diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzjofk" "b/data_all_eng_slimpj/shuffled/split2/finalzzjofk" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzjofk" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nNonlinear wave phenomena occur in various systems in physics,\nengineering, biology and geosciences \\cite{Ba,CV,Ha,Te,Majda,\nPedlosky}. At the macroscopic level, wave phenomena may be\nmodeled by hyperbolic wave type partial differential equations. We\nconsider the following non-autonomous wave equations with\nnonlinear damping, on a bounded domain $\\Omega$ in $\\mathbb{R}^3$,\nwith smooth boundary $\\partial \\Omega$:\n\\begin{equation}\\label{1.1}\nu_{tt}+h(u_t)-\\Delta u+f(u,t)=g(x,\\,t)\\quad x\\in\\Omega\n\\end{equation}\nsubject to the boundary condition\n\\begin{equation}\\label{1.2}\nu |_{\\partial \\Omega}=0,\n\\end{equation}\nand the initial conditions\n\\begin{equation}\\label{1.3}\nu(x,0)=u_{0}(x),\\quad u_t(x,0)=v_0(x).\n\\end{equation}\nHere $h$ is the nonlinear damping function, $ f$ is the\nnonlinearity, $ g$ is a given external time-dependent forcing, and\n$\\Delta = \\partial_{x_1x_1}+\\partial_{x_2x_2} + \\partial_{x_3x_3} $ is the Laplace\noperator.\n\nEquation \\eqref{1.1} arises as an evolutionary mathematical model in\nvarious systems. For example, (i) modeling a continuous Josephson\njunction with specific $h, g$ and $ f $ \\cite{LSC}; (ii) modeling a\nhybrid system of nonlinear waves and nerve conduct; and (iii) when\n$h(u_t)=ku_t$ and $f(u)=|u|^ru$, the equation \\eqref{1.1} models a\nphenomenon in quantum mechanics \\cite{Cha,CY,GM,Te}.\n\nFor the autonomous case of \\eqref{1.1}, i.e., when $f$ and $g$ do\nnot depend on time $t$ explicitly, the asymptotic behaviors of the\nsolutions have been studied extensively in the framework of global\nattractors; see, for example, \\cite{ACH,BV,Ba,CV,Ha} for the\nlinear damping case, and \\cite{CL1,CL2,CL3,Fe,SYZ1} for the\nnonlinear damping case.\n\nIn this paper, we consider the non-autonomous case, especially\nwhen the damping $h$ is nonlinear and when the nonlinearity $f$\nhas critical exponent (see below). For a non-autonomous dynamical\nsystem like \\eqref{1.1}-\\eqref{1.3}, the solution map does not\ndefine a semigroup and instead, it defines a two-parameter\n\\emph{process}, or \\emph{cocycle}. Pullback attractors are\nappropriate geometric objects for describing asymptotic dynamics\nfor cocycles. We will briefly introduce basic concepts for\nnon-autonomous dynamical systems in \\S 3.\nWe will discuss the asymptotic dynamics of \\eqref{1.1}-\\eqref{1.3}\nvia pullback attractors of the corresponding cocycle. This\ndynamical framework allows us to handle more general\nnon-autonomous time-dependency; for example, the external force\n$g$ needs to be neither almost periodic nor translation compact in\ntime.\n\n\nOur basic assumptions about nonlinear damping $h$, nonlinearity\n$f$ and forcing $g$ are as follows. Let $g(x,t)$ be in the space $\nL_{loc}^2(\\mathbb{R};L^2(\\Omega))$, of locally square-integrable\nfunctions, and assume that the functions $h$ and $f$ satisfy the\nfollowing conditions:\n\\begin{equation}\\label{1.4}\nh\\in \\mathcal{C}^1(\\mathbb{R}),\\quad h(0)=0,\\quad h~\\text{\\rm\nstrictly increasing},\n\\end{equation}\n\\begin{equation}\\label{1.5}\n\\liminf_{|s|\\to \\infty}h'(s)>0,\n\\end{equation}\n\\begin{equation}\\label{1.6}\n|h(s)|\\leq C_1(1+|s|^p),\n\\end{equation}\nwhere $p \\in [1,\\, 5)$ which will be given precisely later; $f\\in\n\\mathcal{C}^1(\\mathbb{R}\\times \\mathbb{R};\\,\\mathbb{R})$ and\nsatisfies\n\\begin{equation}\\label{1.7}\nF_s(v,s)\\leqslant \\delta^2 F(v,s) +C_{\\delta},\\quad F(v,s)\\geqslant\n-mv^2-C_m,\n\\end{equation}\n\\begin{equation}\\label{1.8}\n|f_v(v,s)|\\leq C_2(1+|v|^{q}),~|f_s(v,s)|\\leq C_3(1+|v|^{q+1}),\n\\end{equation}\n\\begin{equation}\\label{1.9}\nf(v,s)v-C_4F(v,s)+mv^2\\geqslant -C_m,\\quad \\forall ~(v,s)\\in\n\\mathbb{R}\\times \\mathbb{R},\n\\end{equation}\nwhere $0\\leqslant q\\leqslant 2$, $F(v,s)=\\int_0^vf(w,s)dw$ and\n$\\delta$, $m$ are sufficiently small which will be determined in\n$Lemma$ \\ref{l5.3}. The number $q=2$ is called the \\emph{critical\nexponent}, since the nonlinearity $f$ is not compact in this case\n(i.e., for a bounded subset $B\\subset H_0^1(\\Omega)$, in general,\n$f(B)$ is not precompact in $L^2(\\Omega)$). This is an essential\ndifficulty in studying the asymptotic behavior even for the\nautonomous cases \\cite{ACH,BV,Ba,CL1,CL2,CL3,Fe,SYZ1}. The\nassumptions \\eqref{1.4}-{\\eqref{1.6} on $h$ are similar to those in\n\\cite{CL3,Fe,Kh,SYZ1} for the autonomous cases, while the assumption\n$1 \\leq p<5$ is due to the need for estimating\n$\\int_{\\Omega}g(u_t)u$ by $\\int_{\\Omega}g(u_t)u_t$ and\n$\\int_{\\Omega}|\\nabla u|^2$ via Sobolev embedding. Finally, the\nassumptions \\eqref{1.7}-\\eqref{1.9} are similar to the conditions\nused in Chepyzhov \\& Vishik \\cite{CV} for non-autonomous cases\n but with linear damping.\n\n\\medskip\n\nLet us recall some recent relevant research in this area.\n\nThe existence of pullback attractors are established for the\nstrongly dissipative non-autonomous dynamical systems such as those\ngenerated by parabolic type partial differential equations, e.g.,\nthe non-autonomous 2D Navier-Stokes equation and some non-autonomous\nreaction diffusion equations; see\n\\cite{ARS,CLR,CLR1,CR,Ch,CD,CKS,LLR} and the references therein.\nHowever, the situation for the hyperbolic wave type systems is less\nclear. For the linear damping case $h(v)=kv$ with a constant $k>0$\nand $q<2$ (subcritical), Chepyzhov \\& Vishik \\cite{CV} have obtained\nthe existence of a uniform absorbing set when $g$ is translation\nbounded in time (i.e., $g\\in L^2_b(\\mathbb{R};L^2(\\Omega))$), and\nthe existence of a uniform attractor when $g$ is translation compact\nin time (i.e., $g\\in L^2_c(\\mathbb{R};L^2(\\Omega))$).\n\nUnder the assumptions that $g$ and $\\partial_t g$ are both in the space\nof bounded continuous functions\n$\\mathcal{C}_b(\\mathbb{R},L^2(\\Omega ))$, $h$ has bounded positive\nderivative, and furthermore, $f$ is of critical growth (i.e.,\n$q=2$), Zhou \\& Wang \\cite{ZW} have proved the existence of kernel\nsections and obtained uniform bounds of the Hausdorff dimension of\nthe kernel sections. Caraballo et al. \\cite{CKR} have discussed\nthe pullback attractors for the cases of linear damping and\nsubcritical nonlinearity ($q<2$) .\n\n\n As in the autonomous case, some kind of compactness of the\ncocycle is a key ingredient for the existence of pullback\nattractors of cocycles. The corresponding compactness assumption\nin Cheban \\cite{Ch} is that the cocycle has a compact attracting\nset. Recently, Caraballo et al \\cite{CLR} have established a\ncriterion for the existence of pullback attractors via pullback\nasymptotic compactness, and illustrated their results with the 2D\nNavier-Stokes equation.\n\nFor the autonomous \\emph{linearly} damped wave equations, Ball\n\\cite{Ba} proposed a method to verify the asymptotic\ncompactness for the corresponding solution semigroup. This\nso-called energy method has been generalized \\cite{LS, MRW}\n to some non-autonomous cases. However, for our problem, due\nto the nonlinear damping, it appears difficult to apply the\nmethod of Ball \\cite{Ba}. Moreover, a decomposition technique\n\\cite{ACH,CV,Fe,Ha,PZ,SY} has been successfully applied to\nverify the asymptotic smoothness of the corresponding solution\nsemigroup for autonomous wave equations.\n\n\nIn this paper, after some preliminaries, we first introduce the\n\\emph{pullback $\\kappa-$contraction} concept, a generalization of\n$\\kappa-$contraction from autonomous systems to non-autonomous\nsystems. Then we establish a criterion for the existence of\npullback attractors, in terms of pullback $\\kappa-$contraction or\npullback asymptotic compactness. This criterion is for a class of\n``weakly\" continuous cocycles (i.e., the so-called norm-to-weak\ncontinuous cocycles; see \\S 3 below). Thirdly, we show that the\npullback $\\kappa-$contraction is not equivalent to the pullback\nasymptotic compactness, unless the cocycle mapping has a nested\nbounded pullback absorbing set (see $Definition$ \\ref{d3.2}\nbelow). This fact is different from the autonomous semigroup\ncases. Moreover, we propose a technique for verifying pullback\nasymptotic compactness. Finally, we apply these results to show\nthe existence of pullback attractors for the non-autonomous\nhyperbolic wave system \\eqref{1.1}-\\eqref{1.3}.\n\n\n\nDue to the difference between the cases $p=1$ and $10|\\text{ A has a finite open cover of sets of\ndiameter} < \\delta\\}.\n\\]\n\\end{definition}\n\nIf $A$ is a nonempty, unbounded set in $X$, then we define\n$\\kappa(A)=\\infty$.\n\nThe properties of $\\kappa(A)$, which we will use in this paper, are\ngiven in the following lemmas:\n\\begin{lemma}(\\cite{Ha,SY}) \\label{l2.2}\nThe Kuratowski measure of non-compactness $\\kappa(A)$ on a\ncomplete metric space $X$ satisfies the following properties:\n\\begin{enumerate}\n\\item[(1)]~$\\kappa(A)=0$ if and only if $\\bar{A}$ is compact,\nwhere $\\bar{A}$ is the closure of $A$;\n\n\\item[(2)]~$\\kappa(\\bar{A})=\\kappa(A)$, $\\kappa(A\\cup B)=\\max\\{\\kappa(A),\\,\\kappa(B)\\}$;\n\n\\item[(3)]~If $A\\subset B$, then $\\kappa(A)\\leqslant \\kappa(B)$;\n\n\\item[(4)]~If $A_t$ is a family of nonempty, closed, bounded sets\ndefined for $t>r$ that satisfy $A_t \\subset A_s$, whenever $s \\leq\nt$, and $\\kappa(A_t)\\to 0$, as $t \\to \\infty$, then\n$\\underset{t>r}{\\cap}A_t$ is a nonempty, compact set in $X$.\n\\end{enumerate}\nIf in addition, $X$ is a Banach space, then the following estimate\nis valid:\n\\begin{enumerate}\n\\item[(5)]~$\\kappa(A+B)\\leqslant \\kappa(A)+\\kappa(B)$\\quad for any\nbounded sets $A,B$ in $ X$.\n\\end{enumerate}\n\\end{lemma}\n\n\\subsection{Some useful properties for nonlinear damping function}\n\nIn the following, we will recall some simple properties of the\nnonlinear damping function $h$, which will be used later.\n\n\\begin{lemma}(\\cite{Fe,Kh})\\label{l2.3}\nLet $h$ satisfy \\eqref{1.4} and \\eqref{1.5}. Then for any $\\delta\n>0$, there exists a constant $C_{\\delta}$ depending on $\\delta$\nsuch that\n\\[\n|u-v|^2\\leqslant \\delta + C_{\\delta}(h(u)-h(v))(u-v)\\quad \\text{for\nall}~u,v\\in \\mathbb{R}.\n\\]\n\\end{lemma}\n\nMoreover, condition \\eqref{1.6} implies that\n\\[\n|h(s)|^{\\frac{1}{p}}\\leqslant C(1+|s|).\n\\]\nTherefore, we have\n\\[\n|h(s)|^{\\frac{p+1}{p}}=|h(s)|^{\\frac{1}{p}}\\cdot |h(s)|\\leqslant\nC(1+|s|)|h(s)|\\leqslant C|h(s)|+Ch(s)\\cdot s.\n\\]\nCombining this estimate with the Young's inequality and\n\\eqref{1.4}, we further obtain that\n\\begin{equation}\\label{2.1}\n|h(s)|^{\\frac{p+1}{p}}\\leqslant C(1+h(s)\\cdot s)\\quad \\text{for\nall}~s\\in \\mathbb{R},\n\\end{equation}\nwhere the constant $C$ is independent of $s$.\n\n\n\n\n\n\n\n\\section{Criterion for the existence of pullback attractors}\n\n\nIn this section, we first recall a few basic concepts for\nnon-autonomous dynamical systems, including pullback\n$\\kappa-$contraction, pullback asymptotic compactness and pullback\nattractor. Then we present criteria for existence of pullback\nattractors, in terms of $\\kappa-$contraction or pullback\nasymptotic compactness.\n\n Let $X$ be a complete metric space, which is the state space for\n a non-autonomous dynamical system (NDS).\n As in \\cite{BCD,Ch,CKS}, we define a\nnon-autonomous dynamical system in terms of a cocycle mapping\n$\\phi$: $\\mathbb{R}^+\\times \\Sigma\\times X \\to X$ which is driven\nby an \\emph{autonomous} dynamical system $\\theta$ acting on a\nparameter space $\\Sigma$. In details, $\\theta=\\{\\theta_t\\}_{t\\in\n\\mathbb{R}}$ is a autonomous dynamical system on $\\Sigma$, i.e., a\ngroup of homeomorphisms under composition on $\\Sigma$ with the\nproperties\n\\begin{enumerate}\n\\item[(i)] $\\theta_0(\\sigma) = \\sigma$ for all $\\sigma \\in \\Sigma$;\n\n\\item[(ii)] $\\theta_{t+\\tau}(\\sigma) = \\theta_t(\\theta_{\\tau}(\\sigma))$ for\nall $t,\\tau\\in \\mathbb{R}$.\n\\end{enumerate}\nThe cocycle mapping $\\phi$ satisfies\n\\begin{enumerate}\n\\item[(i)] $\\phi(0, \\sigma; x) = x$ for all $(\\sigma,\\, x)\\in \\Sigma \\times X$;\n\n\\item[(ii)] $\\phi(s + t,\\, \\sigma; x) = \\phi(s, \\theta_t(\\sigma); \\phi(t, \\sigma;\nx))$ for all $s, t\\in \\mathbb{R}^+$ and all $(\\sigma, x)\\in \\Sigma\n\\times X$.\n\\end{enumerate}\n\nSometimes we say $\\phi$ is a cocycle with respect to (w.r.t.)\n$\\theta$ and denote this by $(\\phi, \\theta)$.\n If,\nin addition, the mapping $\\phi(t, \\sigma; \\cdot):~X\\to X$ is\ncontinuous for each $\\sigma \\in \\Sigma$ and $t\\geqslant 0$, then\nwe call $\\phi$ is a {\\it continuous cocycle}. If the mapping\n$\\phi(t, \\sigma; \\cdot):~X\\to X$ is \\emph{norm-to-weak continuous}\nfor each $\\sigma \\in \\Sigma$ and $t\\geqslant 0$, that is, for each\n$\\sigma \\in \\Sigma$ and $t\\geqslant 0$, norm convergence $x_n \\to\nx$ in $X$ implies weak convergence $\\phi(t, \\sigma;\nx_n)\\rightharpoonup \\phi(t, \\sigma; x)$, then we call $\\phi$ is a\n{\\it norm-to-weak continuous cocycle}. A continuous cocycle is\nobviously also a norm-to-weak continuous cocycle.\n\nFor convenience, hereafter, we will use the following notations:\n\\[\n\\mathcal{B}\\overset{\\vartriangle}{=}\\{B~|~B~\\text{is bounded\nin}~X\\}; \\quad\n\\phi(t,\\,\\sigma;\\,B)\\overset{\\vartriangle}{=}\\{\\phi(t,\\,\\sigma;\\,x_0)~\n|~x_0 \\in B\\}.\n\\]\n\n\\begin{definition}\\label{d3.1} (\\cite{Ch})\nA family of bounded sets $\\mathscr{B} = \\{B_\\sigma\\}_{\\sigma\\in\n\\Sigma}$ of $X$ is called a bounded pullback absorbing set for the\ncocycle $\\phi$ with respect to (w.r.t.) $\\theta$, if for any\n$\\sigma \\in \\Sigma$ and any $B\\in \\mathcal{B}$, there exists\n$T=T(\\sigma, B)\\geqslant 0$ such that\n\\[\n\\phi(t,\\, \\theta_{-t}(\\sigma); B)\\subset B_{\\sigma}\\quad \\text{for\nall}~t\\geqslant T.\n\\]\n\\end{definition}\n\n\n\n\\begin{definition}\\label{d3.4}(\\cite{Ch})\n(\\textbf{Pullback attractor})\\\\\nA family of nonempty compact sets $\\mathscr{A} =\n\\{\\mathcal{A}_\\sigma\\}_{\\sigma\\in \\Sigma}$ of $X$ is called a\npullback attractor for the cocycle $\\phi$ w.r.t. $\\theta$, if for\nall $\\sigma \\in \\Sigma$, it satisfies\n\\begin{enumerate}\n\\item[(i)] $\\phi(t, \\sigma; \\mathcal{A}_{\\sigma}) = \\mathcal{A}_{\\theta_t(\\sigma)}$ for all $t \\in\n\\mathbb{R}^+$ \\quad ($\\phi-$invariance);\n\n\\item[(ii)] $\\lim\\limits_{t\\to +\\infty} dist_X(\\phi(t;\n\\theta_{-t}(\\sigma); B),\\,\\mathcal{A}_{\\sigma}) = 0$ for all bounded\nset $B\\subset X$.\n\\end{enumerate}\nOften, $\\mathcal{A}_\\sigma$ is called a fiber at parameter $\\sigma\n\\in \\Sigma$.\n\\end{definition}\n\n\\begin{definition}\\label{d3.5}(\\cite{Ch})\nLet $\\phi$ be a cocycle w.r.t. $\\theta$ on $\\mathbb{R}^+\\times\n\\Sigma \\times X$, and let $B\\in \\mathcal{B}$. We define the\npullback $\\omega$-limit set $\\omega_{\\sigma}(B)$ as follows\n\\[\n\\omega_{\\sigma}(B)=\\bigcap_{s\\geqslant\n0}\\overline{\\bigcup_{t\\geqslant s}\\phi(t, \\theta_{-t}(\\sigma);\nB)}, \\;\\;\\;\\; \\sigma \\in \\Sigma ,\n\\]\nwhere $\\overline{A}$ means the closure of $A$ in $X$.\n\\end{definition}\n\nIf the parameter space $\\Sigma$ contains only one element\n$\\sigma_0$ and $\\theta_t(\\sigma_0)\\equiv\\sigma_0$ for all $t\\in\n\\mathbb{R}$, then $\\varphi$ reduces to a semigroup and all the\nconcepts in $Definitions$ \\ref{d3.1}-\\ref{d3.5} coincide with the\ncorresponding concepts in autonomous systems. Especially, in the\nautonomous case, the pullback attractor coincides with the global\nattractor; see \\cite{BV,Robinson,SY,Te, Ilyin}. Moreover,\nChepyzhov \\& Vishik \\cite{CV} define the concept of kernel\nsections for non-autonomous dynamical systems, which correspond to\nthe fibers $\\mathcal{A}_{\\sigma}$ in the above Definition\n\\ref{d3.4} of a pullback attractor. Furthermore, similar to the\nautonomous cases, we have also the following equivalent\ncharacterization about the pullback $\\omega$-limit set.\n\n\n\\begin{lemma}\\label{l3.6} (\\cite{Ch})\nFor any $B\\subset \\mathcal{B}$ and any $\\sigma\\in \\Sigma$, $x_0\\in\n\\omega_{\\sigma}(B)$ if and only if there exist $\\{x_n\\}\\subset B$\nand $\\{t_n\\}\\subset \\mathbb{R}^+$ with $t_n \\to +\\infty$ as $n\\to\n\\infty$, such that\n\\[\n\\phi(t_n, \\theta_{-t_n}(\\sigma); x_n) \\to x_0\\quad \\text{as}~n\\to\n\\infty.\n\\]\n\\end{lemma}\n\n\nNow we define the pullback $\\kappa$-contracting cocycle in terms\n of the Kuratowski non-compactness measure:\n\\begin{definition}\\label{d3.7} (\\textbf{$\\kappa$-contracting cocycle})\\\\\nLet $\\phi$ be a cocycle w.r.t. $\\theta$ on $\\mathbb{R}^+\\times\n\\Sigma \\times X$. Then $\\phi$ is called pullback\n$\\kappa$-contracting if for any $\\varepsilon>0$, $\\sigma\\in\n\\Sigma$ and any $B\\in \\mathcal{B}$, there is a $T=T(\\varepsilon,\n\\sigma, B)\\geqslant 0$ such that\n\\[\n\\kappa_{_X}\\left(\\phi(t, \\theta_{-t}(\\sigma); B)\\right) \\leqslant\n\\varepsilon \\quad \\text{for all}~t\\geqslant T.\n\\]\n\\end{definition}\n\nFrom the definitions above, we have the following basic fact.\n\\begin{lemma}\\label{c3.8}\nLet $\\phi$ be a cocycle w.r.t. $\\theta$ on $\\mathbb{R}^+\\times\n\\Sigma \\times X$. If $\\phi$ has a pullback attractor, then $\\phi$\nhas a bounded pullback absorbing set and $\\phi$ is pullback\n$\\kappa$-contracting.\n\\end{lemma}\n\nWe introduce another definition, needed for characterizations of\nexistence of pullback attractors later.\n\\begin{definition}\\label{d3.2}\n(\\textbf{Nested pullback absorbing set})\\\\\nA family of bounded sets $\\mathscr{B} = \\{B_\\sigma\\}_{\\sigma\\in\n\\Sigma}$ of $X$ is called a nested bounded pullback absorbing set\nfor $\\phi$ w.r.t. $\\theta$ if $\\mathscr{B}$ is a bounded pullback\nabsorbing set, and, moreover, $B_{\\sigma}$ satisfy the nested\nrelation: $B_{\\theta_{-t}(\\sigma)}\\subset B_{\\sigma}$ for any\n$t\\geqslant 0$ and any $\\sigma \\in \\Sigma$.\n\\end{definition}\n\n\\begin{remark}\nThis nested relation appears in some systems arising in physical\napplications.\n For example, the non-autonomous systems considered in \\cite{CKR,Ch,\nCV} have nested bounded pullback absorbing sets.\n\n\\end{remark}\n\n\nIn the following, we will present some characterizations for the\npullback $\\kappa$-contracting cocycles.\n\\begin{lemma}\\label{l3.9}\nLet $\\phi$ be a $\\kappa$-contracting cocycle w.r.t. $\\theta$ on\n$\\mathbb{R}^+\\times \\Sigma \\times X$ and have a nested bounded\npullback absorbing set $\\mathscr{B}=\\{B_{\\sigma}\\}_{\\sigma \\in\n\\Sigma}$. Then for every $\\sigma\\in \\Sigma$, every bounded\nsequence $\\{x_n\\}_{n=1}^{\\infty}\\subset X$ and every time sequence\n$\\{t_n\\}\\subset \\mathbb{R}^+$ with $t_n\\to +\\infty$ as $n \\to\n\\infty$, we have\n\\begin{enumerate}\n\\item[(i)] $\\{\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\,\nx_n)\\}_{n=1}^{\\infty}$ is pre-compact in $X$;\n\n\\item[(ii)] all clusters of $\\{\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\,\nx_n)\\}_{n=1}^{\\infty}$ are contained in\n$\\omega_{\\sigma}(B_{\\sigma})$, that is, if\n\\begin{equation*}\n\\phi(t_{n_j},\\,\\theta_{-t_{n_j}}(\\sigma);\\, x_{n_j}) \\to x_0 \\quad\n\\text{as}~j \\to \\infty,\n\\end{equation*}\nthen $x_0\\in \\omega_{\\sigma}(B_{\\sigma})$;\n\n\\item[(iii)] $\\omega_{\\sigma}(B_{\\sigma})$ is nonempty and compact in $X$.\n\\end{enumerate}\n\\end{lemma}\n\n\n{\\bf Proof.} $(i).$ Denote $\\{x_n\\}_{n=1}^{\\infty}$ by $B$. For\nany $\\varepsilon>0$ and for each $\\sigma \\in \\Sigma$, by the\ndefinition of pullback $\\kappa$-contracting cocycle, we know that\nthere exists a $T_0=T_0(\\varepsilon,\\,\\sigma,B_{\\sigma}) >0$ such\nthat\n\\begin{equation}\\label{3.1}\n\\kappa_{_X}\\left(\\phi(t, \\theta_{-t}(\\sigma); B_{\\sigma})\\right)\n\\leqslant \\varepsilon \\quad \\text{for all}~t\\geqslant T_0\n\\end{equation}\nand there exists also a $T_1=T_1(\\varepsilon, \\, \\sigma,B)$ such\nthat\n\\begin{equation}\\label{3.2}\n\\phi(t+T_1,\\,\\theta_{-(t+T_1)}(\\theta_{-T_0}(\\sigma));\\, B) \\subset\nB_{\\theta_{-T_0}(\\sigma)} \\subset B_{\\sigma}\\quad \\text{for all}~\nt\\geqslant 0.\n\\end{equation}\nHence, for any $t\\geqslant 0$, we have\n\\begin{align}\\label{3.3}\n\\phi(t+T_1+T_0,\\,& \\theta_{-(t+T_1+T_0)}(\\sigma);\\, B)\n\\nonumber \\\\\n& = \\phi(T_0,\\,\\theta_{-T_0}(\\sigma);\\,\n\\phi(t+T_1,\\,\\theta_{-(t+T_1)}(\\theta_{-T_0}(\\sigma)); B))\n\\nonumber \\\\\n& \\subset \\phi(T_0,\\,\\theta_{-T_0}(\\sigma);\\,\nB_{\\theta_{-T_0}(\\sigma)}) \\nonumber \\\\\n& \\subset \\phi(T_0,\\,\\theta_{-T_0}(\\sigma);\\, B_{\\sigma}),\n\\end{align}\nand then\n\\begin{equation}\\label{3.4}\n\\bigcup_{t\\geqslant T_0+T_1}\\phi(t, \\theta_{-t}(\\sigma);\\, B)\n\\subset \\phi(T_0,\\,\\theta_{-T_0}(\\sigma);\\, B_{\\sigma}).\n\\end{equation}\n\nTherefore, combining \\eqref{3.1} and \\eqref{3.4}, we have\n\\begin{equation}\\label{3.5}\n\\kappa_{_X}\\left(\\bigcup_{t\\geqslant T_0+T_1}\\phi(t,\n\\theta_{-t}(\\sigma);\\, B)\\right) \\leqslant \\varepsilon.\n\\end{equation}\n\nThen by the properties $(1),(2)$ of $Lemma$ \\ref{l2.2} and\n$\\{\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\, x_n)\\}_{n=n_0}^{\\infty}\n\\subset \\bigcup_{t\\geqslant T_0+T_1}\\phi(t, \\theta_{-t}(\\sigma);\\,\nB)$ for some $n_0$, we know that\n$\\kappa_X(\\{\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\,\nx_n)\\}_{n=1}^{\\infty}) \\leqslant \\varepsilon$. Hence by the\narbitrariness of $\\varepsilon$ and property $(1)$ of $Lemma$\n\\ref{l2.2}, we conclude that\n$\\{\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\, x_n)\\}_{n=1}^{\\infty}$ is\npre-compact in $X$.\n\n$(ii).$ Let $x_0$ be a cluster of\n$\\{\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\, x_n)\\}_{n=1}^{\\infty}$, we\nneed to show that $x_0\\in \\omega_{\\sigma}(B_{\\sigma})$. Without\nloss of generality, we assume that\n$\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\, x_n) \\to x_0$ as $n\\to\n\\infty$.\n\nWe claim first that for each sequence $\\{s_m\\}_{m=1}^{\\infty}\n\\subset \\mathbb{R}^+$ satisfying $s_m \\to \\infty$ as $m\\to \\infty$,\nwe can find two sequences $\\{t_{n_m}\\}_{m=1}^{\\infty}\\subset\n\\{t_{n}\\}_{n=1}^{\\infty}$ and $\\{y_m\\}_{m=1}^{\\infty}\\subset\nB_{\\sigma}$ satisfying $t_{n_m} \\to \\infty$ as $m\\to \\infty$, such\nthat\n\\begin{equation}\\label{3.6}\n\\phi(s_m,\\,\\theta_{-s_m}(\\sigma);\\,y_m)=\\phi(t_{n_m},\\,\\theta_{-t_{n_m}}(\\sigma);\\,x_{n_m}).\n\\end{equation}\nIndeed, for each $m\\in \\mathbb{N}$, we can take $n_m$ so large that\n$t_{n_m} \\geqslant s_m$ and\n\\[\ny_m\\overset{\\vartriangle}{=} \\phi(t_{n_m}-s_m,\\,\n\\theta_{-(t_{n_m}-s_m)} (\\theta_{-s_m}(\\sigma)); \\,x_{n_m}) \\in\nB_{\\theta_{-s_m}(\\sigma)} \\subset B_{\\sigma}.\n\\]\nTherefore,\n\\begin{align}\\label{3.7}\n\\phi(t_{n_m},\\,& \\theta_{-t_{n_m}}(\\sigma);\\, x_{n_m}) \\nonumber \\\\\n& = \\phi(s_m+(t_{n_m}-s_m),\\,\n\\theta_{-(s_m+(t_{n_m}-s_m))}(\\sigma);\\, x_{n_m})\n\\nonumber \\\\\n& = \\phi(s_m,\\,\\theta_{-s_m}(\\sigma);\\,\\phi(t_{n_m}-s_m,\\,\n\\theta_{-(t_{n_m}-s_m)}(\\theta_{-s_m}(\\sigma));\\,x_{n_m})) \\nonumber \\\\\n& = \\phi(s_m,\\,\\theta_{-s_m}(\\sigma);\\,y_m).\n\\end{align}\n\nHence,\n\\[\n\\lim\\limits_{m\\to \\infty} \\phi(s_m,\\,\\theta_{-s_m}(\\sigma);\\,y_m) =\n\\lim\\limits_{m\\to \\infty}\\phi(t_{n_m},\\,\\theta_{-t_{n_m}}\n(\\sigma);\\,x_{n_m}) =x_0,\n\\]\nand $y_m\\in B_{\\sigma}$ for each $m\\in \\mathbb{N}$, which implies,\nby the definition of $\\omega_{\\sigma}(B_{\\sigma})$, that $x_0 \\in\n\\omega_{\\sigma}(B_{\\sigma})$.\n\n$(iii).$ The fact that $\\omega_{\\sigma}(B_{\\sigma})$ is\nnonempty is obvious. Substitute $B$ by $B_{\\sigma}$ in\n\\eqref{3.2}-\\eqref{3.5}, we obtain that there exists a\n$T_2=T_2(\\varepsilon,B_{\\sigma},\\sigma)$ such that\n\\begin{equation*}\n\\kappa_{_X}\\left(\\overline{\\bigcup_{t\\geqslant T_0+T_2}\\phi(t,\n\\theta_{-t}(\\sigma);\\, B_{\\sigma})}\\right) =\n\\kappa_{_X}\\left(\\bigcup_{t\\geqslant T_0+T_2}\\phi(t,\n\\theta_{-t}(\\sigma);\\, B_{\\sigma})\\right) \\leqslant \\varepsilon.\n\\end{equation*}\nThen by the definition of pullback $\\omega$-limit set and property\n$(4)$ of $Lemma$ \\ref{l2.2}, we know that\n$\\omega_{\\sigma}(B_{\\sigma})$ is compact in $X$. $\\hfill\n\\blacksquare$\n\nA criterion for the existence of pullback attractors is then\nobtained by means of $\\kappa$-contraction.\n\n\\begin{theorem}\\label{t3.10}\n(\\textbf{Sufficient condition for existence of pullback attractors})\\\\\n Let\n$\\phi$ be a continuous cocycle w.r.t. $\\theta$ on\n$\\mathbb{R}^+\\times \\Sigma \\times X$. Then $(\\phi, \\theta)$ has a\npullback attractor provided that\n\\begin{enumerate}\n\\item[(i)]\\ $(\\phi, \\theta)$ has a nested bounded pullback absorbing set $\\mathscr{B}=\\{B_{\\sigma}\\}_{\\sigma \\in \\Sigma}$;\n\n\\item[(ii)]\\ $(\\phi, \\theta)$ is pullback $\\kappa$-contracting.\n\\end{enumerate}\n\\end{theorem}\n{\\bf Proof.} For any $\\sigma \\in \\Sigma$, we consider a family of\n $\\omega$-limit sets $\\mathscr{B}=\\{B_{\\sigma}\\}_{\\sigma\\in\n\\Sigma}$:\n\\begin{equation*}\n\\omega_{\\sigma}(B_{\\sigma})=\\bigcap_{s\\geq\n0}\\overline{\\bigcup_{t\\geq s}\\phi(t, \\theta_{-t}(\\sigma);\nB_{\\sigma})}, \\quad \\sigma \\in \\Sigma.\n\\end{equation*}\n\nBy $Lemma$ \\ref{l3.9} we know that $\\omega_{\\sigma}(B_{\\sigma})$ is\nnonempty and compact in $X$ for each $\\sigma \\in \\Sigma$.\n\nIn the following, we will prove that\n$\\mathscr{A}=\\{\\omega_{\\sigma}(B_{\\sigma})\\}_{\\sigma \\in \\Sigma}$\nis a pullback attractor of $(\\phi,\\,\\theta)$, which will be\naccomplished in two steps.\n\n{\\it Claim 1. For each $\\sigma\\in \\Sigma$ and any $B\\in\n\\mathcal{B}$, we have\n\\begin{equation*}\n\\lim\\limits_{t\\to +\\infty} dist_X(\\phi(t,\\, \\theta_{-t}(\\sigma);\nB),\\,\\omega_{\\sigma}(B_{\\sigma})) = 0.\n\\end{equation*}\n}\n\nIn fact, if {\\it Claim 1} is not true, then there exist\n$\\varepsilon_0>0$, $\\{x_n\\}_{n=1}^{\\infty}\\subset B$ and $\\{t_n\\}$\nwith $t_n\\to +\\infty$ as $n\\to \\infty$, such that\n\\begin{equation}\\label{3.8}\ndist_X(\\phi(t_n,\\, \\theta_{-t_n}(\\sigma);\\,\nx_n),\\,\\omega_{\\sigma}(B_{\\sigma})) \\geqslant \\varepsilon_0 \\quad\n\\text{for}~n=1,2,\\cdots.\n\\end{equation}\n\nHowever, thanks to $Lemma$ \\ref{l3.9}, we know that\n$\\{\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\,x_n)\\}_{n=1}^{\\infty}$ is\npre-compact in $X$. Without loss of generality, we assume that\n\\begin{equation}\\label{3.9}\n\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\,x_n) \\to x_0 \\quad \\text{as}~n\n\\to \\infty.\n\\end{equation}\nThen $x_0 \\in \\omega_{\\sigma}(B_{\\sigma})$, which is a contraction\nwith \\eqref{3.8}. This complete the proof of {\\it Claim 1}.\n\n{\\it Claim 2. $\\mathscr{A}=\\{\\omega_{\\sigma}(B_{\\sigma})\\}_{\\sigma\n\\in \\Sigma}$ is $\\phi$ invariant, that is,\n\\begin{equation*}\n\\phi(t,\\,\\sigma;\\, \\omega_{\\sigma}(B_{\\sigma}))=\n\\omega_{\\theta_t(\\sigma)}(B_{\\theta_t(\\sigma)})\\quad \\text{for\nall}~t\\geqslant 0,~\\sigma\\in \\Sigma.\n\\end{equation*}\n}\n\nWe first take $x \\in \\phi(t,\\,\\sigma;\\,\n\\omega_{\\sigma}(B_{\\sigma}))$.