in our sample is Arp 270 (also NGC 3395\/3396). This comprises two spiral galaxies of comparable mass \\citep{Hern_01}, separated by 12\\,kpc at a distance of 28 Mpc (assuming \\ensuremath{H_{\\mathrm{0}}}\\ = 75\\,km s$^{-1}$\nMpc$^{-1}$, and accounting for Virgocentric in-fall). These galaxies are connected by an optical bridge which is thought to have formed during the systems first perigalactic passage which occurred approximately 5$\\times$10$^8$ years ago.\n \nA 20 ks \\emph{Chandra}\\ observation of the system was made in 2001 and is discussed in detail in \\citet{Brassington_05}, the contours of adaptively smoothed 0.3$-$8.0\\,keV X-ray contours overlaid on an optical image are shown in Figure \\ref{fig:arp270_optic_con}. From this observation 16 point sources are detected, 7 of which are classified as {\\em Ultraluminous X-ray Sources} (ULX's), with \\ensuremath{L_{\\mathrm{X}}}\\ $\\ge 1\\times$10$^{39}$erg s$^{-1}$ \\citep{Soria_05b}. The diffuse gas emits at a global temperature of $\\sim$0.5 keV and shows no evidence of hot gaseous outflows as are seen in later stage systems (RP98). The galaxy pair, although in a very early stage of interaction, already show increased levels of \\ensuremath{L_{\\mathrm{FIR}}}\\ compared to quiescent galaxies, indicating that there is enhanced star formation taking place. The numerical simulations of \\citet{Mihos_96} suggest that a system such as Arp 270 will coalesce in $\\sim$650 Myrs.\n\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{images\/arp270_opt_xcon.ps}\n \\hspace{0.1cm}\n \\caption{The early merger system Arp 270. Contours of adaptively smoothed 0.3$-$8.0\\,keV X-ray data from {\\em Chandra} ACIS-S overlaid on an optical image from the Palomar 5m telescope. }\n \\label{fig:arp270_optic_con}\n\\end{figure}\n\n\\subsection{The Mice}\n\nThe Mice (also Arp 242, NGC 4676A\/B) is another early stage merger system, lying second in the evolutionary sequence proposed by \\citet{Toomre_72}. At a distance of 88 Mpc, the two detached spiral galaxies are again connected by a tidal bridge and, in addition to this feature, exhibit large tidal tails, a consequence of the galaxies interacting as they have passed each other. \\citet{Read_03} reports on a 30.5\\,ks \\emph{Chandra}\\ observation of the system, the adaptively smoothed 0.2$-$10.0\\,keV X-ray contours overlaid on an optical image are shown in Figure \\ref{fig:mice_opt_con}. \n\nFive ULXs are detected in association with the galaxies, these sources have been found to be coincident with regions of ongoing star formation, both within the nuclei of the galaxies and also in the tidal tails. The structure of the diffuse X-ray gas in both nuclei suggests that this system is at a more advanced stage of evolution than Arp 270. This is indicated by the morphology of the gas, a soft, thermal plasma, which extends out of the minor axis of both galaxies, suggesting that these features are starburst driven winds. As in Arp 270, the Mice emits an enhanced level of \\ensuremath{L_{\\mathrm{FIR}}}, again indicating that this system has enhanced star formation taking place, this emission is particularly high in the nuclei of the galaxies. \n\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{images\/mice_opt_xcon.ps}\n \\hspace{0.1cm}\n \\caption{The Mice, contours of adaptively smoothed 0.2$-$10.0\\,keV X-ray data from {\\em Chandra} ACIS-S overlaid on an optical image from WFPC2 on board the {\\em HST}. }\n \\label{fig:mice_opt_con}\n\\end{figure}\n\n\\subsection{The Antennae}\n\nThe Antennae, NGC 4038\/4039, or Arp 244, is probably the most famous example of a galaxy pair undergoing a major merger, and, lying at a distance of 19 Mpc, is also the nearest. A deep integrated \\emph{Chandra}\\ observation of 411\\,ks has been made of this system, enabling the nature of both the point source population and the diffuse gas to be investigated \\citep{Fabbiano_04, Zezas_04}, Fig \\ref{fig:ant_opt_con} shows the full band (0.3$-$6.0 keV) X-ray contours overlaid on an optical image. \n\nDue to the depth of the observation, sources were detected down to a luminosity of 2$-$5 $\\times$10$^{37}$erg s$^{-1}$. This resulted in a detection of 120 point sources, 12 of these have been confirmed as ULXs. The integrated observation comprises 7 separate pointings, enabling the variability of the point sources to be investigated.\n\nOut of the 12 ULXs, 4 emit below 1 $\\times$10$^{39}$ erg s$^{-1}$, the lower luminosity threshold for a source to be classified as a ULX, in at least one observation. Further, one of the sources was observed in only one pointing, indicating that it is likely to be a transient, providing further evidence that ULXs are a heterogeneous class, comprising of contributions from X-ray binary systems, transient sources and also, possibly, intermediate mass black holes (IMBHs). \n\nThe quality of the data has also lead to detailed mapping of the diffuse gas, investigating both the temperature and metallicity variation of the ISM. 21 separate regions of the diffuse gas have been analysed, spectral fitting of these regions reveal that there is a variation in temperature of the diffuse gas from 0.2 to 0.9 keV and metallicities vary from regions of sub-solar abundances to areas of emission displaying super-solar abundances, notably in both the nuclear regions and two hotspots in the northern loop of the disc (R1 and R2 reported in \\citet{Fabbiano_03b}). \n\nThe morphology of the gas reveals that there are large scale diffuse features; two large faint X-ray loops extending to the South of the system and a low-surface-brightness halo in the region surrounding the stellar discs extending out to $\\sim$18\\,kpc from the nucleus of NGC 4039. The two loops have temperatures ranging from 0.29 to 0.34 keV and the low surface brightness halo has a nominal temperature of 0.23 keV. This cooler larger scale emission may be the aftermath of a superwind, possibly from the first encounter of the two systems, which took place $\\sim$2$-$5 $\\times$10$^8$ years ago. The star formation rates within the two nuclei are 2.1 \\ensuremath{\\Msol~\\pyr}\\ and 1.7 \\ensuremath{\\Msol~\\pyr}\\ and in the region where the discs overlap has been found to be 5.0 \\ensuremath{\\Msol~\\pyr}\\ \\citep{Mihos_93}.\n\n\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{images\/ant_opt_xcon.ps}\n \\hspace{0.1cm}\n \\caption{The Antennae, contours of adaptively smoothed 0.3$-$6.0\\,keV X-ray data from {\\em Chandra} ACIS-S overlaid on an optical image from the UK Schmidt telescope. }\n \\label{fig:ant_opt_con}\n\\end{figure}\n\n\\subsection{Mkn 266}\n\nThe next system within this sample, Markarian 266 (also NGC 5256), is a double nucleus system with a large gaseous envelope. Here, a summary of the galaxy properties are presented, with the full analysis and results of a recent \\emph{Chandra}\\ observation detailed in section \\ref{sec:mkn266}. \n\nThis system lies at a distance of 115 Mpc and has an X-ray luminosity of 7.32 $\\times$10$^{41}$erg s$^{-1}$. The two nuclei, originating from the progenitors, are thought to comprise a LINER with a powerful starburst component, to the north of the system, and a Seyfert type 2 to the South \\citep{Mazzarella_88}. In addition to these sources, an area of enhanced emission has been detected between these nuclei, it is thought that this is caused by the collision of the two discs. Prior to a 20\\,ks \\emph{Chandra}\\ observation of Mkn 266, made in 2001, this feature had not been seen in X-rays before. But, due to {\\em Chandra's} superior spatial resolution, it is now possible to distinguish this feature. This can be seen in Figure \\ref{fig:mkn_266_opt_con}, where the adaptively smoothed, full band X-ray contours (0.3$-$8.0\\,keV) overlaid on an image from WFPC2, on board The {\\em Hubble Space Telescope (HST)} are shown.\n\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{images\/mkn266_opt_xcon.ps}\n \\hspace{0.1cm}\n \\caption{Mkn 266, contours of adaptively smoothed 0.3$-$8.0\\,keV X-ray data from {\\em Chandra} ACIS-S overlaid on an optical image from WFPC2 on board the {\\em HST}. }\n \\label{fig:mkn_266_opt_con}\n\\end{figure}\n\nIn addition to the central emission from the colliding galaxies, a large X-ray gas cloud can be seen to the North of the system. From a \\emph{ROSAT}\\ observation it was proposed that this feature is a superwind, driven by the effect of the starburst's supernovae and stellar winds from the centre of the system \\citep{Wang_97}. \\citet{Kollatschny_98} argue that an X-ray `jet', arising from a centrally powered superwind, is unlikely due to both the non-radial geometry of the emission and also the bright X-ray emission of the feature. They suggest that the most plausible mechanism to explain the `jet' is from excitation by hot post-shock gas, although they do not conclude where the energy to power this feature would arise from. From the \\emph{Chandra}\\ observation we speculate that the northern emission observed could also arise from star formation taking place in a tidal arm that has been stripped from the galaxy during an earlier interaction. The exact nature of this emission is discussed in detail in section \\ref{sec:mkn266_northern}.\n\n\\subsection{NGC 3256}\n\nThe most X-ray luminous system in our sample, NGC 3256, is a powerful ultraluminous infrared galaxy (ULIRG), lying at a distance of 56 Mpc. This merger system, like Mkn 266, has one common gaseous envelope containing the nuclei from its two parent galaxies. \\citet{Lira_02} report on a 28\\,ks \\emph{Chandra}\\ observation made in 2000. Contours of adaptively smoothed 0.2$-$8.0\\,keV X-ray emission overlaid on an optical image of the system are shown in Figure \\ref{fig:ngc3256_opt_con}. \n\nThe total X-ray luminosity of NGC 3256 is 7.87$\\times$ 10$^{41}$ \\ensuremath{\\erg~\\ps}\\ with $\\sim$70\\% of the X-ray luminosity arising from the diffuse emission. 14 discrete X-ray sources were detected in this system, all of which have been classified as ULXs. Due to the high source detection threshold of this observation (\\ensuremath{L_{\\mathrm{X}}}=1.4$\\times$10$^{39}$erg s$^{-1}$) fainter point sources were not detected. Both galaxy nuclei are clearly detected in X-rays. The Northern nucleus is a site of intense star formation, and UV spectra \\citep{Lipari_00} show strong absorption lines, implying the presence of massive young stars. The Southern nucleus is heavily obscured and appears to be less active then the Northern nucleus. It has been suggested that it hosts an AGN, although there is no clear evidence of this provided in the X-ray data. The soft diffuse emission of the system can be described by two thermal components with a harder tail. The thermal plasma components exhibit temperatures of 0.6 keV and 0.9 keV and the hard component is thought to arise from a contribution from the lower luminosity X-ray point source population. \n\nA kinematic study of the system \\citep{English_03} suggests that NGC 3256 is currently experiencing the starburst that just precedes the final core collision and, given the close proximity of the two nuclei, it is likely that this system has undergone more than one perigalactic approach. Assuming that the two tidal tails formed during the last closest-encounter, the time that has elapsed since then can be estimated. This characteristic timescale was calculated to be 500 Myr and it is thought that coalescence will take place in $\\sim$200 Myr.\n\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{images\/ngc3256_opt_xcon.ps}\n \\hspace{0.1cm}\n \\caption{NGC 3256, contours of adaptively smoothed 0.3$-$8.0\\,keV X-ray data from {\\em Chandra} ACIS-S overlaid on an optical image from WFPC2 on board the {\\em HST}. }\n \\label{fig:ngc3256_opt_con}\n\\end{figure}\n\n\\subsection{Arp 220}\n\nThe double nuclear system, Arp 220 (IC 4553\/4, UGC 09913) is a prototypical ULIRG, lying at a distance of 76 Mpc. The two nuclei have a separation of less than 0.5 kpc and are just at the point of coalescence. This complicated, dusty system has the highest value of \\ensuremath{L_{\\mathrm{FIR}}}\\ in our sample, which arises from the large reservoir of hot gas within the system. The mechanism by which this gas is heated is still currently under debate. Both the presence of a heavily shrouded AGN and intense nuclear starbursts have been suggested, along with a combination of both these energy sources contributing to the heating mechanism. A 60 ks \\emph{Chandra}\\ observation of the system was made in 2000, and from these data both the nuclear and the extended emission have been analysed \\citep{Clements_02,McDowell_03}. Figure \\ref{fig:arp220_opt_con} shows the 0.2$-$1.0 keV adaptively smoothed X-ray emission, overlaid on an optical image.