\n\nThen there is a $y\\in \\omega_{\\sigma}(B_{\\sigma})$ such that\n$x=\\phi(t,\\,\\sigma;\\,y)$, and by the definition of $y$, there exist\n$\\{y_n\\}\\subset B_{\\sigma} \\subset B_{\\theta_t(\\sigma)}$ and $t_n$\nwith $t_n \\to \\infty$ as $n \\to \\infty$ such that\n$y=\\lim\\limits_{n\\to\n\\infty}\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\,y_n)$.\n\nTherefore, by the continuity of $\\phi$, as $n \\to \\infty$,\n\\begin{equation}\\label{3.10}\n\\phi(t_n+t,\\,\\theta_{-(t_n+t)}(\\theta_t(\\sigma));\\,y_n) =\n\\phi(t,\\,\\sigma;\\,\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\,y_n)) \\to\n\\phi(t,\\,\\sigma;\\,y)=x.\n\\end{equation}\n\nOn the other hand, from $Lemma$ \\ref{l3.9}, we know that\n$\\{\\phi(t_n+t,\\,\\theta_{-(t_n+t)}(\\theta_t(\\sigma));\\,y_n)\\}_{n=1}^{\\infty}$\nis pre-compact in $X$. Without loss of generality, we assume that\n\\begin{equation*}\n\\phi(t_n+t,\\,\\theta_{-(t_n+t)}(\\theta_t(\\sigma));\\,y_n) \\to x_0\n\\in \\omega_{\\theta_t(\\sigma)}(B_{\\theta_t(\\sigma)}) \\quad\n\\text{as} ~n\\to \\infty.\n\\end{equation*}\nThen by the uniqueness of limitation, we have $x=x_0$, which\nimplies that $x\\in\n\\omega_{\\theta_t(\\sigma)}(B_{\\theta_t(\\sigma)})$, and thus\n\\begin{equation}\\label{3.11}\n\\phi(t,\\,\\sigma;\\,\\omega_{\\sigma}(B_{\\sigma})) \\subset\n\\omega_{\\theta_t(\\sigma)}(B_{\\theta_t(\\sigma)}).\n\\end{equation}\nNow, we only need to prove the converse inclusion relation.\n\nLet $z\\in \\omega_{\\theta_t(\\sigma)}(B_{\\theta_t(\\sigma)})$. Then\nthere exist $\\{z_n\\}\\subset B_{\\theta_t(\\sigma)}$ and $t_n$ with\n$t_n \\to \\infty$ as $n \\to \\infty$ such that $z=\\lim\\limits_{n\\to\n\\infty}\\phi(t_n,\\,\\theta_{-t_n}(\\theta_t(\\sigma));\\,z_n)$.\n\nSince $\\{z_n\\}\\subset B_{\\theta_t(\\sigma)}$ is bounded, from\n$Lemma$ \\ref{l3.9}, we know that\n$\\{\\phi(t_n-t,\\,\\theta_{-(t_n-t)}(\\sigma);\\,z_n)\\}_{n=1}^{\\infty}$\nis pre-compact in $X$. Without loss of generality, we assume that\n$\\phi(t_n-t,\\,\\theta_{-(t_n-t)}(\\sigma);\\,z_n) \\to x_0 \\in\n\\omega_{\\sigma}(B_{\\sigma})$ as $n \\to \\infty$. Then by the\ncontinuity of $\\phi$, we have\n\\begin{align}\\label{3.12}\n\\phi(t,\\,\\sigma;\\,x_0)\\leftarrow\n\\phi(t,&\\,\\sigma;\\,\\phi(t_n-t,\\,\\theta_{-(t_n-t)}(\\sigma);\\,z_n))\n\\nonumber \\\\\n& =\\phi(t_n, \\, \\theta_{-(t_n-t)}(\\sigma);\\,z_n) \\nonumber \\\\\n& = \\phi(t_n, \\, \\theta_{-(t_n)}(\\theta_t(\\sigma));\\,z_n) \\to z.\n\\end{align}\nHence, $z= \\phi(t,\\,\\sigma;\\,x_0)$ with $x_0\\in\n\\omega_{\\sigma}(B_{\\sigma})$, which implies\n\\begin{equation}\\label{3.13}\n\\omega_{\\theta_t(\\sigma)}(B_{\\theta_t(\\sigma)}) \\subset\n\\phi(t,\\,\\sigma;\\,\\omega_{\\sigma}(B_{\\sigma})).\n\\end{equation}\n\nCombining \\eqref{3.11} and \\eqref{3.13} we know that {\\it Claim 2}\nis true.\n\nFrom {\\it Claim 1} and {\\it Claim 2}, we complete the proof of\n$Theorem$ \\ref{t3.10}. $\\hfill \\blacksquare$\n\n\\begin{remark}\\label{r3.11}\nIn the proof of $Theorem$ \\ref{t3.10}, the continuity of the cocycle\n$\\phi(t,\\,\\sigma;\\,\\cdot):\\, X\\to X$ can be replaced by the\n``weaker\" continuity; see \\eqref{3.10} and \\eqref{3.12}. That is,\nthe above proof holds for norm-to-weak continuous cocycles; see\n\\cite{ZYS} for autonomous cases.\n\\end{remark}\n\n\n\nSimilar to the definition in Caraballo et al \\cite{CLR}, we define\nthe following {\\it pullback asymptotic compactness} for NDS.\n\\begin{definition}\\label{d3.12}\nLet $\\phi$ be a cocycle w.r.t. $\\theta$ on $\\mathbb{R}^+\\times\n\\Sigma \\times X$. Then $\\phi$ is called pullback asymptotically\ncompact, if for each $\\sigma\\in \\Sigma$, every bounded\nsequence $\\{x_n\\}_{n=1}^{\\infty}$, and every time sequence\n$\\{t_n\\}\\subset \\mathbb{R}^+$ with $t_n\\to +\\infty$ as $n \\to\n\\infty$, $\\{\\phi(t_n,\\, \\theta_{-t_n}(\\sigma); \\,\nx_n)\\}_{n=1}^{\\infty}$ is pre-compact in $X$.\n\\end{definition}\n\nIn the framework of pullback attractors, the pullback asymptotic\ncompactness may not be equivalent to the $\\kappa$-contraction if\nthe cocycle only has a general bounded pullback absorbing set. In\nfact, we need this bounded pullback absorbing set to satisfy an\nadditional nesting condition; see the next theorem.\n\nFrom $Lemma$ \\ref{l3.9} we know that if $\\phi$ has a nested\nbounded pullback absorbing set, then $\\phi$ being pullback\n$\\kappa$-contracting implies that $\\phi$ being pullback\nasymptotically compact; furthermore, in the proof of $Theorem$\n\\ref{t3.10}, we note that we indeed only used the pullback\nasymptotic compactness. This, combining with $Lemma$ \\ref{c3.8},\nimplies the following criterion.\n\n\\begin{theorem} \\label{t3.13}(\\textbf{Criterion for existence of pullback\nattractor})\\\\ Let $\\phi$ be a norm-to-weak continuous cocycle\nw.r.t. $\\theta$ on $\\mathbb{R}^+\\times \\Sigma \\times X$ such that\n$(\\phi, \\theta)$ has a nested bounded pullback absorbing set. Then\n$(\\phi, \\theta)$ has a pullback attractor if and only if $(\\phi,\n\\theta)$ is pullback $\\kappa$-contracting, or equivalently,\n$(\\phi, \\theta)$ is pullback asymptotically compact.\n\\end{theorem}\nThat is, under the assumption that $(\\phi, \\theta)$ has a nested\nbounded pullback absorbing set, pullback $\\kappa$-contraction is\nequivalent to pullback asymptotic compactness.\n\nOn the other hand, the authors in \\cite{CLR} have proven that\n$(\\phi, \\theta)$ has a pullback attractor provided that $(\\phi,\n\\theta)$ is pullback asymptotically compact and has a bounded\npullback absorbing set (see $Theorem$ 7 of \\cite{CLR}). In fact,\nfrom the definition of pullback attractor, $Lemma$ \\ref{c3.8},\nTheorems \\ref{t3.10} and \\ref{t3.13}, we observe that these\nconditions are also necessary. We summarize this result in the\nfollowing theorem.\n\n\\begin{theorem}\\label{t3.14}\n(\\textbf{Another criterion for existence of pullback attractor})\\\\\nLet $\\phi$ be a norm-to-weak continuous cocycle w.r.t. $\\theta$ on\n$\\mathbb{R}^+\\times \\Sigma \\times X$. Then $(\\phi, \\theta)$ has a\npullback attractor if and only if $(\\phi, \\theta)$ is pullback\nasymptotically compact and has a bounded pullback absorbing set.\n\\end{theorem}\n\n$Theorem$ \\ref{t3.13} and $Theorem$ \\ref{t3.14} show that pullback\nasymptotically compact is stronger than pullback\n$\\kappa-$contracting to some extent, which is different from the\nautonomous cases. Note also that $Theorem$ 3.14 is a slight\nimprovement of $Theorem$ 7 in \\cite{CLR}, from continuous cocycles\nto ``weakly\" continuous cocycles (i.e., norm-to-weak continuous\ncocycles).\n\nAlthough we only use the pullback asymptotic compactness in our\nlater applications in \\S 5, we think that the pullback\n$\\kappa-$contraction criterion for existence of pullback\nattractors for ``weakly\" continuous cocycles (i.e.,\n norm-to-weak continuous cocycles) is of independent interest and will be\nuseful for other non-autonomous dynamical systems. Another reason\nto present the $\\kappa-$contraction criterion here is that we like\nto highlight a difference with the \\emph{autonomous} systems: In\nnon-autonomous systems, the pullback asymptotic compactness\ncriterion and the pullback $\\kappa-$contraction criterion, for\nexistence of pullback attractors, are not equivalent unless when\nthere exists a \\emph{nested} bounded absorbing set ($Theorem$\n\\ref{t3.13}).\n\n\n\\medskip\n\nWe also remark that the definitions and results in this\nsection can be expressed in the framework of \\emph{processes},\ninstead of cocycles, as in \\cite{CV}.\n\n\n\n\n\n\\section{A technical method for verifying pullback asymptotic compactness}\n\nWe now present a convenient method for verifying the pullback\nasymptotic compactness for the cocycle generated by\n\\emph{non-autonomous} hyperbolic type of equations, in order to\napply $Theorem$ \\ref{t3.13} to obtain existence of pullback\nattractors in the next section. This method is partially motivated\nby the methods in \\cite{CL2,CL3,Kh} in some sense; see also in\n\\cite{SYZ2}. In \\cite{CL3}, the authors present a general abstract\nframework for asymptotic dynamics of \\emph{autonomous} wave\nequations.\n\n\\begin{definition}(\\cite{SYZ2})\n Let $X$ be a Banach space and $B$ be a bounded subset of $X$.\nWe call a function $\\psi(\\cdot,\\cdot)$, defined on $X\\times X$, a\ncontractive function on $B\\times B$ if for any sequence\n$\\{x_n\\}_{n=1}^{\\infty}\\subset B$, there is a subsequence\n$\\{x_{n_k}\\}_{k=1}^{\\infty} \\subset \\{x_n\\}_{n=1}^{\\infty}$ such\nthat\n\\begin{equation*}\n\\lim_{k\\to \\infty}\\lim_{l\\to \\infty}\\psi(x_{n_k},\\,x_{n_l})=0 .\n\\end{equation*}\nWe denote the set of all contractive functions on $B\\times B$ by\n$Contr(B)$.\n\\end{definition}\n\n\\begin{theorem}\\label{t4.2}\n\\textbf{(Technique for verifying pullback asymptotic compactness)}\n\\\\\nLet $\\phi$ be a cocycle w.r.t. $\\theta$ on $\\mathbb{R}^+\\times\n\\Sigma \\times X$ and have a nested bounded pullback absorbing set\n$\\mathscr{B}=\\{B_{\\sigma}\\}_{\\sigma \\in \\Sigma}$. Moreover, assume\nthat for any $\\varepsilon >0$ and each $\\sigma\\in \\Sigma$, there\nexist $T=T(B_{\\sigma}, \\varepsilon)$ and\n$\\psi_{T,\\,\\sigma}(\\cdot,\\cdot) \\in Contr(B_{\\sigma})$ such that\n\\begin{equation*}\n\\|\\phi(T,\\,\\theta_{-T}(\\sigma);x)-\\phi(T,\\,\\theta_{-T}(\\sigma);y)\\|\\leqslant\n\\varepsilon + \\psi_{T,\\,\\sigma}(x,\\,y) \\quad \\text{for all}~x,y\\in\nB_{\\sigma},\n\\end{equation*}\nwhere $\\psi_{T,\\,\\sigma}$ depends on $T$ and $\\sigma$. Then $\\phi$\nis pullback asymptotically compact in $X$.\n\\end{theorem}\n\n{\\bf Proof.} Let $\\{y_n\\}_{n=1}^{\\infty}$ be a bounded sequence of\n$X$ and $\\{t_n\\}\\subset \\mathbb{R}^+$ with $t_n \\to \\infty$ as\n$n\\to \\infty$}. We need to show that\n\\begin{align}\\label{4.1}\n\\text{$\\{\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\,y_n)\\}_{n=1}^{\\infty}$\nis precompact in $X$ for each $\\sigma\\in \\Sigma$}.\n\\end{align}\n\nIn the following, we will prove that\n$\\{\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\,y_n)\\}_{n=1}^{\\infty}$ has a\nconvergent subsequence via diagonal methods (e.g., see \\cite{Kh}).\n\nTaking $\\varepsilon_m>0$ with $\\varepsilon_m \\to 0$ as $m\\to\n\\infty$.\n\nAt first, for $\\varepsilon_1$, by the assumptions, there exist\n$T_1=T_1(\\varepsilon_1)$ and $\\psi_{1}(\\cdot,\\cdot) \\in\nContr(B_{\\sigma})$ such that\n\\begin{equation}\\label{4.2}\n\\|\\phi(T_1,\\,\\theta_{-T_1}(\\sigma);\\,x)-\\phi(T_1,\\,\\theta_{-T_1}(\\sigma);\\,y)\\|\\leqslant\n\\varepsilon_1 + \\psi_1(x,\\,y) \\quad \\text{for all}~x,y\\in\nB_{\\sigma},\n\\end{equation}\nwhere $\\psi_1$ depends on $T_1$ and $\\sigma$.\n\nSince $t_n\\to \\infty$, for such fixed $T_1$, without loss of\ngenerality, we assume that $t_n$ is so large that\n\\begin{equation}\\label{4.3}\n\\phi(t_n-T_1,\\,\\theta_{-(t_n-T_1)}(\\theta_{-T_1}(\\sigma));\\,y_n) \\in\nB_{\\theta_{-T_1}(\\sigma)} \\subset B_{\\sigma}\\quad \\text{for\neach}~n=1,2,\\cdots.\n\\end{equation}\nSet\n$x_n=\\phi(t_n-T_1,\\,\\theta_{-(t_n-T_1)}(\\theta_{-T_1}(\\sigma));\\,y_n)$.\nThen from \\eqref{4.2} we have\n\\begin{align}\\label{4.4}\n\\|\\phi(t_n,\\,\\theta_{-t_n}&(\\sigma);\\,y_n)-\\phi(t_m,\\,\\theta_{-t_m}(\\sigma);\\,y_m)\\|\n\\nonumber \\\\\n&= \\|\\phi(T_1,\\theta_{-T_1}(\\sigma); x_n)\n-\\phi(T_1,\\theta_{-T_1}(\\sigma);x_m)\\| \\leqslant \\varepsilon_1 +\n\\psi_1(x_n,\\,x_m).\n\\end{align}\n\nDue to the definition of $Contr(B_{\\sigma})$ and\n$\\psi_1(\\cdot,\\cdot) \\in Contr(B_{\\sigma})$, we know that\n$\\{x_n\\}_{n=1}^{\\infty}$ has a subsequence\n$\\{x^{(1)}_{n_k}\\}_{k=1}^{\\infty}$ such that\n\\begin{equation}\\label{4.5}\n\\lim_{k\\to \\infty}\\lim_{l\\to\n\\infty}\\psi_1(x^{(1)}_{n_k},\\,x^{(1)}_{n_l}) \\leqslant\n\\frac{\\varepsilon_1}{2},\n\\end{equation}\nand similar to \\cite{Kh}, we have\n\\begin{align*}\n\\lim_{k\\to \\infty}\\sup_{p\\in \\mathbb{N}}&\n\\|\\phi(t^{(1)}_{n_{k+p}},\\,\\theta_{-t^{(1)}_{n_{k+p}}}(\\sigma);\\,y^{(1)}_{n_{k+p}})\n-\\phi(t^{(1)}_{n_{k}},\\,\\theta_{-t^{(1)}_{n_{k}}}(\\sigma);\\,y^{(1)}_{n_{k}})\\|\n\\nonumber \\\\\n&\\leqslant \\lim_{k\\to \\infty}\\sup_{p\\in \\mathbb{N}}\\limsup_{l\\to\n\\infty}\n\\|\\phi(t^{(1)}_{n_{k+p}},\\,\\theta_{-t^{(1)}_{n_{k+p}}}(\\sigma);\\,y^{(1)}_{n_{k+p}})\n-\\phi(t^{(1)}_{n_{l}},\\,\\theta_{-t^{(1)}_{n_{l}}}(\\sigma);\\,y^{(1)}_{n_{l}})\\|\n\\nonumber \\\\\n& \\qquad + \\limsup_{k\\to \\infty}\\limsup_{l\\to \\infty}\n\\|\\phi(t^{(1)}_{n_{k}},\\,\\theta_{-t^{(1)}_{n_{k}}}(\\sigma);\\,y^{(1)}_{n_{k}})-\\phi(t^{(1)}_{n_{l}},\\,\\theta_{-t^{(1)}_{n_{l}}}(\\sigma);\\,y^{(1)}_{n_{l}})\\|\n\\nonumber \\\\\n&\\leqslant \\varepsilon_1+\\lim_{k\\to \\infty}\\sup_{p\\in\n\\mathbb{N}}\\lim_{l\\to\n\\infty}\\psi_1(x^{(1)}_{n_{k+p}},\\,x^{(1)}_{n_l}) +\\varepsilon_1+\n\\lim_{k\\to \\infty}\\lim_{l\\to\n\\infty}\\psi_1(x^{(1)}_{n_k},\\,x^{(1)}_{n_l}),\n\\end{align*}\nwhich, combining with \\eqref{4.4} and \\eqref{4.5}, implies that\n\\begin{align*}\n\\lim_{k\\to \\infty}\\sup_{p\\in \\mathbb{N}}\n\\|\\phi(t^{(1)}_{n_{k+p}},\\,\\theta_{-t^{(1)}_{n_{k+p}}}(\\sigma);\\,y^{(1)}_{n_{k+p}})\n-\\phi(t^{(1)}_{n_{k}},\\,\\theta_{-t^{(1)}_{n_{k}}}(\\sigma);\\,y^{(1)}_{n_{k}})\\|\n\\leqslant 4\\varepsilon_1.\n\\end{align*}\n\nTherefore, there is a $K_1$ such that\n\\begin{align*}\n\\|\\phi(t^{(1)}_{n_{k}},\\,\\theta_{-t^{(1)}_{n_{k}}}(\\sigma);\\,y^{(1)}_{n_{k}})\n-\\phi(t^{(1)}_{n_{l}},\\,\\theta_{-t^{(1)}_{n_{l}}}(\\sigma);\\,y^{(1)}_{n_{l}})\\|\n\\leqslant 5\\varepsilon_1 \\quad \\text{for all}~k,l\\geqslant K_1.\n\\end{align*}\n\nBy induction, we obtain that, for each $m\\geqslant 1$, there is a\nsubsequence \\linebreak\n$\\{\\phi(t^{(m+1)}_{n_{k}},\\,\\theta_{-t^{(m+1)}_{n_{k}}}(\\sigma);\\,y^{(m+1)}_{n_{k}})\\}_{k=1}^{\\infty}$\nof\n$\\{\\phi(t^{(m)}_{n_{k}},\\,\\theta_{-t^{(m)}_{n_{k}}}(\\sigma);\\,y^{(m)}_{n_{k}})\\}_{k=1}^{\\infty}$\nand certain $K_{m+1}$ such that\n\\begin{equation*}\n\\|\\phi(t^{(m+1)}_{n_{k}},\\,\\theta_{-t^{(m+1)}_{n_{k}}}(\\sigma);\\,y^{(m+1)}_{n_{k}})-\\phi(t^{(m+1)}_{n_{l}},\\,\\theta_{-t^{(m+1)}_{n_{l}}}(\\sigma);\\,y^{(m+1)}_{n_{l}})\\|\n\\leqslant 5\\varepsilon_{m+1} \\quad \\text{for all}~k,l\\geqslant\nK_{m+1}.\n\\end{equation*}\n\nNow, we consider the diagonal subsequence\n$\\{\\phi(t^{(k)}_{n_{k}},\\,\\theta_{-t^{(k)}_{n_{k}}}(\\sigma);\\,y^{(k)}_{n_{k}})\\}_{k=1}^{\\infty}$.\nSince for each $m\\in \\mathbb{N}$,\n$\\{\\phi(t^{(k)}_{n_{k}},\\,\\theta_{-t^{(k)}_{n_{k}}}(\\sigma);\\,y^{(k)}_{n_{k}})\\}_{k=m}^{\\infty}$\nis a subsequence of\n$\\{\\phi(t^{(m)}_{n_{k}},\\,\\theta_{-t^{(m)}_{n_{k}}}(\\sigma);\\,y^{(m)}_{n_{k}})\\}_{k=1}^{\\infty}$,\nthen,\n\\begin{equation*}\n\\|\\phi(t^{(k)}_{n_{k}},\\,\\theta_{-t^{(k)}_{n_{k}}}(\\sigma);\\,y^{(k)}_{n_{k}})-\n\\phi(t^{(l)}_{n_{l}},\\,\\theta_{-t^{(l)}_{n_{l}}}(\\sigma);\\,y^{(l)}_{n_{l}})\\|\n\\leqslant 5\\varepsilon_{m} \\quad \\text{for all}~k,l\\geqslant\n\\max\\{m,K_{m}\\},\n\\end{equation*}\nwhich, combining with $\\varepsilon_m \\to 0$ as $m\\to \\infty$,\nimplies that\n$\\{\\phi(t^{(k)}_{n_{k}},\\,\\theta_{-t^{(k)}_{n_{k}}}(\\sigma);\\,y^{(k)}_{n_{k}})\\}_{k=1}^{\\infty}$\nis a Cauchy sequence in $X$. This shows that\n$\\{\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\,y_n)\\}_{n=1}^{\\infty}$ is\nprecompact in $X$ for each $\\sigma\\in \\Sigma$. We thus complete\nthe proof.\n $\\hfill \\blacksquare$ \\\\\n\nNote that the nested properties and the contractive properties are\nonly used in \\eqref{4.3} and \\eqref{4.2} respectively. We have a\nsimilar corollary for the cocycle without nested pullback absorbing\nset, the proof is similar to that for $Theorem$ \\ref{t4.2} above.\n\\begin{corollary}\\label{c4.3}\nLet $\\phi$ be a cocycle w.r.t. $\\theta$ on $\\mathbb{R}^+\\times\n\\Sigma \\times X$ and have a bounded pullback absorbing set\n$\\mathscr{B}=\\{B_{\\sigma}\\}_{\\sigma \\in \\Sigma}$. Moreover, assume\nthat for any $\\varepsilon >0$ and each $\\sigma\\in \\Sigma$, there\nexist $T=T(\\sigma, \\varepsilon)$ and $\\psi_{T,\\,\\sigma}(\\cdot,\\cdot)\n\\in Contr(B_{\\theta_{-T}(\\sigma)})$ such that\n\\begin{equation*}\n\\|\\phi(T,\\,\\theta_{-T}(\\sigma);x)-\\phi(T,\\,\\theta_{-T}(\\sigma);y)\\|\\leqslant\n\\varepsilon + \\psi_{T,\\,\\sigma}(x,\\,y) \\quad \\text{for all}~x,y\\in\nB_{\\theta_{-T}(\\sigma)},\n\\end{equation*}\nwhere $\\psi_{T,\\,\\sigma}$ depends on $T$ and $\\sigma$. Then $\\phi$\nis pullback asymptotically compact in $X$.\n\\end{corollary}\n\n\\section{Pullback attractors for a non-autonomous wave equation}\n\nIn this section, we prove the existence of the pullback attractor\nfor the non-autonomous wave system \\eqref{1.1}-\\eqref{1.3}, by\napplying $Theorems$ \\ref{t3.13} and \\ref{t3.14}. We use the method\n(via contractive functions) in \\S 4 to verify the pullback\nasymptotic compactness. This method appears to be very efficient for\nnon-autonomous wave or hyperbolic equations, while the approach in\n\\cite{CLR}, which is an energy method and is different from ours, is\nvery appropriate for some non-autonomous parabolic equations or wave\nequations with \\emph{linear} damping, e.g., see \\cite{MRW}. In fact,\nthe approach in \\cite{CLR} is an energy method and may be seen as a\nnon-autonomous generalization of Ball's method \\cite{Ba}.\n\n\n\n\n\\subsection{Mathematic setting}\n\nWe consider the non-autonomous wave system \\eqref{1.1}-\\eqref{1.3}\non the \\emph{state space} $X=H_0^1(\\Omega)\\times L^2(\\Omega)$.\n For each $g_0\\in L_{loc}^2(\\mathbb{R};\\,L^2(\\Omega))$,\nwe denote $\\{g_0(s+t)|t\\in \\mathbb{R}\\}$ by $\\mathcal{H}_1(g_0)$.\nFor $f_0(v,s)$ satisfying \\eqref{1.8}-\\eqref{1.9}, we similarly\ndenote $\\mathcal{H}_2(f_0)=\\{f_0(\\cdot,s+t)|t\\in \\mathbb{R}\\}$.\n\n\nLet $\\Sigma=\\mathcal{H}_2(f_0)\\times \\mathcal{H}_1(g_0)$ be the\nparameter space. We define the driving system $\\theta_t$:\n$\\Sigma\\to \\Sigma$ by\n\\begin{equation}\\label{5.1}\n\\theta_t(f_0(\\cdot),\\,g_0(\\cdot))=(f_0(t+\\cdot),\\,g_0(t+\\cdot)),\n\\;\\; t\\in \\mathbb{R}.\n\\end{equation}\n\nThen, system \\eqref{1.1}-\\eqref{1.3} is rewritten as the following\nsystem\n\\begin{equation}\\label{5.2}\n\\begin{cases}\nu_{tt}+h(u_t)-\\Delta u+f(u,t+s)=g(x,\\,t+s), (x,t)\\in \\Omega\\times \\mathbb{R}^+,\\\\\nu(x,\\,t)|_{\\partial \\Omega}=0, \\\\\nu(x,0)=u_{0}(x),\\quad u_t(x,0)=v_0(x),\n\\end{cases}\n\\end{equation}\nwhere $s\\in \\mathbb{R}$ means the initial symbol, corresponding to\nsome $\\sigma \\in \\Sigma$.\n\nApplying monotone operator theory or Faedo-Galerkin method, e.g.,\nsee \\cite{CV,Li,Sh}, it is known that conditions\n\\eqref{1.4}-\\eqref{1.9} guarantee the existence and uniqueness of\nstrong solution and generalized solution for\n\\eqref{1.1}-\\eqref{1.3}, and the time-dependent terms make no\nessential complications.\n\n\\begin{lemma} \\label{l5.1} (\\textbf{Well-posedness}) \\\\\nLet $\\Omega$ be a bounded subset of $\\mathbb{R}^3$ with smooth\nboundary, and assume that either Assumption I or Assumption II\nholds. Then the non-autonomous system \\eqref{5.2} has a unique\nsolution $(u(t),\\,u_t(t))\\in\n\\mathcal{C}(\\mathbb{R}^+;\\,H_0^1(\\Omega)\\times L^2(\\Omega))$ and\n$\\partial_t^2u(t)\\in L^2_{loc}(\\mathbb{R}^+;\\,H^{-1}(\\Omega))$ for\nany initial data $x_0=(u^{0},\\,u^{1})\\in H_0^1(\\Omega)\\times\nL^2(\\Omega)$ and any initial symbol $\\sigma\\in \\Sigma$.\n\\end{lemma}\n\nBy $Lemma$ \\ref{l5.1}, we can define the cocycle as follows:\n\\begin{equation}\\label{5.3}\n\\begin{cases}\n\\phi:\\quad \\mathbb{R}^+\\times \\Sigma\\times X & \\to X, \\\\\n(t,\\sigma,(u^{0}(x),u^{1}(x))) & \\to\n(u^{\\sigma}(t),u^{\\sigma}_t(t)),\n\\end{cases}\n\\end{equation}\nwhere $(u^{\\sigma}(t),u^{\\sigma}_t(t))$ is the solution of\n\\eqref{1.1} corresponding to initial data $(u^{0}(x),u^{1}(x))$ and\nsymbol $\\sigma=(f_0(s+\\cdot),g_0(s+\\cdot))$; and for each\n$(t,\\,\\sigma)\\in \\mathbb{R}^+\\times \\Sigma$, the mapping\n$\\phi(t,\\,\\sigma;\\,\\cdot): X\\to X$ is continuous.\n\n \\emph{Hereafter, we always denote by $(\\phi,\\,\\theta)$ the\ncocycle defined in \\eqref{5.1} and \\eqref{5.3}.}\n\nWe now prove the following main result.\n\n\\begin{theorem}\\label{t5.2} (\\textbf{Existence of pullback\nattractor})\\\\\nLet $\\Omega$ be a bounded domain of $\\mathbb{R}^3$ with smooth\nboundary. Then under either Assumption I or Assumption II, the\nNDS $(\\phi,\\,\\theta)$ generated by the weak solutions of\n\\eqref{1.1}-\\eqref{1.3} has a pullback attractor\n$\\mathscr{A}=\\{\\omega_{\\sigma}(B_{\\sigma})\\}_{\\sigma \\in \\Sigma}$.\n\\end{theorem}\n\n\nWe need a few lemmas before proving this theorem.\n\n\n\n\n\n\\subsection{Pullback absorbing sets} \\label{pullback}\n\n\nIn the following, we deal only with the strong solutions of\n\\eqref{1.1}. The generalized solution case then follows easily by\na density argument. We begin with the following existence result\non a bounded pullback absorbing set.\n\\begin{lemma} \\label{l5.3}(\\textbf{Pullback absorbing set})\\\\\nLet $\\Omega$ be a bounded domain of $\\mathbb{R}^3$ with smooth\nboundary. Then under either Assumption I or Assumption II, the NDS\n$(\\phi,\\,\\theta)$ has a bounded pullback absorbing set\n$\\mathscr{B}=\\{B_{\\sigma}\\}_{\\sigma\\in \\Sigma}$.\n\\end{lemma}\n{\\bf Proof.} For each $\\sigma \\in \\Sigma$, we know that $\\sigma$\nis corresponding to some $s_0$ satisfying that\n$\\sigma=(f(v,s_0+t),g(x,s_0+t))$, and\n$\\phi(t,\\theta_{-t_0}(\\sigma);\\,x_0)$ is the solution of the\nfollowing equation at time $t$:\n\\begin{equation}\\label{5.4}\n\\begin{cases}\nu_{tt}+h(u_t)-\\Delta u+f(u,t-t_0+s_0)=g(x,\\,t-t_0+s_0),\\quad (x,t)\\in \\Omega\\times \\mathbb{R}^+,\\\\\n(u(0),\\,u_t(0))=x_0,\\\\\nu|_{\\partial \\Omega}=0.\n\\end{cases}\n\\end{equation}\n\nUnder {\\it Assumption $II$}, we can repeat what have done in the\nproof of [$Theorem$ 1, Haraux\\cite{Ha2}], to obtain that there exist\na $\\rho$ (which depends only on\n$\\|g\\|_{_{L^{\\infty}(\\mathbb{R},L^2(\\Omega))}}$ and the coefficients\nin \\eqref{1.4}-\\eqref{1.9}) and a $T$ (which depends only on\n$\\|g\\|_{_{L^{\\infty}(\\mathbb{R},L^2(\\Omega))}}$, the coefficients in\n\\eqref{1.4}-\\eqref{1.9} and the radius of $B$) such that for any\n$\\sigma\\in \\Sigma$,\n\\begin{equation}\\label{5.5}\n\\|\\phi(t,\\,\\theta_{-t_0}(\\sigma);\\,x_0)\\|_{_X} \\leqslant \\rho \\quad\n\\text{for all}~T\\leqslant t \\leqslant t_0~\\text{and}~x_0 \\in B.\n\\end{equation}\nHence, for {\\it Assumption $II$}, we can take $B_{\\sigma}\\equiv\n\\{x\\in X~|~\\|x\\|_{_X} \\leqslant \\rho\\}$ for each $\\sigma$.\n\nUnder {\\it Assumption $I$}, we can use the methods as that in the\nproof of Chepyzhov and Vishik [\\cite{CV}, $Lemma$ 4.1, $Proposition$\n4.2, P 121-123], obtain also that there exist $C_{\\beta}$ and\n$\\beta$ (which depend only on the coefficients in\n\\eqref{1.4}-\\eqref{1.9} and $C_0$) and a $T$ (which depends only on\n$\\int_{-\\infty}^{s_0}e^{\\beta}\\int_{\\Omega}|g(x,s)|^2dxds$, the\ncoefficients in \\eqref{1.4}-\\eqref{1.9} and the radius of $B$) such\nthat\n\\begin{equation}\\label{5.6}\n\\|\\phi(t,\\,\\theta_{-t_0}(\\sigma);\\,x_0)\\|^2_{_X} \\leqslant\n\\rho_{_{\\sigma,\\beta}}=C_{\\beta}(1+e^{-\\beta\ns_0}\\int_{-\\infty}^{s_0}e^{\\beta s}\\int_{\\Omega}|g(x,s)|^2dxds)\\quad\n\\forall~T\\leqslant t \\leqslant t_0,~x_0 \\in B.\n\\end{equation}\nTherefore, under {\\it Assumption $I$}, we can take\n$B_{\\sigma}=B^{\\beta}_{\\sigma}= \\{x\\in X~|~\\|x\\|^2_{_X} \\leqslant\n\\rho_{_{\\sigma,\\beta}}\\}$. $\\hfill \\blacksquare$\n\n\\begin{remark}\\label{r5.4}\nFrom \\eqref{5.5} we know that under {\\it Assumption $II$}, the NDS\n$(\\phi,\\,\\theta)$ has a nested bounded pullback absorbing set\n$\\mathscr{B}=\\{B_{\\sigma}\\}_{\\sigma\\in \\Sigma}$.\n\\end{remark}\n\n\n\n\n\\subsection{Pullback asymptotic compactness}\n\nWe new prove the pullback asymptotic compactness.\n\n\\begin{lemma} \\label{l5.5} (\\textbf{Pullback asymptotic compactness}) \\\\\nUnder either Assumption $I$ or $II$, for any bounded sequence\n$\\{x_n\\}_{n=1}^{\\infty}\\in \\mathcal{B}$ and $\\sigma\\in \\Sigma$,\nthe sequence\n$\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\,x_n\\}_{n=1}^{\\infty}$ is\nprecompact in $X$.\n\\end{lemma}\nThe idea for the proof is similar to that in Chueshov \\& Lasiecka\n\\cite{CL1,CL2,CL3} and Khanmamedov \\cite{Kh}; see also in\n\\cite{SYZ2} for linear damping and autonomous cases.\n\n\n\n\nIn order to prove this lemma on pullback asymptotic compactness, we\nneed to derive a few energy inequalities; see\n\\eqref{5.19}-\\eqref{5.22} below.\n\\medskip\n\nWe first present some preliminaries and notations.\n\nFor each $\\sigma \\in \\Sigma$, we know that $\\sigma$ is corresponding\nto some $s_0$ such that $\\sigma=(f(v,s_0+t),g(x,s_0+t))$. For any\n$x_0^i=(u_0^i,\\,v_0^i)\\in X$ ($i=1,2$), let\n$(u_i(t),u_{i_t}(t))=\\phi(t,\\theta_{-t_0}(\\sigma);\\,x_0^i)$ be the\ncorresponding solution of the following equation at time $t$:\n\\begin{equation}\\label{5.7}\n\\begin{cases}\nu_{tt}+h(u_t)-\\Delta u+f(u,t-t_0+s_0)=g(x,\\,t-t_0+s_0),\\quad (x,t)\\in \\Omega\\times \\mathbb{R}^+,\\\\\n(u(0),\\,u_t(0))=x_0^i,\\\\\nu|_{\\partial \\Omega}=0.\n\\end{cases}\n\\end{equation}\nFor convenience, we introduce notations\n\\begin{equation*}\nf_i(t)=f(u_i(t),\\,t-t_0+s_0),\\quad h_i(t)=h(u_{i_t}(t)),\\quad\nt\\geqslant 0, ~~ i=1,2,\n\\end{equation*}\nand\n\\begin{equation*}\nw(t)=u_1(t)-u_2(t).\n\\end{equation*}\n\nThen $w(t)$ satisfies\n\\begin{equation}\\label{5.8}\n\\begin{cases}\nw_{tt}+h_1(t)-h_2(t)-\\Delta w+ f_1(t)-f_2(t)=0,\\\\\nw|_{\\partial \\Omega}=0,\\\\\n(w(0),\\,w_t(0))=(u_0^1,\\,v_0^1)-(u_0^2,\\,v_0^2).\n\\end{cases}\n\\end{equation}\n\nWe also define an energy functional\n\\begin{equation}\\label{5.9}\nE_w(t)=\\frac 12\\int_{\\Omega}|w(t)|^2+\\frac 12 \\int_{\\Omega}|\\nabla\nw(t)|^2.\n\\end{equation}\n\nSince the pullback attractors obtained in $Lemma$ \\ref{l5.3} are\ndifferent for \\emph{Assumption I} and $II$, in the following we will\ndeduce different estimations.\n\nWe first deal with the case corresponding to \\emph{Assumption II}:\n\n\\emph{Step 1} Multiplying \\eqref{5.8} by $w_t(t)$, and integrating\nover $[s,\\,T]\\times \\Omega$, we obtain\n\\begin{align}\\label{5.10}\nE_w(T) +\\int_s^T\\int_{\\Omega}(h_1(\\tau)-h_2(\\tau))w_t(\\tau)dxd\\tau\n+\\int_s^T\\int_{\\Omega}(f_1(\\tau)-f_2(\\tau))w_t(\\tau)dxd\\tau =\nE_w(s),\n\\end{align}\nwhere $0\\leqslant s \\leqslant T \\leqslant t_0$. Then\n\\begin{align}\\label{5.11}\n\\int_s^T \\int_{\\Omega}(h_1(\\tau)-h_2(\\tau))w_t(\\tau)dxd\\tau\n\\leqslant E_w(s)-\n\\int_s^T\\int_{\\Omega}(f_1(\\tau)-f_2(\\tau))w_t(\\tau)dxd\\tau.\n\\end{align}\nCombining with $Lemma$ \\ref{l2.3}, we get that for any $\\delta\n>0$,\n\\begin{align}\\label{5.12}\n\\int_s^T\\int_{\\Omega}|w_t(\\tau)|^2dxd\\tau \\leqslant |T-s|\\delta\n\\cdot mes(\\Omega) +C_{\\delta}E_w(s) -C_{\\delta}\\int_s^T\n\\int_{\\Omega}(f_1-f_2)w_txd\\tau.\n\\end{align}\n\n\\emph{Step 2} Multiplying \\eqref{5.8} by $w(t)$, and integrating\nover $[0,\\,T]\\times \\Omega$, we get that\n\\begin{align}\\label{5.13}\n\\int_0^T & \\int_{\\Omega}|\\nabla w(s)|^2dxds + \\int_{\\Omega}w_t(T)\\cdot w(T) \\nonumber \\\\\n& = \\int_0^T\\int_{\\Omega}|w_t(s)|^2dxds -\n\\int_0^T\\int_{\\Omega}(h_1-h_2)w + \\int_{\\Omega}w_t(0)\\cdot w(0) -\n\\int_0^T\\int_{\\Omega}(f_1-f_2)w.\n\\end{align}\n\nTherefore, from \\eqref{5.12} and \\eqref{5.13}, we have\n\\begin{align}\\label{5.14}\n2\\int_0^T&E_w(s)ds \\nonumber \\\\\n& \\leqslant 2\\delta T mes(\\Omega)+2 C_{\\delta}E_w(0) - 2 C_{\\delta}\\int_0^T\\int_{\\Omega}(f_1-f_2)w(t) dxds \\nonumber \\\\\n& \\quad - \\int_{\\Omega}w_t(T)w(T) + \\int_{\\Omega}w_t(0)w(0)-\n\\int_0^T\\int_{\\Omega}(h_1-h_2)w-\\int_0^T\\int_{\\Omega}(f_1-f_2)w.\n\\end{align}\n\nIntegrating \\eqref{5.10} over $[0,\\,T]$ with respect to $s$, we have\nthat\n\\begin{align}\\label{5.15}\nTE_w(T)&+\\int_0^T\\int_s^T\\int_{\\Omega}(h_1(\\tau)-h_2(\\tau))w_t(\\tau) dxd\\tau ds \\nonumber \\\\\n&=-\\int_0^T\\int_s^T\\int_{\\Omega}(f_1-f_2)w_tdxd\\tau ds + \\int_0^TE_w(s) ds \\nonumber \\\\\n& \\leqslant - \\int_0^T\\int_s^T\\int_{\\Omega}(f_1-f_2)w_tdxd\\tau ds+ \\delta T mes(\\Omega)+C_{\\delta}E_w(0) \\nonumber \\\\\n& \\quad -C_{\\delta}\\int_0^T\\int_{\\Omega}(f_1-f_2)w_tdxds -\\frac 12 \\int_{\\Omega}w_t(T)w(T) +\\frac 12\\int_{\\Omega}w_t(0)w(0) \\nonumber \\\\\n& \\quad -\\frac 12 \\int_0^T\\int_{\\Omega}(h_1-h_2)w-\\frac12 \\int_0^T\n\\int_{\\Omega}(f_1-f_2)w.\n\\end{align}\n\n\\emph{Step 3} We will deal with $\\int_0^T\\int_{\\Omega}(h_1-h_2)w$.\nMultiplying \\eqref{5.7} by $u_{i_t}(t)$, we obtain\n\\begin{align*}\n\\frac 12\\frac{d}{dt}\\int_{\\Omega}(|u_{i_t}|^2+|\\nabla\nu_i|^2)+\\int_{\\Omega}h(u_{i_t})u_{i_t}+\\int_{\\Omega}f(u_i,t+s_i)u_{i_t}=\\int_{\\Omega}g_iu_{i_t},\n\\end{align*}\nwhich, combining with the existence of bounded uniformly absorbing\nset, implies that\n\\begin{align}\\label{5.16}\n\\int_0^T\\int_{\\Omega}h(u_{i_t})u_{i_t} \\leqslant M_T,\n\\end{align}\nwhere the constant $M_T$ depends on $T$ (which is different from the\nautonomous cases). Then, noticing \\eqref{2.1}, we obtain that\n\\begin{equation}\\label{5.17}\n\\int_0^T\\int_{\\Omega}|h(u_{i_t})|^{\\frac{p+1}{p}}dxds \\leqslant M_T.\n\\end{equation}\n\nTherefore, using H$\\ddot{o}$lder inequality, from \\eqref{5.17} we\nhave\n\\begin{align*}\n|\\int_0^T\\int_{\\Omega}h_iw| \\leqslant\nM_T^{\\frac{p}{p+1}}\\left(\\int_0^T\\int_{\\Omega}|w|^{p+1}\n\\right)^{\\frac{1}{p+1}},\n\\end{align*}\nwhich implies that\n\\begin{align}\\label{5.18}\n|\\int_0^T\\int_{\\Omega}(h_1-h_2)w| \\leqslant\n2M_T^{\\frac{p}{p+1}}\\left(\\int_0^T\\int_{\\Omega}|w|^{p+1}\n\\right)^{\\frac{1}{p+1}}.\n\\end{align}\n\nHence, combining \\eqref{5.15} and \\eqref{5.18}, we obtain that\n\\begin{align*}\nE_w(T)& \\leqslant \\delta mes(\\Omega)-\\frac{1}{T} \\int_0^T\\int_s^T\\int_{\\Omega}(f_1(\\tau)-f_2(\\tau))w_t(\\tau)dxd\\tau ds+ \\frac{C_{\\delta}}{T}E_w(0) \\nonumber \\\\\n& \\quad -\\frac{C_{\\delta}}{T}\\int_0^T\\int_{\\Omega}(f_1(s)-f_2(s))w_t(s)dxds -\\frac{1}{2T} \\int_{\\Omega}w_t(T)w(T) +\\frac{1}{2T}\\int_{\\Omega}w_t(0)w(0) \\nonumber \\\\\n& \\quad +\\frac 1T\nM_T^{\\frac{p}{p+1}}\\left(\\int_0^T\\int_{\\Omega}|w(s)|^{p+1}dxds\n\\right)^{\\frac{1}{p+1}}-\\frac{1}{2T} \\int_0^T\n\\int_{\\Omega}(f_1(s)-f_2(s))w(s)dxds\n\\end{align*}\nfor any $0\\leqslant T\\leqslant t_0$.\n\nWe define\n\\begin{align}\\label{5.19}\n&\\psi_{T,\\,\\delta,\\,\\sigma}(x_0^1,\\,x_0^2) \\nonumber \\\\\n&=-\\frac{1}{T} \\int_0^T\\int_s^T\\int_{\\Omega}(f_1(\\tau)-f_2(\\tau))w_t(\\tau)dxd\\tau ds \\nonumber \\\\\n& \\quad -\\frac{C_{\\delta}}{T}\\int_0^T\\int_{\\Omega}(f_1(s)-f_2(s))w_t(s)dxds -\\frac{1}{2T} \\int_{\\Omega}w_t(T)w(T) \\nonumber \\\\\n& \\quad +\\frac 1T\nM_T^{\\frac{p}{p+1}}\\left(\\int_0^T\\int_{\\Omega}|w(s)|^{p+1}dxds\n\\right)^{\\frac{1}{p+1}}-\\frac{1}{2T} \\int_0^T\n\\int_{\\Omega}(f_1(s)-f_2(s))w(s)dxds.\n\\end{align}\nThen we have\n\\begin{equation}\\label{5.20}\nE_w(T) \\leqslant \\delta mes(\\Omega)\n+\\frac{1}{2T}\\int_{\\Omega}w_t(0)w(0)+\n\\frac{C_{\\delta}}{T}E_w(0)+\\psi_{T,\\,\\delta,\\,\\sigma}(x_0^1,\\,x_0^2)\n\\end{equation}\nfor any $\\delta >0$, $0\\leqslant T\\leqslant t_0$.\n\n\\medskip\n\nFor the case corresponding to \\emph{Assumption I}:\n\nSince under our general assumption \\eqref{1.10}, as shown in\n\\eqref{5.6}, the pullback attractors may not satisfy the nested\nproperties. Inspired partly by the results in \\cite{CLR,LLR} we will\ndeduce different estimations by the same methods; see \\eqref{5.21}\nand \\eqref{5.22} below.\n\nRepeat \\emph{Step 1} and \\emph{Step 2} above, and just replace the\nmultipliers $w_t(t)$ and $w(t)$ by $e^{\\beta t}w_t(t)$ and $e^{\\beta\nt}w(t)$ respectively, and take into account $\\beta0$, $0\\leqslant T\\leqslant t_0$, where\n$\\alpha=(C_0+\\beta)\/(C_0-\\beta)$ and\n\\begin{align}\\label{5.22}\n&\\psi'_{T,\\,\\sigma}(x_0^1,\\,x_0^2)\n\\nonumber \\\\\n&=-\\frac{e^{-T\\beta}}{T} \\int_0^T\\int_s^T\\int_{\\Omega}e^{\\beta\n\\tau}(f_1(\\tau)-f_2(\\tau))w_t(\\tau)dxd\\tau ds\n+\\frac{\\alpha}{2T}e^{-\\beta\nT}\\int_{\\Omega}w_t(0)w(0)\\nonumber \\\\\n& \\quad -\\frac{\\alpha}{TC_0}e^{-\\beta\nT}\\int_0^T\\int_{\\Omega}e^{\\beta s}(f_1(s)-f_2(s))w_t(s)dxds -\\frac{C}{T} \\int_{\\Omega}w_t(T)w(T) \\nonumber \\\\\n& \\quad + M_{_{E_w(0), T,\nC_0,\\beta}}\\left(\\int_0^T\\int_{\\Omega}|w(s)|^{2}dxds\n\\right)^{\\frac{1}{2}}.\n\\end{align}\n\nWith the above energy inequalities, we are now\nready to prove pullback asymptotic compactness. \\\\\n\n{\\bf Proof of $Lemma$ \\ref{l5.5}:} We will deal with\n\\emph{Assumption I} and \\emph{Assumption II} separately.\n\n\\emph{Assumption II}:\n\nFor each $\\sigma\\in \\Sigma$, and for any fixed $\\varepsilon>0$, from\n\\eqref{5.19}, we can take $t_0$ large enough such that\n\\begin{equation}\nE_w(t_0) \\leqslant\n\\varepsilon+\\psi_{t_0,\\,\\delta,\\,\\sigma}(x_0^1,\\,x_0^2)~\\text{for\nall}~x_0^1,\\,x_0^2\\in B_{\\sigma}.\n\\end{equation}\nHence, thanks to $Theorem$ \\ref{t4.2} and $Lemma$ \\ref{l5.3}, it is\nsufficiently to prove that the function\n$\\psi_{t_0,\\,\\delta,\\,\\sigma}(\\cdot,\\,\\cdot)$ defined in\n\\eqref{5.19} belongs to $Contr(B_{\\sigma})$ for each fixed $t_0$.\n\nWe observe from equation \\eqref{5.7} (and also see \\cite{Ha1}) that\nfor any $t_0>0$,\n\\begin{equation}\\label{5.24}\n\\bigcup_{t \\in\n[0,\\,t_0]}\\phi(t,\\,\\theta_{-t_0}(\\sigma);B_{\\sigma})~~\\text{is\nbounded in}~ X,\n\\end{equation}\nand the bound depends only on $t_0$ and $\\sigma$.\n\nLet $(u_n,u_{t_n})$ be the corresponding solution of\n$(u^n_0,v_0^n)\\in B_{\\sigma}$ for problem \\eqref{5.7},\n$n=1,2,\\cdots$. From the observation above, without loss of\ngenerality (or by passing to subsequences), we assume that\n\\begin{equation}\\label{5.25}\nu_n \\to u\\quad \\star-\\text{weakly in} ~ L^{\\infty}(0,t_0;\\,\nH_0^1(\\Omega)),\n\\end{equation}\n\\begin{equation}\\label{5.26}\nu_n \\to u\\quad \\text{in} ~L^{p+1}(0,t_0;\\, L^{p+1}(\\Omega)),\n\\end{equation}\n\\begin{equation}\\label{5.27}\nu_{n_t} \\to u_t\\quad \\star-\\text{weakly in} ~ L^{\\infty}(0,t_0;\\,\nL^2(\\Omega)),\n\\end{equation}\n\\begin{equation}\\label{5.28}\nu_n \\to u\\quad \\text{in} ~ L^2(0,t_0;\\, L^2(\\Omega))\n\\end{equation}\nand\n\\begin{equation}\\label{5.29}\nu_n(0) \\to u(0)~~\\text{and}~~u_n(t_0) \\to u(t_0)\\quad \\text{in} ~\nL^4(\\Omega).\n\\end{equation}\nHere we have used the compact embeddings $H_0^1 \\hookrightarrow\nL^{4}$ and $H_0^1 \\hookrightarrow L^{p+1}$ (since $1 \\leq p<5$).\n\nNow, we will deal with each term in \\eqref{5.19} one by one.\n\nFirst, from \\eqref{5.24}, \\eqref{5.29} and \\eqref{5.26} we get that\n\\begin{equation}\\label{5.30}\n\\lim_{n\\to \\infty}\\lim_{m\\to \\infty}\\int_{\\Omega}\n(u_{n_t}(t_0)-u_{m_t}(t_0))(u_n(t_0)-u_m(t_0)) dx = 0,\n\\end{equation}\n\\begin{equation}\\label{5.31}\n\\lim_{n\\to \\infty}\\lim_{m\\to\n\\infty}\\int_0^{t_0}\\int_{\\Omega}|u_n(s)-u_m(s)|^{p+1}dxds=0,\n\\end{equation}\nand from \\eqref{1.8} and \\eqref{5.28}, we further have\n\\begin{equation}\\label{5.32}\n\\lim_{n\\to \\infty}\\lim_{m\\to \\infty}\\int_0^{t_0}\\int_{\\Omega}\n(f(u_n(s),s-t_0+s_0)-f(u_m(s),s-t_0+s_0))(u_n(s)-u_m(s))dxds = 0.\n\\end{equation}\n\nSecond, note that\n\\begin{align*}\n\\int_0^{t_0}\\int_{\\Omega}&\n(u_{n_t}(s)-u_{m_t}(s))(f(u_n(s),s-t_0+s_0)-f(u_m(s),s-t_0+s_0))dxds \\nonumber \\\\\n& = \\int_0^{t_0}\\int_{\\Omega}\nu_{n_t}(s)f(u_n(s),s-t_0+s_0)+\\int_0^{t_0}\\int_{\\Omega}\nu_{m_t}(s)f(u_m(s),s-t_0+s_0)\\nonumber \\\\\n&\\qquad -\\int_0^{t_0}\\int_{\\Omega} u_{n_t}(s)f(u_m(s),s-t_0+s_0)-\\int_0^{t_0}\\int_{\\Omega} u_{m_t}(s)f(u_n(s),s-t_0+s_0)\\nonumber \\\\\n& =\\int_{\\Omega} F(u_n(t_0),s_0)-\\int_{\\Omega}\nF(u_n(0),-t_0+s_0)-\\int_0^{t_0}\\int_{\\Omega}F_s(u_n(\\tau),\\tau-t_0+s_0)dxd\\tau\n\\nonumber \\\\\n&\\qquad+\\int_{\\Omega} F(u_m(t_0),s_0)-\\int_{\\Omega}\nF(u_m(0),-t_0+s_0) -\\int_0^{t_0}\\int_{\\Omega}F_s(u_m(\\tau),\\tau-t_0+s_0)dxd\\tau \\nonumber \\\\\n&\\qquad -\\int_0^{t_0}\\int_{\\Omega}\nu_{n_t}(s)f(u_m(s),s-t_0+s_0)-\\int_0^{t_0}\\int_{\\Omega}\nu_{m_t}(s)f(u_n(s)s-t_0+s_0).\n\\end{align*}\nBy \\eqref{5.25}, \\eqref{5.27}, \\eqref{5.29} and \\eqref{1.8},\ntaking first $m\\to \\infty$ and then $n\\to \\infty$, we obtain that\n\\begin{align}\\label{5.33}\n\\lim_{n\\to \\infty}&\\lim_{m\\to \\infty}\\int_0^{t_0}\\int_{\\Omega}\n(u_{n_t}(s)-u_{m_t}(s))(f(u_n(s),s-t_0+s_0)-f(u_m(s),s-t_0+s_0))dxds \\nonumber \\\\\n&=\\int_{\\Omega} F(u(t_0),s_0)-\\int_{\\Omega}\nF(u(0),-t_0+s_0)-\\int_0^{t_0}\\int_{\\Omega}F_s(u(\\tau),\\tau-t_0+s_0)dxd\\tau\n\\nonumber \\\\\n&\\qquad+\\int_{\\Omega} F(u(t_0),s_0)-\\int_{\\Omega}\nF(u(0),-t_0+s_0) -\\int_0^{t_0}\\int_{\\Omega}F_s(u(\\tau),\\tau-t_0+s_0)dxd\\tau \\nonumber \\\\\n&\\qquad -\\int_0^{t_0}\\int_{\\Omega}\nu_tf(u(s),s-t_0+s_0)-\\int_0^{t_0}\\int_{\\Omega} u_{t}f(u(s),s-t_0+s_0) \\nonumber \\\\\n&=0.\n\\end{align}\nSimilarly, we have\n\\begin{align*}\n\\int_s^{t_0}\\int_{\\Omega}&\n(u_{n_t}(\\tau)-u_{m_t}(\\tau))(f(u_n(\\tau),\\tau-t_0+s_0)-f(u_m(\\tau),\\tau-t_0+s_0))dxd\\tau \\nonumber \\\\\n& =\\int_{\\Omega} F(u_n(t_0),s_0)-\\int_{\\Omega}\nF(u_n(s),s-t_0+s_0)-\\int_s^{t_0}\\int_{\\Omega}F_s(u_n(\\tau),\\tau-t_0+s_0)dxd\\tau\n\\nonumber \\\\\n&\\quad+\\int_{\\Omega} F(u_m(t_0),s_0)-\\int_{\\Omega}\nF(u_m(s),s-t_0+s_0) -\\int_s^{t_0}\\int_{\\Omega}F_s(u_m(\\tau),\\tau-t_0+s_0)dxd\\tau \\nonumber \\\\\n&\\quad -\\int_s^{t_0}\\int_{\\Omega}\nu_{n_t}f(u_m(\\tau),\\tau-t_0+s_0)-\\int_s^{t_0}\\int_{\\Omega}\nu_{m_t}f(u_n(\\tau),\\tau-t_0+s_0).\n\\end{align*}\nSince $|\\int_s^{t_0}\\int_{\\Omega}\n(u_{n_t}(\\tau)-u_{m_t}(\\tau))(f(u_n(\\tau),\\tau-t_0+s_0)-f(u_m(\\tau),\\tau-t_0+s_0))dxd\\tau|$\nis bounded for each fixed $t_0$, by the Lebesgue dominated\nconvergence theorem, we finally have\n\\begin{align}\\label{5.34}\n\\lim_{n\\to \\infty}&\\lim_{m\\to\n\\infty}\\int_0^{t_0}\\int_s^{t_0}\\int_{\\Omega}\n(u_{n_t}(\\tau)-u_{m_t}(\\tau))(f(u_n(\\tau),\\tau-t_0+s_0)-f(u_m(\\tau),\\tau-t_0+s_0))dxd\\tau ds \\nonumber \\\\\n&= \\int_0^{t_0}\\left(\\lim_{m\\to \\infty}\\lim_{n\\to\n\\infty}\\int_s^{t_0}\\int_{\\Omega}\n(u_{n_t}(\\tau)-u_{n_t}(\\tau))\\right.\n\\nonumber \\\\\n&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad\n(f(u_n(\\tau),\\tau-t_0+s_0)-f(u_m(\\tau),\\tau-t_0+s_0))dxd\\tau\n\\Big)ds \\nonumber \\\\\n&=\\int_0^{t_0} 0ds=0.\n\\end{align}\n\nHence, from \\eqref{5.30}-\\eqref{5.34}, we see that\n$\\psi_{t_0,\\delta,\\sigma}(\\cdot,\\,\\cdot)\\in Contr(B_{\\sigma})$.\n\n\\emph{Assumption I}:\n\nFrom the definition of $\\rho_{_{\\sigma,\\beta}}$ (see \\eqref{5.6}),\nwe have the conclusion: {\\it for each $\\sigma$ and for any\n$\\varepsilon\n>0$, we can take $t_0$ large enough such that $e^{-\\beta\nt_0}\\rho_{_{\\theta_{-t_0}(\\sigma),\\beta}}\\leqslant \\varepsilon$}.\n\nHence, from $Corollary$ \\ref{c4.3} and \\eqref{5.21}, we only need to\nverify that the function $\\psi'_{t_0,\\,\\sigma}(\\cdot,\\,\\cdot)$\ndefined in \\eqref{5.22} belongs to\n$Contr(B_{\\theta_{-t_0}(\\sigma)})$. To this end, we notice that\n$e^{\\beta t}$ is bounded in $[0,t_0]$ and $\\bigcup_{t \\in\n[0,\\,t_0]}\\phi(t,\\,\\theta_{-t_0}(\\sigma);B_{\\theta_{-t_0}(\\sigma)})$\nis bounded in $X$. The remainder is just a repeat of that for\n$\\psi_{t_0,\\,\\delta,\\,\\sigma}(\\cdot,\\,\\cdot)$ above.\n\nThis completes the proof of $Lemma$ \\ref{l5.5}.\n $\\hfill \\blacksquare$\n\n\n\n\n\\subsection{Existence of pullback attractors}\n\nNow we complete the proof of the main result.\n\n {\\bf Proof of\n$Theorem$ \\ref{t5.2}} \\ From $Lemma$ \\ref{l5.3} and $Lemma$\n\\ref{l5.5}, we see that the conditions of $Theorems$ \\ref{t3.13} and\n\\ref{t3.14} are all satisfied respectively and thus we imply the\nexistence of the pullback attractor. $\\hfill \\blacksquare$\n\n\\begin{remark} \\label{kappa}\nIn this section, we obtain the pullback asymptotic compactness for\nthe non-autonomous wave system \\eqref{1.1}-\\eqref{1.3} by the\ntechnique presented in \\S 4. This technique is different from the\nmethod in \\cite{CLR}. Due to the existence of nested bounded\npullback absorbing set for \\emph{Assumption II}(Lemma \\ref{l5.3}),\nthe pullback $\\kappa-$contraction is equivalent to pullback\nasymptotic compactness (see $Theorem$ \\ref{t3.13}). Thus, in\nprinciple, we could also use the pullback $\\kappa-$contraction\ncriterion to conclude the existence of pullback attractor, using the\ndecomposition method as in \\cite{Fe,SYZ1} (popular for autonomous\nsystems).\n\n\\end{remark}\n\n\n\n\n\n\n\\section{Some remarks}\n\nIn this paper, we discuss the asymptotic behavior of solutions in\nthe framework of pullback attractors. Another interesting question\nis forward attractors; see \\cite{Ch,CKS} for general discussions\nor \\cite{CKR} for practical applications to wave equations with\ndelays. However, for the forward attraction property to hold, one\nusually needs some uniformity about the time-dependent terms\n(i.e., about the symbol spaces \\cite{CV}). As discussed in\ndetails in \\cite{CLR1,LLR}, for general non-autonomous\ndissipative systems, how to obtain the forward attraction\nproperties is an open problem if without this uniformity\nassumption.\n\nFor our problem, under the \\textbf{Assumption II} in \\S 1, we\nindeed obtain a bounded uniformly absorbing set in the sense of\n\\cite{CV} in $Lemma$ \\ref{l5.3} (or see Haraux\\cite{Ha1}). If we\nassume further that $g$ satisfies some additional conditions, e.g.,\n$g$ is translation compact or $g\\in\nW^{1,\\infty}(\\mathbb{R};\\,L^2(\\Omega))$, then by the same method, we\ncan verify the family of processes (see \\cite{CV} for more details)\ncorresponding to the non-autonomous wave system\n\\eqref{1.1}-\\eqref{1.3} is uniformly asymptotically compact and thus\nhas a uniform (w.r.t. $\\sigma \\in \\Sigma$) attractor in the sense of\n\\cite{CV}. However, for the case of \\textbf{Assumption I}\n in \\S 1, it appears difficult to discuss the forward\nattraction for $g$ satisfying only \\eqref{1.10}.\n\nFor the \\emph{autonomous} case of \\eqref{1.1}-\\eqref{1.3},\nrecently, Chueshov \\& Lasiecka \\cite{CL3} have shown a general\nresult for the existence of global attractor, and they allow $p\n=5$, i.e., the so-called critical interior damping. In their\n\\emph{autonomous} case, it is true that for all $0\\leqslant\ns\\leqslant t$,\n\\begin{equation}\\label{6.1}\n\\int_s^t\\int_{\\Omega}h(u_t)u_tdxd\\tau \\leqslant C_R,\n\\end{equation}\nwhere $C_R$ depends only on the norm of initial data, but\nindependent of time instants $s$ and $t$. However, for our\nnon-autonomous case, this constant may depend on time instants $s$\nand $t$ (e.g., see \\eqref{5.16},\\eqref{5.17}), and thus in our\nproofs, we require (at least, technically) that the growth order of\n$h$ to be strictly less than $5$: $p<5$.\n\nMoreover, in the present paper, we use the {\\it pullback asymptotic\ncompactness} to obtain the existence of pullback attractors of\nnon-autonomous hyperbolic systems. This is mainly based on a\ntechnical method for verifying {\\it pullback asymptotic compactness}\nin \\S 4. However, for other non-autonomous systems\nor using other techniques (e.g., the decomposition method), the\npullback $\\kappa-$contraction criterion may be more appropriate\nfor proving the existence of pullback attractors.\n\nFinally, we point out that all the contents in this paper can be\nexpressed by the framework of \\emph{processes}, instead of\ncocycles, as in \\cite{CKR,CR,CLR1,LLR}; see also \\cite{CV}\n for more results about processes.\n\\\\\n\n\n\\noindent{\\bf \\Large Acknowledgement}\n\nThe authors would like to thank Edriss Titi and the referees for\nhelpful comments and suggestions.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Problem Statement}\n\nThe simulation begins with the reactive gas at rest with the initial condition\n$\\rho_0=p_0=Y_0=1$ and $\\mathbf{v_0}=0$.\nThe domain lies in $x \\in [-3,57]$ and $y \\in [-3,12]$ and reflecting slip walls\nare present on all walls except the exit $x=57$.\nThe heat addition is limited to a circle of radius $R=2$ centered at the origin. \nThe simulation uses a heat of reaction $q=15$, specific heat ratio $\\gamma=1.4$,\nactivation energy $E=13.8$, and pre-exponential factor $B=35$.\nHeat is added between $t_a=0.5$ and $t_b=5.25$.\n\n\n\nThe Parallel Adaptive Wavelet-Collocation Method\n(PAWCM) is used to capture the wide range of scales that are present\n\\cite{Kevlahan2005, Vasilyev2000}.\nThe PAWCM combines second generation wavelets with a prescribed\n error threshold parameter $\\epsilon$ to determine which grid points\nare necessary in order to achieve a prescribed level of accuracy.\nThe hyperbolic solver developed for the PAWCM is used to maintain \nnumerical stability and reduce spurious oscillations across jump\ndiscontinuities~\\cite{Regele2009}. \nThe effective grid resolution for the simulation is $15360\\times3072$.\n\n\n\n\n\n\nFigure \\ref{fig:2D_j9} demonstrates the indirect detonation initiation\nprocess by presenting a series of snapshots of temperature contours\ncorresponding to the same events shown in the video. \nHeat is deposited in a circle in the bottom left hand corner\nfrom $0.5\\le t \\le 5.25$. The rapid deposition of heat creates compression waves\nthat propagate away from the initially heated region. \nBefore $t=2$, the reactants inside the deposition region are\nconsumed in a chemical explosion, which adds additional heat to the deposition\nregion. \nIt is difficult to discern from the contour at $t=2$, but the interface \nbetween the burning and reactive gas forms a rippled surface. It is thought that \nthis is a result of the Darrieus-Landau instability at the burning gas interface\nbecause once the reactants are consumed the growth of surface fluctuations ceases.\n\\begin{figure}[tb]\n \\centering\n \\includegraphics[width=0.35\\textwidth]{figures\/legend3}\n \\begin{tabular}{cc}\n \\includegraphics[width=0.465\\textwidth]{figures\/T020_j9-crop_t-red} &\n \\includegraphics[width=0.465\\textwidth]{figures\/T140_j9-crop_t-red} \\\\\n \\includegraphics[width=0.465\\textwidth]{figures\/T035_j9-crop_t-red} &\n \\includegraphics[width=0.465\\textwidth]{figures\/T160_j9-crop_t-red} \\\\\n \\includegraphics[width=0.465\\textwidth]{figures\/T050_j9-crop_t-red} &\n \\includegraphics[width=0.465\\textwidth]{figures\/T180_j9-crop_t-red} \\\\\n \\includegraphics[width=0.465\\textwidth]{figures\/T070_j9-crop_t-red} &\n \\includegraphics[width=0.465\\textwidth]{figures\/T200_j9-crop_t-red} \\\\\n \\includegraphics[width=0.465\\textwidth]{figures\/T100_j9-crop_t-red} &\n \\includegraphics[width=0.465\\textwidth]{figures\/T220_j9-crop_t-red} \\\\\n \\includegraphics[width=0.465\\textwidth]{figures\/T120_j9-crop_t-red} &\n \\includegraphics[width=0.465\\textwidth]{figures\/T240_j9-crop_t-red} \\\\\n \\includegraphics[width=0.46\\textwidth]{figures\/axesbottom2} &\n \\includegraphics[width=0.46\\textwidth]{figures\/axesbottom2}\n \\end{tabular}\n\\caption{Sequence of temperature contours demonstrate the multidimensional indirect \ndetonation formation process for times $2\\le t \\le 24$.}\n\\label{fig:2D_j9}\n\\end{figure}\n\n\nAt some point shortly after $t=2$ when the compression waves first reflect off\nthe left and bottom walls the compression waves become fully discontinuous shock\nwaves. The reflected and transmitted shocks form Mach stems that propagate in\nthe positive $x$- and $y$-directions. The reflected waves impinge on the \nburnt-unburnt gas interface and induce Richtmyer-Meshkov instabilities, which then\nincrease the fluctuation magnitude at the material interface. This can be initially\nobserved in the $t=3.5$ contour.\n\nAt about $t=2.5$, a second explosion occurs in the lower left corner\nwhen the shock waves reflect off the bottom and left walls and \nraise the pressure in that region in a duration short enough\nthat the temperature rises with pressure. The reactive gas explodes once it has\nreached a sufficient temperature. \n\nIn the $t=5$ frame, the original outward propagating shock wave has reflected off the\nleft and bottom walls and the Mach stems are clearly visible in the temperature\ncontour. On the left wall, the leading edge of the shock wave is just about to\nreflect off the upper wall, starting in the upper left corner.\nWhen reflection occurs on the upper boundary, a hot spot appears in the upper\nleft-hand corner of the channel, characterized by substantial local inertial\nconfinement. This hot spot releases heat and generates compression waves that \npropagate away from the hot spot location. \n\nAt $t=7$, Fig.~\\ref{fig:2D_j9} shows the reflected wave re-enters the reacted region and\nis refracted, which induces an additional longitudinal component to the wave direction.\nThe transverse waves compress and heat previously unreacted fuel pockets, which ignite\nand help produce additional longitudinal waves, as well as sustaining the transverse \nwaves the reverberate off the top and bottom walls.\n\nKelvin-Helmholtz roll-up instabilities are clearly visible in frames $t=[10, 12, 14]$\nat the burnt-unburnt gas interface with a fairly high level of detail. The \nexistence of such a detailed interface serves as an indicator that any numerical\ndiffusion present in the algorithm has been minimized to the point that these \nfeatures are possible to capture.\n\nAt about $t=14$ the heat release rate by the preheated gas begins to escalate.\nThis acceleration in\nheat release can be observed in the temperature contour sequence as the rapid \nconsumption of fuel starting at $t=14$ and ending at $t=24$ with the formation of \nthe over-driven detonation wave emerging from the lead shock front. \n\n\n\\section*{Acknowledgements}\nJ.D.R. would like to thank Guillaume Blanquart for the use of his computational\nresources to perform this simulation.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIn the centers of many late-type galaxies one finds massive stellar clusters, the nuclear star clusters \n\\citep{Phillips_96,Matthews_97,Carollo_98,Boeker_02}. The nuclear clusters are central light overdensities on a scale of about 5~pc \\citep{Boeker_02}. \nAlso the central light concentration \nof the Milky Way \\citep{Becklin_68}, is a nuclear star cluster \\citep{Philipp_99,Launhardt_02}. \nNuclear clusters are comparably dense as globular clusters, but are typically more massive \\citep{Walcher_05}.\nIn some galaxies the clusters coexist with a supermassive black hole (SMBH), see e.g. \\citet{Graham_09}.\nThe formation mechanism of nuclear stars cluster is debated \\citep{Boeker_10}.\nThere are two main scenarios: on the one hand formation of stars in dense star clusters, which are possibly globular clusters, followed by cluster \ninfall \\citep{Tremaine_75,Andersen_08,Capuzzo_08}. On the other hand in situ star formation from the cosmological gas inflow \n\\citep{Milsavljevic_04,Emsellem_08}.\n\n\nDue to the proximity of the center of the Milky Way the nuclear cluster of the Milky \nWay can\n be observed in much higher detail than any other nuclear cluster \\citep{Genzel_10}. It is useful to shed light on the properties and the origin\nof nuclear clusters in general. \nThe access is hampered by the high foreground\n extinction of A$_{\\mathrm{Ks}}=$2.42 \\citep{Fritz_11}. \n Therefore the light profile is uncertain: \\citet{Becklin_68,Haller_96,Philipp_99} find a central light excess with a size of at least 400$\\arcsec$\non top of the bulge. In contrast, \\citet{Graham_09,Schoedel_11a} claim that the nuclear \ncluster transits at 150$\\arcsec$ to the bulge. On a larger scale \\citep{Launhardt_02} find\nthat there is another stellar component between nuclear cluster and bulge,\nan edge-on disk of 3$^{\\circ}$ length, the nuclear disk, which is flattened by a factor five. It corresponds roughly to the central molecular zone \\citet{Launhardt_02}. \nThe flattening is qualitatively confirmed in\n \\citet{Catchpole_90} and \\citet{Alard_01}. Further in, the flattening is less well constrained.\n\\citet{Vollmer_03} mention an ellipsoid of 1.4:1 in the range of about 200$\\arcsec$.\nWithin 70$\\arcsec$ the light distribution seems to be circular \\citep{Schoedel_07} although a quantification is \nmissing therein. \nThe majority of the stars in the central R$\\approx$2.5 pc are older than 5 Gyrs \n\\citep{Blum_02,Pfuhl_11}. Only in the center (r$\\approx\\,$0.4 pc) the light\n is dominated by 6 Myrs old stars \\citep{Forrest_87,Krabbe_91,Paumard_06,Bartko_09} \nwith a top-heavy IMF \\citep{Bartko_10,Lu_13}. \n\n\n\nThe mass of the SMBH is well-determined to be $ 4.3 \\times10^6$ M$_{\\odot}$ with an error of less than 10\\% \n\\citep{Gillessen_09,Ghez_09}. In contrast the mass of the nuclear cluster is less well constrained. The mass within \nr$\\leq1$pc is $ 10^6$ M$_{\\odot}$ with about 50 \\% systematic uncertainty \\citep{Genzel_10}. In the central parsec the potential is dominated by the SMBH which makes measurements of the additional stellar mass \nmore difficult. \nFurther, possibly due to the surprising core in the profile of the stellar distribution \\citep{Buchholz_09,Do_09,Bartko_10}\nalso recent Jeans modeling attempts \\citep{Trippe_08,Schoedel_09a} to recover the right SMBH mass fail. This possibly biases also the stellar mass determination there. \nAs a result also the newer works of \\citet{Trippe_08} and \\citet{Schoedel_09a} have still about \n50 \\% stellar mass uncertainty in the central parsec, similar to \\citet{Haller_96} and \\citet{Genzel_96} who used fewer radial velocities.\nThe mass determination outside the central parsec is mainly based on relatively few radial velocities of late-type stars, either \nmaser stars \\citep{Lindqvist_92b,Deguchi_04}, or stars with CO band-heads \\citep{Rieke_88,McGinn_89}. With the absence of proper motion information outside the center the extent of anisotropy is there also not well constrained.\nAlso radial velocities of gas in the circumnuclear disk (CND) were used for mass determinations outside the central \nparsec \\citep{Genzel_85,Serabyn_85,Serabyn_86}.