\n\nThe system hosts 4 ULXs, including the nuclei, no other point sources are detected. Three distinct regimes of large diffuse gas structures are observed; the circumnuclear region, a plume region and, further out, diffuse lobe regions. The circumnuclear region has both compact, hard nuclear emission that resides in the 1 kpc at the centre of the system and softer, more extended emission. It has been suggested that the hard component of this region arises from a significant point source population, emitting at a luminosity lower than $\\sim 5 \\times 10 ^{39}$ \\ensuremath{\\erg~\\ps}, the sensitivity threshold for this observation. Although, this could also be a AGN, albeit one emitting at a low luminosity.\n\nThe plume regions extend to the northwest and southeast of the system with a projected tip to tip length of $\\sim$10 kpc, these features are clearly seen in Figure \\ref{fig:arp220_opt_con}. The spectrum of the plumes include both hot (1$-$5 keV) and cooler (0.25 keV) thermal contributions. It is likely that these regions are associated with a superwind extending from the nuclear region as a consequence of the vigorous star formation that is taking place at the centre of the system. Beyond the plumes, two large, low surface brightness lobe regions have been observed extending from 10$-$15 kpc on either side of the nuclear region. These regions have been found to be cooler than the plumes (0.2$-$1.0 keV), although, due to the low number of counts, higher temperature gas residing in these regions cannot be ruled out. \n\nIt was noted by \\citet{Heckman_96} that the plumes and lobes are ``misaligned'' by 25\\ensuremath{^\\circ}$-$30\\ensuremath{^\\circ}, they suggest that this could be due to a change in orientation of the system as the encounter has progressed. \\citet{McDowell_03} propose that this misalignment is actually a consequence of the lobes being produced not by the superwinds in the system, but are a product of the merger itself. Tentative evidence to support this scenario is presented in an INTEGRAL observation by \\citet{Colina_04}.\n\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{images\/arp220_opt_xcon.ps}\n \\hspace{0.1cm}\n \\caption{Arp 220, contours of adaptively smoothed 0.2$-$1.0\\,keV X-ray data from {\\em Chandra} ACIS-S overlaid on an optical image from the Palomar 5m telescope. }\n \\label{fig:arp220_opt_con}\n\\end{figure}\n\n\\subsection{NGC 7252}\n\nNGC 7252, the first example of a merger-remnant galaxy in our sample, is a proto-typical merger galaxy at a distance of 63 Mpc. The central part of the system, a single relaxed body, displays an r$^{1\/4}$ optical surface brightness profile typical of elliptical galaxies \\citep{Schweizer_82}. In addition to this relaxed nucleus, the galaxy exhibits complex loops and ripples, notably two long tidal tails, indicative of the merger history of this system. From both {\\em UBVI} images taken with the WCFPT instrument on the {\\em HST} \\citep{Miller_97} and N-body simulations \\citep{Hibbard_95}, it has been estimated that nuclear coalescence took place $\\sim$1 Gyr ago.\n\nA 28 ks \\emph{Chandra}\\ observation, along with an \\emph{XMM-Newton}\\ observation is reported in \\citet{Nolan_04}. Figure \\ref{fig:ngc_7252_opt_con} shows the 0.3$-$7.0 keV adaptively smoothed \\emph{Chandra}\\ X-ray contours overlaid on an optical image. A total of nine ULXs are detected within the optical confines of this system, a further 5 sources are also detected but at a lower significance ($<3\\sigma$). The hot, diffuse gas of the system has been found to be fairly symmetrical and has an X-ray luminosity of 2.42$\\times$10$^{40}$ \\ensuremath{\\erg~\\ps}. This is low when compared to luminosities from typical elliptical galaxies (\\ensuremath{L_{\\mathrm{X}}}\\ $\\sim$10$^{41-42}$ \\ensuremath{\\erg~\\ps}). The low luminosity of this system is possibly due to the young age of the galaxy and, over much longer timescales, X-ray halo regeneration may increase the mass and hence luminosity of the gas within this system to levels seen in typical elliptical galaxies. From spectral modelling of the \\emph{XMM-Newton}\\ data the X-ray emission from the nuclear region is found to emit at 0.72 keV and 0.36 keV. There is also a harder contribution that can be fitted with a power law, and it is expected that this is due to a lower luminosity point source population. \n\nDuring the nuclear coalescence of NGC 7252 the star formation rate at the centre of the galaxy would have been massively enhanced. Now, as the reserve of gas becomes depleted, the star formation rate has fallen to one third of the value it would have been at its peak \\citep{Mihos_93}, although its \\ensuremath{L_{\\mathrm{FIR}}}\\ is still enhanced when compared to normal quiescent galaxies.\n\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{images\/ngc7252_opt_xcon.ps}\n \\hspace{0.1cm}\n \\caption{NGC 7252, contours of adaptively smoothed 0.3$-$7.0\\,keV X-ray data from {\\em Chandra} ACIS-S overlaid on an optical image from the LCO 2.5m telescope. }\n \\label{fig:ngc_7252_opt_con}\n\\end{figure}\n\n\\subsection{Arp 222}\n\nArp 222, or NGC 7727, is a post merger galaxy at a slightly more advanced stage of evolution than NGC 7252 \\citep{Georgakakis_00}. A CO observation \\citep{Crabtree_94} indicates that these systems are very similar in morphology and both host young cluster populations, however, Arp 222 seems to have a much smaller molecular gas content than NGC 7252. From a K-band observation \\citep{Rothberg_04} it has been found that the optical surface brightness profile of Arp 222 follows the de Vaucouleurs r$^{1\/4}$ law, indicating that violent relaxation has taken place since the merger. From analysing the discrete structure of the galaxy, plumes extending from the nucleus were identified, these features indicate that the system has still not fully relaxed into a mature elliptical galaxy.\n\nA 19 ks \\emph{Chandra}\\ observation of the post merger-remnant was carried out in 2001, and analysis and results from this observation are given in section \\ref{sec:arp222}. Figure \\ref{fig:arp222_opt_con} shows the 0.3$-$8.0 keV adaptively smoothed X-ray emission overlaid on an optical image. From this observation 15 point sources were detected, two of which are classified as ULXs. The X-ray luminosity of the diffuse gas in this system is 6.52 $\\times$10$^{39}$\\ensuremath{\\erg~\\ps}, much lower than the X-ray luminosity emitted from NGC 7252. This is likely to be due to the smaller gas content of Arp 222, which, as can be seen in Figure \\ref{fig:arp222_opt_con}, does not extend to the optical confines of the galaxy. From spectral modelling the global temperature of the diffuse gas in this system has been found to be 0.60 keV.\nThe \\ensuremath{L_{\\mathrm{FIR}}}\\ value of the system is much lower than that of the previous systems and is similar to values seen in elliptical galaxies. \n\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{images\/arp222_opt_xcon.ps}\n \\hspace{0.1cm}\n \\caption{Arp 222 Contours of adaptively smoothed 0.3$-$8.0\\,keV X-ray data from {\\em Chandra} ACIS-S overlaid on an optical image from the UK Schmidt telescope. }\n \\label{fig:arp222_opt_con}\n\\end{figure}\n\n\\subsection{NGC 1700}\n\nThe final system in our sample, NGC 1700, is a protoelliptical galaxy with a best age estimate of $\\sim$3 Gyr \\citep{Brown_00}. This galaxy possesses a kinematically distinct core and boxy isophotes at larger radii, it also contains two symmetrical tidal tails, a good indicator that this system formed through the merger of two, comparable mass, spiral galaxies. It has been suggested that it is the presence of these tidal features that causes the 'boxiness' of the galaxy \\citep{Brown_00}. A 42 ks \\emph{Chandra}\\ observation was made of the system in 2000, and the results of this are reported in \\citet{Statler_02}. Figure \\ref{fig:ngc1700_opt_con} shows the 0.3$-$0.8 keV adaptively smoothed X-ray emission overlaid on an optical image of NGC 1700.\n\nFrom this observation, 36 point sources are detected, 6 of which are classified as ULXs \\citep{Diehl_06}. The diffuse X-ray gas is well modelled by a single temperature thermal plasma at a temperature of 0.43 keV and has an X-ray luminosity of 1.47$\\times 10^{41}$ \\ensuremath{\\erg~\\ps}, similar to that of a mature elliptical galaxies. The change in morphology of the diffuse X-ray gas from the central elliptical region to the outer boxy region, can clearly be seen in Figure \\ref{fig:ngc1700_opt_con}. \\citet{Statler_02} suggest that the flattening of the isophotes is a consequence of an elliptical-spiral interaction, not a spiral-spiral merger as suggested by \\citet{Brown_00}. They argue that an interaction involving a preexisting elliptical with a hot ISM, would lead to the channelling of the already hot gas into the systems common potential well. This gas, at sufficiently low densities, would then settle into a rotationally flattened cooling disc, as is observed. Currently, neither these two scenarios, nor the suggestion that the system could have formed from a 3-body interaction \\citep{Statler_96}, can be ruled out. \n\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{images\/ngc1700_opt_xcon.ps}\n \\hspace{0.1cm}\n \\caption{NGC 1700 Contours of adaptively smoothed 0.3$-$0.8\\,keV X-ray data from {\\em Chandra} ACIS-S overlaid on an optical image from the UK Schmidt telescope. }\n \\label{fig:ngc1700_opt_con}\n\\end{figure}\n\n\\section{{\\em Chandra} Observations and Data Analysis of Mkn 266 and Arp 222}\n\\label{sec_data_red}\n\nObservations of Mkn 266 and Arp 222 were carried out with the ACIS-S camera on board the {\\em Chandra} X-ray Observatory. Mkn 266 was observed on 2nd November 2001 with a total observation time of 19.7\\,ks and Arp 222 was observed on 18th December 2001 with a total observation time of 19.0\\,ks. The initial data processing to correct for the motion of the\nspacecraft and apply instrument calibration was carried out with the\nStandard Data Processing (SDP) at the {\\em Chandra} X-ray Center\n(CXC). The data products were then analysed using the CXC CIAO\nsoftware suite (v3.2)\\footnote{http:\/\/asc.harvard.edu\/ciao} and\nHEASOFT (v5.3.1). The data were reprocessed, screened for bad pixels, and time filtered to remove periods of high background (when\ncounts deviated by more than 5$\\sigma$ above the mean). This\nresulted in a corrected exposure time of 18.5ks for Mkn 266 and 18.8\\,ks for Arp 222. The full data analysis and results from these two observations are detailed in the following subsections.\n\n\\subsection{Mkn 266}\n\\label{sec:mkn266}\n\\subsubsection{Overall X-ray Structure}\n\\label{mkn:sec_struc}\n\nA 0.3$-$8.0\\,keV (from here on referred to as `full band') {\\em Chandra}\nimage was created from the cleaned events file and adaptively smoothed\nusing the CIAO task {\\em csmooth} which uses a smoothing kernel to\npreserve an approximately constant signal to noise ratio across the\nimage, which was constrained to be between 2.6 and 4. In Figure\n\\ref{fig:mkn_rgb} both the optical image from the {\\em HST} with the full band X-ray contours overlaid (left) and the `true colour' image of the galaxy system (right) are shown. The `true colour' image was created by combining three separate smoothed images in three energy bands; 0.3$-$0.8 keV, 0.8$-$2.0 keV and 2.0$-$8.0 keV, using the same smoothing scale for each image. These energy bands correspond to red, green and blue respectively. From these images it can be seen that the system is comprised of two separate regions of X-ray emission. To the South of the images the central emission from the interacting galaxies can be seen. This comprises of the two nuclei from the progenitor galaxies, a LINER to the North and a Seyfert 2 to the South, contained within a common gaseous envelope and a region of enhanced emission between the two nuclei. This region is coincident with a radio source reported in \\citet{Mazzarella_88}. It is likely that this enhanced emission is due to the interaction of the two galaxy discs. To the North of the images diffuse gas is detected and shows some correlation with emission seen in the {\\em HST} image. The nature of this emission is discussed in detail in section \\ref{sec:mkn266_northern}.\n\nFrom the `true colour' image it can be seen that the north-east nucleus emits in all three energy bands, whilst the south-west nuclear emission is not as hard. There is also some enhanced X-ray emission in the central region between the two nuclei. The X-ray emission surrounding these nuclei appears to be soft and diffuse with some suggestion of a super-bubble to the south-east of the system.\n\n\\begin{figure*}\n \\begin{minipage}{0.5\\linewidth}\n \\vspace{0.7cm}\n \\includegraphics[width=\\linewidth]{images\/mkn266_main.ps}\n\n\n \\end{minipage}\\vspace{0.05\\linewidth}\n \\begin{minipage}{0.38\\linewidth}\n\n \\hspace{0.1\\linewidth}\n \\includegraphics[width=\\linewidth]{images\/mkn_rgb.ps}\n\n \\end{minipage}\n \\caption{Left, contours of adaptively smoothed 0.3$-$8.0\\,keV X-ray data from {\\em Chandra} ACIS-S overlaid on an optical image from WFPC2 on board the {\\em HST}. The black box indicates the plate boundary of the {\\em HST} image. Right shows the `true colour' image of Mkn 266. Red corresponds to 0.3$-$0.8 keV, green to 0.8$-$2.0 keV and blue to 2.0$-$8.0 keV.}\n \\label{fig:mkn_rgb}\n\\end{figure*}\n\n\\subsubsection{Point Source Spatial and Spectral Analysis}\n\\label{mkn:sec_source_ps}\n\nDiscrete X-ray sources were detected using the CIAO tool {\\em\nwavdetect}. This was run on the full band\nimage, over the 2, 4, 8, 16 pixel wavelet scales (where pixel\nwidth is 0.1\\ensuremath{^{\\prime\\prime}}), with a significance threshold of\n$2.5\\times10^{-5}$, which corresponds to one spurious source over a\n200 $\\times$ 200 pixel grid, the size of our image. Only 4 sources were\ndetected in this range; the two nuclei and two regions in the diffuse northern feature. Both the regions in the northern emission contained less than 30 counts and had large detection regions of r$\\ge$3.5\\arcsec\\ and so were not defined as point sources. Instead, a spectrum was extracted from a region file containing all the northern emission. These two detections, along with the extraction and background extraction regions for the 2 nuclei, are shown in Figure \\ref{fig:mkn_reg}.\n\nThe two detected nuclear sources were extracted using the source\nregion files created by {\\em wavdetect}. The size of each region was\nselected to ensure that as many source photons as possible were\ndetected whilst minimising contamination from nearby sources and\nbackground. The background files were defined as a source free annulus surrounding and concentric with each source region\nfile, to account for the variation of diffuse emission, and to\nminimise effects related to the spatial variation of the CCD response. \n\nThe source spectra for the two nuclei were created using the CIAO tool {\\em psextract} and fitted in XSPEC (v11.3.1). Due to the low number of counts in this observation the Cash statistic \\citep{Cash_79} was used in preference to \\ensuremath{\\chi^2}\\ when modelling the data. Both sources were well fitted with an absorbed thermal model plus an absorbed power law, the absorption component was fixed at the value out of our Galaxy (1.68 $\\times 10 ^{20} $ atom cm$^{-2}$). The data were restricted to 0.3$-$6.0 keV, as energies below this have calibration uncertainties, and the spectra presented here do not have significant source flux above 6.0 keV. The parameters from these best fit models can be seen in Table \\ref{tab:mkn266_ps}; where columns 2 and 3 give the right ascension and declination (J2000), column 4 the count rate, column 5 the source significance, column 6 the Galactic value of \\ensuremath{N_{\\mathrm{H}}}, column 7 $kT$, column 8 metallicity, column 9 power law photon index ($\\Gamma$) and columns 10 and 11 give the observed and intrinsic (i.e. corrected for absorption) luminosities. \n\n\\begin{figure}\n \\hspace{0.1\\linewidth}\n \\includegraphics[width=0.8\\linewidth]{images\/mkn_reg.ps}\n \\hspace{0.1cm}\n \\caption{Adaptively smoothed 0.3$-$8.0\\,keV X-ray data from {\\em Chandra} ACIS-S of Mkn 266, with {\\em wavdetect} point source denoted with an ``X''. Extraction and background region files for source A and B are also shown. }\n \\label{fig:mkn_reg}\n\\end{figure}\n\n\\begin{table*}\n\\centering\n\\caption[]{Sources detected in the 0.3$-$6.0 keV band within the Mkn 266 with a summary of point source spectral fits. Errors on the spectral\nfits parameters are given as 1\\( \\sigma \\) for 1 interesting parameter\nfrom XSPEC. An F denotes that the value has been frozen.\n\\label{tab:mkn266_ps}}\n\\begin{tabular}{c@{}c@{}c@{}c@{}c@{}c@{}cccc@{}c@{}}\n\\noalign{\\smallskip}\n\\hline\nSource & RA & Dec. & Count Rate & Sig. & \\multicolumn{4}{c}{Spectral Fit } & \\multicolumn{2}{c}{Luminosity (0.3$-$6.0 keV) } \\\\\n & & & & & \\ensuremath{N_{\\mathrm{H}}} & kT & Z & $\\Gamma$ \n & \\multicolumn{2}{c}{(10\\(^{41} \\) erg s\\(^{-1} \\)) } \\\\ \n & & &($\\times 10^{-2}$ count s$^{-1}$) & ($\\sigma$) & ($\\times 10 ^{20} $ atom cm$^{-2}$) & (keV) & (\\ensuremath{\\mathrm{Z}_{\\odot}}) & & Observed & Intrinsic \\\\ \n \\\\ \\hline\n \n \n\nA & 13:38:17.84 & +48:16:41.2 & 1.90$\\pm$0.11 & 41.8 & 1.68 F & 0.80$^{+0.07}_{-0.11}$ & 0.59$^{+1.50}_{-0.24}$ & 0.02$^{+0.25}_{-0.21}$ & 3.41 $\\pm 0.17$ & 3.47 $\\pm 0.17$ \\\\\nB & 13:38:17.36 & +48:16:32.5 & 0.90$\\pm$0.08 & 18.7 & 1.68 F & 0.88$^{+0.08}_{-0.10}$ & 0.30 F & 1.13$^{+0.49}_{-0.64}$ & 0.98 $\\pm 0.07$ & 1.04 $\\pm 0.07$ \\\\ \n\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{table*}\n\nFrom this table it can be seen that source A, the LINER, exhibits a remarkably flat power law of 0.02$^{+0.25}_{-0.21}$. It is likely that part of the contribution to this component arises from unresolved point sources but the very flat slope of this fit suggests that some of the hard component is either heavily absorbed or dominated by reflection \\citep{Levenson_04,Matt_00}. By including an absorber for the power law component we can allow for this extra absorption, however, due to the low number of counts, when including this component the fit becomes unconstrained.\n\nIt is likely that the cause of this heavy obscuration is due to the strong starburst in this region. An observation by \\citet{Sanders_86} found that Mkn 266 is rich in CO, indicating that this system contains a massive, warm reservoir of molecular gas to fuel this starburst, and the dynamic conditions that concentrate this material also serve to further obscure the LINER. The MEKAL component of the fit to this source arises from this starburst, which contributes 5.7$\\times$10$^{40}$\\ensuremath{\\erg~\\ps}\\ to the total intrinsic luminosity of source A. \n\nSource B, the southern nucleus, emits at a temperature of 0.88 keV and has a power law slope of 1.13. This thermal component is likely to arise from the diffuse emission surrounding the nucleus, and the value of $\\Gamma$ is consistent with the interpretation of this nucleus being a Seyfert 2 \\citep{Cappi_05}. In previous X-ray surveys it has been found that up to $\\sim$75\\% of Seyfert 2 objects are heavily obscured with \\ensuremath{N_{\\mathrm{H}}}\\ $\\ge$10$^{23}$atom cm$^{-2}$ \\citep{Risaliti_99}. With the data we have from this observation the two component model describes the spectrum well, and, due to the limited number of counts, a more complex model with an additional absorption component would over-fit the data. In addition to this, \\citet{Cappi_05} have found from a recent survey of nearby Seyfert galaxies, that these sources possess the entire range of \\ensuremath{N_{\\mathrm{H}}}\\ from 10$^{20}$atom cm$^{-2}$ to 10$^{24}$atom cm$^{-2}$, fairly continuously. Indicating that, although the southern nucleus could be heavily obscured, it is also possible that the two component model described here is good indication of the properties of this source. \n\\subsubsection{Diffuse Emission Spatial and Spectral Analysis}\n\\label{mkn:sec_source_dif}\n\nFrom the `true colour' image of Mkn 266 it is clear that the galaxy contains significant amounts of diffuse gas, in both the northern feature and also surrounding the central galaxy. To investigate the nature of this diffuse emission, spectra were extracted from these two separate regions using the CIAO tool {\\em acisspec}, and fitted in XSPEC. Once again, due to the low number of counts, the Cash statistic was used when modelling the data. \n\nThe diffuse gas contained within the galaxy is not well described by a single temperature fit, and is better modelled with two temperature components. It is likely that the hotter gas contributing to this model arises from the enhanced emission seen between the two nuclei. To investigate this, the extraction region was divided into two separate parts; one, smaller region, to probe the impact area, where it is likely that the two discs have collided, and a second, larger region, covering the rest of the galaxy, but which excludes the inner impact region and the northern diffuse region. These, along with the extraction region for the northern diffuse emission and the associated background regions, are shown in Figure \\ref{fig:mkn_diff}. The two separate regions, diffuse 1 and diffuse 2, are both well described with single component MEKAL fits, exhibiting the same temperatures that were fitted in the two temperature model. The northern region is also well described with a single component MEKAL fit. The parameters from the best fit models for these regions, along with their X-ray luminosities, are shown in Table \\ref{tab:mkn_diff}.\n\n\\begin{figure}\n \\hspace{0.1\\linewidth}\n \\includegraphics[width=0.8\\linewidth]{images\/mkn_diff.ps}\n \\hspace{0.1cm}\n \\caption{Adaptively smoothed 0.3$-$8.0\\,keV X-ray data from {\\em Chandra} ACIS-S of Mkn 266, with extraction and background region for the diffuse gas shown. The background region files are selected to be source free annuli, surrounding and concentric with each extraction region file.}\n \\label{fig:mkn_diff}\n\\end{figure}\n\n\\begin{table*}\n\\centering\n\\caption[]{Summary of the spectral fits of the diffuse gas regions in Mkn 266. Errors on the spectral\nfits parameters are given as 1\\( \\sigma \\) for 1 interesting parameter\nfrom XSPEC. An F denotes that the value has been frozen.\n\\label{tab:mkn_diff}}\n\\begin{tabular}{cccccc}\n\\noalign{\\smallskip}\n\\hline\nDiffuse Region & \\ensuremath{N_{\\mathrm{H}}} \t\t\t\t\t& kT \t& Z \t\t& \\multicolumn{2}{c}{Luminosity (0.3$-$6.0 keV)} \\\\\n \t\t& ($\\times 10 ^{20} $ atom cm$^{-2}$) \t& (keV) & (\\ensuremath{\\mathrm{Z}_{\\odot}})\t& \\multicolumn{2}{c}{(10$^{40}$ erg s$^{-1}$)} \\\\\n\t\t&\t\t\t\t\t&\t&\t\t& Observed & Intrinsic \\\\ \n \\hline\n \n \n\nDiffuse 1 \t& 1.68 F \t\t\t\t& 0.52$^{+0.06}_{-0.06}$ & 0.23$^{+0.05}_{-0.03}$ \t& 9.88 $\\pm 0.50$ & 10.92 $\\pm 0.50$ \\\\\nDiffuse 2\t& 1.68 F \t\t\t\t& 1.07$^{+0.06}_{-0.09}$ & 0.22$^{+0.15}_{-0.08}$ \t& 8.10\t$\\pm 0.49$ & 9.83 $\\pm 0.53$ \\\\\nDiffuse 3 \t& 1.68 F \t\t\t\t& 0.30$^{+0.02}_{-0.03}$ & 0.30F\t \t\t& 6.10 $\\pm 0.49$ & 6.90 $\\pm 0.55$ \\\\\n\n\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{table*}\n\nFrom this table it can be seen that the diffuse gas within the galaxy system, diffuse 1, exhibits a temperature of 0.52 keV, a fairly typical temperature for gas in interacting galaxies (RP98). The impact region, diffuse 2, has a higher temperature of 1.07 keV. This is, within errors, the same temperature attained from the higher temperature contribution from the two component MEKAL fit of the combined spectrum for diffuse 1 and diffuse 2, indicating that the higher temperature from this fit was a consequence of the gas arising from this impact area. \n\nThe northern emission appears spectrally distinct from the surrounding diffuse gas and has been found to emit at a cooler temperature of 0.30 keV. With this \\emph{Chandra}\\ data it has also been found that this region has a much lower luminosity than the one derived in \\citet{Kollatschny_98}, who report on a \\emph{ROSAT}\\ HRI observation of the galaxy and find the X-ray luminosity of this region to be 3.1$\\times$10$^{41}$ \\ensuremath{\\erg~\\ps}, as opposed to 6.9$\\times$10$^{40}$ \\ensuremath{\\erg~\\ps}, as reported in this work. However, in \\citet{Kollatschny_98} this X-ray luminosity is derived from assuming that the relative number of HRI counts is directly proportional to their share of the integrated X-ray flux, with the total luminosity of the system being derived from the \\emph{ROSAT}\\ PSPC observation of the system and consequently has large uncertainties associated with it.