\n\n\nThus, although the central cluster of the Milky Way is the closest\nnuclear cluster, its mass and luminosity profiles are still\npoorly constrained. Here and in \\citet{Chatzopoulos_14} we improve\nthe constraints on these parameters. In this paper, we first present\nimproved observational data: we extend the area for which all\nthree stellar velocity components are measured, to r$\\approx 4$ pc. We also\nconstruct a surface density map of the nuclear cluster out to r$_{\\mathrm{box}}= 1000\\arcsec$. Then we\npresent a first analysis of the new data, using simple isotropic\nspherical Jeans models. With these assumptions, the analysis is\nrelatively fast and we can easily investigate several systematic effects. Because these models do not fully match the nuclear cluster, we employ\nmore detailed axiymmetric models in \\citet{Chatzopoulos_14} which\nfit the data well.\n\n\nIn Section~\\ref{sec:dataset} we present our data, and describe the extraction of velocities in Section~\\ref{sec:deriv_vel}. In Section~\\ref{sec:lum_prop} we derive \nthe surface\ndensity properties of the nuclear cluster\n and fit it with empirical models. In Section~\\ref{sec:an_method} we describe the kinematic properties mostly qualitatively and\nuse Jeans modeling to estimate the mass of the nuclear cluster.\nWe discuss our results in Section~\\ref{sec:discussion} and conclude in Section~\\ref{sec:conclusions}. Where a distance to the GC needs to be \nassumed, we\nadopt $R_0=8.2\\,$kpc \\citep{Reid_93,Genzel_10,Gillessen_13,Reid_14}.\nThroughout this paper we define the projected distance from Sgr~A* as R and\nthe physical distance from Sgr~A* as r \\citep{Binney_08}.\n\n\n\n\n\n\\section{Data Set}\n\\label{sec:dataset}\n\n\nIn this section we describe the observations used for deriving proper \nmotions, radial velocities, and the luminosity properties of the nuclear \ncluster. \n\n\\subsection{High Resolution Imaging}\n\\label{sec:highres_im}\n\n\nFor deriving proper motions and for determining the stellar \ndensity profile in the center (R$_{\\mathrm{box}}\\approx20\\arcsec$)\nwe use adaptive optics images with a resolution of\n$\\approx 0.080\\arcsec$.\nIn the central parsec we use the same NACO\/VLT images \n\\citep{Lenzen_03,Rousset_03} as described in\n\\citet{Trippe_08} and \\citet{Gillessen_09}. We add images obtained in \nfurther epochs since then, in the 13 mas\/pixel scale matching to the\n\\citet{Gillessen_09} data set and in the 27 mas\/pixel scale extending \nthe \\citet{Trippe_08} data set.\nThe images are listed in Appendix~\\ref{sec:ap_prom_mot}.\n\nFor obtaining proper motions outside\nthe central parsec we use adaptive optics images covering a larger field of view. These \nare four epochs of NACO\/VLT images,\none epoch of MAD\/CAMCAO at the VLT \\citep{Marchetti_04,Amorim_06} and \none epoch of\nHokupa'a+Quirc \\citep{Graves_98,Hodapp_96} Gemini North images, see Appendix~\\ref{sec:ap_prom_mot}. \nMost images cover the Ks-band, some are obtained with H-band or narrower \nfilters within the K-band.\nThe VLT images are flat-fielded, bad pixel corrected and sky subtracted.\nIn case of the Gemini data we use the publicly available images\\footnote{Based on the Data Set of the Gemini North Galactic Center Demonstration Science.}. \nThese \nimages are combinations of reduced images with nearly the same pointings.\n\n\n \n\\subsection{Wield Field Imaging}\n\\label{sec:wide_im}\n\n\nTo obtain the structural properties of the nuclear cluster\noutside of the central R$_{\\mathrm{box}}=20\\arcsec$ we use two additional\ndata sets. Here, high resolution is less important, but area coverage and extinction correction are the keys.\n\n\\begin{enumerate}\n\\item Closer to the center we use HST WFC3\/IR data\\footnote{\nBased on observations made with the NASA\/ESA Hubble Space Telescope, obtained from the Data Archive at the Space Telescope Science Institute, \nwhich is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are \nassociated with program 11671 (P.I. A. Ghez).\n}.\nThe central R$_{\\mathrm{box}}\\approx68\\arcsec$ around Sgr~A* are covered in the filters M127, M139 and M153. We use the images in M127 and M153 in our analysis.\n We optimize the data reduction compared to the pipeline in order to achieve Nyquist-sampled final pixels. We use MultiDrizzle to combine the different images in the same filter, with a drop size parameter of 0.6 with boxes and a final pixel \nsize of 60 mas. These choices achieve a high resolution and still samples the image homogeneously enough to avoid pixels without flux.\nWe do not subtract the sky background from the images, since it is \ndifficult to find a source-free region from where one could estimate it.\nSince we use only point source fluxes this does not affect our analysis. \nWe change the cosmic removal parameters to 7.5, 7, 2.3 and 1.9 \nto avoid removal of actual sources. Due to the brightness of the GC sources cosmics are only of minor importance.\nThe final images have an effective resolution of 0.15$\\arcsec$.\n\\item On a larger scale we use the public VISTA Variables in the Via\n Lactea Survey (VVV) data obtained with VIRCAM \\citep{Saito_12}. We use from data release 1 the central tile 333 in H and Ks-band. \nThe data contain flux calibrated, but not \nbackground subtracted images. The resolution is about 1$\\arcsec$. These images cover more than one square degree around the GC. We only\nuse R$_{\\mathrm{box}}\\approx1000\\arcsec$. \nIn the center the crowding is severe and nearly all sources are saturated in the Ks-band. This is not a limitation, since \nthere we can use the higher resolution images from NACO and WFC3\/IR.\n\\end{enumerate}\n\n\n\\subsection{Spectroscopy}\n\nFor obtaining spectra of the stars we use data cubes obtained with the \nintegral field spectrometer SINFONI \\citep{Eisenhauer_etal2003, \nBonnet_03}.\nWe use data with the combined H+K-band (spectral resolution 1500) and \nK-band (spectral resolution 4000) grating.\nThe spatial scale of the data varies between the smallest pixel scale \n($12.5\\,$mas pixel$^{-1}$ $\\times$ $25\\,$mas pixel$^{-1}$) and the \nlargest scale $125\\,$mas pixel$^{-1}$ $\\times$ $250\\,$mas pixel$^{-1}$. \nThe spatial resolution of the data correspondingly is between 70 \nmas and 2$\\arcsec$. It has been matched at the time of the \nobservations to the stellar density that increases steeply toward\nSgr~A*. Thus, we can detect many sources in the center, and can sample \nalso large areas at large radii.\nWe apply the standard data reduction SPRED \\citep{Abuter_06, \nSchreiber_04} for SINFONI data, including detector calibrations (such as \nbad pixel\ncorrection, flat-fielding, and distortion correction) and cube \nreconstruction. The wavelength scale is calibrated with emission line \nlamps and finetuned with atmospheric\nOH lines.\nFinally, we correct the data for the atmosphere by dividing with a late \nB-star spectrum and multiplying with a blackbody of the same temperature.\n\n\n\n\\section{Determination of Velocities}\n\\label{sec:deriv_vel}\n\nThe primary constraint for estimating the mass of the nuclear cluster \n comes from velocity data. \nAll our velocity dispersions are derived from individual velocities. Wrongly estimated errors for the velocities cause a bias in the calculated dispersions. Therefore, realistic error estimates are essential in our analysis. \n\n\n\n\n\n\n\\subsection{Proper Motions}\n\\label{sec:proper_motion}\n\nWe combine in this work proper motions obtained from four different data sets: central field, \nextended field, large field and outer field, see Figure~\\ref{fig:_vel_map}. \\citet{Trippe_08} used the extended field.\n In the central field the stellar crowding is high and a different analysis is needed than further out. Also, \n the number of epochs and their similarity decrease from inside out \nwhich requires a more careful and separated error analysis further out.\nWe give here a short overview of the data and methods used.\nThe details are explained in Appendix~\\ref{sec:ap_prom_mot}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.99 \\columnwidth,angle=-90]{f1.eps} \n\\caption{ \nDistribution of stars with radial velocities and proper motions. \nFor the proper motions we combine four different data sets: the central field in the central 2$\\arcsec$. The extended field from 2$\\arcsec$ outwards \nto $\\approx$20$\\arcsec$, the large field from there outwards to $\\approx$40$\\arcsec$ and a separate, outer \nfield in the north. \nThe yellow lines define our \ncoordinate system, shifted Galactic coordinates l$^*$\/b$^*$ \\citep{Deguchi_04, Reid_04}\n where the center is shifted to \nSgr~A*. \n} \n\\label{fig:_vel_map}\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{itemize}\n\\item In the central (R$\\leq 2\\arcsec$) we use the same method for astrometry as in \\citet{Gillessen_09}, see Appendix~\\ref{sec:pm_center}. \nWe now track stars out to r$_{\\mathrm{box}}\\approx2\\arcsec$. With this increased \nfield of view, we more than double the number of stars compared to \\citet{Gillessen_09}.\nWe use 79 stars with a median magnitude of m$_\\mathrm{Ks}=15.45$.\nSome of the old, late-type stars have significant accelerations \\citep{Gillessen_09}, but the curvature of their orbits\nis not important compared to their linear motion, and thus the linear motion approximation is sufficient.\nDue to the sample size the Poisson error of the dispersions dominates all other dispersion errors.\n\\item In the radial range between a box radius of 2$\\arcsec$ and 20$\\arcsec$ (extended field) we expand slightly on the data and method used by \n\\citet{Trippe_08}. \nWe do not change the field of view and the selection of well isolated stars compared to \\citet{Trippe_08}. \nThe number of stars with velocities has increased a bit due to the addition of new images.\nIn total we have dynamics for 5813 stars in this field. The median magnitude is m$_\\mathrm{Ks}=15.76$.\nWe obtain a proper motion dispersion of $\\sigma_{\\mathrm{1D}}=2.677 \\pm 0.018$ mas\/yr using all\n stars and averaging the two dimensions. \nThe error includes only Poisson noise, which is likely the dominating error, because we measure identical dispersion values and errors for the bright and faint half of \nthe sample, see Appendix~\\ref{sec:tri_fie}. \n\\item Outside a box radius of 20$\\arcsec$ (large field) nearly no proper motions were available, with the exception of a small area in \\citet{Schoedel_09a}, \nwhich however the\nauthors did not use for their analysis. From the images with sufficiently good AO correction we obtain velocities for 3826 stars, see \nAppendix~\\ref{sec:big_fie}. The median magnitude is m$_\\mathrm{Ks}=15.30$.\nFor the proper motion dispersion we obtain $\\sigma_{\\mathrm{1D}}=2.330 \\pm 0.019$ mas\/yr. \nThe comparison of the dispersion for fainter and brighter stars shows that the error on the dispersion (after subtracting the velocity errors in \nquadrature) of the fainter stars is 2$\\,\\sigma$ larger than for the brighter stars. Since this is barely significant the errors again include only Poisson noise. \n\\item For expanding our coverage of proper motions to 78$\\arcsec$ we use the outer field. These data do not cover the full circle \naround Sgr~A*, but only a selected field. We have two epochs for this field: \nGemini data from 2000 and NACO data from the 29$^{\\mathrm{th}}$ May\n 2011. \nWe exclude in this field stars with m$_\\mathrm{Ks}>15.6$, because we obtain a higher velocity dispersion for them than for brighter\n stars, see Appendix~\\ref{sec:out_fie}.\nIn that way we select 633 proper motion stars with a median magnitude of m$_\\mathrm{Ks}=14.79$.\nFrom these stars we obtain $\\sigma_{\\mathrm{1D}}=1.918 \\pm 0.038$ mas\/yr using both dimensions together.\nWe assume again that the Poisson error dominates over other error sources for the stars selected.\n\\end{itemize}\n\n\n\n\\subsection{Radial Velocities}\n\\label{sec:rad_vel}\n\nSince we can resolve the stellar population in the GC, we measure velocities of single stars and obtain\n dispersions by binning stars together. \nWe use two different sources for the velocities (see also Appendix~\\ref{sec:app_rv}):\n\n\n\\begin{itemize}\n\\item Within R$<95\\arcsec$ \nwe use our SINFONI data to measure the radial velocities of the late-type giants in the GC, see Appendix~\\ref{sec:app_our}\n for the details. We obtain radial velocities for 2513 stars.\nThe median velocity error of these velocities is about 8 km\/s as \nwe obtain from comparing independent measurements for multiple covered stars. \nThe line-of-sight velocity dispersion of these stars is $\\sigma_z=102.2 \\pm 1.4$ km\/s.\nThe velocity errors have less than 1$\\,\\sigma$ influence on the measured dispersion. \nWhen we take into account the uneven distribution in $l^*$ (Figure~\\ref{fig:_vel_map}) with binning (to avoid the influence of rotation)\nwe obtain as the total radial motion of the nuclear cluster \n$6.1 \\pm 3.8$ km\/s. By definition it should be 0.\n This indicates that also our velocity calibration is probably correct. \nIn conclusion it seems likely that Poisson errors are dominating the dispersion uncertainty for our radial velocity sample, and we only include those in the final analysis.\n\n\n\\item\nOutside of 110$\\arcsec$ \n we use radial velocities from the literature \\citep{Lindqvist_92a,Deguchi_04} out to $\\approx\\,$3000$\\arcsec$ (Appendix~\\ref{sec:app_maser}).\nBoth of them used maser radial velocities. We match the two samples and use each star only once. \nAs a side product of this matching, we confirm that the typical velocity uncertainty is less than 3 km\/s as stated in \\citet{Lindqvist_92a}\nand \\citet{Deguchi_04}.\nDue to the big position errors in these radio data it is difficult to find the corresponding IR stars.\nWe therefore exclude from the combined list the eleven stars that overlap spatially with the areas in which we obtained spectra, with the aim of \navoiding using stars twice. Overall we use 261 radial velocities outside the central field. \n\\end{itemize}\n\nThe radial velocity stars sample is not identical with the proper motion stars sample. The majority of the faint proper motions stars have\nno radial velocities and many of the outer radial velocity stars have no proper motion coverage. The radial velocity stars have a median magnitude of m$_{\\mathrm{Ks}}=13.66$, while the proper motion stars are in the median 1.85 magnitudes fainter. We thus use the common (e.g. \\citet{Vandeven_06,Trippe_08,Marel_10}) approach of using different stars for proper motions and radial velocities. Otherwise we would have only 1840 stars in both samples, with an inhomogeneous spatial distribution. The difference in magnitude is likely not a problem, because the stars in both samples are giants. Their different luminosities are mainly caused by different evolutionary stages, not by different ages and masses. Thus, mass dependent effects, like mass segregation, affect both samples in same way and we can safely use all 10368 proper motion and all 2774 radial velocity stars.\n\n\n\\subsection{Sample Cleaning}\n\\label{sec:out_rej}\n\n\nIn order to probe the gravitational potential of the GC out to more than 100$\\arcsec$ we \nneed to use a stellar population in dynamical equilibrium extending over a sufficiently large radial range. \nThe late-type population in the GC is suited.\nA small fraction of the stars in our sample does not belong to this population and is therefore excluded as far as possible from our analysis. We have the following three exclusion criteria (see Appendix~\\ref{sec:app_clean} for the details):\n\\begin{itemize}\n\\item\n\nThe young stars follow different radial profiles \\citep{Bartko_10} and have different dynamics. \nWe thus exclude the young stars from our sample. These are the early-type stars, the WR-, O- and B-stars \\citep{Paumard_06,Bartko_09,Bartko_10}\nand the red supergiant IRS7,\n that has the same age as the WR\/O-stars of around 6 Myrs \\citep{Blum_02,Pfuhl_11}.\nThese stars are the most important contamination, especially close to Sgr A*. Due to\nmissing spectra we cannot clean our proper motion sample\ncompletely from these stars. However, we choose the selection criteria such (Appendix~\\ref{sec:app_early})\nthat in all radial ranges not more than about 4\\% of the stars are young. \n\\item\nWe also exclude stars, which due to their low extinction \nbelong to the Galactic disk or bulge, similar to what is done in \\citet{Buchholz_09}, see Appendix~\\ref{sec:app_foreg}.\nBecause in some areas images in a second filter are missing we cannot clean our sample\ntotally from foreground stars.\nSince these stars are not clustered this pollution is nowhere important. \nIntegrated we include maybe about 1\\% foreground stars in the GC sample.\n\\item\nExtreme outliers in velocity are visible in some subsamples (Appendix~\\ref{sec:app_fast}). About 1\\% of the proper motion stars outside of the\n extended field\nare outliers probably caused by measurement flukes. \nIn the maser sample about 5\\% are outliers (\\citet{Lindqvist_92a,Deguchi_04}\n and Section~\\ref{sec:fast_stars}). (See Appendix~\\ref{sec:app_fast} for our procedure of outlier identification.) Their high velocities probably indicate that they belong to another population. \nThe other samples are free from obvious outliers.\n\\end{itemize}\n\n\n\n\\section{Luminosity Properties of the Nuclear Cluster}\n\\label{sec:lum_prop}\n\n\nFor obtaining masses from Jeans modeling it is necessary to know the space density distribution of the tracer population\n\\citep{Binney_08}.\nIn principle the dynamics alone constrain the tracer distribution \\citep{Trippe_08}. However, that constraint is so weak that strong priors are necessary.\nBetter constraints are possibly by using the surface density \\citep{Binney_08,Genzel_96,Schoedel_09a,Das_11}.\n Given the inhomogeneous and spatially incomplete nature of our dynamics sample, it would be very \ncumbersome (if not impossible) to derive the spatial distribution of tracers from that data set. \n\nIt is easier to derive the tracer surface density of our tracers from other more complete data set.\nThere are two possibilities:\n \\begin{itemize}\n\\item One can use the luminosity profile of the cluster. The usual assumption of a constant mass to light ratio will fail, however, because of the \nyoung stars in the center that dominate the luminosity. Hence, spectral information is needed in addition.\n\\item Another method for the GC cluster is to extract the surface (stellar) density profile. \nIt is also necessary to correct for the early-type stars that contribute an important fraction of all stars in the center.\n\\end{itemize}\n\nWith these data we can obtain the density profile also in radial ranges where we have only very few velocities. Due to projection effects these radii are also important. \nIn Section~\\ref{sec:number_density} we obtain density maps, the radial profile of the nuclear cluster and also the profile in direction of and \nperpendicular to the Galactic plane. \nIn the Jeans modeling (Section~\\ref{sec:jeans_modelling}) we \nwill actually fit our radial density and dynamics data at once, since also the dynamics yield a weak constraint on the \ntracer profile. However, in order to make this a problem with few parameters, we identify beforehand (Section~\\ref{sec:fitting_profile}) a functional form that gives an\nsatisfying description of the radial distribution of the velocity tracers. \nIn Section~\\ref{sec:flattening} we use the density maps to separate nuclear cluster and nuclear disk.\nFinally, we obtain in Section~\\ref{sec:total_luminosity} the total luminosity of the nuclear cluster.\n\n\n\n\n\\subsection{Deriving Density Profiles}\n\\label{sec:number_density}\n\nFor deriving the light properties we use three different data sources (Section~\\ref{sec:dataset}).\n In case of the star density we use for all areas the dataset with the highest resolution. In case of the flux density\n we omit the WFC3\/IR data because it is obtained in different filters. \n\n\n\nWe use the following steps to derive the stellar distribution:\n\\begin{itemize}\n\\item We exclude very bright foreground stars and GC clusters like the Arches. \n\\item We correct the star counts for completeness if necessary, see Appendix~\\ref{sec:obt_lum}.\n\\item We exclude stars younger than 10 Myrs from our sample.\n\\item We correct for extinction using two NIR filters. The resulting map is still patchy in some areas because of the optical depth being too high for\n correction.\n\\item We create masks to exclude the emission from these areas. \nThe masks are defined such that the maps appear smooth and symmetric\nin $|l^*|$ and $|b^*|$. \nSince only few pixels are masked out, the overall bias is small.\n\n\\item The areas excluded are masked out for two-dimensional fitting. They are replaced with the average of the other areas at the same $|l^*|$ and $|b^*|$ for creation of radial profiles and for visualization.\n\\end{itemize}\nNot all steps are necessary for all data subsets, and the procedures are described in detail in Appendix~\\ref{sec:obt_lum}.\n\nIn Figure~\\ref{fig:_surf_bri2} we present the profiles obtained.\nThe profiles obtained from the integrated flux and from stellar number counts are \nsimilar but not identical within the errors.\nOn the one hand, in the case of the flux profile the assumption of screen extinction is a simplification.\nOn the other hand due to the high source density in the GC, magnitudes of point sources are less reliable than the extended flux in the GC.\nTo be conservative we use \nboth profiles to fit the mass in our Jeans modeling (Section~\\ref{sec:jeans_modelling}) and \nwe include the scatter between the obtained masses\nin the mass error budget.\nSplitting the number \ncounts based profile into a $|l^*|$ and $|b^*|$ component (Appendix~\\ref{app:meas_flat} and Figure~\\ref{fig:_surf_bri5b}) yields profiles similar to the model of \\citet{Launhardt_02}.\n\n \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f2.eps}\n\\caption{\nRadial distribution of stars. We construct the radial stellar\/flux density profile from \nNACO, WFC3\/IR and VISTA images (in order of increasing field of view).\n} \n\\label{fig:_surf_bri2}\n\\end{center}\n\\end{figure}\n \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f3.eps}\n\\caption{ Stellar density in and orthogonal to the Galactic plane. \nDue to the limited number counts, the density, especially close to the center (R$<68\\arcsec$), is not only measured exactly in that planes, see Appendix~\\ref{app:meas_flat}.\nFrom \\citet{Launhardt_02} we show the model presented in their Figure~12, scaled to our data.\nTheir model does not include Galactic Disk and bulge, which are included in our data. \n} \n\\label{fig:_surf_bri5b}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Spherical Fitting of the Stellar Density Profile}\n\\label{sec:fitting_profile}\n\n\nWe now fit the density profile assuming spherical symmetry. \nWe relax this assumption in Section~\\ref{sec:flattening}. \n For the Jeans \nmodeling we need a space density profile. Nevertheless, here we firstly parametrize the projected density in order to compare our data set with the \nliterature before we fit the space density. Often we use power laws for first comparisons. The power law of the projected density is defined\nin the following way: $\\Sigma(R)=R^{-\\Delta} $, and the power law of the space density as: $\\rho(r)=r^{-\\delta} $. $\\Delta$ and $\\delta$ are our definitions for power law\nslopes. We use $\\delta_\\mathrm{L}$ for flux and star \ncounts and $\\delta_\\mathrm{M}$ for mass.\n\n\\subsubsection{Nuker models}\n\\label{sec:fitting_profile_nuk}\n\nUsually projected surface densities in the GC were fitted with (broken) power laws \\citep{Genzel_03b}. A generalization to two slopes is the Nuker \nprofile \\citep{Lauer_95}: \n\n \n\\begin{equation}\n\\begin {split}\n\\Sigma(R)=\\Sigma(R_b) 2^{(\\beta-\\Gamma)\/\\alpha} \n\\left(\\frac{R}{R_b}\\right)^{-\\Gamma} \\left[1+\\left(\\frac{R}{R_b}\\right)^{\\alpha}\\right]^{(\\Gamma-\\beta)\/\\alpha}\n\\end{split}\n\\label{eq:nuker_prof}\n\\end{equation}\n\n\nTherein, $R_b$ and $\\Sigma_b$ are the break radius and the density at the break radius. The exponent $\\Gamma$ is the inner (usually flatter) power law \nslope, \nand $\\beta$ is the outer (usually steeper) power law slope. The parameter $\\alpha$ is the sharpness of the transition; a large value of $\\alpha$ \nyields a very sharp\ntransition, essentially a broken power law. Using $\\alpha=100$ (fixed) our fits can be compared with the literature.\n\nDue to the break at 220$\\arcsec$ (Figure~\\ref{fig:_surf_bri2}), we restrict the Nuker fits to $r<220\\arcsec$ (Table~\\ref{tab:_surf_fit1}).\nRows 1 and 2 in Table~\\ref{tab:_surf_fit1} \n give our fits for stellar number counts and the flux profile, respectively. Row 1 can be directly \n compared to the \nliterature. \\citet{Buchholz_09} conducted the largest area (r$<20\\arcsec$) study so far. They obtained\n$\\Gamma=-0.17\\pm0.09 $ and $\\beta=0.70\\pm0.09$ for $R_b=6.0\\arcsec$. This fit is broadly consistent with our data, although we obtain \nfor this radial range no clear sign for a power law break. The break radius (6$\\arcsec$) of \\citet{Buchholz_09} is smaller than ours, although these authors do not \ncite an error. \nThe binned data of \\citet{Buchholz_09} look similar to our data. The same is true for the data in \\citet{Bartko_10} who do not attempt to fit the\n profile. \nThese works find a very weak increase of the density\nwith radius inside of $\\approx5\\arcsec$ and then a somewhat stronger decrease of the density with radius further out. This is also the \ncase in \\citet{Do_09,Do_13}. \n\\citet{Do_13} used data out to 14$\\arcsec$ and find a single power law with a slope of $\\Delta_\\mathrm{L}=0.16\\pm0.07$. \n\n\nOur data seem to be consistent with a break radius for the late-type stars around 20$\\arcsec$. When counting stars regardless of their age, a smaller break radius of about 8$\\arcsec$ is found \\citep{Genzel_03b,Schoedel_07}. \nFor our large radial coverage, a single power law obviously fails to fit our data, even when we restrict our data to the \nrange for which we have spectral classifications ($R<90\\arcsec$). The best fit broken power law has then $\\chi^2$\/d.o.f$=11.43\/20$ \n(Row~3 in Table~\\ref{tab:_surf_fit1}), while a single power law yields $\\chi^2$\/d.o.f$=53.12\/22$. \n\n\nThis indicates that the \ntransition between the two slopes is softer than for a broken power law. \nWe also test whether a very small $\\alpha$ (very soft transition) can be excluded. However, there are nearly no differences in the $\\chi^2$ for small $\\alpha$; all $\\alpha$ have nearly the same probability. Therefore, we give only upper limits for $\\alpha$: they are $1.15$ and $0.65$ for the flux and star count data, respectively.\nFigure~\\ref{fig:_surf_fit1} shows our data together with some fits.\n\n\nThe older literature used single power law profiles to describe their flux density profiles. Between 20$\\arcsec$ and 220$\\arcsec$ our data can be fit relatively well by a \nsingle power law of $\\Delta_\\mathrm{L}=0.765 \\pm 0.018$ (stellar density) and of $\\Delta_\\mathrm{L}= 0.915 \\pm 0.015$ (flux density). These slopes show again that the two data sets are not\n consistent.