\n\n\\subsubsection{Nature of the Northern Emission}\n\\label{sec:mkn266_northern}\n\nThe origin of the diffuse emission to the north of the system is still the subject of debate. There have been a number of suggestions as to how this feature has been formed. From HRI and PSPC observations with \\emph{ROSAT}, \\citet{Wang_97} suggest that it is an outflow, driven by the mechanical energy of the supernovae and stellar winds in the starburst. A subsequent paper \\citep{Kollatschny_98}, investigating both the HRI observation and optical B,V,R-images of Mkn 266, argued that the scenario proposed by \\citet{Wang_97} is unlikely, due to both the high luminosity of the `jet' and also its non-radial geometry. They instead suggest that the mechanism that gives rise to the `jet' is excitation by hot post-shock gas, although they do not conclude where the energy to power this feature would arise from.\n\nA tridimensional spectrophotometric study by \\citet{Ishigaki_00} investigated the H{$\\alpha$}, [O~\\textsc{iii}] and [S~\\textsc{ii}] emission-lines within Mkn 266. Figure \\ref{fig:mkn_266_halpha} shows the H{$\\alpha$}\\ contours from this observation, overlaid on the full band, adaptively smoothed, \\emph{Chandra}\\ X-ray emission. As can be seen, the enhanced regions of H{$\\alpha$}\\ in the northern region are coincident with the two X-ray regions detected with {\\em wavdetect} (see Figure \\ref{fig:mkn_reg}). Some caution should be exercised when interpreting this plot due to the uncertainty in the astrometry, which we conservatively estimate to be 2\\ensuremath{^{\\prime\\prime}}\\ (1.1 kpc). Even so, the correlation between the X-ray and H{$\\alpha$}\\ is striking, suggesting that this region to the north of the system is a site of star formation, possibly a tidal arm that has been stripped from the southern progenitor during the merger process. This interpretation is strengthened by the fact that both the {\\em HST} (Figure \\ref{fig:mkn_rgb}) and H{$\\alpha$}\\ emission seem to connect with the dust lanes around the southern nucleus \\citep{Ishigaki_00}.\n\n\\begin{figure}\n \\includegraphics[width=0.95\\linewidth]{images\/mkn266_halpha.ps}\n \\hspace{0.1cm}\n \\caption{Adaptively smoothed 0.3$-$8.0\\,keV X-ray data from {\\em Chandra} ACIS-S of Mkn 266. H{$\\alpha$}\\ contours from \\citet{Ishigaki_00} are overlaid. }\n \\label{fig:mkn_266_halpha}\n\\end{figure}\n\nAs shown in the previous section, the luminosity arising from the diffuse gas within the northern feature is 6.9$\\times$10$^{40}$ \\ensuremath{\\erg~\\ps}. Given that our spectral fit to the data indicates that this emission arises from a thermal component, and both the {\\em HST} and H{$\\alpha$}\\ emission indicate that there is star formation within this region, the origin of these X-rays could come from supernovae. Making some assumptions about the geometry of the emitting region of this northern feature the thermal energy of the gas can be derived. First the volume, $V$, of the diffuse northern emission has been assumed to be an ellipsoid, with symmetry about the longer axis. The fitted emission measure is equal to $\\eta n_\\mathrm{e}^{2}V$\nand can be used to infer the mean particle number density $n_\\mathrm{e}$, with\nthe filling factor, $\\eta$, assumed to be 1. This factor represents the fraction of volume filled by the emitting gas. Although we have assumed this to be 1 in our calculations, there is evidence from hydrodynamical simulations to suggest this value could be $\\le$2 per cent \\citep{Strickland_00b}. The mean electron density is then used to derive the total gas mass $M_\\mathrm{gas}$, which then leads to the thermal energy $E_\\mathrm{th}$ of the hot gas. From these assumptions $E_\\mathrm{th}$ has been calculated to be between 6.72$\\times 10^{54}$ \\ensuremath{\\mbox{erg}}\\ (for $\\eta$=0.02) and 3.36$\\times 10^{56}$ \\ensuremath{\\mbox{erg}}\\ (for $\\eta$=1).\n\nBy deriving the supernova rate, \\ensuremath{r_{\\mathrm{SN}}}, for this region, the energy arising from supernovae can be calculated. \\citet{Davies_00} calculate \\ensuremath{r_{\\mathrm{SN}}}\\ for the LINER in Mkn 266, using the relation derived in \\citet{Condon_90} \n\\begin{equation}\nL_{\\mathrm{NT}}[\\mathrm{WHz^{-1}}]\\sim1.3\\times 10^{23}(\\upsilon[\\mathrm{GHz}])^{-0.8}r_{\\mathrm{SN}}[\\mathrm{yr^{-1}}],\n\\label{eq:sn}\n\\end{equation}\nand the 20 cm radio continuum \\citep{Mazzarella_88}. Where $L_{\\mathrm{NT}}$ is the calculated power and $\\upsilon$ is the frequency of the observation. From this they calculate \\ensuremath{r_{\\mathrm{SN}}}\\ to be 0.45 SNyr$^{-1}$. \n\nFor the northern diffuse emission, \\citet{Mazzarella_88} find a power of 8.9$\\times 10^{21}$WHz$^{-1}$, using this value and the above equation we find \\ensuremath{r_{\\mathrm{SN}}}\\ =0.05 SNyr$^{-1}$ for this region. From \\citet{Mattila_01} it has been found that it is more appropriate to use \\ensuremath{L_{\\mathrm{FIR}}}\\ to calculate \\ensuremath{r_{\\mathrm{SN}}}\\ in starburst galaxies but, as we only have \\ensuremath{L_{\\mathrm{FIR}}}\\ for the whole system, we would greatly overestimate the supernova rate, and consequently, we will use the rate of 0.05 SNyr$^{-1}$ we have derived from the 20 cm continuum data.\n\nFrom \\citet{Davies_00}, models of the star formation timescales have been derived for the LINER. And, although these models give a fairly poorly constrained age (20-500 Myr), by making the assumption that this is a proxy for the timescales of star formation taking place in the northern diffuse emission, the number of supernovae formed in this time can be estimated. Given that we also make the assumption that a massive star takes $\\sim 1\\times$10$^{7}$ yr to evolve into a supernova, we calculate the number of SNe formed in this time to be $5\\times10^5 - 2.5\\times10^7$. If each supernova releases $\\sim1\\times 10^{51}$ \\ensuremath{\\mbox{erg}}, the total thermal energy available will be between $\\sim5\\times 10^{56}$ \\ensuremath{\\mbox{erg}}\\ to $\\sim2.5\\times 10^{58}$ \\ensuremath{\\mbox{erg}}, thus demonstrating that the observed luminosity in the northern diffuse emission could arise from the contribution of star formation alone. \n\nThe reason that both \\citet{Wang_97} and \\citet{Kollatschny_98} prefer an outflow scenario to that of tidal stripping to explain this northern feature is due to the presence of optical emission in a highly excited state within this feature. These line ratios, coupled with the electron temperature calculated for this region, cannot be explained by thermal collisional ionisation, and are likely to arise through either photoionisation or shock excitation. \\citet{Wang_97} and \\citet{Kollatschny_98} both suggest that these shocks arise from a `jet', outflowing from the central galaxy. However, \\citet{Ishigaki_00} propose that the northern emission is photoionised by radiation from the Seyfert nucleus and therefore, excitation by shocks is not required to explain the high excitation emission lines that are observed. \n\nThe present study adds further evidence to this debate. With the higher resolution provided by \\emph{Chandra}, the structure of the northern emission can now be resolved. From this it can be seen that the northern feature is curved, and the X-ray emission appears to be `clumpy', with the two most intense areas of X-ray emission coincident with the H{$\\alpha$}\\ emission (see Figure \\ref{fig:mkn_266_halpha}). These are not features which indicate a superwind. Furthermore, the morphology of this region traces that of the optical emission, which is highly suggestive that these two features are connected. Consequently, we propose that the most likely source of this X-ray emission is star formation taking place within a tidal arm that has been stripped out from one of the progenitors during the merger of the two galaxies.\n\n\\subsubsection{The South East Extension}\n\nTo the south east of the central galaxy region there is some suggestion of X-ray extension, which appears to be coincident with filaments seen in the {\\em HST} image, just beginning to break out of the galactic disc (Figure \\ref{fig:mkn_rgb}). Given the low count statistics in this observation we cannot extract spectra for this region alone. But, from the morphology of the X-ray emission, coupled with the H{$\\alpha$}\\ emission, there is some indication that this region is a site of star formation just on the point of break-out from the gaseous envelope of the central system. This scenario is preferred to the one suggested in the case of the northern emission as there is additional evidence of an outflow feature from the starburst region around the LINER \\citep{Ishigaki_00}. This outflow could be beginning to sweep the dust out of the galaxy, indicating that this system is in the stage just prior to the outbreak of large-scale galactic winds throughout the system. Of course, alternatively, this feature could be a small scale version of the northern emission, particularly given its coincidence with optical emission seen with the {\\em HST}. However, without spectral information to investigate the nature of this object, neither scenario can be ruled out.\n\n\\subsection{Arp 222}\n\\label{sec:arp222}\n\\subsubsection{Overall X-ray Structure}\n\nA full band adaptively smoothed \\emph{Chandra}\\ image of Arp 222 was produced using the same tools and techniques as described in section \\ref{mkn:sec_struc}. In Figure \\ref{fig:222_gal} both the optical image from the UK Schmidt telescope, with the smoothed full band X-ray contours overlaid (left), and the `true colour' image (right) of Arp 222 is shown. The true colour image was produced using the same methods as described in section \\ref{mkn:sec_struc}.\n\n\\begin{figure*}\n \\begin{minipage}{0.58\\linewidth}\n \\vspace{0.7cm}\n \\includegraphics[width=\\linewidth]{images\/arp222_main.ps}\n\n\n \\end{minipage}\\vspace{0.03\\linewidth}\n \\begin{minipage}{0.385\\linewidth}\n\n \\includegraphics[width=\\linewidth]{images\/arp222_rgb.ps}\n\n \\end{minipage}\n \\caption{Left, contours of adaptively smoothed 0.3$-$8.0\\,keV X-ray data from {\\em Chandra} ACIS-S overlaid on an optical image from the UK Schmidt telescope. Right shows the `true colour' image of Arp 222. Red corresponds to 0.3$-$0.8 keV, green to 0.8$-$2.0 keV and blue to 2.0$-$8.0 keV.\n \\label{fig:222_gal}}\n\\end{figure*}\n\nFrom these images it can be seen that there is some diffuse gas at the centre of the galaxy, but this emission does not extend out to the optical confines of the system. In addition to the diffuse gas, a number of point sources are found through out the galaxy. From the `true colour' image, the white appearance of the nucleus indicates that it emits in all three energy bands, whilst the other point sources can be seen to be softer.\n\n\\subsubsection{Spatial and Spectral Analysis}\n\nX-ray point sources were searched for using the CIAO tool {\\em wavdetect}. This was run on the full band image, over the 2, 4, 8, 16 pixel wavelet scales (where pixel width is 0.5\\ensuremath{^{\\prime\\prime}}), with a significance threshold of 2.8$\\times$10$^{-6}$, which corresponds to one spurious source over a 600 $\\times$ 600 pixel grid, the size of our image. 24 sources were detected in this full band range, these were then limited to those that lie within the $D_{25}$ ellipse of the galaxy, reducing the number to 15 detected sources. These regions, along with the $D_{25}$ ellipse are shown in Figure \\ref{fig:arp222_ps}. From using the \\emph{Chandra}\\ Deep Field South number counts \\citep{Giacconi_01} we estimate 2 to 3 of these sources to be background objects. \n\n\\begin{figure}\n\\hspace{0.1\\linewidth}\n \\includegraphics[width=0.8\\linewidth]{images\/arp222_reg.ps}\n \\hspace{0.1cm}\n \\caption{Adaptively smoothed 0.3$-$8.0\\,keV X-ray data from {\\em Chandra} ACIS-S of Arp 222, with point sources detected by {\\em wavdetect} indicated. The D$_{25}$ ellipse is also shown.