\n\\citet{Becklin_68,Allen_83},\n \\citet{Haller_96} obtain a slope of $\\Delta_\\mathrm{L}=0.8$, while \\citet{Philipp_99} obtain a flatter slope of\n$\\Delta_\\mathrm{L}=0.6$. \n\n\nIn contrast to these data the flux profiles from \\citet{Graham_09}\\footnote{ \\citet{Graham_09} used the profile from \\citet{Schoedel_08a} constructed from public \n2MASS images.} and \\citet{Schoedel_11a} \ndo not follow a single power law, they flatten at about 80$\\arcsec$ \nindicating that the \n bulge dominates already there. Likely this bright bulge is an artifact caused by a missing sky subtraction.\nThe floor level in the profiles of \\citet{Graham_09,Schoedel_11a}, much higher than the bulge floor in the COBE\/DIRBE data of \\citet{Launhardt_02}, fits\nto typical K-backgrounds observation on earth.\nLike \\citet{Vollmer_03} for 2MASS data we subtract the light level towards dark clouds as sky. \n None of \\citet{Becklin_68,Allen_83,Haller_96,Philipp_99,Graham_09,Schoedel_11a} use two color information to correct for extinction in contrast to our work.\nThe single power law fit of \\citet{Catchpole_90} to extinction corrected\nstar counts with a slope of $\\Delta_\\mathrm{L}=1.1$ in the radial range from 140$\\arcsec$ to 5700$\\arcsec$\n is not well comparable to our data set due to the different radial range. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f4.eps} \n\\caption{Nuker fits (with $\\alpha=1$) to the late-type stars surface density, either their star \ndensity or their flux density.\nWe present here fits to the inner data (r$<220\\arcsec$, filled dots). The outer data outside the break is presented by \nopen dots.\nWe also plot the fits of \\citet{Buchholz_09} and \\citet{Do_13}.\n} \n\\label{fig:_surf_fit1}\n\\end{center}\n\\end{figure}\n\n\n\\begin{deluxetable*}{lllllllll} \n\\tabletypesize{\\scriptsize}\n\\tablecolumns{9}\n\\tablewidth{0pc}\n\\tablecaption{Nuker fits of the surface density profile \\label{tab:_surf_fit1}}\n\\tablehead{ No. &data source & radial range & $\\alpha$ & $\\chi^2$\/d.o.f & R$_b$ [$\\arcsec$] & $\\Sigma$(R$_b$) & $\\beta$ & $\\Gamma$ }\n\\startdata\n1&star density & R$<220\\arcsec$ & 100 \t &26.66\/25\t& 23 $\\pm$ 3 & 0.86 $\\pm$ 0.09\t & 0.771 $\\pm$ 0.018 & 0.277 $\\pm$ 0.054 \\\\\n2&flux density & R$<220\\arcsec$ & 100 \t &74.11\/50\t& 24 $\\pm$ 2 & 71.2 $\\pm$ 4.6 & 0.934 $\\pm$ 0.016\t & 0.304 $\\pm$ 0.030 \\\\\n3&star density & R$<90\\arcsec$ & 100 \t &11.43\/20\t& 13 $\\pm$ 3 & 1.11 $\\pm$ 0.15\t & 0.645 $\\pm$ 0.040 & 0.186 $\\pm$ 0.084 \\\\\n4&star density & R$<220\\arcsec$ & 1\t\t &15.28\/25\t& 21 $\\pm$ 14 & 0.84 $\\pm$ 0.28\t & 0.972 $\\pm$ 0.089\t & 0.059 $\\pm$ 0.146 \\\\\n5&flux density & R$<220\\arcsec$ & 1\t\t &50.71\/50\t& 10 $\\pm$ 3 & 110 $\\pm$ 20 & 1.048 $\\pm$ 0.040\t & -0.163 $\\pm$ 0.148 \\\\\n\\enddata\n\n\\end{deluxetable*}\n\n\n\n\n\\subsubsection{$\\gamma$-models}\n\\label{sec:fitting_profile_gam}\n\nFor our Jeans-modeling (Section~\\ref{sec:jeans_modelling}) we need a parametrization of the space density profile $\\rho_{\\mathrm{N}}(r)$. This is connected to the observable surface density profile $\\Sigma(R)$\nby the following projection integral:\n\\begin{equation}\n\\begin {split}\n\\Sigma(R)=2 \\int_{R}^{\\infty} \\rho_{\\mathrm{N}}(r) r dr\/\\sqrt{r^2-R^2} \n\\end{split}\n\\label{eq:abel_1}\n\\end{equation}\n\nWe use the spherical $\\gamma$-model \\citep{Dehnen_93} (This is equivalent to the $\\eta$-model of \\citet{Tremaine_94} under the transformation $\\gamma=3-\\eta$.):\n\n \\begin{equation}\n\\begin {split}\n\\rho_{\\mathrm{N}}(r)= \\frac{3-\\gamma}{4\\,\\pi}\\frac{L}{r^{\\gamma}}\\frac{a}{(r+a)^{4-\\gamma}}\n\\end{split}\n\\label{eq:eta}\n\\end{equation}\n\n\nTherein $L$ is the total flux, respectively the star counts. \n$a$ is the scale of the core of the model. The density slope is $-\\gamma$ within the core and -4 at $\\infty$.\n The $\\gamma$-model has a known positive distribution function which we employ in \n\\citet{Chatzopoulos_14}.\nMore complex profiles, like for example a three-dimensional Nuker model, can also fit the data. The projected profiles are nearly independent of the parametrization in that case, e.g. for the star counts\nthe $\\Delta\\chi^2$ between the $\\gamma$-model fit and the Nuker fit is only 1.3. However, the Nuker model has more degrees of freedom. Further, complex profiles contain poorly constrained parameters (e.g. $\\alpha$ in Nuker) for\nour data set; they overfit the data.\n\nThe GC light profile has two breaks (Figure~\\ref{fig:_surf_fit1}).\nThe breaks in the profiles around 200$\\arcsec$ are probably a sign of a two component nature of the nuclear\nlight distribution, as suggested by \\citet{Launhardt_02}. \nThey call the inner component the nuclear (stellar) cluster and the outer one the nuclear (stellar) disk,\nin analogy to other galaxies.\nIn contrast, \\citet{Serabyn_96} assumed that the central\n active star forming zone inside the inactive bulge of the Milky Way consists of a single component,\na central stellar cluster of R$=100\\,$pc.\n\nDue to the breaks we cannot fit the full data range with one $\\gamma$-model. We use instead two independent \n$\\gamma$-models. The use of two models is the main reason that we cannot use the Nuker model since many parameters are then ill determined. \nStill the central slope of the outer component is difficult to determine also for $\\gamma$ models\nfrom the data and we fix it to be flat by setting $\\gamma_\\mathrm{outer}=0$. A smaller value would correspond to a central depression, and values $>0.5$ \ncan create profiles in which the outer component dominates again at very small distances. \n We obtain the fits presented in Row 1 and 2 in Table~\\ref{tab:_surf_fit2}.\n \n\n\n\n\n\\begin{deluxetable*}{lllllllll} \n\\tabletypesize{\\scriptsize}\n\\tablecolumns{9}\n\\tablewidth{0pc}\n\\tablecaption{$\\gamma$-model fits of the surface density profile \\label{tab:_surf_fit2}}\n\\tablehead{ No. &data source & radial range & $\\chi^2\/d.o.f.$ &$L_\\mathrm{inner}$ & $a_\\mathrm{inner}$ & $\\gamma_\\mathrm{inner}$ & $K_\\mathrm{outer}$ & $a_\\mathrm{outer}$ }\n\\startdata\n1&star density & all R &29.1\/55 & $6.73\\pm0.88 \\cdot10^4$ stars &$194\\pm 33$'' & $0.90\\pm0.11$ &$7.05\\pm1.49 \\cdot10^6$ stars & $3396\\pm 458$'' \\\\\n2&flux density & all R &220.3\/204 & $3.42\\pm0.14 \\cdot10^6$ mJy & $117\\pm 10$'' & $0.76\\pm0.08$ & $5.48\\pm0.38 \\cdot10^8$ mJy &$3711\\pm 158$''\\\\\n3&star density & R$<220\\arcsec$ & 18.8\/26 & $28.8\\pm3.4 \\cdot10^4$ stars &$626\\pm 93$'' & $1.14\\pm0.06$ &- & - \\\\ \n4&flux density & R$<220\\arcsec$ &96.87\/51 & $14.52\\pm0.83 \\times10^6$ mJy & $511\\pm 44$'' & $1.21\\pm0.04$ &- & -\\\\\n\\enddata\n\\end{deluxetable*}\n\n \nThe two best fitting profiles for stars and light are similar (Figure~\\ref{fig:_densities_1}) to each other, but again not consistent within their errors\nas already noticed during the Nuker profile fits.\nThe central slope of the cluster is consistently $\\gamma_\\mathrm{inner}=0.83\\pm0.12$.\nHowever, the small error is a consequence of the functional form that we use. Other functions like Nuker(r) yield a range of inner slopes that\nappears to be consistent with the uncertainty reported by \\citet{Do_09}.\n\nThe outer $\\gamma$-model is needed inside of 220$\\arcsec$. Row 3 and 4 in \nTable~\\ref{tab:_surf_fit2} show fits with a single component. \nThe resulting \n $\\gamma_\\mathrm{inner}$ are unrealistically large ($>1$) when comparing with \\citet{Do_09}.\n To quantify the mutual consistence of the two data set, we fit the two data sets with the best single $\\gamma$ fit of the other allowing only the scaling to change. \nThe star density data gives $\\chi^2\/d.o.f.$=70.0\/28; the flux density data $\\chi^2\/d.o.f.$=194.0\/53.\n\n \n\n \n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f5a.eps} \n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f5b.eps} \n\\caption{Space and projected density fits in comparison with the star and flux density data. \nFor illustration purposes the stellar density data and fits are shifted by a factor 105. We fit two $\\gamma$-models. We also show the models of\n\\citet{Genzel_96,Trippe_08,Schoedel_09a},\n and \\citet{Do_13b}.\n} \n\\label{fig:_densities_1}\n\\end{center}\n\\end{figure}\n\n\n\nThe space density models in the literature do not describe our data well, even when we restrict the comparison to the inner 220$\\arcsec$, \nsee Figure~\\ref{fig:_densities_1}. The density model of \\citet{Schoedel_09a} is a bad fit to both of our data sets ($\\chi^2\/d.o.f.=83.37\/28$ and $\\chi^2\/d.o.f.=494.67\/53$, for star and flux density, respectively).\nThe reason is the combination of a large break radius with a relatively small outer slope. It thus overestimates the density further out and underestimates it in the center. \nThe models of \\citet{Trippe_08} ($\\chi^2\/d.o.f=117.46\/28$ and $\\chi^2\/d.o.f.=142.71\/53$) and especially \\citet{Genzel_96} ($\\chi^2\/d.o.f.=323.04\/28$ and $\\chi^2\/d.o.f.=165.99\/53$) fit especially our flux data better. \nStill their average $\\chi^2\/d.o.f.$ is worse than our average $\\chi^2\/d.o.f.$\nwhen we fit both data sets with the same parameters\nThe reason is that the core radii of \\citet{Genzel_96,Trippe_08} are small. \nThe different profiles of \\citet{Do_13b}, which are very similar to each other, do not fit our data at all, because the outer slope of the cluster is much too steep and the core too large.\nThese discrepancies with our fit are not surprising, since none of these works used such a large radial coverage as we. \\citet{Schoedel_09a} did not fit the density model, it is a fixed input to their modeling. They were guided by recent literature. \\citet{Trippe_08} only fit dynamic data.\n\\citet{Do_13b} fit density data together with dynamic data, but only inside 12$\\arcsec$. That covers essentially only the core.\nThis shows that it is important to use density data over a scale larger than the core of the nuclear cluster since otherwise\nthe properties of the outer profile, which have influence on the core parameters, cannot be determined accurately.\n\n\n\n\n\\subsection{Flattening of the Cluster}\n\\label{sec:flattening}\n\n\n\nOur profiles show that the axis ratio ($q=b\/a$) of the nuclear cluster increases with radius, see Figure~\\ref{fig:_surf_bri5b}.\nWe measure the flattening binwise, see Appendix~\\ref{app:meas_flat}, and Table~\\ref{tab:_flat1}. \nIn the inner most bin, $|l^*|<68\\arcsec$, $q=0.80\\pm0.04$. There is some indication in the data (Figure~\\ref{fig:_surf_bri5b}), that the flattening is smaller around 40$\\arcsec$ than around 20$\\arcsec$. That dip is consistent with noise. We do\nnot find a relevant a systematic error source, see Appendix~\\ref{app:flat_syst_err}.\nThe flattening increases further outside of our field, as it is visible in large scale IRAC data and in\n \\citet{Launhardt_02}. \nThey model the inner r$_{\\mathrm{box}}=2^{\\circ}$ of the GC and obtain an axis ratio of 0.2 at \naround $l^*=3100\\arcsec$. \n\n\n\n\\begin{deluxetable}{ll} \n\\tabletypesize{\\scriptsize}\n\\tablecolumns{2}\n\\tablewidth{0pc}\n\\tablecaption{flattening profile \\label{tab:_flat1}}\n\\tablehead{ $|l^*|$ range & $q=b\/a$ }\n\\startdata\n 0 to 68$\\arcsec$ & $0.80\\pm0.04$\\\\\n 68 to 130$\\arcsec$ & $0.63\\pm0.03$\\\\\n 130 to 248$\\arcsec$ & $0.58\\pm0.05$\\\\\n 248 to 473$\\arcsec$ & $0.45\\pm0.04$\\\\\n 473 to 1030$\\arcsec$ & $0.32\\pm0.03$\\\\\n\\enddata\n\\end{deluxetable}\n\n\n\n\\begin{figure*}\n\\begin{center}\n\n \\includegraphics[width=1.19 \\columnwidth]{f6_com7.eps}\n\\caption{Map of the stellar density in the inner r$_{\\mathrm{box}}=1000\\arcsec$. All panels use the same\ncolor scale. Upper left: stellar density from VISTA\/WFC3\/NACO star\ncounts, corrected for completeness and extinction. To the fitted surface brightness data we added smooth contours for illustration.\nUpper right: GALFIT fit to the data. The fit consists of two components, which are\nshown in the lower two panels. Lower left: the central component, a n$=1.46$ Sersic profile, with $q=0.80$ slightly flattened;\nlower right: a Nuker profile ($q=0.26$), which is not well constrained since it extends well outside of our field of view.\n} \n\\label{fig:2dim-map}\n\\end{center}\n\\end{figure*}\n\n\nWe can use our two dimensional data to distinguish between nuclear cluster and nuclear disk, where the outer disk component is \nused to estimate the background for the cluster in the following manner.\nIn the central 68$\\arcsec$ we use the \ndensity profiles shown in Figure~\\ref{fig:_surf_bri5b} in the corresponding quadrants.\nFurther out, we use the VISTA data. Figure~\\ref{fig:2dim-map} shows the resulting map.\nFor finding an empirical description we use GALFIT \\citep{Peng_02}.\nIn the first fits we fix the centers to the known location, and enforce alignment of the\n inner component with the Galactic Plane, but not of the outer component. (Because we measure the density only in two sectors inside of 68$\\arcsec$ alignment with the Galactic Plane is enforced there by construction.) In this fit the outer component is aligned within 1.1$\\pm1.3^{\\circ}$. The uncertainty of that angle is obtained by four trials, in each we mask out the lower\/upper half in l\/b. In contrast to many other parameters that angle is robust. \n Thus, the misalignment is not significant and we fix the angle to be $0^{\\circ}$.\nOur result is consistent with \\citet{Schoedel_14} who obtain both for the nuclear disk, and the nuclear cluster\nalignment with about 2$^{\\circ}$ uncertainty.\n\n \n\nWe fit the data with Sersic \\citep{Sersic_68} and Nuker profiles. Which of them is used is not important, similar sizes and flattening are obtained with both.\nHowever, it is difficult to disentangle the inner and the outer component. Small changes in $\\chi^2\/d.o.f.$ result in large differences in most parameters.\nMainly because the flattening has a local minimum around $40\\arcsec$, a free fit results in a large Sersic index for the outer components. In that case the outer component contributes relevantly again in the center, which reduces the flattening of the inner component. Then the nuclear cluster has \n$q\\approx0.91$. \nHowever, it is physically unlikely that outer component is more important in the very center than slightly further out. Since in the case of our $\\gamma$-profiles (Section~\\ref{sec:fitting_profile_gam}) the outer component does not contribute relevantly in the center, we assume that the inside 68$\\arcsec$ measured flattening is identical with flattening of the nuclear cluster as a whole. \nTo obtain this flattening we use for the nuclear disk a Nuker profile with a flat core, $\\alpha=0.6$, $\\beta=3$, $\\Gamma=0$. This set of parameters approximately models a projected $\\gamma$ model\nin its outer and inner slope (Section~\\ref{sec:fitting_profile_gam}). The fit obtained for the disk is q$=0.264$. \nIn the center we use a Sersic model to enable comparison with the recent literature. \nUnder these assumptions we obtain for the inner component $q=0.80\\pm0.05$, n$=1.46\\pm0.05$, R$_e=127 \\pm 10\\arcsec$, and an integrated count uncertainty of 10\\%. In contrast to many other parameters, the Sersic parameter is well constrained. It is in all cases between 1.4 and 1.6.\n In the very center the outer component contributes 13\\% of the star counts of the inner component.\n The outer component dominates outside about 104$\\arcsec$.\n \nOur half light radius of 5.0 pc is slightly larger than the preferred value of \\citet{Schoedel_14} (R$_e=4.2\\pm0.4$). However, when the outer Sersic is also free, in their two Sersic fits they obtain $6\\pm0.2$ pc. Our axis ratio is somewhat larger then their value of $q=0.71\\pm0.02$ but within the uncertainties. Our Sersic index n is smaller than their of\nn$=2\\pm0.2$. Since they were not able to use the central parsec in their fit, it is likely that our fit is better in that region. The very center is important for the Sersic index, thus our n is probably better. \n Overall from comparing our and their different fits it is probable that both works underestimate the systematic error in component fitting. The main reason for that fact is the existence of two not clearly separated components in the GC and the high extinction towards the region.\n\n\\subsection{Luminosity of the Nuclear Cluster}\n\\label{sec:total_luminosity}\n\n\n\nTo obtain the Ks-luminosity in the GC \nwe integrate the flux of the old stars (Figure~\\ref{fig:_surf_bri2}).\nThe total extinction-corrected flux within R$<100\\arcsec$ is $1052\\pm200\\,$Jy. The absolute error of 20\\% contains the uncertainty of the extinction law toward the GC (\\citealt{Fritz_11}: 11\\%), the calibration uncertainty (7\\%), and 14\\% for the differences between\n the stellar density and flux density profiles. The latter we obtain from the scatter between the star counts profile and the flux profile scaled to each other. \n \nTo estimate the total luminosity of the nuclear cluster, we need to estimate its size. We use again different methods, to estimate\nthe systematic error. Firstly, we use two-dimensional decomposition\nof the star counts in nuclear cluster and nuclear disk (Section~\\ref{sec:flattening}).\\footnote{We do not consider other Galactic components, they are very minor in the center. In the model of \\citet{Launhardt_02} (their Figure~2) these contribute 0.35 mJy\/''$^2$ before extinction correction, while we have 2 mJy\/''$^2$ in total at R$=100\\arcsec$. Extinction corrected, their contribution is even smaller.}\nThe fraction of star counts from the inner component is 67\\% and the fraction the nuclear cluster which is within 100$\\arcsec$ is 44\\%.\nThat leads to a total luminosity of 1599 Jy. Using instead the one dimensional $\\gamma$ decomposition of Section~\\ref{sec:fitting_profile_gam} of the counts leads\nto 5058 Jy. The $\\gamma$-model fit of the flux implies 3420 Jy, see Table~\\ref{tab:_surf_fit2}. \nThe main reason for the different fluxes are the different models: a Sersic can, when it is as here close to exponential, decay fast outside its characteristic radius, while a $\\gamma$-model decays only with its fixed power law of -3.\n\n\nWe obtain absolute luminosities using R$_0=8.2$ kpc and M$_\\odot(Ks)=3.28$.\nThereby, we use the 3420 Jy as best estimate for the total flux and the other two values for the error range.\nWe obtain \n$7.4^{+3.5}_{-3.9} \\times 10^7\\,$L$_{\\odot}$ \nfor the total nuclear cluster, consistent with the estimate of $ 6\\pm 3 \\times 10^7\\,$L$_{\\odot}$ of\n\\citet{Launhardt_02} and the $4.1\\pm0.4 \\times 10^7\\,$L$_{\\odot}$ at 4.5~$\\micron$ of \\citet{Schoedel_14}.\n\n\nThe young O(B)-stars in the center, which are not included in our sample, add 25 Jy in the Ks-band. Although they are\nirrelevant for the light in the Ks-band, this is different for bolometric measurement:\nthe bolometric luminosity of the young stars is about L$_{UV}\\approx10^{7.5}\\,L_{\\odot}$ and $M_{\\mathrm{bol}}\\approx-14.1$ \\citep{Genzel_10,Mezger_96} \nand thus larger than what we obtain for the old stars, $M_{\\mathrm{bol}}\\approx-13.4$ within R$<100\\arcsec$.\nAlso, the young stars are concentrated on a more than $\\approx1000$ times \n smaller volume than the old stars.\n\n\n\n\n\\section{Kinematic Analysis}\n\\label{sec:an_method}\n\n\nWe now use our kinematic data to characterize the properties of the data and to obtain a rough mass estimate from it.\nIn Section~\\ref{sec:_anisotropy},~\\ref{sec:fast_stars} and \\ref{sec:rotation} we discuss anisotropy, \nfast stars, and rotation, respectively. In Section~\\ref{sec:binning} we explain and justify the binning\nused in our Jeans modeling and in \\citet{Chatzopoulos_14}.\nIn this work we use isotropic spherical symmetric Jeans modeling \\citep{Binney_08} \nas illustrative models of what can be derived from our data (Section~\\ref{sec:jeans_modelling}). \nWith that relatively simple model we can also explore\nmany systematic error sources easily.\nIn \\citet{Chatzopoulos_14} we use two-integral modeling with self-consistent rotation \\citep{Hunter_93},\nwhich allows us to include intrinsic flattening and rotation.\n\n\n\n\\subsection{Velocity Anisotropy}\n\\label{sec:_anisotropy}\n\n \n\nThe cluster could be anisotropic. One type of anisotropy is radial anisotropy, \nwhich would manifest itself as a difference between the dispersions in tangential and radial direction\n \\citep{Leonard_89,Schoedel_09a}. \n\nWe obtain an estimate of the anisotropy from the proper motions from $\\beta'_{\\mathrm{pm}}(R)=1-[\\sigma_\\mathrm{tan\\,PM}(R)\/\\sigma_\\mathrm{rad\\,PM}(R)]^2$, where $\\sigma_\\mathrm{tan}$ and $\\sigma_\\mathrm{rad}$ are the radial and tangential dispersions of the proper motions, respectively.\n$\\beta'_{\\mathrm{pm}}$ has the advantage that it follows directly from measured properties \nwithout modeling\nand does not depend on $R_0$.\nUneven angular sampling of stars together with the flattening which causes $\\sigma_{l^*}>\\sigma_{b^*}$ (Figure~\\ref{fig:_std_v}) can mimic anisotropy in the following way:\nconsider a radial bin that is only covered close to the $b^*$-axis. Then, $\\sigma_{\\mathrm{tan}}\\approx \n\\sigma_{l^*}$ and $\\sigma_{\\mathrm{rad}}\\approx \\sigma_{b^*}$. Since $\\sigma_{l^*}>\\sigma_{b^*}$, $\\beta'_{\\mathrm{pm}}<0$ implies tangential anisotropy.\nThe arguments also hold for uneven angular sampling.\n\\citet{Do_13b} had outside the center only covered close to the $b^*$ axis\nthe Galactic Plane. As expected $\\sigma_{\\mathrm{tan}}>\\sigma_{rad}$ in these data. That is the reason that their fit \nprefers $\\beta_\\infty<0$. \n\n\nTo avoid \na spurious influence of the flattening on the anisotropy we firstly restrict the analysis here to \nr$<$40$\\arcsec$ for having full (not necessarily even) angular coverage.\nSecondly, to even out density fluctuations, we obtain the dispersions by taking the average of the dispersions in two angular bins: one within $\\phi<45^{\\circ}$ to the Galactic Plane and the other with $\\phi> 45^{\\circ}$. To even out coverage fluctuations, we give the two bins equal weight.\n We obtain $\\beta'_{\\mathrm{pm}}=-0.040 \\pm 0.022$ (Figure~\\ref{fig:_disp_aniso}). \nThe scatter between the bins is consistent with the Poisson errors. Similar to what was found by \\citet{Schoedel_09a} we see that \n $\\sigma_{\\mathrm{tan}}$ is somewhat larger in the center. \nHowever, since $\\chi^2\/d.o.f.=33.71\/39$\n shows that this is not significant. \n\\citet{Schoedel_09a} suggested that the increase in the center could be due to pollution with early-type stars, \nwhich are in average on more tangential orbits \\citep{Genzel_00,Bartko_09,Bartko_10}. We test this hypothesis by using only stars with late-type \nspectra. Integrated over the full field this yields $\\beta'_{\\mathrm{pm}}=0.045 \\pm 0.049$. \nThat is smaller than for all stars, but not significantly different. \n\nBetween 2'' and 5'' $\\beta'_{\\mathrm{pm}}=-0.463 \\pm 0.170$\nwhen using all stars and $\\beta'_{\\mathrm{pm}}=-0.134 \\pm 0.207$ when using only late-type stars. Since these two values\nare again consistent, pollution is probably not important in this radial range. \n\nThe anisotropy parameter $\\beta$ is defined in 3D coordinates (r) \\citep{Binney_08}. $\\beta$ can be only estimated in full modeling which also needs to account for projection effects. \nIn such model, $R_0$ would also need to be fit and it would also be necessary \\citep{Chatzopoulos_14} to use a flattening model to account for the different effect of the flattening on v$_z$, v$_{l^*}$, and v$_{b^*}$. That is beyond the scope of this paper. \nThe deprojected anisotropy parameter $\\beta$ is more different from 0 than $\\beta'_{\\mathrm{pm}}$ \\citep{Marel_10}. Due to the core-like density profile the difference between $\\beta'$ and $\\beta_{\\mathrm{pm}}$ can be large especially in the center. Because of projection effects, full modeling is required to constrain the\nradial dependency of the anisotropy \\citep{Marel_10}.\nWhile the overall radial anisotropy is small, other deviations from prefect isotropy exist; see Section~\\ref{sec:rotation}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f7.eps} \n\\caption{\nRadial and tangential dispersions as function of the radius. The points are slightly offset in R from each other for better visibility.\n} \n\\label{fig:_disp_aniso}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Rotation and dynamic main axis}\n\\label{sec:rotation}\n\n\nIn Section~\\ref{sec:flattening} we determined the major and minor axes of the\ndistribution of stars in the nuclear cluster. Here we use the new kinematic\ndata to find the rotation axis of the system. First, we use the\nproper motion data.\n\n\\citet{Trippe_08} interpreted the difference between the velocity\ndispersions $\\sigma_l$ and $\\sigma_b$ (Figure~\\ref{fig:_std_v}) as a sign of rotation.\nHowever, $\\sigma_l>\\sigma_b$ is globally required for any axisymmetric star\ncluster flattened parallel to the Galactic plane. This is independent\nof whether the extra kinetic energy along the Galactic plane is due to\nnet rotation or to higher in-plane velocity dispersions; reversing Lz\nfor any orbit does not change $\\sigma_l$. Furthermore, spherical clusters\nwith rotation \\citep{Lynden-Bell_60} do not show $\\sigma_l>\\sigma_b$. Therefore\nthe difference between $\\sigma_l$ and $\\sigma_b$ is ultimately due to the\nflattening, even though most of the flattening of the nuclear cluster is\nin fact generated by additional rotational kinetic energy\n\\citep{Chatzopoulos_14}.\n\nHowever, in any case we can follow the approach of \\citet{Trippe_08}\nto find the kinematic major axis of the nuclear cluster. To this end, we bin the motions in angle (their Figure~7). The pattern is sinusoidal but with more variation close to the peaks. Thus, we use for fitting following function: \\begin{equation} \nf(\\theta)=a+b\\times(|\\theta-\\phi|)^c\\end{equation} \nTherein, theta is the angle relative to the line to the east;\n$\\phi$ the position of the maximum; $a$ is the constant floor, b the peak parameter; a big $b$ implies that the\nmaximum has small width than the minimum.\nWe obtain $b=3.97\\pm0.49$ and $\\phi=58.8\\pm1.2^{\\circ}$\nconsistent with the Galactic plane ($\\phi=58.6^{\\circ}$).\n\n\nLastly, we use the radial velocities, whose\n gradient in the mean radial velocity as a function of $|l^*|$ \\citep{Trippe_08} can only be explained \n by rotation. For fitting the Galactic plane we aim to find the angle for which the radial velocity is constant along the coordinate $x''$, which we obtain by the rotation. We assume cylindrical rotation, since we ignore $y''$. The angle is the rotation axis, rotated by 90$^\\circ$. \n The advantage of that angle compared to the rotation axis is that it is not necessary to fit at the same time for the possible complex rotation curve, because vertical to it the velocity is identical everywhere. We obtain $53.7\\pm4.0^{\\circ}$ broadly consistent with the Galactic plane.\n \nDue to the finding of \\citet{Feldmeier_14} that radial velocity field is not only a function of l$^*$ and b$^*$ as expected, we divided our velocities in radial bins (Figure~\\ref{fig:_rot_axis}). \nWe can confirm their findings mostly. At medium radii (between 24 and 90$\\arcsec$) the radial velocity plane follows an angle of $45.1\\pm4.2^{\\circ}$. The value is 3.2~$\\sigma$ different from the Galactic Plane. That agrees well with the measurement of \\citet{Feldmeier_14}. Further out and further in the rotation axis aligns with the Galactic plane within the partly large error. \nWe do not find deviations of $\\sigma_z$ in these bins.\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f8.eps} \n\\caption{Orientation of the major (flattening\/rotation) axis.\n} \n\\label{fig:_rot_axis}\n\\end{center}\n\\end{figure} \n\n\n\n\n\nAlthough it is likely that the rotation is not a function of l$^*$ only, we assume it here to ease comparison with \nmost of the literature.\nFor illustration we bin the radial velocity data (Section~\\ref{sec:rad_vel}) in a less crowded way (Figure~\\ref{fig:_v_rad_fit1}).\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f9.eps}\n\\caption{Average radial velocities from our data and the literature.\n We assume symmetry of the rotation pattern and reverse the sign of the radial velocities\nFor the maser data we use the velocities from \\citet{Lindqvist_92a} and \\citet{Deguchi_04}. \n The data from \\citet{Trippe_08} overlaps within $|l^*|<27\\arcsec$ largely with our data. Further out \\citet{Trippe_08} utilized \na subsets of the dataset of \\citet{McGinn_89}. We also plot the data of \\citet{Rieke_88}.\nWe fit the good (reddish) data without binning by a polynomial for illustration.\n} \n\\label{fig:_v_rad_fit1}\n\\end{center}\n\\end{figure} \n\n\nInside of 27$\\arcsec$ our velocities are consistent with the velocities of \\citet{Trippe_08}, since the data set is largely identical.\nOutside of 27$\\arcsec$ our new SINFONI radial velocities are on average smaller than \nthe velocities of \\citet{Trippe_08}, who used only a subset of the velocities in \\citet{McGinn_89}, a problem already pointed out\nby \\citet{Schoedel_09a}. Surprisingly, our new high resolution data do not yield the average of the velocities reported by \\citet{McGinn_89},\nbut agree roughly with the lower end of values found. These lower end of values agrees with the \n velocities of single bright stars by \\citet{Rieke_88}.\n\n\n The maser data of \\citet{Lindqvist_92a} and \\citet{Deguchi_04} agree with the lower velocity data of \\citet{McGinn_89} and our CO band head\nvelocities. \\citet{Schoedel_09a} suggested that the differences in the radial velocities in the literature could be a sign of two populations in the GC.\nHowever, in our data we find no sign for any population dependence of the rotation pattern.\nPossibly, the difference in radial velocities in \\citet{McGinn_89} and \\citet{Rieke_88} is an indication that the velocity calibration\nof these old CO band head measurements was more difficult than assumed back then. \nOverall we are confident that our smaller rotation of the cluster compared to \\citet{Trippe_08} is correct and is not population dependent.\n\n\\subsection{Binning}\n\\label{sec:binning}\n\n\nFor simplicity, we choose to bin our data. The loss of information \\citep{Merritt_94,Feigelson_12,Scott_92} is small, since we use\na large amount of data and the variations between the bins are smaller than the errors.\nWe assume symmetry relative to the Galactic plane, as supported by most observations, see Section~\\ref{sec:rotation}.\nThere might be little deviations in the radial velocities. However, since we fit second moments and not velocities and dispersions the impact is very small. \nAn edge-on flattened system has different symmetry properties in proper motion and in radial velocity. That is another reason for our different binnings which is explained in \n Appendix~\\ref{app_binning}. The same bins are also used by \\citet{Chatzopoulos_14}.\nWe have tried also different binnings, which bin only in r and bin proper motions and radial velocities together. \nOur test includes bins of nearly equal size in r, log(r) and nearly equally populated bins.\nThe Jeans masses varies usually by less than the formal fitting errors, sometimes slightly more.\nSince systematic errors are much larger than the fitting error (thus also larger than the binning error) it is therefore not necessary to include binning terms in the errors.\nWe show in Figure~\\ref{fig:_std_v} the binned dispersion data in all three dimensions (l$^*$, b$^*$, z).\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f10a.eps} \n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f10b.eps} \n\\caption{Binned velocity dispersion used for Jeans modeling. The upper panel presents the proper motion data, \nthe lower one the radial velocity data. \n} \n\\label{fig:_std_v}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\subsection{Jeans Modeling}\n\\label{sec:jeans_modelling}\n\n\nIt is obvious from the dispersion difference (Figure~\\ref{fig:_std_v}) in the two proper motion axes\nthat the nuclear cluster is not a spherical, isotropic system, which would have the dispersions in all directions. \nAs shown in \\citet{Chatzopoulos_14}, this difference is caused by\nthe flattening of the NSC; their Figure~10 shows how $\\sigma_l>\\sigma_b>\\sigma_z$\nover the whole range of radii for their favored axisymmetric models.\nWhile we cannot include anisotropy totally, it is very likely minor given our constrains of Section~\\ref{sec:_anisotropy} and the fact that an isotropic rotator fits the data well \\citep{Chatzopoulos_14}.\nThus, anisotropic spherical modeling is at most only a small improvement compared to isotropic spherical modeling. \nTherefore, we use the most simple kind of Jeans modeling \\citep{Binney_08} assuming isotropy and spherical symmetry for mass and light.\nThat simple model is mainly used for illustration of what can be derived from the data. In a simple model,\ntests for other effects are also easier and faster, in contrast to more complex models.\n\n\n Two variants of isotropic Jeans modeling were used in the past for the GC:\n\\begin{itemize}\n\\item \\citet{Genzel_96} and \\citet{Trippe_08} parametrized the deprojected dispersion.\n\\item \\citet{Schoedel_09a} used a direct mass parametrization.\n\\end{itemize}\nWe follow here \\citet{Schoedel_09a}. \nSince the existence of the SMBH is shown with orbits \\citep{Schoedel_02}, is\nadvantageous when it can be parametrized directly.\n\n\n\n\\subsubsection{Relation of dispersion and mass}\n\nTo relate the measured dispersion to the mass we use the following equation derived from the Jeans equation \\citep{Schoedel_09a,Binney_08}:\n\n\\begin{equation}\n\\begin {split}\n\\sigma_P^2(R)\/G=\\frac{ \\int^{\\infty}_R dr\\, r^{-2} (r^2-R^2)^{1\/2} n(r) M(r)} {\n\\int^{\\infty}_R dr\\, r (r^2-R^2)^{-1\/2} n(r) \\,\\, ,\n}\n\\end{split}\n\\label{eq:jeans_par_is}\n\\end{equation}\n\nand one needs to choose a parametrization for $M(r)$. The assumption which has\nthe fewest degrees of freedom is a constant mass to light ratio, \n or a theoretically predicted profile of that \nratio \\citep{Luetzgendorf_12}. \nThe GC data sets have been rich enough to \nleave the shape of the extended mass\ndistribution as a free parameter \\citep{Schoedel_09a}. \nGiven that the presence of the central SMBH is well established in the GC, one\ncan add the SMBH mass explicitly ($M_{\\bullet}$) to $M(r)$, as done in \\citet{Schoedel_09a}.\n\n\n\\subsubsection{Projection}\n\nIt is necessary to deproject the observed radii $R$ into\ntrue 3D radii $r$, for which one needs to know the space tracer density distribution $n(r)$, see Section~\\ref{sec:fitting_profile}. \nThis is done by two projection integrals, namely equation~(\\ref{eq:abel_1}) and the following integral:\n\n\\begin{equation}\n\\begin {split}\n\\Sigma(R) \\sigma_P(R)^2=2 \\int_{R}^{\\infty} n(r) \\sigma(r)^2 r dr\/\\sqrt{r^2-R^2} \n\\end{split}\n\\label{eq:abel_2}\n\\end{equation}\n\n\n\\subsubsection{Averaging the 3D-dispersions}\n\n\n\nIn the Jeans equation a one-dimensional dispersion $\\sigma$ is used. \nWe have data in all three dimensions and thus need to average them. We use proper motions and radial velocities separately\nsince they cover different areas and have different errors. \n\\begin{itemize}\n\n\\item \nFor the radial velocities, there is significant rotation at large radii (Figures~\\ref{fig:_v_rad_fit1}). \nTo roughly take care of rotation we use instead of $\\sigma_z$, $\\langle v^2_{z}\\rangle^{1\/2}$ (see e.g. \\citet{Tremaine_02,Kormendy_13}) in Equation~\\ref{eq:jeans_par_is}. \nOur approach is an approximation to axisymmetric modeling, which we use in \\citet{Chatzopoulos_14}.\n\n\\item By definition of our coordinates, there\nis no rotation term in the proper motions. Still a non-zero mean velocity can appear locally. \nThus we also use $\\langle v^2\\rangle^{1\/2}$.\nThe impact of using this measurement instead of $\\sigma$ is very small.\nWe combine both dimensions in each bin to \n$\\langle v^2_{\\mathrm{pm}}\\rangle=1\/2\\times(\\langle v^2_{l}\\rangle+\\langle v^2_{b}\\rangle)$ \nto reduce the impact of the flattening. \n\\end{itemize}\n\n\n\n\\subsubsection{Errors of the Dispersions}\n\nCalculating the error of the dispersion with $\\delta\\sigma=\\sigma\\times 1\/\\sqrt{2N}$ is problematic when there\nare only a few stars in a bin (especially as in our maser radial velocity data), since the weight of the individual bins will scale inversely with the \ndispersion value. Thus, low dispersion values have too large weights. \nTo obtain errors which do not bias the result, we determine the errors with an iterative procedure: we fit the binned profile $\\sigma(r)$ with an empirical function \n(a fourth order log-log-polynomial). In the first iteration we weight the points according to the observed $\\sigma\\times 1\/\\sqrt{2N}$.\nIn the further iterations we use the value of the polynomial fit ($\\sigma_\\mathrm{fit}$) to obtain the errors: $\\delta\\sigma=\\sigma_\\mathrm{fit}\\times 1\/\\sqrt{2N}$. \n With these errors we repeat the fit and get refined error \nestimates. After four iterations the procedure converges. \nWe use the same method also for the proper motions. Due to the higher star numbers the effect of this correction\nis smaller there.\n\nAs result our errors are slightly correlated between the different bins. The effect is small.\nAn estimate of its size can be obtained by comparing our masses when using different binning.\nThese differences are smaller than the formal fitting errors, and thus negligible compared to the\nsystematic errors.\n\n\n\n\\subsubsection{Tracer Distribution Profiles}\n\nWe fit our data using three different types of profiles for the three-dimensional tracer distribution:\n\\begin{itemize}\n\\item For most causes we fit a double $\\gamma$-profile (Section~\\ref{sec:fitting_profile}).\n\\item For checking the robustness of these, we also use a single $\\gamma$-profile which we fit only to our inner density data. \n\\item For comparison with the literature, we also use the single component tracer models of\n\\citet{Genzel_96,Trippe_08};\n and \\citet{Schoedel_09a}. \n\\end{itemize}\n\n\n\\subsubsection{Mass Parametrization}\n\n\nOur mass model contains the point mass SMBH at the center and an extended component made of stars. It is justified to ignore gas clouds, since even the most\n massive structure,\nthe circumnuclear disk, has a mass of only a few times $10^4$ M$_{\\odot}$ \\citep{Mezger_96,Launhardt_02,Requena_12}.\nWe use two different ways of \nparametrization for the extended\nmass:\\begin{itemize}\n\\item We use a power law, similar to \\citet{Schoedel_09a}: \n\\begin{equation}\n\\begin{split}\nM(r)=M_{\\mathrm{100}\\arcsec}\\times(r\/100\\arcsec)^{\\delta_\\mathrm{M}}\n\\end{split}\n\\label{eq:mass_param1}\n\\end{equation}\nThe fact that the integrated mass is not finite for $r\\rightarrow\n\\infty$ is not a problem, since our tracer profile $n(r)$ falls more rapidly than $M(r)$. \n\\item We use a constant mass to light ratio for the extended mass:\n\\begin{equation}\n\\begin{split}\nM(r)=M\/L\\times L(r)\n\\end{split}\n\\label{eq:mass_param2}\n\\end{equation}\nThe light is either flux or the star counts (Section~\\ref{sec:lum_prop}).\n\\end{itemize}\n\nFor normalization of each \nwe choose 100$\\arcsec$, since that mass is well determined from our data. \n\n\n\\subsubsection{Fitting}\n\nWe fit simultaneously the surface density and the projected dispersion data. \nTo the surface density we fit the density model, the double $\\gamma$ model (Section~\\ref{sec:fitting_profile_gam}). \nThe dispersion data depends on both the mass model and the density model, which in that case is the tracer model. Since we fit both at once both model components are constrained by both data sets.\nWhen we use for the mass the power law model, the fit values of the density model are consistent\nwith the fits which only use the surface density (Table~\\ref{tab:_surf_fit2}).\nThis confirms that our fits converge well. For constant M\/L the parameters for the best fitting density model\ndiffer often by more than 1$\\,\\sigma$ from the ones when we fit only the surface density. The reason is, that in the second case the density model depends more on the dispersions, because due to constant M\/L the influence of the dispersion on the density model is grater than in the other case.\n Usually, the density model is less concentrated in the full fit than in the density only fit. That is an expression of the low dispersion problem in the center\n(Section~\\ref{sec:mass_smbh}).\nHowever, the density parameter differences are not very big and do not influence the obtained masses relevantly. \nIn one of our fits the central slope ($\\gamma_{\\mathrm{inner}}$) of the inner component of the tracer density is slightly (by only 0.02)\nsmaller than 0.5. This is problematic, because a slope smaller than 0.5 is not possible in the spherical isotropic case\nwhen a central point mass is dominating \\citep{Schoedel_09a}. \nTo retain self-consistency in our simple isotropic models we fix the smaller slope to 0.5 in that case. The fix has no influence on the obtained masses. Also, the inner slope of the outer component is not well constrained and\nwe fix it usually to 0. \nBoth restrictions have no relevant influence on the masses obtained.\nIn the fitting we optimize $\\chi^2$ \\citep{Press_86}: \n\\begin{equation} \n\\chi^2=\\sum\\limits_{i=1}^n (\\frac{x_i-\\mu_i}{\\sigma_i})^2\n\\end{equation}\nTherein $\\mu_i$ is the function which we optimize; it consists of two function, one for density and one for the dispersion. The density function consists of two terms of Equation~\\ref{eq:eta} projected with Equation~\\ref{eq:abel_1}, while the dispersion function consists of Equation~\\ref{eq:abel_2} using one of our two types of extended mass parameterizations (power law or constant M\/L). In both cases, the dispersion function is projected with Equation~\\ref{eq:jeans_par_is}.\n$x_i$ and $\\sigma_i$\nare the values and errors of the observables which are obtained in bins. \nWe assume Gaussian errors.\n\n\nWe present the fits corresponding to the various choices of how to set up the Jeans model of the nuclear cluster in Table~\\ref{tab:_mass_iso1}. We do not show the cluster properties since they do not differ by much. The\n robustness of the results can be assessed by comparing the different fits. For the fitting itself we first use routine mpfit \\citep{Markwardt_09}. \nSecondly, we also fit the data by using Markoff-Chain Monte Carlo (MCMC) simulations following the method of \\citet{Tegmark_04,Gillessen_09}. Our modification is that we start in the previously found minimum. We never find a better minimum\n than the starting point. \n The errors are usually rather similar with both methods. There is usually some asymmetry, i.e. the errors\n are larger on the positive side. For mass properties that asymmetry is small. It is larger for some\n of the less well defined tracer density parameters. Such asymmetry cannot be found by mpfit.\n In Table~\\ref{tab:_mass_iso1} we give as best value the median value of the accepted MCMC values. As error we give half of the difference between the 84.1\\% and the 15.9\\% quantile. Such errors encompass the central 1~$\\sigma$ range. \n \nInitially, we fit with a free SMBH mass (Rows 1 and 2 in Table~\\ref{tab:_mass_iso1}). However, especially for the power law mass model \n which has one parameter more, the cluster shape, the resulting mass of the SMBH is smaller than the direct mass measurements \nby means of stellar orbits \\citep{Ghez_09,Gillessen_09}. \n We discuss the reason for the small SMBH mass in Section~\\ref{sec:mass_smbh}. The direct cause \nfor the too small SMBH mass is that models with a realistic black hole mass (M$_\\bullet=4.17 \\times 10^6M_{\\odot}$) predict a larger dispersion within $r<10\\arcsec$ than we measure. That implies that our model is in some aspect incomplete near the black hole.\nIn the following we fix the central mass to M$_\\bullet=4.17 \\times 10^6M_{\\odot}$, corresponding to our fixed distance of $R_0=8.2\\,$kpc, and we neglect the small distance independent uncertainty of 1.5\\% \\citep{Gillessen_09}.\n\n\n\n\\begin{deluxetable*}{llllllllll} \n\\tabletypesize{\\scriptsize}\n\\tablecolumns{10}\n\\tablewidth{0pc}\n\\tablecaption{Jeans model fits \\label{tab:_mass_iso1}}\n\\tablehead{No.& mass & tracer & tracer &tracer & dynamics & M$_\\bullet$ & M$_{100\\arcsec}$& $\\delta_\\mathrm{M}$&$\\chi^2$\/d.o.f \\\\\n & model & model & source &range & range & [$10^6 M_{\\odot}$] & [$10^6 M_{\\odot}$] & &}\n\n\\startdata\n1 & power law & double $\\gamma$ & stars&all & all R & $ 2.26\\pm 0.26$ & $9.28 \\pm 0.48 $ & $ 0.92 \\pm 0.04 $&185.43\/182 \\\\ \n2 & M\/L$=$const & double $\\gamma$ & stars&all & all R & $ 4.37\\pm 0.13$ & $5.17 \\pm 0.24 $ & & 276.16\/184 \\\\ \n\\hline\n\\bf 3 &\\bf power law &\\bf double $\\bf\\gamma$ &\\bf stars&\\bf all &\\bf 10$\\arcsec<$R$<$100$\\arcsec$ &\\bf 4.17 &\\bf 5.81 $\\pm$ \\bf 0.26 &\\bf 1.21 $\\pm$ \\bf 0.05 &\\bf145.83\/137 \\\\ \n\\bf 4 &\\bf power law &\\bf double $\\gamma$ &\\bf flux &\\bf all &\\bf 10$\\arcsec<$R$<$100$\\arcsec$ & \\bf 4.17 & \\bf 6.17 $\\pm$ \\bf 0.23 & \\bf 1.25 $\\pm$ \\bf 0.04 &\\bf 349.51\/286\\\\\n\\bf 5 &\\bf M\/L$=$const &\\bf double $\\gamma$ &\\bf stars &\\bf all &\\bf 10$\\arcsec<$R$<$100$\\arcsec$ & \\bf 4.17 &\\bf 5.62 $\\pm$ 0.17 &&\\bf 156.97\/138 \\\\ \n\\bf 6 & \\bf M\/L$=$const &\\bf double $\\gamma$ &\\bf flux &\\bf all &\\bf 10$\\arcsec<$R$<$100$\\arcsec$ &\\bf 4.17 &\\bf 6.81 $\\pm$ 0.16 &&\\bf 342.58\/287 \\\\\n\\hline\n7 & power law & double $\\gamma$ &stars& all & R $<100\\arcsec$ & $4.17 $ & $4.98 \\pm 0.29 $ & $1.33 \\pm 0.05 $&199.69\/163 \\\\ \n8 & power law &double $\\gamma$ & flux &all & R $<100\\arcsec$ & $4.17 $ & $5.35 \\pm 0.23 $ & $1.37 \\pm 0.04 $ &435.79\/312\\\\\n\\hline \n9& M\/L$=$const & double $\\gamma$ &stars& all & R $<100\\arcsec$ & $4.17 $ & $5.48 \\pm 0.16$ &&195.82\/164 \\\\ \n10 & M\/L$=$const & double $\\gamma$ & flux &all & R $<100\\arcsec$ & $4.17 $ & $6.70 \\pm 0.18$ &&414.04\/313 \\\\ \n\n11 & power law & double $\\gamma$ &stars& all & R$>$10$\\arcsec$ & $4.17 $ & $6.14 \\pm 0.18 $ & $1.12 \\pm 0.03 $&182.36\/157 \\\\ \n12 & power law & double $\\gamma$ & flux &all & R$>$10$\\arcsec$ & $4.17 $ & $6.57 \\pm 0.17 $ & $1.12 \\pm 0.03 $&399.37\/306 \\\\ \n13 & M\/L$=$const & double $\\gamma$ &stars& all & R$>$10$\\arcsec$ & $4.17 $ & $5.58 \\pm 0.15 $ &&240.18\/158 \\\\\\ \n14 & M\/L$=$const & double $\\gamma$ & flux &all & R$>$10$\\arcsec$ &$4.17 $ & $6.65 \\pm 0.16 $ &&433.36\/307 \\\\ \n15 & power law & double $\\gamma$ &stars& all & all R & $4.17 $ & $5.59 \\pm 0.17 $ & $1.19 \\pm 0.03 $&247.59\/183 \\\\ \n16 & power law & double $\\gamma$ & flux &all & all R & $4.17 $ & $6.03 \\pm 0.17 $ & $1.19 \\pm 0.03 $&507.08\/332 \\\\ \n17 & M\/L$=$const & double $\\gamma$ &stars& all & all R & $4.17 $ & $5.47 \\pm 0.14 $ &&277.14\/185 \\\\ \n18 & M\/L$=$const & double $\\gamma$ & flux &all & all R & $4.17 $ & $6.56 \\pm 0.15 $ &&500.64\/333 \\\\ \n\n\\hline\n19 &power law & single $\\gamma$ & stars&R$<220\\arcsec$ & 10$\\arcsec<$R$<100\\arcsec$ & $4.17 $ & $5.91\\pm 0.21$ & $1.32 \\pm 0.08 $&126.27\/108 \\\\ \n20 &power law & single $\\gamma$ & flux &R$<220\\arcsec$ & 10$\\arcsec<$R$<100\\arcsec$ & $4.17 $ & $6.27\\pm 0.18$ & $1.36 \\pm 0.07 $ &207.12\/132 \\\\\n21 & M\/L$=$const & single $\\gamma$ & stars&R$<220\\arcsec$ & 10$\\arcsec<$R$<100\\arcsec$ & $4.17 $ & $6.23\\pm 0.16$ &&130.58\/108 \\\\ \n22 & M\/L$=$const & single $\\gamma$ & flux & R$<220\\arcsec$ & 10$\\arcsec<$R$<100\\arcsec$ & $4.17 $ & $6.92\\pm 0.15$ &&205.30\/134 \\\\\n\\hline\n23 & power law & Sch\\\"odel+ 2009 & stars& R$<220\\arcsec$ & 10$\\arcsec<$R$<100\\arcsec$ & $4.17 $ & $5.79\\pm 0.25$ & $1.16 \\pm 0.05 $&195.97\/110\\\\\n24 & power law & Trippe+ 2008 & stars& R$<220\\arcsec$ & 10$\\arcsec<$R$<100\\arcsec$ & $4.17 $ & $7.22\\pm 0.17$ & $1.20 \\pm 0.04 $&453.40\/110\\\\\n25 &power law & Genzel+ 1996 & stars& R$<220\\arcsec$ & 10$\\arcsec<$R$<100\\arcsec$ & $4.17 $ & $6.75\\pm 0.18$ & $1.12 \\pm 0.04 $&237.05\/110\\\\\n26 & M\/L$=$const & Sch\\\"odel+ 2009 & stars&R$<220\\arcsec$ & 10$\\arcsec<$R$<100\\arcsec$& $4.17 $ & $4.66\\pm 0.09$& &218.72\/111 \\\\\n27 & M\/L$=$const & Trippe+ 2008 & stars&R$<220\\arcsec$ & 10$\\arcsec<$R$<100\\arcsec$ & $4.17 $ & $7.56\\pm 0.15$& &444.73\/111 \\\\ \n28 & M\/L$=$const & Genzel+ 1996 & stars& R$<220\\arcsec$& 10$\\arcsec<$R$<100\\arcsec$ & $4.17 $ & $6.19\\pm 0.12$ &&243.69\/111 \\\\\n\\hline\n29& M\/L$=$const & double $\\gamma$ &stars& all & R $<27\\arcsec$ & $3.37\\pm0.16 $ & $7.31 \\pm 0.42$ && 91.93\/120 \n\\enddata\n\\tablecomments{\nJeans model fitting of our dynamics and density data, assuming different mass and tracer models, and different selections for the data. For the fitted surface density (tracer) we use two data sources and restrict us sometimes to\na subset of the available data range (Column 3 and 4). The dynamics data consists of $\\langle v^2\\rangle$ in all three dimensions. We restrict it partly radially.\nThe mass model includes in all cases a central point mass. M$_{100\\arcsec}$ is the nuclear cluster mass within 100$\\arcsec$. \nIf no error is given for a parameter it is fixed. \nThe literature tracer models are from\n\\citet{Trippe_08,Schoedel_09a},\n and \\citet{Genzel_96}.\n}\n\\end{deluxetable*}\n\n\\subsubsection{Results}\n\n\n The fits which meet our assumption best (Table~\\ref{tab:_mass_iso1}) are in the rows 3, 4, 5, and 6. They use\na range of 10$\\arcsec10\\arcsec$}\n\nOutside of 10$\\arcsec$ most fast stars are unbound to SMBH and nuclear cluster mass (Figure~\\ref{fig:_esc_mass}). These stars are detected in all three velocity dimensions: we have 5, 6, 5, with $|v|>275$ km\/s in either\/or z, $l^*$ and $b^*$, respectively among the stars with all velocity components measured outside of 7$\\arcsec$ in the central and extended sample.\nSome of these stars were already discussed in \n\\citet{Reid_07} and \\citet{Genzel_10}. Our improved mass estimate for the nuclear cluster reinforces the statement that these stars\nare not bound to the GC. Note that placing a star out of the plane of the sky enlarges the discrepancy \\citep{Reid_07}, \ne.g. for a star at 40$\\arcsec$ the escape velocity decreases out to $z=200\\arcsec$ and then increases only very slowly to about 165 km\/s at 3600$\\arcsec$, \nwhich is still less than in the plane of the sky. The fact that the maximum velocity decreases for $R>8\\arcsec$ in the same\nway as the median velocity (Figure~\\ref{fig:_max_vel}) excludes that the fast stars are foreground objects, for which \nthe velocity would not depend on radius. The extinction appears to be normal for the fast stars.\n\n\n\\citet{Reid_07} also discussed binaries as a solution for the high velocities. This is excluded by our data set since for some stars the high velocity \nis dominated by long-term proper motions measurements, which cover much more time than what one orbital period would need to be.\n\nFor testing whether the Hills-mechanism is important, the directions of motion of the fast stars can help. We use therefore the two sided K-S~test on the distribution of proper motion vectors in polar coordinates comparing fast stars with all stars. Fast stars are again selected in radial bins to avoid influence of radial trends.\nWe obtain for different selections borders within the fastest 1\\% that the two distributions are identical, probability usually 0.5, in one case 0.07. Thus the fast stars are not preferentially moving away from the black hole.\nThat makes it unlikely that the Hills mechanism is the main mechanism.\nThe comparably large number of fast stars makes that also unlikely.\nProbably as already advocated by \\citet{Reid_07}, the best solution is that these stars are on very eccentric orbits in the large scale potential.\nTherein they can obtain higher velocities than the escape velocity determined from the local mass distribution.\n\n\n\nSince the database of \\citet{Genzel_10} contained many bright, unbound stars, these authors suggested that preferentially young, \nrelaxed, bright TP-AGB stars are on these unbound orbits. This finding might be affected by low number statistics \nand a bias. Bright stars have smaller velocity\nerrors and are therefore easier to identify as significantly unbound. In our sample and excluding the database of \\citet{Genzel_10}, \nwe see no evidence that bright stars \nare dynamically distinct from the other stars. This also holds for the subsample of medium old TP-AGB stars from \\citet{Blum_02}. Also \n\\citet{Pfuhl_11} found that their two samples of younger and older giants show consistent dynamics.\n\n\\subsubsection{Velocity histograms}\n\nIn Figure~\\ref{fig:_velocity_his} we show the velocity histograms for the three dimensions. In case of the proper motion samples we show\n$R>7\\arcsec$. For the radial velocities, we show two bins, 7$\\arcsec100\\arcsec$. In the latter\nbin (consisting of the maser stars) we have subtracted off the rotation. \n\\begin{figure}\n\\begin{center}\n\n \\includegraphics[width=0.70 \\columnwidth,angle=-90]{f17.eps}\n\\caption{Velocity histograms of our dynamics sample.} \n\\label{fig:_velocity_his}\n\\end{center}\n\\end{figure}\n\n\n\n We see deviations from Gaussian distributions in many aspects:\n\n\\begin{itemize}\n\\item The central part of distributions in $l^*$ and $b^*$ are not as peaked as Gaussians (Figure~\\ref{fig:_velocity_his}).\nIn $l^*$ the distribution has a flatter peak than a Gaussian \\citep{Trippe_08,Schoedel_09a}.\nIn $b^*$ the distribution has a slightly, but significantly ($>5\\,\\sigma$), steeper peak than a Gaussian.\nThe flatter peak in $l^*$ is actually a signature of the flattening of the cluster, as shown in \\citet{Chatzopoulos_14}.\n\\item The varying ratio of maximal and median velocity (Figure~\\ref{fig:_max_vel}) is indicative of non-Gaussian wings\nwithin 7$\\arcsec$. \n\\citet{Trippe_08} did not see that\nbecause they did not radially subdivide their sample. Furthermore our sample contains\ntwelve more stars in the central 2$\\arcsec$ with v$_{\\mathrm{3D}}>460$ km\/s than the one from \\citet{Trippe_08}. \nIn that work the only such fast star was S111. \nThe high-velocity wings in all three dimensions are mainly caused by the presence of the SMBH.