}\n \\label{fig:arp222_ps}\n\\end{figure}\n\nSpectra for the point sources were extracted using the CIAO tool {\\em psextract} with region and background region files selected in the same way as described in section \\ref{mkn:sec_source_dif}. Of the 15 detected sources, only one, source 5, emitted sufficient counts to be modelled individually, the other 14 sources were fitted simultaneously. The spectrum of the diffuse gas was extracted with the CIAO tool {\\em acisspec}, from a region file centred on source 5, with a radius of 45\\ensuremath{^{\\prime\\prime}}. The background region file was defined to be a source free annulus surrounding and concentric with the region file. Once again, due to the low number of counts, the Cash statistic was used in preference to \\ensuremath{\\chi^2}. Both the combined sources and source 5 were well described by single component, power law models. And, from this combined fit model, the luminosities for each individual source were derived. The diffuse gas was well fitted by a single component MEKAL model. These spectral fits are summarised in Table \\ref{tab:arp222}, where column 1 gives the source identification number, columns 2 and 3 give the right ascension and declination (J2000), column 4 the Galactic value of \\ensuremath{N_{\\mathrm{H}}}, column 5 $kT$, column 6 metallicity, column 7 $\\Gamma$ and columns 8 and 9 give the observed and intrinsic luminosities. \n\n\\begin{table*}\n\\centering\n\\caption[]{A summary of the best fit parameters for the diffuse gas and point sources detected in the 0.3$-$6.0 keV band within Arp 222. Errors on the spectral fits parameters are given as 1\\( \\sigma \\) for 1 interesting parameter\nfrom XSPEC. An F denotes that the value has been frozen.\n\\label{tab:arp222}}\n\\begin{tabular}{cccc@{}cccc@{}c@{}}\n\\noalign{\\smallskip}\n\\hline\nSource & RA & Dec. & \\multicolumn{4}{c}{Spectral Fit } & \\multicolumn{2}{c}{Luminosity (0.3$-$6.0 keV) } \\\\\n & & & \\ensuremath{N_{\\mathrm{H}}} & kT & Z &$\\Gamma$ \n & \\multicolumn{2}{c}{(10\\(^{39} \\) erg s\\(^{-1} \\)) } \\\\ \n & & & ($\\times 10 ^{20} $ atom cm$^{-2}$) & (keV) & (\\ensuremath{\\mathrm{Z}_{\\odot}}) & & Observed & Intrinsic \\\\ \n \\\\ \\hline\n \n \n\n1 & 23:40:00.61 & -12:17:10.2 & 2.75 F & - & - & 1.72F & 0.99 $\\pm 0.17$ & 1.07 $\\pm 0.17$ \\\\\n2 & 23:39:57.01 & -12:16:33.0 & 2.75 F & - & - & 1.72F & 0.31 $\\pm 0.16$ & 0.32 $\\pm 0.16$ \\\\ \n3 & 23:39:55.95 & -12:16:47.3 & 2.75 F & - & - & 1.72F & 0.19 $\\pm 0.07$ & 0.21 $\\pm 0.08$ \\\\\n4 & 23:39:54.37 & -12:16:59.1 & 2.75 F & - & - & 1.72F & 0.45 $\\pm 0.11$ & 0.48 $\\pm 0.12$ \\\\\n5 & 23:39:53.67 & -12:17:31.6 & 2.75 F & - & - & 2.15$^{+0.37}_{-0.36}$ & 1.74 $\\pm 0.21$ & 1.97 $\\pm 0.24$ \\\\\n6 & 23:39:52.84 & -12:17:36.8 & 2.75 F & - & - & 1.72F & 0.37 $\\pm 0.11$ & 0.40 $\\pm 0.12$ \\\\\n7 & 23:39:52.97 & -12:18:01.5 & 2.75 F & - & - & 1.72F & 0.20 $\\pm 0.09$ & 0.21 $\\pm 0.09$ \\\\\n8 & 23:39:52.67 & -12:17:21.1 & 2.75 F & - & - & 1.72F & 0.43 $\\pm 0.11$ & 0.46 $\\pm 0.12$ \\\\\n9 & 23:39:51.67 & -12:18:19.6 & 2.75 F & - & - & 1.72F & 0.52 $\\pm 0.14$ & 0.54 $\\pm 0.15$ \\\\\n10 & 23:39:51.31 & -12:18:30.0 & 2.75 F & - & - & 1.72F & 0.35 $\\pm 0.11$ & 0.37 $\\pm 0.11$ \\\\\n11 & 23:39:51.15 & -12:17:48.8 & 2.75 F & - & - & 1.72F & 0.38 $\\pm 0.11$ & 0.42 $\\pm 0.13$ \\\\\n12 & 23:39:50.57 & -12:16:53.7 & 2.75 F & - & - & 1.72F & 0.28 $\\pm 0.09$ & 0.30 $\\pm 0.10$ \\\\\n13 & 23:39:51.18 & -12:15:57.6 & 2.75 F & - & - & 1.72F & 0.30 $\\pm 0.12$ & 0.31 $\\pm 0.13$ \\\\\n14 & 23:39:49.36 & -12:18:24.4 & 2.75 F & - & - & 1.72F & 0.38 $\\pm 0.11$ & 0.40 $\\pm 0.12$ \\\\\n15 & 23:39:45.63 & -12:17:41.2 & 2.75 F & - & - & 1.72F & 0.21 $\\pm 0.12$ & 0.55 $\\pm 0.13$ \\\\\n\\\\\nDiffuse & 23:39:53.67 & -12:17:31.6 & 2.75 F & 0.60$^{+0.07}_{-0.06}$ & 0.08$^{+0.06}_{-0.02}$ & - & 5.48 $\\pm$ 0.28 & 6.52 $\\pm$ 0.33 \\\\\n\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{table*}\n\nFrom this table it can be seen that the best-fit spectral index for the combined sources is $\\Gamma$=1.72, a typical value for XRBs in, what can be described as, a low\/hard state \\citep{Soria_03}. For the central source, source 5, the best fit model has shown that the spectral state is much softer, with $\\Gamma$=2.15. This value is consistent with it also being an XRB, but this time in a high\/soft state \\citep{Colbert_04} This source, along with source 1, has been found to be a ULX. Given the quiescent nature of the galaxy, and that the time since the last episode of widespread star formation has been shown to be $\\sim$1.2 Gyr ago \\citep{Georgakakis_00}, this indicates that these sources are likely to arise from low mass X-ray binaries (LMXRBs), not HMXRBs as has been seen in younger merger systems within this sample.\n\nThe amount of diffuse gas within the system is smaller than in mature elliptical galaxies, and, as mentioned previously, does not extend out to the optical confines of the system. The temperature of the gas, 0.6 keV, has been found to be comparable to that of other interacting systems, but the diffuse gas exhibits a lower X-ray luminosity (6.52$\\times 10^{39}$ \\ensuremath{\\erg~\\ps}) than the other systems within this sample. The low gas content of Arp 222 is further seen from CO observations of the galaxy \\citep{Crabtree_94}. From this \\emph{Chandra}\\ observation it can be seen that this system is X-ray faint and does not currently resemble a mature elliptical galaxy, although may at a greater dynamical age.\n\n\\section{The Evolution of Merging Galaxies}\n\\label{sec_evol}\n\n\\subsection{Multi-wavelength Evolution Sequence}\n\nTo gain a greater understanding of the merger process of galaxy pairs, in addition to the X-ray luminosity of each system, B-band, K-band and FIR luminosities have been obtained to study how each of these luminosity diagnostics evolve along the chronological sequence. In the case of systems where the X-ray luminosity has been obtained from the literature, the stated band has been converted into the 0.3$-$6.0 band used in this paper by assuming canonical models of the two X-ray components. For the diffuse gas, a MEKAL model with a gas temperature of 0.5 keV has been assumed, and for the point source population a power law model with a photon index of 1.5 has been adopted.\n\nThese luminosities are given in Table \\ref{tab:merger:props} where; column 1 gives the system name, column 2 the distance to the object, column 3 the merger age, with nuclear coalescence being defined as 0 Myr, column 4 the total (0.3$-$6.0 keV) intrinsic X-ray luminosity from the \\emph{Chandra}\\ observation, column 5 gives the percentage of luminosity arising from the diffuse gas (\\%{\\ensuremath{L_{\\mathrm{diff}}}}), column 6 \\ensuremath{L_{\\mathrm{FIR}}}, column 7 \\ensuremath{L_{\\mathrm{B}}}\\ and column 8 \\ensuremath{L_{\\mathrm{K}}}.\n\n\\begin{table*}\n\n\\centering\n\n\\caption[]{The fundamental properties of all the systems within this sample. Columns are explained in the text.\n\\label{tab:merger:props}}\n\\begin{tabular}{cccccccc}\n\\noalign{\\smallskip}\n\\hline\n\nGalaxy \t\t& Distance\t& Merger Age\t& \\ensuremath{L_{\\mathrm{X}}}\t\t\t& \\%L$_{diff}$\t& Log \\ensuremath{L_{\\mathrm{FIR}}}\t\t\t& Log \\ensuremath{L_{\\mathrm{B}}} \t& Log \\ensuremath{L_{\\mathrm{K}}} \\\\\nSystem \t\t& \t\t&\t\t& (0.3$-$6.0 keV)\t&\t\t& \t\t\t\t&\t\t&\t \\\\\n \t\t& (Mpc)\t\t& (Myr)\t\t& ($\\times10^{40}$\\ensuremath{\\erg~\\ps}) & \t\t& (\\ensuremath{\\erg~\\ps}) \t\t& (\\ensuremath{\\erg~\\ps})\t& (\\ensuremath{\\erg~\\ps}) \\\\\n \n\\noalign{\\smallskip}\n\\hline\n\nArp 270\t\t& 28\t\t& -650 \t\t& 2.95\t\t\t&\t28\t&43.63\t\t\t\t&43.72\t\t&43.37\t \\\\\nThe Mice\t& 88\t\t& -500 \t\t& 6.01\t\t\t&\t31\t&44.12\t\t\t\t&44.07\t\t&44.04\t \\\\\nThe Antennae\t& 19\t\t& -400 \t\t& 6.97\t\t\t&\t54\t&43.95\t\t\t\t&43.98\t\t&44.64\t \\\\\nMkn 266\t\t& 115\t\t& -300 \t\t& 73.18\t\t\t& \t41\t&44.75\t\t\t\t&44.30\t\t&44.38\t \\\\\nNGC 3256\t& 56\t\t& -200 \t\t& 87.70\t\t\t&\t80\t&45.19\t\t\t\t&44.42\t\t&45.22\t \\\\\nArp 220\t\t& 76\t\t& 0 \t\t& 23.70\t\t\t&\t44\t&45.50\t\t\t\t&43.97\t\t&44.78\t \\\\\nNGC 7252\t& 63\t\t& 1000 \t\t& 6.17\t\t\t&\t31\t&44.00\t\t\t\t&44.40\t\t&44.61\t \\\\ \nArp 222\t\t& 23\t\t& 1200 \t\t& 1.46\t\t\t&\t45\t&$\\geq$42.37\t\t&43.92\t\t&44.84\t \\\\\nNGC 1700\t& 54\t\t& 3000 \t\t& 17.58\t\t\t&\t83\t&42.91\t\t\t\t&44.32\t\t&45.15\t \\\\\n\n\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{table*}\n\nThe FIR luminosities are calculated using the\nexpression \\citep{Devereux_89}\n\\begin{equation}\n\\label{eq:lfir}\n\\ensuremath{L_{\\mathrm{FIR}}} =3.65 \\times 10^5[2.58S_{60 \\mu m}+S_{100\\mu m}]D^2\\ensuremath{\\mathrm{L}_{\\odot}},\n\\label{equ:lfir}\n\\end{equation}\nwith {\\em IRAS} 60- and 100-$\\mu$m fluxes taken from the {\\em IRAS}\nPoint Source Catalogue \\citep{Moshir_90}. The optical (B) luminosities\nwere calculated as in \\citet{Tully_88}\n\\begin{equation}\n\\mathrm{log} \\ensuremath{L_{\\mathrm{B}}}\\, (\\ensuremath{\\mathrm{L}_{\\odot}}) =12.192-0.4B_\\mathrm{T}+2\\mathrm{log}D,\n\\end{equation}\nwhere $B_\\mathrm{T}$ is the blue apparent magnitude and $D$ is the\ndistance in Mpc. Values of blue apparent magnitude were taken from\n\\citet{Dev_91} (the value for The Mice was taken from NGC 2000.0\n\\citep{Dreyer_88}). The values of \\ensuremath{L_{\\mathrm{K}}}\\ are derived using the relation given in \\citet{Seigar_05}\n\\begin{equation}\n\\mathrm{log} \\ensuremath{L_{\\mathrm{K}}}=11.364-0.4K_\\mathrm{T}+\\mathrm{log}(1+Z)+2\\mathrm{log}D,\n\\end{equation}\nwhere $K_\\mathrm{T}$ is the K-band apparent magnitude, $Z$ is the galaxy redshift and $D$ is the distance in Mpc. Apparent K-band magnitudes were taken from the 2MASS survey.\n\nTo calculate the value of \\%{\\ensuremath{L_{\\mathrm{diff}}}} for each system, the contribution to the luminosity of the diffuse gas that arises from unresolved low luminosity point sources had to be estimated and removed. This was done in a number of ways; in the case of systems for which we have access to the reduced data (The Mice, Mkn 266 and Arp 222), an additional power law component with a photon index of 1.5 was included in the spectral fit of the diffuse emission. It was then assumed that the luminosity of this component arises from unresolved point sources within the system. Additionally, for Mkn 266, it was assumed that the MEKAL component of the point source fits arises from the diffuse gas surrounding the LINER and Seyfert. In the cases of Arp 270 and NGC 7252, estimates of the contribution of the unresolved point sources to the total diffuse luminosity has already been made in \\citet{Brassington_05} and \\citet{Nolan_04}, respectively.\n\nFor the systems we have taken from the literature, the `Universal Luminosity Function' (ULF), derived by \\citet{Grimm_03}, was used to predict the flux from unresolved low luminosity sources. The predicted XRB luminosity function is\n\\begin{equation}\nN(>L)=5.4\\mathrm{SFR}(L_{38}^{-0.61}-210^{-0.61}),\n\\label{equ:sfr}\n\\end{equation}\nwhere $L_{38}$ = $L\/10^{38}$ erg s$^{-1}$ and the factor 210 arises\nfrom the upper luminosity cut-off of 2.1$\\times 10^{40}$ erg\ns$^{-1}$. \nSFR was estimated from the population of brighter point sources detected above the completeness limit from the \\emph{Chandra}\\ observations. \nIn the case of The Antennae, point sources were detected down to a luminosity of 5$\\times10^{37}$erg s$^{-1}$ and therefore no correction was required for this system.\n\nAs mentioned previously, one of the main difficulties in compiling an evolutionary sequence such as this, is assigning an age to each of these systems. The way in which these estimates were made is described in section \\ref{sec:sample}, but another important point that was not discussed is that when constructing an evolutionary sample, the absolute timescale of the merger process must be considered. From N-body simulations \\citep{Mihos_96} it is clear that from the first initial strong interaction between equal-mass mergers, through to their nuclear coalescence, takes $\\sim$700 Myr. What is not so well defined is the amount of time that elapses between coalescence and the relaxation of the galaxy into a system resembling a mature elliptical galaxy. Within the sample presented here, this issue has been addressed by selecting a greater dynamical range than has previously been studied. By doing this it is hoped that the transition between young merger remnants to relaxed mature ellipticals can be observed, and therefore a more complete picture of the merger process can be obtained. \n\nAlthough these merger systems are being compared to establish a single evolutionary sequence, it should be remembered that nine individual systems, not nine examples of one merger at different stages of its evolution, are being looked at. Consequently, although the trends that have been identified here should be a good indicator of how X-ray emission evolves during the merger process, due to the careful selection criteria outlined in section \\ref{sec:sample}, it is likely that other merger pair systems will exhibit different X-ray properties. These variations are likely to arise due to the unique interaction parameters associated with each merger system, as well as the variation of gas content and mass of individual galaxies within each system.\n\nWith the multi-wavelength luminosities that have been collected for each system (Table \\ref{tab:merger:props}), the evolution of the galaxy properties along the merger process has been investigated. The activity levels, a proxy for star formation normalised by galaxy mass, is indicated by {\\ensuremath{L_{\\mathrm{FIR}}}}\/{\\ensuremath{L_{\\mathrm{K}}}}. To investigate the variation of {\\ensuremath{L_{\\mathrm{X}}}}, scaled by galaxy mass for each system, both the B-band and the K-band luminosities, {\\ensuremath{L_{\\mathrm{X}}}}\/{\\ensuremath{L_{\\mathrm{B}}}} and {\\ensuremath{L_{\\mathrm{X}}}}\/{\\ensuremath{L_{\\mathrm{K}}}} have been used. Both of these have been used as normalisation values as \\ensuremath{L_{\\mathrm{B}}}\\ can become greatly enhanced during periods of star formation due to the presence of young stars, indicating that \\ensuremath{L_{\\mathrm{K}}}\\ is likely to provide a more reliable tracer of galaxy mass. Therefore, by plotting \\ensuremath{L_{\\mathrm{X}}}\\ against both of these values, how great this effect is can be observed. \n\nThese ratios are shown in Figure \\ref{fig:evol}, where, not only the luminosity ratios, but also the percentage of luminosity arising from diffuse gas (\\%{\\ensuremath{L_{\\mathrm{diff}}}}), as a function of merger age have been plotted. {\\ensuremath{L_{\\mathrm{FIR}}}}\/{\\ensuremath{L_{\\mathrm{K}}}} (solid line), {\\ensuremath{L_{\\mathrm{X}}}}\/{\\ensuremath{L_{\\mathrm{B}}}} (dot-dash line) and {\\ensuremath{L_{\\mathrm{X}}}}\/{\\ensuremath{L_{\\mathrm{K}}}} (dashed line) have been normalised to the typical spiral galaxy, NGC 2403. Whilst \\%{\\ensuremath{L_{\\mathrm{diff}}}} (dotted line) is plotted on the right-hand y-axis of the plot, as an absolute value, where \\%{\\ensuremath{L_{\\mathrm{diff}}}} for NGC 2403 is 12\\%. The horizontal lines to the right of the plot indicate {\\ensuremath{L_{\\mathrm{FIR}}}}\/{\\ensuremath{L_{\\mathrm{K}}}}, {\\ensuremath{L_{\\mathrm{X}}}}\/{\\ensuremath{L_{\\mathrm{B}}}}, {\\ensuremath{L_{\\mathrm{X}}}}\/{\\ensuremath{L_{\\mathrm{K}}}} and \\%{\\ensuremath{L_{\\mathrm{diff}}}} for NGC 2434, a typical elliptical galaxy \\citep{Diehl_06}. \n\n\\begin{figure*}\n \\includegraphics[width=\\linewidth]{images\/evolution.ps}\n \\caption {The evolution of X-ray luminosity in merging galaxies. Shown are {\\ensuremath{L_{\\mathrm{FIR}}}}\/{\\ensuremath{L_{\\mathrm{K}}}} (solid line), {\\ensuremath{L_{\\mathrm{X}}}}\/{\\ensuremath{L_{\\mathrm{B}}}} (dot-dash line), {\\ensuremath{L_{\\mathrm{X}}}}\/{\\ensuremath{L_{\\mathrm{K}}}} (dashed line) and \\%{\\ensuremath{L_{\\mathrm{diff}}}}, plotted as a function of merger age, where 0 age is defined to be the time of nuclear coalescence. All luminosity ratios ({\\ensuremath{L_{\\mathrm{FIR}}}}\/{\\ensuremath{L_{\\mathrm{K}}}}, {\\ensuremath{L_{\\mathrm{X}}}}\/{\\ensuremath{L_{\\mathrm{B}}}} and {\\ensuremath{L_{\\mathrm{X}}}}\/{\\ensuremath{L_{\\mathrm{K}}}}) are normalised to the spiral galaxy NGC 2403. \\%{\\ensuremath{L_{\\mathrm{diff}}}} is plotted on a linear scale, shown on the right-hand y-axis of the plot and is an absolute value. \\%{\\ensuremath{L_{\\mathrm{diff}}}} for NGC 2403 is 12\\%. The horizontal lines to the right of the plot indicate {\\ensuremath{L_{\\mathrm{FIR}}}}\/{\\ensuremath{L_{\\mathrm{K}}}}, {\\ensuremath{L_{\\mathrm{X}}}}\/{\\ensuremath{L_{\\mathrm{B}}}}, {\\ensuremath{L_{\\mathrm{X}}}}\/{\\ensuremath{L_{\\mathrm{K}}}} and \\%{\\ensuremath{L_{\\mathrm{diff}}}} for NGC 2434, a typical elliptical galaxy.}\n \\label{fig:evol}\n\\end{figure*}\n\nThe system NGC 2403 was selected to represent a typical spiral galaxy. This system was chosen due to its close proximity (D=3.2 Mpc), the fact that it does not have a high level of star-forming activity and that it does not host an AGN \\citep{Schlegel_03}. NGC 2434, was selected to represent a typical elliptical galaxy. At a distance of 22 Mpc, this system is close enough to enable the point source population to be disentangled from the hot gas, allowing this diffuse emission to be modelled \\citep{Diehl_06}, this systems also exhibits the quiescent levels of star formation normally associated with such galaxies. Table \\ref{tab:typical} shows the properties for both of these systems, with columns as in Table \\ref{tab:merger:props}.\n\n\\begin{table*}\n\n\\centering\n\n\\caption[]{Properties of the typical spiral (NGC 2403) and the typical elliptical (NGC 2434) used in Figure \\ref{fig:evol}. Columns as in Table \\ref{tab:merger:props}.\n\\label{tab:typical}}\n\\begin{tabular}{ccccccc}\n\\noalign{\\smallskip}\n\\hline\n\nGalaxy \t\t& Distance\t& \\ensuremath{L_{\\mathrm{X}}}\t\t\t& \\%L$_{diff}$\t& Log \\ensuremath{L_{\\mathrm{FIR}}}\t& Log \\ensuremath{L_{\\mathrm{B}}} \t& Log \\ensuremath{L_{\\mathrm{K}}} \\\\\nSystem \t\t& \t\t&(0.3$-$6.0)\t\t\t&\t\t& \t\t&\t\t&\t \\\\\n \t\t& Mpc\t\t&($\\times10^{40}$\\ensuremath{\\erg~\\ps})& \t\t& (\\ensuremath{\\erg~\\ps}) \t\t& (\\ensuremath{\\erg~\\ps})\t& (\\ensuremath{\\erg~\\ps}) \\\\\n \n\\noalign{\\smallskip}\n\\hline\n\nNGC 2403\t& 3\t\t& 0.29\t\t\t&\t12\t& 42.21\t\t& 43.22\t\t&43.49\t \\\\\nNGC 2434\t& 22\t\t& 4.82\t\t\t&\t76\t& 41.96\t\t& 43.54\t\t&44.49\t \\\\\n\n\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{table*}\n\nAnother factor that must be considered when plotting these luminosity ratios is the AGN that is hosted by Mkn 266. This is the only system within the sample that has a confirmed AGN, and as such, it is expected to be significantly more luminous than the other systems. Therefore, the values of {\\ensuremath{L_{\\mathrm{X}}}}\/{\\ensuremath{L_{\\mathrm{B}}}}, {\\ensuremath{L_{\\mathrm{X}}}}\/{\\ensuremath{L_{\\mathrm{K}}}} and \\%{\\ensuremath{L_{\\mathrm{diff}}}} have been calculated for Mkn 266, both including and excluding the contribution from the Seyfert. As shown in Table \\ref{tab:mkn266_ps}, this nucleus is not a powerful AGN, instead it is close to the Seyfert-LINER borderline, and therefore, the total value of \\ensuremath{L_{\\mathrm{X}}}\\ only reduces by 14\\% if it is excluded, with the value of \\%{\\ensuremath{L_{\\mathrm{diff}}}} rising to 48\\%. When the reduced luminosity is used to derive the luminosity ratios for Mkn 266, the change in Figure \\ref{fig:evol} is very small, and the overall trends exhibited by these ratios are preserved.\n\n\\subsection{Arp 220: the Normalisation of the Universal Luminosity Function}\n\\label{sec:ulf}\n\n\\citet{Grimm_03} considered a number of SFR indicators for each galaxy within their sample. This was done, as there is considerable scatter in the SFR estimates obtained from different indicators. To calculate an `adopted' SFR for each galaxy, indicators that deviated significantly from the other values were disregarded, and the remaining indicators were averaged to give a final value used in subsequent calculations. \n\nIn the present work, the SFR was estimated from the brighter point sources, detected from the \\emph{Chandra}\\ observations. To further investigate the normalisation of the ULF, SFR indicators, derived from \\ensuremath{L_{\\mathrm{FIR}}}, using the expression \\citep{Rosa_02}\n\\begin{equation}\n\\mathrm{SFR}_\\mathrm{FIR}=4.5 \\times 10^{-44}L_\\mathrm{FIR}(\\mathrm{erg~s}^{-1}),\n\\end{equation}\nhave also been used to estimate the luminosity arising from the unresolved point sources. This ULF correction, using the SFR indicator derived from \\ensuremath{L_{\\mathrm{FIR}}}, has been calculated for the most active system within our sample, Arp 220. From this correction, a luminosity of 5.05$\\times$10$^{41}$erg s$^{-1}$, has been derived for the low luminosity (\\ensuremath{L_{\\mathrm{X}}}\\ $\\le 5\\times$10$^{39}$erg s$^{-1}$) point sources. This value is almost two times greater than the {\\em total} observed luminosity of Arp 220 (see Table \\ref{tab:merger:props}), indicating that using \\ensuremath{L_{\\mathrm{FIR}}}\\ as a SFR in the ULF must greatly overestimate the point source population for this system. To ensure that the value calculated in equation \\ref{equ:lfir} was consistent with other observations, additional \\ensuremath{L_{\\mathrm{FIR}}}\\ values were obtained (Log \\ensuremath{L_{\\mathrm{FIR}}}=45.86, \\citet{David_92} and Log \\ensuremath{L_{\\mathrm{FIR}}}=45.54, \\citet{Liu_95}). Averaging these values gives Log \\ensuremath{L_{\\mathrm{FIR}}}\\ = 45.73 for Arp 220, similar to the value given in Table \\ref{tab:merger:props}. \n\nTo investigate this further, we compare the observed population of brighter sources with that predicted using the ULF. From the \\emph{Chandra}\\ observation, 4 sources with \\ensuremath{L_{\\mathrm{X}}} $\\ge 5\\times$10$^{39}$erg s$^{-1}$ are detected in Arp 220. Using the SFR derived from \\ensuremath{L_{\\mathrm{FIR}}}\\ in equation \\ref{equ:sfr}, the bright point source population is estimated to be ten times larger, with 41 discrete sources predicted. This demonstrates that, in the case of Arp 220, when using \\ensuremath{L_{\\mathrm{FIR}}}\\ as the SFR indicator, the ULF greatly overestimates the whole point source population. {\\ensuremath{L_{\\ensuremath{\\mathrm{H}{\\alpha}}}}} values for this galaxy were also used to calculate the SFR of the system \\citep{Young_96,Colina_04} but, due to the very dusty nature of this object, are subject to large extinction effects, leading to greatly reduced SFRs with values as little as 0.55, which, given the merger status of this system, is unlikely to be a true measure of the SFR.\n\nTo investigate the reliability of using the \\ensuremath{L_{\\mathrm{FIR}}}\\ SFR indicator to normalise the ULF, the detected point source populations (with \\ensuremath{L_{\\mathrm{X}}}\\ $\\le 2.1\\times10^{40}$ erg s$^{-1}$) from the \\emph{Chandra}\\ observations were used to derive SFRs from equation \\ref{equ:sfr} for a number of galaxies in our sample; these values are plotted against the \\ensuremath{L_{\\mathrm{FIR}}}\\ derived SFR values in Figure \\ref{fig:sfr}. Only seven of the nine systems have been included in this figure, as the ULF is only a measure of the HMXRB population, and, it is likely that the stellar populations from the two older merger-remnants are dominated by LMXRBs.\n\\begin{figure}[h]\n \\hspace{0.8cm}\n \\includegraphics[width=0.8\\linewidth]{images\/sfr_ratio.ps}\n \\caption{A plot of the SFR indicated by \\ensuremath{L_{\\mathrm{FIR}}}\\ and the SFR derived from equation \\ref{equ:sfr}, with the solid line indicating unity. Arp 222 and NGC 1700 have not been included in this plot as their point source populations are likely to be dominated by LMXRBs.}\n \\label{fig:sfr}\n\\end{figure}\nFrom Figure \\ref{fig:sfr}, it can be seen that at high levels of the\n\\ensuremath{L_{\\mathrm{FIR}}}\\ SFR indicator, the SFR derived from equation \\ref{equ:sfr} is\nmuch smaller, indicating that, in very active star-forming systems,\nthe prediction of the point source population, when using the \\ensuremath{L_{\\mathrm{FIR}}}\\\nSFR value, does not represent the observed X-ray population well. \n\nThis relationship between \\ensuremath{L_{\\mathrm{X}}}\\ and SFR was also investigated in\n\\citet{Gilfanov_04b}, where active systems in the HDF-N were used to\nprobe the ULF at high values of SFR. From this work it was found that,\nfor higher levels of star formation, the observed X-ray luminosities\nfrom these galaxies do follow the \\ensuremath{L_{\\mathrm{X}}}-SFR relation. However, the\ncalculated SFR for all of these systems were derived from 1.4 GHz, {\\ensuremath{L_\\nu}} SFR\nindicators only, not \\ensuremath{L_{\\mathrm{FIR}}}\\ SFR indicators. This suggests that the discrepancy\nbetween \\ensuremath{L_{\\mathrm{FIR}}}\\ SFR, and the observed values derived from the\n\\emph{Chandra}\\ observations, are a consequence of the normalisation\nindicator chosen, and do not arise from non-linearity of the ULF at high values of SFR.