\n Therefore, because of the higher number of fast stars S111 is less special than discussed in \\citet{Trippe_08}. \n\\item From the 274 stars in the $R>100\\arcsec$-bin, roughly 13 are in the high (radial) velocity wings of the distribution (Figure~\\ref{fig:_velocity_his}). \nMost of these outliers were already noted by \\citet{Lindqvist_92b} and \\citet{Deguchi_04}.\n\\end{itemize}\n\n\n\\begin{deluxetable*}{lllllllll}\n\\tabletypesize{\\scriptsize}\n\\tablecolumns{9}\n\\tablewidth{0pc}\n\\tablecaption{Fast stars \\label{tab:_fast star}}\n\\tablehead{ID & R.A. [$\\arcsec$] & Dec. [$\\arcsec$] & v$_{\\mathrm{R.A.}}$ [mas\/yr]&v$_{\\mathrm{Dec.}}$ [mas\/yr] & v$_{\\mathrm{z}}$ [km\/s] &v$_{\\mathrm{3D}}$ [km\/s] & v$_{\\mathrm{esc}}$ [km\/s] & Comment}\n\\startdata\n770 & -1.11 & -0.91 & $-2.78 \\pm 0.02$ & $-7.72 \\pm 0.01$ & $-741 \\pm 5$ &$807 \\pm 5$ & 792 & S111\\\\\n569 & 0.13 & 3.08 & $13.22 \\pm 0.04$ & $-3.6 \\pm 0.05$ & $-92 \\pm 6$ &$540 \\pm 7$ & 541 & \\\\\n4 & 5.68 & -6.33 & $2.96 \\pm 0.09$ & $2.58 \\pm 0.08$ & $-312 \\pm 14$ &$347 \\pm 14$ & 326 & IRS 9\\\\\n4258 & 2.66 & 13.61 & $-8.48 \\pm 0.15$ & $-1.67 \\pm 0.09$ & $-315 \\pm 45$& $458 \\pm 46$ & 255 & fastest v$_{\\mathrm{3D}}$, R$>7\\arcsec$ \\\\\n899 & -8.37 & -12.21 & $-1.98 \\pm 0.15$ & $11.03 \\pm 0.24$ & &$435 \\pm 11$ & 247 & fastest v$_{\\mathrm{2D}}$, R$>7\\arcsec$\\\\ \n903 & 14.11 & 7.45 & $-1.58 \\pm 0.04$ & $-0.98 \\pm 0.13$ & $379 \\pm 24$ &$385 \\pm 24$ & 238 & only high v$_{\\mathrm{z}}$\\\\\n787 & -6.8 & 18.35 & $-5.96 \\pm 0.13$ & $7.36 \\pm 0.2$ & $-91 \\pm 11$ &$379 \\pm 15$ & 215 & runaway candidate\\\\\n & & & & & & & & \\citep{Schoedel_09a}\\\\\n\\enddata\n\\tablecomments{Extraordinarily fast stars sorted by their projected distance from Sgr~A*. The errors of\nthe positions are smaller than 5 mas. v$_{\\mathrm{esc}}$ assumes $R_0=8.2\\,$kpc.\n}\n\\end{deluxetable*}\n\nTable~\\ref{tab:_fast star} gives an overview of the unusually fast stars (R$\\leq20\\arcsec$, further out extremely fast stars are not reliably measured) in our sample. ID 787 is the star which \n\\citet{Schoedel_09a} called a runaway candidate.\nWe find a proper motion that is 3.8$\\,\\sigma$ smaller, and derive a rather normal radial velocity from its late-type spectrum, \nmaking this star considerably less noteworthy. \nThe difference in proper motion is probably due to our better distortion correction at the edge of the field of view\n(Section~\\ref{sec:mass_smbh}). \nThe star with the highest 3D velocity is the giant ID 4258, but it is (projected) close to another bright late-type star, such that\nthe radial velocity error might well be larger than indicated.\nThe outlier fraction appears to be smaller than in the maser sample (Figure~\\ref{fig:_velocity_his}), but at smaller radii the dispersion is higher and thus\nthe identification of outliers is more difficult there. \n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\n\n\nWe discuss our results in the context of previous results for the nuclear cluster of the Milky Way and of other galaxies.\n\n\n\\subsection{Central Low Dispersion Problem and SMBH Mass}\n\\label{sec:mass_smbh}\n\n\nRow~1 in Table~\\ref{tab:_mass_iso1} shows that fitting for the SMBH mass with a power law profile yields a mass which\nis much smaller than the estimates from stellar orbits \\citep{Gillessen_09,Ghez_09}. \nWe now look closer at that problem.\nIn order to reduce rotation or flattening influence in the result, \n we use here only the data in the central 27$\\arcsec$, where the assumption of \nspherical symmetry is fulfilled approximately. \nThe same effect also occurred in \nearlier works \\citep{Genzel_96,Trippe_08}. However, recently \\citet{Schoedel_09a} and \\citet{Do_13b} obtained \nhigher masses from Jeans modeling of old stars.\n\n\\citet{Do_12} obtained from the three-dimensional motions\nof 248 late-type stars in the central 12$\\arcsec$ masses between M$_\\bullet=3.77^{+0.62}_{-0.52}\\times 10^6 M_{\\odot}$ and M$_\\bullet=5.76^{+1.76}_{-1.26}\\times 10^6 M_{\\odot}$, consistent with\nthe orbit-based estimates considering the R$_0$ of the fits. They obtained higher values with anisotropic spherically symmetric Jeans modeling than with isotropic spherically symmetric Jeans modeling. These fits obtain isotropy in the center, but tangential anisotropy further out. The latter is probably spurious, see Section~\\ref{sec:_anisotropy}.\nThe lowest value is obtained in isotropic modeling.\n\nThe break radius of \\citet{Do_13b} is large and inconsistent with our data, see Figure~\\ref{fig:_densities_1}.\nWe test its influence on the obtained mass.\nWe therefore select from our data the bins inside of 12$\\arcsec$ and fit the SMBH while fixing all other parameters to fit number 1 of \\citet{Do_13b}. We obtain M$_\\bullet=3.77\\pm0.09\\times 10^6 M_{\\odot}$ fully consistent with \\citet{Do_13b}. (Our error is smaller because we had to fix most parameters.) Thus, our motions are consistent with the motions of \\citet{Do_13b}. Also the binning has no relevant influence. \nThe SMBH mass increases to about $4.5\\times10^6 M_{\\odot}$ when fitting our profiles within 220$\\arcsec$ with a single Nuker profile. In this and the previous fit the cluster mass is zero. When the cluster is included with constant M\/L we obtain M$_\\bullet=3.33\\pm 0.00|_{\\mathrm{tracer\\,profile}} \\pm0.34|_{\\mathrm{dispersion}}\\times 10^6 M_{\\odot}$ with the Do profile and \n M$_\\bullet=2.86\\pm0.10|_{\\mathrm{tracer\\,profile}}\\pm0.23|_{\\mathrm{dispersion}}\\times 10^6 M_{\\odot}$ \n with our density data. Note that for checking whether the tracer profile is important, only the error which is caused by the tracer profile is relevant. Thus, the reason for the high mass\n of \\citet{Do_13b} is twofold: the tracer profile and the extended mass. \n In conclusion, not only high quality dynamic data is important but also good data of the surface density profile. Further, all components, also the extended mass need to be fit, also in the inner 12$\\arcsec$ due to projection effects.\n\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.74 \\columnwidth,angle=-90]{f18a.eps} \n\\includegraphics[width=0.74 \\columnwidth,angle=-90]{f18b.eps} \n\\includegraphics[width=0.74 \\columnwidth,angle=-90]{f18c.eps} \n\\includegraphics[width=0.74 \\columnwidth,angle=-90]{f18d.eps} \n\\caption{M$_{\\mathrm{SMBH}}$ correlation for model 2 and 29 (Table~\\ref{tab:_mass_iso1}).\nThe top row shows the correlation with cluster mass, the bottom the correlation with the inner slope $\\gamma_\\mathrm{in}$. \nBoth models use the star counts as tracer density and assume a constant mass to light ratio. The left (model 2) use the full radial range\nof the dynamics. The right (model 29) use only the data inside of 27$\\arcsec$.\n The dark gray shows the 1$\\,\\sigma$ area from MCMC, the light gray the 1$\\,\\sigma$ area.\n} \n\\label{fig:corr_plot3}\n\\end{center}\n\\end{figure*}\n\n\n\\citet{Schoedel_09a} obtained from isotropic Jeans modeling $M_\\bullet=3.55 \\times 10^6 \\,M_\\odot$, larger than \nour estimate and the earlier works. For their assumed distance of $R_0=8\\,$kpc the SMBH mass of \\citet{Gillessen_09} \nis within the 90\\% probability interval of \\citet{Schoedel_09a}, but in their anisotropic modeling it is excluded by 99\\%. It is interesting to clarify the differences between our analysis and \\citet{Schoedel_09a}, using their publicly available data set. \n\\begin{itemize}\n\\item Using the \\citet{Schoedel_09a} data, their tracer profile and our two mass parameterizations we get results consistent with what these authors found. For example with the power law and R$_0=8\\,$ kpc we obtain M$_\\bullet=3.33\\pm0.56\\times 10^6 M_{\\odot}$. With our normal R$_=8.2$ kpc the mass increases to M$_\\bullet=3.59\\pm0.60\\times 10^6 M_{\\odot}$.\nHence, the details of the modeling only play a minor role. \n\\item \nUsing our dynamical data within 27$\\arcsec$ and the tracer profile of \\citet{Schoedel_09a} we retrieve $M_\\bullet=2.88\\pm0.48\\times10^6 M_{\\odot}$ repeating the previous fit with R$_0=8.2$ kpc. That decrease in mass is not because we add also radial velocities, in the opposite: \nwhen we use only our proper motion the mass decreases slightly to $M_\\bullet=2.63\\pm0.60\\times10^6 M_{\\odot}$. When we fit our density data inside the break with a single Nuker profile, in which $\\gamma=0.5$ but the rest is free, we get $M_\\bullet=2.34\\pm0.60\\times10^6 M_{\\odot}$. \nThus, the use of the \\citet{Schoedel_09a} tracer profile increases that SMBH mass by about $0.5 \\times10^6$ $M_{\\odot}$.\nThe use of different proper motion data sets changes the mass by about $0.7 \\times10^6$ $M_{\\odot}$.\nTherefore, differences in the tracer profiles are besides the dispersion data\nthe main reason for the differences in the mass obtained between our work and \\citealt{Schoedel_09a}.\n\\end{itemize}\n\nWe now investigate the most important reason, namely the higher dispersions in the data of \\citealt{Schoedel_09a}.\n \n\\begin{enumerate}\n\\item The largest relative differences occur within $R<2.5\\arcsec$. There, \\citet{Schoedel_09a} still have a few early-type stars\ncontaminating their sample, since they used the spectroscopy-based, but slightly outdated star list in \\citet{Paumard_06} and \nthe photometric identifications of \\citet{Buchholz_09}. Four certainly early-types are contained in their sample.\nFor example, the well-known fast early-type\nstar S13 \\citep{Eisenhauer_05}, for which an orbit is known, is part of their sample. In contrast, we use in the central 2.5$\\arcsec$ \nonly stars which we positively identify spectroscopically as late-type stars. The issue is critical in the center,\nsince the early-type stars are more concentrated toward the SMBH than the late-type stars and thus\nshow a higher dispersion in the center. We obtain inside of 1.2$\\arcsec$ $\\sigma=9.44\\pm1$ mas\/yr using the data of \\citet{Schoedel_09a} and $\\sigma=6.92\\pm0.69$ mas\/yr from our proper motions.\nRemoving the early-type stars from the \\citet{Schoedel_09a} sample\nyields a central dispersion consistent with our value.\n\\item At R$>$15$\\arcsec$ differences occur again in the proper motion data. This could be caused by differences in the distortion correction which is more important at large radii.\nAn imperfect distortion correction enlarges dispersions artificially, and thus a smaller measured value is more likely to be correct. \n\\end{enumerate}\nOverall, we believe that our dispersions are more reliable than the ones of \\citet{Schoedel_09a}. This is also supported by the fact that we get an extended mass of $M_{100\\arcsec}=5.65\\pm 2.0 \\times10^6 M_\\odot$, consistent with our best value, \nwhen we restrict our dynamics data to the range R$<$27$\\arcsec$ (for fixed SMBH mass with the count profile and constant M\/L) in contrast\nto \\citealt{Schoedel_09a} (Section~\\ref{sec:oth_GC}).\n\n Why do our and other attempts of Jeans-modeling fail to recover the right SMBH mass? \nThe direct cause is that our measured dispersion within 10$\\arcsec$ is smaller than the dispersion in our (isotropic) models\nwhich obtain the right black hole, see Figure~\\ref{fig:_fits_iso3}. \nWhat is the reason for the dispersion difference and thus of the low mass?\n\n\n\\begin{itemize}\n\\item Neglecting anisotropy is probably not the reason. While \\citet{Do_13b} obtain higher SMBH masses in that case, that is mainly caused by the R$_0$-M$_{\\bullet}$ relation. \\citet{Do_13b} obtain R$_0=8.92$ kpc in their free fit, a distance larger than all recent measurements \\citep{Genzel_10}. While the data of \\citet{Schoedel_09a} is problematic in some radial ranges like the very center, we agree about anisotropy: we both get that the cluster is in the inner part slightly (not significant) tangential anisotropy. Central tangential anisotropy decreases mass estimates, see e.g. \\citet{Genzel_00}. Therefore, \\citet{Schoedel_09a} obtain a slightly smaller SMBH mass with anisotropic modeling. We therefore assume that the inclusion of anisotropy likely would decrease the obtained SMBH mass slightly.\n\n\\item The binning and the dispersion errors are probably not the reason. Our different binnings obtain all masses between $3.25\\times10^6$ and $3.38\\times10^6$ M$_{\\odot}$ for the SMBH using the count profile, a constant mass to light ratio and dynamics within 27$\\arcsec$.\nThat makes it also less likely that we underestimated the dispersion errors, since in each of the binnings we use a slightly different\nbias correction, but the results are consistent. The errors are easy to calculate, especially in the center, which is most relevant for the SMBH mass, since measurement errors are significantly smaller than\n Poisson errors.\n\n\n\\item For checking whether we can recover the correct SMBH mass for another $R_0$, we use, as previously, the SMBH mass-$R_0$ relation of \\citet{Gillessen_09} for another $R_0$. Even for an unrealistically large value of\n$R_0=9\\,$kpc the SMBH mass-$R_0$ relation is not recovered. Thus, the SMBH mass underestimation is largely independent of the distance.\n\n\\item The power law extended mass parametrization \n yields a smaller mass than using a constant M\/L: \n$M_\\bullet=(2.68\\pm 0.51) \\times 10^6~M_\\odot$ for a free power law slope and $M_\\bullet=(2.98 \\pm 0.17) \\times 10^{6}~M_\\odot$ for a Bahcall-Wulf cusp of $\\delta_M=1.25$.\nWhile a constant mass to light yields $M_\\bullet=(3.37 \\pm 0.16) \\times 10^{6}~M_\\odot$. Hence, a core in the stellar mass\ninstead of a Bahcall-Wulf cusp may be part of the solution.\n\n\\item \\citet{Trippe_08} and \\citet{Genzel_10} argued that a central core-like structure introduces a bias toward low SMBH masses.\nThis, however is only applicable when a tracer profile without core is used for cored data. If the correct profile contains a core and\nis modeled as such, the central core only increases the error on the central mass. In our models such an uncertainty is included via the free \ndensity profile, yet the SMBH mass falls outside of the error band. In Figure~\\ref{fig:corr_plot3}\nit is visible that the correlation of $\\gamma_{\\mathrm{in}}$ and the SMBH mass is weak. It is not possible to obtain the right mass by changing $\\gamma$ within our model.\nStill, the fact that our flux and star count profiles are not consistent indicates that their errors are probably underestimated. \nNeither produces the right mass, but maybe the true profile is none of them. A profile with a large enough core radius might\nproduce the right mass. Another possibility is that our functional form is inadequate. Perhaps a profile works with a large radial transition region, in which the profile has a constant slope, \nsomewhat steeper than in the center.\n\n\\item \nIntroducing a flat true core is unlikely to solve the issue. It is not only inconsistent with the apparent isotropy, \nit also under predicts the number of late-type stars with orbits. \n\n\\item Also the flattening influences the SMBH mass, both by the tracer profile, and by reducing the dispersion in most dimensions,\nsee \\citet{Chatzopoulos_14}. Also well inside of 27$\\arcsec$ the dispersion is different in l and b, indicating that \nflattening is important also there.\nWe test that case by fitting the data within 27$\\arcsec$ with the flattened 2-integral model of \\citet{Chatzopoulos_14}.\nWe obtain $M_\\bullet=(3.54\\pm 0.18) \\times 10^6~M_\\odot$. This error also includes the distance uncertainty.\n\\end{itemize}\n\nSummarizing, we have tested several potential solutions to the mass bias. For some we have shown quantitatively that they alone \ncannot correct the bias. Others would requires modeling which goes beyond this work. \nLikely, several together are necessary\nfor a solution.\nThe absence of a dark cusp and another more complex tracer profile\nare maybe the most likely solution.\n Perhaps maximum likelihood modeling\nis needed for our discrete data set \\citep{Dsouza_13}, in particular for properly incorporating the information the fast stars carry.\n\n\n\n\n\\subsection{Comparison with the M$_\\mathrm{cluster}$ Literature}\n\\label{sec:oth_GC}\n\nOverall, most M$_\\mathrm{cluster}$ values from the literature are similar to our values. \nThe comparison with other works needs some care, since different values for $R_0$ have been used, and hence\nwe scale to $R_0=8.2\\,$kpc.\n We correct the masses in the literature to our distance using a mass scaling with exponent 1 for purely radial velocity based masses, and \nexponent 3 for purely proper motion based ones. \nWe extrapolate from our best estimate $M_{100\\arcsec}$ in two ways: \n on the one hand we use a broken power law with break radius 100$\\arcsec$. The inner slope is $\\delta_\\mathrm{M}=1.232$, the \nouter is $\\delta_\\mathrm{M}=1.126$. (That implies we use inside of 100$\\arcsec$ the average $\\delta_\\mathrm{M}$ of row~3 and 4 of Table~\\ref{tab:_mass_iso1} and outside the average $\\delta_\\mathrm{M}$ of row~11 and 12. The first uses dynamics between 10 and 100$\\arcsec$, the second dynamics outside of 10 $\\arcsec$.)\n On the other hand, we use one of the preferred fits with constant M\/L (Row~5 in Table~\\ref{tab:_mass_iso1})\nMost other mass profiles are similar to one of these cases. \nWe quantify the quality of the comparisons only when errors are given in literature and there is not more than one value in a source without clear preference for one.\n\n\n\\subsubsection{\\citet{Schoedel_09a}}\n\nThe possible mass range of \\citet{Schoedel_09a} within $1\\,$pc is $(0.5-2.2)\\times10^6M_{\\odot}$, similar to our result. \nHowever, when assuming a constant M\/L and isotropy they obtain a mass of $1.6 \\times 10^6 M_{\\odot}$,\nlarger than our value of $0.6\\times 10^6\\,$M$_{\\odot}$ for the same assumptions. At 100$\\arcsec$ their profile yields a\nrather large mass of $16 \\times 10^6\\,$M$_{\\odot}$. \nThe main reason for the difference is their larger dispersion outside of 15$\\arcsec$ (Section~\\ref{sec:mass_smbh}).\n\n\n\\subsubsection{\\citet{Trippe_08}}\nAt 100$\\arcsec$ \\citet{Trippe_08} found a mass that is roughly a factor three larger than ours. \nThe difference is due to the high rotation at large radii as a result of their selective use of data from \\citet{McGinn_89}.\nAt $1\\,$pc where rotation is negligible they found a value of $1.2 \\times 10^6 M_\\odot$ (their Figure~14, gray dashed curve),\nsimilar to our mass in the power law case.\n\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=1.50 \\columnwidth,angle=-90]{f19.eps}\n\\caption{Cumulative mass profile of the GC. The main measurement of this work is the red pentagon at 4 pc, through\nwhich the profiles for the power law or constant M\/L case pass. The latter is slightly preferred. The stellar orbits based value from \\citet{Gillessen_09} is at about 0.002 pc. \n\\citet{Beloborodov_06} used an orbit roulette technique to obtain an enclosed mass.\nIn the work of \\citealt{Schoedel_09a} (gray diamond) no formal error is given, we show the largest range mentioned.\nFor the masses from \\citealt{Trippe_08} (green open circles) no errors are given. The value from \\citet{Serabyn_85} is the violet square.\nFor \\citealt{Serabyn_86} the thick black line is the value and the light gray area gives the error range.\nFrom \\citealt{Lindqvist_92b} (light violet triangles) and \\citealt{McGinn_89} (green stars) we use the Jeans \nmodeling values. In case of \\citealt{Deguchi_04} (light green line) we show their extended mass model fit using the Boltzmann equation, \nadapting to our assumed SMBH mass.\n We plot Jeans-modeling based values from \\citealt{Genzel_96} (pink dots).\nWe also plot the two-integral fit of by \\citet{Chatzopoulos_14} (pink curve).\nWe omit the values from \\citet{Genzel_96} and \\citet{McGinn_89} inside of 0.55 pc, since more accurate values are available there.\n} \n\\label{fig:_mass_lit1}\n\\end{center}\n\\end{figure*}\n\n\n\\subsubsection{Inner Circumnuclear Disk}\n\n\nThe gas in the in circumnuclear disk (CND) can be used to obtain a mass estimate.\n\\citet{Serabyn_85} used the emission of [NeII] 12.8 $\\mu$m from gas streamers \n at the inner edge (including the Western arc) of the CND to obtain its rotation velocity. They found \n $ (3.9 \\pm 0.8) \\times 10^6 M_{\\odot}$ at 1.4 pc. \n That is 1.7$\\,\\sigma$ and 2.4$\\,\\sigma$ less than our constant M\/L and power law estimates. \nTheir mass is smaller than the mass of the SMBH.\nThe projected rotation velocity of the inner CND of $100\\,$km\/s is well-determined \\citep{Genzel_85,Serabyn_85,Guesten_87,Jackson_93,Christopher_05},\nand hence the reason for the discrepancy has to be in the model assumed.\n Conceptually, the mass derivation of \\citet{Serabyn_85} was simple: they assumed circular motion of all the gas in one ring\n with an inclination of $60-70^{\\circ}$. \n\nIn the HCN J-0 data of the inner CND in \\citet{Guesten_87} the \nvelocities in the southern and western parts follow the model of \\citet{Serabyn_85}, but not in the northern and \neastern parts. The latter can be described by a less inclined ring ($\\approx45^{\\circ}$) with an intrinsic rotation velocity of \n$137\\pm8 $ km\/s. This velocity results in a total mass $(6.1\\pm 0.7)\\times10^6 M_{\\odot}$ somewhat larger than our estimates. Using an average inclination \nbetween \\citet{Serabyn_85} and \\citet{Guesten_87} would hence yield a mass estimate very similar to ours.\n\nHowever, \n\\citet{Zhao_09} find an inclination consistent with \\citet{Serabyn_85}. \nFurther \\citet{Zhao_09} show that a single \norbit, even slightly elliptical, cannot fit all gas streamers in the Western arc, and hence the inner CND is likely more complicated than \nassumed by \\citet{Serabyn_85}. \n\n\nAnother possibility to solve the discrepancy between our result and the result\nof \\citet{Serabyn_85} is that the assumption of \\citet{Serabyn_85} that the line width is due to nongravitional processes\nmay be wrong. If the line width is due to asymmetric drift \\citep{Binney_08} the true circular velocity $v_{\\mathrm{c},\\mathrm{true}}$ would be higher\nthan the measured velocity $v_{\\mathrm{c},\\mathrm{obs}}$ \\citep{Kormendy_13}: \\begin{equation}\nv^2_{\\mathrm{c},\\mathrm{true}}=v^2_{\\mathrm{c},\\mathrm{obs}}+x \\times \\sigma^2 \n\\end{equation}\nThe factor $x$ depends on the disk profile and other not well known parameters of the \ninner CND. It is between 1 and 3. \nFrom $\\sigma=35\\pm5\\,$km\/s \\citep{Serabyn_85} follows for $x=$3\nan increase in the mass by 30\\% to $ (5.1 \\pm 1.04) \\times 10^6 M_{\\odot}$. This would be consistent with our constant M\/L estimate of $5.27 \\times 10^6 M_{\\odot}$.\n\n \\subsubsection{Outer Circumnuclear Disk}\n\n\\citet{Serabyn_86} used CO 1-0 in the outer parts of the CND for \nestimating the mass using the same method as \\citet{Serabyn_85}.\nThis measurement is consistent with our mass. The agreement argues\nthat other forces besides of gravity are not relevant for at least the outer gas dynamics.\n\n \\subsubsection{\\citet{Rieke_88} }\n\nAt radii larger than $7\\,$pc the gas velocities get unreliable \\citep{Serabyn_86,Guesten_87}, but stellar velocities are available.\n\\citet{Rieke_88} used a few bright stars and found a dispersion of $75\\,$km\/s between 6$\\arcsec$ and 160$\\arcsec$, independent of radius. With our\ndata we can reject the hypothesis of \\citet{McGinn_89}, proposing that a magnitude effect causes the surprisingly flat\ndispersion curve. Possibly, \\citet{Rieke_88} were limited by the low-number statistics.\n\n\\subsubsection{\\citet{McGinn_89} }\n\nThese authors covered a similar area in size to ours, using integrated velocities and dispersions in large beams.\nBroadly their dispersion data is consistent with our data, and they measure a somewhat stronger radial dispersion trend.\nThey obtain masses similar to our estimates and also find a too small SMBH mass, like most works that apply free Jeans modeling.\nAt the outer edge the masses of \\citet{McGinn_89} are somewhat larger than ours, due to the high rotation velocity in this work\n(Section~\\ref{sec:rotation}).\n\n\n\\subsubsection{\\citet{Lindqvist_92b}}\n\nUsing maser velocities and Jeans modeling out to much larger radii \\citet{Lindqvist_92b} obtained \nmasses consistent with our model. This is not surprising, since we include their velocities in our data set.\nThe main discrepancy occurs around 17 pc, where their estimate is only half of our extrapolation. \nOn the one hand, our modeling has not much flexibility at these large radii. On other hand, their mass profile had possibly too much freedom since the light profile shows no dip there.\n In both our and the analysis of \\citet{Lindqvist_92b} the deviation from spherical symmetry is not considered,\nwhich might lead to significant changes at large radii.\n\n\\subsubsection{\\citet{Deguchi_04} }\n\n\\citet{Deguchi_04} used also maser velocities, but in a somewhat smaller area than \\citet{Lindqvist_92b}. \nThey prefer a result based on a new method developed from the Boltzmann equation. This method includes rotation, but assumes\nspherical symmetry and a fixed extended mass slope of $\\delta_\\mathrm{M}=$1.25. \nTo compare their extended mass result with\nour work we need to adapt their result to our SMBH mass, which is within their 1.5$\\,\\sigma$ range.\nIn the inner parsec their mass of $(0.90\\pm 0.07)\\times 10^6 M_{\\odot}$ is consistent with our range, but\nat 100$\\arcsec$ their estimate of $(5.06 \\pm 0.39) \\times 10^6 M_{\\odot}$ deviates by about 1.5$\\,\\sigma$ from our estimate. \nDue to their fixed slope their mass deviates also at larger radii from the results in \\citet{Lindqvist_92b}.\nA fixed slope does not describe the stars density in the GC well out to 80 pc.\n\n\\subsubsection{\\citet{Genzel_96}}\n\n\nThis work obtained the mass distribution out to 20 pc from Jeans modeling of radial velocities,\nusing literature radial velocities in combination with newly measured velocities in the center. Also these authors\nunder predict the SMBH mass. Beyond that, their results are consistent with ours.\n\n\\subsubsection{\\citet{Chatzopoulos_14}}\n\nAs discussed also in Section~\\ref{sec:jeans_modelling}, the modeling of \\citet{Chatzopoulos_14} obtain a larger mass than we do due to the flattening. That mass is also larger than most other estimates between 2 and 50 pc. The reason is usually the same - the other models are not flattened. For example, that is the reason why \\citet{Deguchi_04} obtain a smaller mass.\n\n\\subsubsection{Works Using the Light Distribution}\n\nIt is possible to obtain mass estimates of the extended mass by using the flux and its distribution of the GC.\nIt is however difficult due to \nfollowing systematic uncertainties in M\/L: the star formation history, the extinction toward the GC and the IMF of the stars. Thus, in the past \\citet{McGinn_89,Lindqvist_92b,Launhardt_02} used a M\/L obtained at a selected radius with dynamics\ntogether with the flux profile to estimate the mass of the nuclear cluster. Thus the broad agreement of them with our work shows that the mass of \\citet{McGinn_89,Lindqvist_92b,Genzel_96,Genzel_97} agrees with our mass.