\n\nThere are a number of reasons why the SFRs inferred from the X-ray observations and \\ensuremath{L_{\\mathrm{FIR}}}\\ values might differ so greatly for systems in the most active phases. It could simply be that, for dusty, active systems such as these, \\ensuremath{L_{\\mathrm{FIR}}}\\ is a poor tracer of SFR, becoming greatly enhanced due to the large amounts of gas residing in the galaxy. This would indicate that it is the \\ensuremath{L_{\\mathrm{FIR}}}\\ SFR indicator, not the SFR from the ULF, that is incorrect. However, this explanation is unlikely, as studies have shown that, whilst {\\ensuremath{L_{\\ensuremath{\\mathrm{H}{\\alpha}}}}} and {\\ensuremath{L_{\\mathrm{UV}}}} are susceptible to extinction caused by the presence of dust in star-forming regions, \\ensuremath{L_{\\mathrm{FIR}}}\\ does not vary with gas content, and scales well with both extinction corrected {\\ensuremath{L_{\\mathrm{UV}}}} and radio continuum emission in dusty starbursts \\citep{Buat_96,Buat_92}.\n\nAlternatively, the discrepancy between the predicted and observed values could be a consequence of the age of the stellar population. If the starburst has only recently taken place, it is plausible that, whilst \\ensuremath{L_{\\mathrm{FIR}}}\\ has had enough time to become massively enhanced, the X-ray point source population has not yet evolved into HMXRBs. The consequence of this would be that the X-ray luminosity arising from the point source population is lower than the SFR, indicated by \\ensuremath{L_{\\mathrm{FIR}}}, would predict. \\citet{Wilson_05} recently reported on an optical study of Arp 220 which has detected two populations of young massive star clusters. The age of the youngest of these populations has been found to be 5$-$10 Myr, indicating that enough time has elapsed since the last starburst for HMXRBs to form ($\\sim1-$10 Myr). It is therefore unlikely that a lag between \\ensuremath{L_{\\mathrm{FIR}}}\\ and \\ensuremath{L_{\\mathrm{X}}}\\ is the reason for the difference in the SFR values shown in Figure \\ref{fig:sfr}.\n\nAnother reason that the SFR derived from equation \\ref{equ:sfr} is lower than expected, when compared to the \\ensuremath{L_{\\mathrm{FIR}}}\\ SFR, could simply be that the X-ray observations are not detecting all the point sources in very active galaxies due to obscuration, and, those that are being detected have greatly reduced luminosities. To investigate this, the optical obscuration in Arp 220 \\citep{Shioya_01} has been converted into an absorbing column density, giving \\ensuremath{N_{\\mathrm{H}}} = 2.2$\\times 10 ^{22} $ atom cm$^{-2}$, a value which is actually somewhat smaller than the modelled value given in \\citet{Clements_02} of 3$\\times 10 ^{22} $ atom cm$^{-2}$. This demonstrates that the stated luminosities in \\citet{Clements_02} are reliable values, and all the point sources with \\ensuremath{L_{\\mathrm{X}}} $\\ge 5\\times$10$^{39}$erg s$^{-1}$ (the stated lower luminosity point source detection threshold) should have been detected in this system. \n\nA final explanation for the difference between the predicted and observed SFR indicators for systems close to the point of nuclear coalescence, could arise from a physical change in the birth of stellar systems. These merging galaxies provide extreme examples of violent star formation, and it is therefore possible that the most recent starburst has resulted in the birth of a stellar population with a different Initial Mass Function (IMF). \n\nThe IMF, the distribution of masses with which stars are formed, has long been a contentious issue, with a variety of different models used to explain the observed stellar populations \\citep{Larson_86,Larson_98}. It was initially proposed that the IMF was a universal function with a power law form, regardless of formation time or local environment \\citep{Salpeter_55}. But, it has also been argued that the IMF varies, and was more top-heavy at earlier times \\citep{Rieke_80}. \n\nFrom the most recent observations it is thought that the Salpeter IMF,\nwith a lower mass flattening between 0.5 \\ensuremath{\\mathrm{M}_{\\odot}}\\ and 1.0 \\ensuremath{\\mathrm{M}_{\\odot}}, can be\nconsidered a reasonable approximation for large regions of starburst\ngalaxies \\citep{Elmegreen_05}, with some suggestion of small\nvariations with environment, with denser and more massive star\nclusters producing more massive stars compared to intermediate mass\nstars \\citep{Elmegreen_04,Shadmehri_04}. This more top-heavy IMF has\nthe effect of generating greater numbers of supernova remnants, black\nholes, and HMXRBs, as well as higher amounts of \\ensuremath{L_{\\mathrm{FIR}}}\\ per unit mass\nof stars formed, therefore leading to an increase in both \\ensuremath{L_{\\mathrm{X}}}\\ and\n\\ensuremath{L_{\\mathrm{FIR}}}. However, the\nlevel of enhancement of \\ensuremath{L_{\\mathrm{FIR}}}, relative to the increase in HMXRBs formed, is\ndependent on the shape of the IMF. \n\nIn the galaxy systems close to the point of nuclear coalescence\npresented in this sample,\nthe X-ray binary population is dominated by HMXRBs, meaning that \\ensuremath{L_{\\mathrm{X}}}\\\nwill scale with the number of massive stars produced, whereas \\ensuremath{L_{\\mathrm{FIR}}}\\\nwill scale\nwith the main sequence luminosity of those stars. As the IMF flattens,\n\\ensuremath{L_{\\mathrm{FIR}}}\\ will rise more steeply, a consequence of the strong mass to\nluminosity dependence, $M \\propto L^{3.5}$. Therefore, when\nusing \\ensuremath{L_{\\mathrm{FIR}}}\\ values to derive the SFR for these system, this value will be\noverestimated.\n\n\n\n\nIf this interpretation is correct, it could explain why there appears to be a deficit of X-ray binaries in the \\emph{Chandra}\\ observations of the systems close to the point of coalescence. In actual fact, what is being seen here is an overestimated SFR, derived from the massively enhanced \\ensuremath{L_{\\mathrm{FIR}}}, as can be seen in Figure \\ref{fig:sfr}.\n\n\n\n\n\n\\section{Discussion: X-ray Evolution}\n\\label{sec:dis}\n\nThe first thing to note from Figure \\ref{fig:evol} is that Arp 270, even though an early stage system, is already exhibiting enhanced star formation activity, as measured by \\ensuremath{L_{\\mathrm{FIR}}}\/\\ensuremath{L_{\\mathrm{K}}}\\ (see \\citet{Read_01}), compared to the typical spiral system, NGC 2403. This activity increases up to the point of nuclear coalescence, after which there is a steady drop in \\ensuremath{L_{\\mathrm{FIR}}}\/\\ensuremath{L_{\\mathrm{K}}}\\ until the merger-remnant systems, $\\sim$1 Gyr after coalescence. From this point the activity value levels off to that of a typical elliptical system. \n\nThe evolution of the X-ray luminosity is different to that of \\ensuremath{L_{\\mathrm{FIR}}}\/\\ensuremath{L_{\\mathrm{K}}}. There is initially a rise, as seen with the star formation activity, but this peaks $\\sim$300 Myr before coalescence takes place, whilst \\ensuremath{L_{\\mathrm{FIR}}}\/\\ensuremath{L_{\\mathrm{K}}}\\ is still increasing. From this peak it drops until the young merger-remnants, as is the case for \\ensuremath{L_{\\mathrm{FIR}}}\/\\ensuremath{L_{\\mathrm{K}}}. But, instead of levelling off, the X-ray luminosity once again begins to rise, with the total X-ray luminosity of the 3 Gyr system beginning to resemble that of a mature elliptical galaxy.\n\nBoth \\ensuremath{L_{\\mathrm{X}}}\/\\ensuremath{L_{\\mathrm{B}}}\\ and \\ensuremath{L_{\\mathrm{X}}}\/\\ensuremath{L_{\\mathrm{K}}}\\ broadly exhibit the same variations. One notable exception is the value of \\ensuremath{L_{\\mathrm{X}}}\/\\ensuremath{L_{\\mathrm{B}}}\\ for Mkn 266. This relatively small value is due the massive enhancement of \\ensuremath{L_{\\mathrm{B}}}\\ from this system's AGN \\citep{Risaliti_04}, which \\ensuremath{L_{\\mathrm{K}}}\\ is not susceptible to, therefore indicating that \\ensuremath{L_{\\mathrm{K}}}\\ is a more reliable tracer of galaxy mass than \\ensuremath{L_{\\mathrm{B}}}, and will exhibit less scatter.\n\nThe evolution of \\%{\\ensuremath{L_{\\mathrm{diff}}}} also shows an increased value for Arp 270, when compared to the typical spiral system. This value then steadily rises, up to a point $\\sim$200 Myr prior to nuclear coalescence. This trend line then drops until the young merger-remnant systems, once again rising, with the 3 Gyr system exhibiting a similar value of \\%{\\ensuremath{L_{\\mathrm{diff}}}} to that of the typical elliptical, NGC 2434.\n\n\\subsection{Previous X-ray Studies}\n\\label{sec:RP98}\n\nIn RP98 it was found that the X-ray luminosity of the sample generally followed the same patterns as the star formation activity, with the peak of \\ensuremath{L_{\\mathrm{X}}}\/\\ensuremath{L_{\\mathrm{B}}}\\ being coincident with nuclear coalescence. Also in the RP98 study, the sample of post-merger systems was limited, with a baseline extending out to only 1.5 Gyr after nuclear coalescence. Consequently, no rise in the X-ray luminosity of post-merger systems was observed, because, as shown in the present study, there is a strong suggestion that systems require a much greater relaxation time before they begin to exhibit properties seen in mature elliptical galaxies. This idea is further strengthened by a study of post-merger ellipticals \\citep{Osul_01}, where a long term trend ($\\sim$10 Gyr) for \\ensuremath{L_{\\mathrm{X}}}\/\\ensuremath{L_{\\mathrm{B}}}\\ to increase with dynamical age was found.\n\nThe reason that the peak in \\ensuremath{L_{\\mathrm{X}}}\\ has been found at two different merger ages in these studies could be due to the different observatories that were used. \\emph{Chandra}\\ has greater spatial resolution than \\emph{ROSAT}, and is able to disentangle background galaxies from the target object with much greater accuracy. It is therefore possible that these objects were included in the previous \\emph{ROSAT}\\ work as diffuse features, leading to a different peak value of \\ensuremath{L_{\\mathrm{X}}}. However, both samples include the famous merger system, The Antennae, and the system at coalescence, Arp 220, and the X-ray luminosities from both of these systems are comparable between \\emph{ROSAT}\\ and \\emph{Chandra}. Indicating that this discrepancy does not arise from a difference between these two observatories. \n\nAnother difference between RP98 and the current work, that could account for the differing merger ages of peak \\ensuremath{L_{\\mathrm{X}}}, is the selection of systems within each study. In our sample we include two systems between The Antennae and Arp 220; Mkn 266 and NGC 3256, in RP98 there is only one, NGC 520. This system is highlighted in RP98 as being X-ray faint, not exhibiting the X-ray properties that one would expect, given its large \\ensuremath{L_{\\mathrm{FIR}}}\/\\ensuremath{L_{\\mathrm{K}}}\\ ratio. They suggest that this galaxy does not appear to be on the same evolutionary path as the rest of their sample and, possibly, will not evolve into an elliptical galaxy.\n\nA recent study of NGC 520 with \\emph{Chandra}\\ \\citep{Read_05}, has indicated that this system comprises one gas-rich and one gas-poor galaxy, not two gas rich spirals as has been selected within this sample. This lack of gas in the second galaxy has resulted in a `half merger' being induced in this system, and hence appears underluminous in X-rays. Therefore, it is likely that the peak X-ray luminosity was missed in the RP98 sample due to the galaxy selection. In the present study, use of the selection criteria outlined in section \\ref{sec:sample} has meant that only systems that are gas rich have been included, ensuring that comparable systems in the merger sequence have been selected. Consequently, the systems presented in this sample are more likely to be representative of the typical merger evolution.\n\n\\subsection{The X-ray Luminosity Peak and the Impact of Galactic Winds}\n\\label{sec:galWs}\n\nFrom the X-ray luminosities of the systems within this sample it has been seen, for the first time, that both \\ensuremath{L_{\\mathrm{X}}}\/\\ensuremath{L_{\\mathrm{K}}}\\ and \\ensuremath{L_{\\mathrm{X}}}\/\\ensuremath{L_{\\mathrm{B}}}\\ peak $\\sim$300 Myr before nuclear coalescence takes place. This result, given that the normalised star formation rate is still increasing at this time, is initially surprising, as one would expect the X-ray luminosity to increase with increasing \\ensuremath{L_{\\mathrm{FIR}}}\/\\ensuremath{L_{\\mathrm{K}}}. However, from looking at the morphology of the hot diffuse gas, it is clear that the systems emitting very high levels of \\ensuremath{L_{\\mathrm{FIR}}}\/\\ensuremath{L_{\\mathrm{K}}}\\ are also experiencing outflows from starburst-driven winds. We propose that this relative reduction in X-ray luminosity, for these very active systems, is a consequence of these large scale diffuse outflows.\n\nStarburst-driven winds are responsible for the transport of gas and energy out of star-forming galaxies. The energetics of these galactic winds were investigated by \\citet{Strickland_00b}, where hydrodynamical simulations were compared to the observations of the galactic wind of M82. From this work it was shown that the majority of the thermal and kinetic energy of galactic winds is in a hot volume-filling component of the gas, which is very difficult to probe due to its low emissivity, a consequence of the low density of this component.\n\nWithin our sample, the system that represents the peak X-ray luminosity is Mkn 266. In section \\ref{sec:mkn266} the results from the \\emph{Chandra}\\ observation of this system are presented and tentative evidence suggesting that this system is just about to experience galaxy wide diffuse outflows is found. If this interpretation is correct, it seems probable that the lower values of \\ensuremath{L_{\\mathrm{X}}}\/\\ensuremath{L_{\\mathrm{B}}}\\ and \\ensuremath{L_{\\mathrm{X}}}\/\\ensuremath{L_{\\mathrm{K}}}, for NGC 3256 and Arp 220, the most active systems in this sample, are a consequence of their extensive galactic winds, which have allowed the density, and hence the emissivity, of the hot gas to drop.\n\n\\subsection{Halo Regeneration in Low \\ensuremath{L_{\\mathrm{X}}}\\ Systems}\n\\label{sec:halo}\n\nAnother interesting result from this survey is the increase in \\ensuremath{L_{\\mathrm{X}}}\\ in the older merger-remnants. In previous studies (e.g. RP98) post-merger systems have been found to be X-ray faint when compared to elliptical galaxies. In our study the merger sequence has been extended to include a 3 Gyr system, and, by doing so, it has been shown that these underluminous systems appear to increase in \\ensuremath{L_{\\mathrm{X}}}\\ as they age. Given that these merger-remnants have been shown to be quiescent, this increase in \\ensuremath{L_{\\mathrm{X}}}\\ is not due to any starburst activity within the system. Coupling this, with the increase in \\%{\\ensuremath{L_{\\mathrm{diff}}}}, indicates that diffuse X-ray gas is being produced, leading to the creation of X-ray haloes, as observed in mature elliptical galaxies. \\citet{Osul_01} investigated the relationship between {\\ensuremath{L_{\\mathrm{X}}}}\/{\\ensuremath{L_{\\mathrm{B}}}} and spectroscopic age in post-merger ellipticals and found that there was a long term trend ($\\sim$10 Gyr) for {\\ensuremath{L_{\\mathrm{X}}}} to increase with time. The mechanism by which the regeneration of hot gas haloes in these galaxies is explained, is one in which an outflowing wind to hydrostatic halo phase is driven by a declining SNIa rate. \\citet{Osul_01} argue that a scenario in which gas, driven out during the starburst, infalls onto the\nexisting halo is not the dominant mechanism in generating X-ray haloes as this mechanism would only take $\\sim$1$-$2 Gyr and would therefore not produce the long-term trend they observe. The time baseline from the sample studied in the present paper is not sufficient to allow us to discriminate between these two possibilities.\n\n\\subsection{The Behaviour of the X-ray Point Source Population}\n\\label{sec:compare}\n\nFrom the \\emph{Chandra}\\ observations in this study, the behaviour of the\npoint source population during the merger process has been\ncharacterised for the first time. In Figure \\ref{fig:lx_lps} the total\nluminosity, as well as its two components, the luminosity arising from\nthe point source population (\\ensuremath{L_{\\mathrm{src}}}) and the diffuse gas contribution\n(\\ensuremath{L_{\\mathrm{diff}}}), have been normalised by \\ensuremath{L_{\\mathrm{K}}}\\ and plotted against star\nformation activity (\\ensuremath{L_{\\mathrm{FIR}}}\/\\ensuremath{L_{\\mathrm{K}}}).\n\n\n\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{images\/point_lfir.ps}\n \\caption{Plot indicating how \\ensuremath{L_{\\mathrm{X}}}\/\\ensuremath{L_{\\mathrm{K}}}\\ and the two components that contribute to \\ensuremath{L_{\\mathrm{X}}}\/\\ensuremath{L_{\\mathrm{K}}}\\ (the point source contribution (\\ensuremath{L_{\\mathrm{src}}}) and the diffuse gas contribution (\\ensuremath{L_{\\mathrm{diff}}})) scale with \\ensuremath{L_{\\mathrm{FIR}}}\/\\ensuremath{L_{\\mathrm{K}}}.}\n \\label{fig:lx_lps}\n\\end{figure}\n\nFrom this figure it can be seen that the total \\ensuremath{L_{\\mathrm{X}}}\/\\ensuremath{L_{\\mathrm{K}}}, whilst\nshowing a general trend of increasing with \\ensuremath{L_{\\mathrm{FIR}}}\/\\ensuremath{L_{\\mathrm{K}}}, exhibits a large\namount of scatter, peaking at the merger system Mkn 266, and then dropping at high values of \\ensuremath{L_{\\mathrm{FIR}}}\/\\ensuremath{L_{\\mathrm{K}}}. Decomposing\n\\ensuremath{L_{\\mathrm{X}}}\/\\ensuremath{L_{\\mathrm{K}}}\\ into separate source and diffuse contributions to\n\\ensuremath{L_{\\mathrm{X}}}, it can be seen that the scatter arises\nprimarily from the diffuse component, whilst the drop in\n\\ensuremath{L_{\\mathrm{X}}}\/\\ensuremath{L_{\\mathrm{K}}}\\ at high \\ensuremath{L_{\\mathrm{FIR}}}\/\\ensuremath{L_{\\mathrm{K}}}, is a consequence of both \\ensuremath{L_{\\mathrm{diff}}}\/\\ensuremath{L_{\\mathrm{K}}}\\ and\n\\ensuremath{L_{\\mathrm{src}}}\/\\ensuremath{L_{\\mathrm{K}}}\\ falling steeply. Using the Kendall Rank coefficient, it was found\nthat the point source component, \\ensuremath{L_{\\mathrm{src}}}\/\\ensuremath{L_{\\mathrm{K}}}, shows a correlation of\n2.5$\\sigma$ with \\ensuremath{L_{\\mathrm{FIR}}}\/\\ensuremath{L_{\\mathrm{K}}}. Fitting a single power law to these data gives\na logarithmic index of 0.93$\\pm$0.28.\n\nHowever, from visual inspection of Figure \\ref{fig:lx_lps}, a single\npower law trend does not represent the behaviour adequately.\nAt high values of \\ensuremath{L_{\\mathrm{FIR}}}\/\\ensuremath{L_{\\mathrm{K}}}, the very active\nsystems in this sample actually exhibit declining values of\n\\ensuremath{L_{\\mathrm{X}}}\/\\ensuremath{L_{\\mathrm{K}}}\\ as \\ensuremath{L_{\\mathrm{FIR}}}\/\\ensuremath{L_{\\mathrm{K}}}\\ increases. In section\n\\ref{sec:ulf}, we proposed the possibility that these systems close to\nnuclear coalescence have a value of \\ensuremath{L_{\\mathrm{FIR}}}\\ enhanced relative to that in\nless extreme environments, due to a change in the IMF. Under this\nhypothesis, the corresponding points in Figure \\ref{fig:lx_lps} will have\nbeen shifted strongly to the right, which could account for the\nnegative slope \nseen in all three curves at high \\ensuremath{L_{\\mathrm{FIR}}}\/\\ensuremath{L_{\\mathrm{K}}}.\nThe idea that the change in behaviour relates to \\ensuremath{L_{\\mathrm{FIR}}}\/\\ensuremath{L_{\\mathrm{K}}}, rather than\n\\ensuremath{L_{\\mathrm{X}}}\/\\ensuremath{L_{\\mathrm{K}}}, is attractive since, given the very different\nmechanisms responsible for generating the source and diffuse\ncomponents of \\ensuremath{L_{\\mathrm{X}}}\\ (X-ray binaries and supernova-heated gas\nrespectively), it is hard to envisage any process which could\nsimultaneously change the behaviour of both.\n\n\n\\section{Conclusions}\n\\label{sec:con}\n\nFrom a sample of nine interacting and merging galaxies, the evolution of X-ray emission, ranging from detached spiral pairs through to merger-remnant systems, has been investigated. As part of this survey, results from the analysis of two \\emph{Chandra}\\ observations, Mkn 266 and Arp 222, have also been presented. Here we summarise the results from these \\emph{Chandra}\\ observations and then draw the main conclusions from the survey:\n\n\\subsection{Mkn 266 and Arp 222}\n\n\\begin{itemize}\n\n\\item{The luminous merger system, Mkn 266, has been shown to contain two nuclei, with X-ray luminosities of 3.47$\\times$10$^{41}$\\ensuremath{\\erg~\\ps}\\ and 1.04$\\times$10$^{41}$\\ensuremath{\\erg~\\ps}. In addition, an area of enhanced X-ray emission has been detected between them. This is coincident with a radio source and is likely a consequence of the collision between the two discs from the progenitors. A region of diffuse emission is detected to the north of the system. It is probable that this arises from a spiral arm that has been stripped out of the system during the merger. To the south east of the nucleus, a region of extended emission has been detected, indicating that this system could be on the verge of large-scale galactic winds breaking out. This system has the highest \\ensuremath{L_{\\mathrm{X}}}\/\\ensuremath{L_{\\mathrm{K}}}\\ ratio of any of the galaxies within our sample.} \n\n\\item{ 15 discrete X-ray sources have been detected in Arp 222, two of which are classified as ULXs. This merger-remnant system has been shown to be X-ray faint when compared to both other systems within this sample and the typical elliptical galaxy NGC 2434. The diffuse gas of Arp 222 has been modelled with a temperature of 0.6 keV and, even though optical and CO observations are consistent with those of elliptical galaxies, the X-ray luminosity of Arp 222 does not resemble that of a mature elliptical.}\n\n\\end{itemize}\n\n\\subsection{The X-ray Evolution of Merging Galaxies}\n\n\\begin{itemize}\n\n\\item{The most striking result from this work is the time at which \\ensuremath{L_{\\mathrm{X}}}\/\\ensuremath{L_{\\mathrm{B}}}\\ and \\ensuremath{L_{\\mathrm{X}}}\/\\ensuremath{L_{\\mathrm{K}}}\\ peak. It was previously believed that this was coincident with nuclear coalescence, but here we find the peak $\\sim$300 Myr before this coalescence takes place. We suggest that subsequent drop in X-ray emission is a consequence of large-scale diffuse outflows breaking out of the galactic discs, reducing the hot gas density and allowing the escape of energy in kinetic form.}\n\n\\item{This study has also demonstrated that, in the systems close to the point of coalescence, \\ensuremath{L_{\\mathrm{FIR}}}\\ is massively enhanced when compared to the X-ray binary luminosity of these systems. We suggest here that the high level of \\ensuremath{L_{\\mathrm{FIR}}}\\ result from a change in the IMF in these exceptional starbursts. With the production of more massive stars compared to intermediate mass stars in these galaxies leading to larger values of \\ensuremath{L_{\\mathrm{FIR}}}\\ per unit mass of stars formed. }\n\n\\item{At a time $\\sim$1 Gyr after coalescence, the merger-remnants in our sample are X-ray faint when compared to typical X-ray luminosities of mature elliptical galaxies. However, we see evidence that these systems will start to resemble typical elliptical galaxies at a greater dynamical age, given the properties of the 3 Gyr system within our sample. This supports the idea that halo regeneration will take place within low {\\ensuremath{L_{\\mathrm{X}}}} merger-remnants. We caution that, with only one older, more relaxed, system within our sample, our conclusions on this point are necessarily tentative. To fully understand how young merger-remnants evolve into typical elliptical systems, the period in which this transformation takes place needs to be studied in greater detail.}\n\n\\end{itemize}\n\n\n\\section{Acknowledgements}\n\nWe thank the \\emph{Chandra}\\ X-ray Center (CXC) Data Systems and Science\nData Systems teams for developing the software used for the reduction (SDP) and analysis (CIAO). We would also like to thank the anonymous referee for helpful comments\nwhich improved this paper, and Steve Diehl for providing us with the X-ray information for NGC 1700 and NGC 2403. \n\nThis publication has made use of data products from the Two Micron All Sky Survey, which is a collaboration between The University of Massachusetts and the Infrared Processing and Analysis Center (JPL\/ Caltech). Funding is provided by the National Aeronautics and Space Administration and the National Science Foundation.\n\nNJB acknowledges the support of a PPARC studentship.\n\n\n\n\n\\label{lastpage}\n\n\\bibliographystyle{mn2e}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}