\nWith the recent agreeing measurements of extinction \\citep{Nishiyama_06b,Fritz_11,Schoedel_09b}\n and star formation history \\citep{Blum_02,Pfuhl_11} two uncertainties in a purely light derived mass are now reduced. \nStill the IMF essentially cannot be constrained using light information alone \\citep{Pfuhl_11}.\n Since the IMF of the GC is an interesting subject, we reverse the argument, \nand use the mass to constrain the IMF (Section~\\ref{sec:mass_to_light}).\n\nRecently, \\citet{Schoedel_14} used their light decomposition to infer the total mass of the nuclear cluster\nof $2.1\\pm0.4\\times10^7\\, M_\\odot$. \nThat is only half of our estimate but within the error for the total mass. The $M\/L=0.5\\pm0.1$ of \\citet{Schoedel_14}\nis nearly identical with our dynamic value (Section~\\ref{sec:mass_to_light}). \nWhile \\citet{Schoedel_14}\nused IRAC2 instead of Ks the impact on M\/L is rather small (at most 38\\%) for the GC star formation history (Section~\\ref{sec:mass_to_light}).\n\n\\subsection{Mass Cusp or Core?}\n\\label{sec:mass_cusp_core}\n\n\n\nThe distribution of old stars does not show the expected central cusp \\citep{Bahcall_76} with the inner $\\delta_\\mathrm{L}=1.25$ slope\n \\citep{Buchholz_09,Do_09,Bartko_10}.\nThe stars follow a shallower slope in the center, \nwhich can either be a true core ($\\delta_\\mathrm{L}=3$) or a shallow cusp ($\\delta_\\mathrm{L}>2$, \\citealt{Do_09}).\nAlso the second case we call core here, since it has much less light in the center than a cusp.\nBy comparing \nour power law mass models (with a central cusp) and the constant M\/L models (with a core)\nwe can constrain the central mass distribution.\n\n\n\\begin{itemize} \n\\item Given that a core can reduce the low dispersion bias, the Jeans modeling of our data \n favors a core over a cusp. The same is found by \\citet{Schoedel_09a},\nwho used a unrealistically large, fixed break radii for light and especially mass. \nOur results show that the preference of a smaller slope in the center is \nnot only due to the large break radius in \\citet{Schoedel_09a}.\n\\item The presence of a warped disk of young stars \\citep{Loeckmann_09a,Bartko_09}\nyields an additional constraint.\n \\citet{Ayse_13} found that with\nan isothermal cusp of \nM($r<0.5$pc)$=10^6\\,$M$_{\\odot}$ and a flattening of q$=0.9$ an initial warped disk is too quickly destroyed \n and would not be observable today. For constant M\/L we obtain a smaller mass of M($r<0.5\\,$pc)$\\approx0.15 \\times 10^6\\,$M$_{\\odot}$, which reduces the torque by a factor five, allowing the disk to survive longer. This conclusion\nis not firm, since the warping need not be primordial, and a reduced central flattening might also yield less\nwarping \\citep{Kocsis_11}. \n\\item The most reliable mass measurement around 1.4 pc apart from \nour analysis, that is\n\\citet{Serabyn_85}, is more consistent with a core than with a cusp.\n\\end{itemize}\n\nOverall, it appears likely that the mass profile shows a central core, but better modeling including a solution of the mass bias \nis necessary. \nAnother route for constraining the mass profile worth follow up is using direct accelerations of stars at radii between 1$\\arcsec$ and \n7$\\arcsec$ with\nGRAVITY \\citep{Eisenhauer_08}.\n\n\nOur preference for a core in the mass profile is interesting for theories which aim at explaining the missing light cusp.\nIn some theories the resolvable stars (giants) form a core while dark components (remnants or main sequence stars) \nform a mass cusp. Thus, our result means that mass segregation is less likely to solve the riddle of missing light cusp. \n\\citet{Keshet_09} derive a similar conclusion from different arguments. Another mechanism that destroys a light cusp, but not a \nmass cusp, is the destruction of giants by collisions \\citep{Dale_09}, and hence our results also disfavor that model.\nOne model that could work is presented in \\citet{Merritt_10} who propose that a \nrelatively recent binary black hole (merger) ejected stars. This process would yield a core both in the\nlight and the mass profile. Still, one would need to fine-tune the timing of such a model, since the early-type stars\nare concentrated toward the center, but the giants and even the younger red (super-) giants with ages down to 20 \nMyrs \\citep{Blum_02,Pfuhl_11} are not.\n\n\n\n\\subsection{Cumulative Mass Profile}\n\\label{sec:cum_mass}\n\nAnother way of expressing our results is the cumulative mass profile. In Figure~\\ref{fig:_nc2} we show three\ncases:\n\\begin{itemize}\n\\item We use a constant M\/L model with the best fitting inner slope of -0.81 (Row 5 in Table~\\ref{tab:_mass_iso1}, model A).\n\\item We use a constant M\/L model with the shallowest possible inner slope of -0.5 (and otherwise row 5 in Table~\\ref{tab:_mass_iso1}, model B).\n\\item We use a power law model with a slope of $\\delta_\\mathrm{M}=1.232$ (model C), the average of row 3 and 4 in Table~\\ref{tab:_mass_iso1}\\footnote{Outside of 100$\\arcsec$, a range not relevant here, model C uses\n $\\delta_\\mathrm{M}=1.126$, the average of rows 11 and 12 in Table~\\ref{tab:_mass_iso1}.}. \n\\end{itemize}\nWe normalize these three models to have the overall best-fitting value of $M_{100\\arcsec} =6.09 \\times 10^6 M_\\odot$. Tabulated values of the profiles can\nbe found in Appendix~\\ref{sec_val_cum_mass}. The largest mass difference between the M\/L$=$const and the power law model is reached\n at about 35$\\arcsec$ with $0.59\\times 10^6 M _{\\odot}$. \nSince this number exceeds our mass error, it is possible to detect dynamically the cusp, provided the low dispersion bias can be solved.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f20.eps}\n\\caption{Cumulative mass profiles, for a power law model or assuming M\/L$_{\\mathrm{Ks}}=$const. The normalization\nis done for $M_{100\\arcsec} = 6.09 \\times 10^6 M_\\odot$ (black square).\nOther masses shown are the ZAMS cumulative mass distribution of the O(B)-star population \\citep{Bartko_10}\nand with a filled disk the CND from\n \\citet{Etxaluze_11,Genzel_85, Mezger_89}, \nand\n \\citet{Requena_12}. The open circle is the CND mass estimate from \\citet{Christopher_05}.\n}\n\\label{fig:_nc2}\n\\end{center}\n\\end{figure}\n\n\nExtrapolating the constant M\/L models down to the regime of the S-stars S2 and S55\/S0-102 \\citep{Meyer_12} \nyields a mass there, which is smaller than that of the individual stars. In this regime, the S-stars dominate, and\ntaking into account their Salpeter-like IMF \\citep{Bartko_10}\nor a potential mass-segregated cusp of stellar remnants \n\\citep{Freitag_06,Hopman_06} \nwould make the dominance even stronger. \nFor power law models it is not clear whether the old stars dominate the mass in the central \narcsecond. \n\nOutside the central arcsecond, the mass of the disks of early-type stars \n\\citep{Bartko_09,Lu_09} may be important. We derive the spatial distribution from Figure~2 in \n\\citet{Bartko_10} and globally deproject to space distances with a factor 1.2 the given projected distances.\nWe estimate the total ZAMS mass of that population to $1.5\\times 10^4\\,M_\\odot$ \\citep{Bartko_10}, \ndominated by the O-stars given the \n top-heavy shape of the IMF. Across the literature \\citep{Paumard_06,Bartko_10,Lu_13} the disk mass is uncertain by a factor 2.5. \nAt $r=2\\arcsec$ the O-stars might be\ncomparable in mass to the old stars, if the constant M\/L model is correct.\n\n\n\nFurther out, the only other component is the circumnuclear disk. Most \npublications \n\\citep{Genzel_85,Etxaluze_11,Mezger_89,Requena_12} agree that its\n mass is a few $10^4\\,M _{\\odot}$ and thus irrelevant compared to the old \nstars. However, the mass of $\\approx 10^6 M_{\\odot}$ found by \\citet{Christopher_05}\nwould be about a third of our mass estimate at that radius. \n\n\nOur results also yield a new estimate for the sphere of influence of the SMBH \\citep{Alexander_05}, the radial range within which\nthe extended mass is equal to the mass of the SMBH.\nIt is completely consistent with previous values \\citep{Alexander_05,Genzel_10}. Using model A and B, \nwe get r$_{\\mathrm{infl}}=76.3\\pm5.5\\arcsec=3.03\\pm0.25\\,$pc. \n\n\n\n\\subsection{Mass to Light Ratio}\n\\label{sec:mass_to_light}\n\nWe now obtain the mass to light ratio from the fits to flux and dynamics with constant M\/L.\nThe mass to light ratio is directly calculated from the output of these models. \nWe obtain $M\/L=0.51 \\pm 0.12 \\, M_{\\odot}\/L_{\\odot,\\mathrm{Ks}}$ using $M_{\\mathrm{Ks}\\,\\odot}=3.28$ \\citep{Binney_98}.\nThe error consists of 19~\\% for the light, 10~\\% for the mass, and 8~\\% for the distance uncertainty.\nThe latter number \nfollows from the distance uncertainty of 4.1\\% \\citep{Gillessen_13} multiplied by the exponent 1.83, which is the scaling M\/L with\ndistance given that M scales like 3.83 with distance (Section~\\ref{sec:jeans_modelling}).\nOur M\/L is consistent with the values in \\citet{Pfuhl_11} and \\citet{Launhardt_02}.\nOur error, however, is smaller thanks to the smaller mass error. With the improved mass to light ratio we can constrain the IMF of the old stars \nfurther. \n\n\n\nFirstly, we use the M\/L to determine the IMF slope $\\alpha$, we assume the respective star formation histories for the old stars (older than 10 Myrs) from \\citet{Pfuhl_11}. Their results \nappear to hold also at the larger radii of interest here \\citep{Blum_02}.\nAlready with the accuracy of M\/L of \\citet{Pfuhl_11} it was possible to discard extreme top-heavy IMFs ($\\alpha>-0.6$)\nlike measured for the O-stars in the GC \\citep{Bartko_10}. With our accuracy also the IMF slope\nof $\\alpha=-0.85$ \\citep{Paumard_06} is excluded.\n\n\nSecondly, we concentrate on variants of the usual -2.3 slope below one solar mass, these are the IMFs of \\citet{Salpeter_55,Kroupa_01,Chabrier_03}. \nTo check the result on dependence of the used stellar population model we use the models of \\citet{Bruzual_03} and \\citet{Maraston_05}.\nIn case of the \\citet{Maraston_05} models we obtain \n $M\/L=0.75 M_{\\odot}\/L_{\\odot,\\mathrm{Ks}}$ and $M\/L=1.28 M_{\\odot}\/L_{\\odot,\\mathrm{Ks}}$ for Kroupa and Salpeter, respectively. \nFor the \\citet{Bruzual_03} models we get $M\/L=1.04 M_{\\odot}\/L_{\\odot,\\mathrm{Ks}}$\nand $M\/L=0.76 M_{\\odot}\/L_{\\odot,\\mathrm{Ks}}$ for Salpeter and Chabrier, respectively. \nThe differences in stellar population models cause the differences of up to 20\\%. (The stellar population model dependences increase with wavelength: at 4.5 $\\mu$m (IRAC2), used by \\citet{Schoedel_14} and \\citet{Feldmeier_14}, the model of \\citet{Bruzual_03} obtains $M\/L=0.71 M_{\\odot}\/L_{\\odot,\\mathrm{IRAC2}}$ for Chabrier and the model of \\citet{Bruzual_03} $M\/L=1.01 M_{\\odot}\/L_{\\odot,\\mathrm{IRAC2}}$ for Kroupa.) \nUsing the star formation history of \\citet{Blum_02} instead of the of \\citet{Pfuhl_11}\ncauses 18\\% smaller M\/L in Ks-band.\n\n\n\nRecently, deviations from the Chabrier IMF were found for different stellar systems:\non the one hand elliptical galaxies seem \nto have a Salpeter or more bottom heavy IMF \nas \\citet{Vandokkum_10,Conroy_12} concluded using population modeling of integrated spectra and as \\citet{Cappellari_06,Cappellari_12}\nconcluded from advanced dynamic modeling.\nAlready a Salpeter IMF is in strong tension with our ratio. A more bottom heavy IMF of e.g. a slope of -2.8 has an even larger M\/L (A factor 1.57 more in M\/L compared to Salpeter for the star formation history assumed in \\citet{Cappellari_12,Conroy_12}\nand is thus firmly excluded.\n\n\nIt seems that (metal rich) globular clusters have M\/L smaller than Kroupa \\citep{Kruijssen_09a,Bastian_10,Sollima_12}. This was found in the case of M31 by \\citet{Conroy_12}\n using population modeling of integrated spectra. In addition, \\citet{Strader_11} found the same by using mass and light measurements.\nPrecisely, \\citet{Strader_11} measured for solar metallicity ($\\mathrm{[Z]}=0\\pm0.3$) $M\/L=0.39 M_{\\odot}\/L_{\\odot,\\mathrm{Ks}}$. \nThis is already lower than our measurement, although the GC contains more medium old stars than these globulars. \nIn an attempt to include these younger stars we reduce in our Kroupa and Chabrier models the mass of the stars $>5$ Gyrs to achieve \n$M\/L=0.39 M_{\\odot}\/L_{\\odot,\\mathrm{Ks}}$\nfor these ages. We do not change the populations for the younger stars.\nThis model has the motivation that it describes a possible formation scenario of the nuclear cluster in which the majority\nthe old stars originate in big metal rich globular clusters \\citep{Capuzzo_08} and later stars formed locally are added to it.\nThis model results in $M\/L=0.31 M_{\\odot}\/L_{\\odot,\\mathrm{Ks}}$ deviating by 2$\\,\\sigma$ from our measurement. \nThis argument against a globular cluster origin of the nuclear cluster is weakened by the fact that\nwhen globulars arrive in the GC, they likely have a different M\/L due to mass segregation. In case of M31 it is argued\nthat the low M\/L is mostly due to a different IMF and not only due evolutionary effects \\citep{Strader_11}. \nHowever, the preferred loss of low mass objects due to internal mass segregation and the outer tidal field\nexists certainly \\citep{Bastian_10,Sollima_12} and should be further enhanced close to the GC. \nThe problem is that, while mass segregation first reduces the M\/L, M\/L is enhanced in late stages during the late remnant dominated state of\nold populations \\citep{Sollima_12,Marks_12}. Thus, without detailed modeling of mass segregation \nthe expected M\/L of a globular cluster dominated nuclear cluster is uncertain.\n\n\nThe GC shows a normal ratio of diffuse light to the total light \\citep{Pfuhl_11}. \nThis ratio measures the ratio of main sequence stars to giants and decreases monotonically from a bottom-heavy\nto a top-heavy IMF. Thus it excludes best a cluster dominated by remnants and giants - a possible state of globular\nclusters inspiraling to the GC.\n\nThe Chabrier IMF common in Galactic disk and bulge \\citep{Bastian_10,Zoccali_00} is close to the M\/L of the medium old and \nold stellar populations of the GC. However, the deviation from it is\n1.7 $\\sigma$ when using the most recent star formation history determination \\citep{Pfuhl_11}. \nThat deviation vanishes when the bias to small masses due to our spherical modeling is corrected, see \\citet{Chatzopoulos_14}. In that case the ratio is $M\/L=0.76 \\pm 0.18 \\, M_{\\odot}\/L_{\\odot,\\mathrm{Ks}}$ matching well the Kroupa\/Chabrier IMF.\nTherefore a globular cluster origin of the nuclear cluster \\citep{Antonini_12,Gnedin_14} is somewhat disfavored by its M\/L and \nL$_{\\mathrm{diffuse}}$\/L$_{\\mathrm{total}}$. \n\n\n\n\\subsection{The Nuclear Cluster of the Milky Way in Comparison}\n\\label{sec:mass_nucl}\n\n\nWe now compare the nuclear cluster of the Milky Way with nuclear clusters in other galaxies.\nThe literature about the mass of other nuclear clusters \\citep{Walcher_05,Barth_09} is sparse and biased toward brighter nuclear clusters. \nWe therefore use mainly the light for comparison, for which ample data from HST imaging are available \\citep{Carollo_02,Boeker_04}.\n\\citet{Boeker_04} studied late-type spiral (Scd to Sm)\nwhile \\citet{Carollo_02} concentrated on early-type spirals (Sa to Sbc).\n For color-correction we use $\\mathrm{'H'}-\\mathrm{Ks}=0.15$ and $\\mathrm{I}-\\mathrm{Ks}=1.1$ for \\citet{Carollo_02} and \\citet{Boeker_04}, respectively. \n We compare the obtained magnitudes \nwith our GALFIT decomposition for the nuclear cluster (Section~\\ref{sec:flattening}), of the double $\\gamma$ model fitting (Section~\\ref{sec:fitting_profile_gam}).\nIn addition we use the cumulative flux as function of radius. The latter is not affected by decomposition uncertainties, (Figure~\\ref{fig:_comp_nc}).\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f21.eps} \n\\caption{\nSizes and luminosities of nuclear clusters. We show the half light radii for clusters in late-type \\citep{Boeker_04} and early-type\nspiral galaxies \\citep{Carollo_02}.\nFor the Milky Way we present the inner component of our GALFIT decomposition (green box), of the $\\gamma$ model fitting (green triangle) and the total cumulative flux as function of the radius.\n} \n\\label{fig:_comp_nc}\n\\end{center}\n\\end{figure}\n\n\nThe size of the nuclear cluster of the Milky Way is typical. \nIt is smaller than many clusters in \\citet{Carollo_02} and larger than most clusters in \\citet{Boeker_04}. \nThis simply might be a consequence from the fact that the galaxy type of the Milky Way is broadly\nbetween the two samples.\n\n\n\nIt is visible in Figure~\\ref{fig:_comp_nc} that the nuclear cluster of the Milky Way has an unusually high surface brightness.\nNot only its characteristic brightness is high, but \nalso the cumulative brightness profile lies above nearly all other clusters. \nWhat is the reason for that?\n\\begin{itemize}\n\\item\nThe young ($\\approx 6$ Myrs) and medium old ($\\approx 200$ Myrs) stars in GC are likely not the reason for this offset. They are subdominant \ncompared to the old stars in the Ks-band \\citep{Blum_02,Pfuhl_11} and Section~\\ref{sec:total_luminosity}.\nFurther many nuclear clusters \\citep{Rossa_06,Walcher_06} contain significant fractions of young or medium old stars.\n Thus, the M\/L of the GC is probably typical.\n\\item\nExtinction is probably not an issue although we correct for it. \nThe spirals in the sample of \\citet{Carollo_02} and \\citet{Boeker_04} are seen more or less face-on. \nFurther, the mean IR color in \\citet{Carollo_02} yields a \nsmall extinction of A$_{160W}\\approx0.2$. Also the spectroscopic study of \\citet{Rossa_06} obtained small extinctions for the\nsame samples.\n\\item\nThe higher resolution of our data is also not the reason, since we compare the characteristic brightness within 4 pc, \na size which can be resolved in most of the galaxies of \n\\citet{Carollo_02} and \\citet{Boeker_04}. \n\\item\nThe nuclear cluster of the Milky Way has a projected mass density of $\\log{\\Sigma_e}=5.38 \\pm 0.17$ [M$_{\\odot}$\/pc$^2$]. This is larger than for all 11 nuclear\nclusters in the sample of \\citet{Walcher_05}. \n\\end{itemize}\nWe conclude that the main reason for high surface light density is the high stellar mass density in the nuclear cluster, \ncompared to other galaxies. \n\n\nFor some near-by galaxies a more detailed comparison is possible.\nFor example M33 shows a central star density which is possibly larger than in the nuclear cluster of the Milky Way \\citep{Lauer_98}.\nM33 does not contain a SMBH \\citep{Merritt_01,Gebhardt_01}, and the old stars - if present at all - are outshined by young stars.\nSome nuclear clusters show flattening and rotation, in case of metallicity the flattening is 16\\% \\citep{Lauer_98} combined with \nsome rotation \\citep{Kormendy_93}. The large nuclear cluster in NGC4244 (R$_e\\approx10\\,$pc) is visually flattened and has \n$v_{\\mathrm{rot}}\/\\sigma\\approx1$ \\citep{Seth_08}. NGC404 is a similar case \\citep{Seth_10}.\nThus the flattening of the Milky Way nuclear cluster \n is in the large range of possible shapes. \n\n \\citet{Carollo_99} finds that nuclear cluster with M$_V\\lesssim-12$\nare typically associated with signs of circumnuclear star formation, like dust lanes and an HII spectrum.\nIn this respect the cluster of the Milky Way with M$_V\\lesssim-13$ seems typical for its nuclear disk.\nPreferentially, bright nuclear clusters are observed together with nuclear disks, \n which might be explained \nby a physical coupling of the two systems. \nSince the GC is dominated by old stars this connection is likely not only valid for the recent star formation event but also for the older\npopulation. \n\n\\section{Conclusions and Summary}\n\\label{sec:conclusions}\n\n\n\nThe nuclear cluster in the Milky Way is by far the closest and thus can be studied in more detail\nthan other nuclear clusters.\nIn this paper we investigate not only its center, but its full size to compare\nit with such clusters in other galaxies. \nTo that aim, we obtain its light profile out to 1000$\\arcsec$ and measure motions in all three dimensions out to 100$\\arcsec$,\nexpanding on the works of \\citet{Trippe_08} and \\citet{Schoedel_09a}.\\\\\n\n\\begin{itemize}\n \\item \nWe construct a stellar density map with sufficient resolution for two dimensional structural analysis out to $r=1000\\arcsec$.\nThis map shows in the central 68$\\arcsec$ an axis ratio of\n $q=0.80\\pm0.04$. \n Further out the flattening increases. \nWe fit this map by two components, the inner is a Sersic, the outer a Nuker profile.\nThe more spherical and smaller component is the nuclear cluster, while the more extended component corresponds to the nuclear disk (stellar analog of the central molecular zone).\nThe decomposition is uncertain because the outer component is relatively bright.\nWith our preferred decomposition the nuclear cluster has a half light radius \nof 127$\\arcsec\\approx5.0\\,$pc, a flattening of 1\/0.80, a Sersic index of n$=1.5\\pm0.1$, \nand a total luminosity of M$_{\\mathrm{Ks}}=-15.5$. The outer component has a flattening of about 1\/0.264 and contributes about 13\\% to the projected central flux.\nThe size and luminosity of the nuclear cluster depend on the functional form of the assumed outer slope, which\nis not well constrained by our data. A flatter profile like a $\\gamma$ model yields r$_e\\approx9\\,$pc and \na total luminosity of M$_{\\mathrm{Ks}}\\approx-16.5$.\n\\item Our dynamical analysis shows that\nthe proper motions are radially \nisotropic out to at least 40$\\arcsec$. In the center the velocity distribution shows strong wings\nas expected due to the presence of the SMBH. \nAssuming that the fastest stars in the center are bound, we set a lower bound to the SMBH mass of \n$3.6\\times 10^6\\,$M$_{\\odot}$.\nAssuming in addition that the black hole mass is known to 1.5\\% \\citep{Gillessen_09} for a given distance, we can estimate\n $R_0=8.53^{+0.21}_{-0.15}$ kpc. Our new radial velocities show that the projected rotation velocity increases only weakly outside of 30$\\arcsec$,\nand thus the nuclear cluster rotates less than assumed in \\citet{Trippe_08}.\n\n\n\\item We use the motions of more than 10000 stars for isotropic, spherically symmetric Jeans modeling. As a tracer profile we use either\n stellar number counts or the light profile, and the mass model is either a power law model or assumes constant M\/L.\nForcing the mass of the SMBH to its known value\n we measure for the extended mass\na power law slope of $\\delta_\\mathrm{M}=1.18\\pm 0.06$, consistent with the light profile. Our best mass estimate\nis obtained at $r_{\\mathrm{3D}}=100\\arcsec$ as an average over the power-law and constant M\/L models. We find\n$M_{100\\arcsec} = (6.09 \\pm 0.53|_{\\mathrm{fix} R_0}\\pm 0.97|_{R_0} ) \\times 10^6 M_{\\odot}$ \nThe error contains\ncontributions from the uncertain surface density data and from the uncertainty in $R_0$. \nDeviations from isotropy and spherical symmetry are not included in our error calculation. The most important\ndeviation is the observed flattening, a model which includes it increases the mass by about 47\\%, see \\citet{Chatzopoulos_14}.\nOur $\\gamma$-modeling yields a total cluster mass of\nM$_\\mathrm{NC}=(4.22 \\pm 0.50|_{\\mathrm{fix} R_0}\\pm 0.67|_{R_0}) \\times 10^7$ M$_{\\odot} $. As in case of the light, the total mass is model dependent. A model with faster outer decay like an exponential, has a smaller total mass than our fit.\n\n \\item The preference for a too small SMBH mass in the Jeans modeling can be interpreted as an argument for a central, core-like structure, unlike\n to the expected cusp \\citep{Bahcall_76}. Hence, the deficit in the center would be present not only in the light, but also in the mass profile.\nThe missing mass cusp makes other explanations for the missing cusp in the light less likely, such as\n mass segregation and giant destruction \n\\citep{Dale_09} less likely.\n However, we think that it is premature to draw this conclusion firmly. One would need to develop a model \nthat fits the surface density data and achieves the right SMBH mass. That needs probably at least one of the \nfollowing elements: (i) a relatively large core-like structure in the tracer distribution,\nwhich possibly can be found by non-parametric fitting of the density profile, \nor (ii) the inclusion of an outer background which contributes a significant number of stars \nalso in the center. \n\n \\item We obtain a mass to light ratio of $M\/L=0.51 \\pm 0.12 \\, M_{\\odot}\/L_{\\odot,\\mathrm{Ks}}$. This\nis 1.7$\\,\\sigma$ smaller than the value for a Chabrier IMF. However, since the mass error does not include all contributions, e.g. the contribution from the flattening is not included, it is consistent with a Chabrier IMF.\n\n \\item The obtained half light radius of the nuclear cluster of 4 to 9 pc is typical compared to extragalactic nuclear clusters. However, it is brighter \nand has a higher light and mass density than most other nuclear clusters. \nPossibly, this large density is connected to the nuclear disk further out whose\nsurface brightness also seems high. \n\n \\item The abundance of young stars and molecular clouds in the nuclear disk and the nuclear cluster supports\n the idea that at least the young and medium old stars in the GC formed in-situ. \n Both the nuclear cluster and the nuclear disk are flattened, although the strength of the flattening varies. The different amount of flattening can be interpreted as an argument for a different origin of the two components, a local origin for the disk and a globular cluster origin for the cluster. The ratio of diffuse light to total light\nappears to be normal, which contradicts a globular cluster origin, since they are likely dominated by stellar remnants when they arrive in the GC. \nFurther, the close association of the bright nuclear cluster with the bright nuclear disk is suggestive of a common origin. Finally, the metallicity of most nuclear cluster stars is close to solar \\citep{Cunha_07,Ryde_15,Do_15} agreeing with an in-situ origin. Only 5 of the 83 stars of \\citet{Do_15} are compatible with the metallicity [Fe\/H]$\\approx-0.75$ (\\citet{Harris_96}, private communication with Christian Johnson) of bulge globular clusters. Because metallicities are an important discriminator, it would be good to confirm the medium resolution metallicities of \\citet{Do_15} with higher resolution spectroscopy.\nIn conclusion, it seems likely that the majority of the nuclear cluster stars originated not in globular clusters but more locally.\n\n\\end{itemize}\n\nTo improve further the constraints on mass, shape, and origin of the nuclear cluster we use the data presented here\nfor axisymmetric modeling in \\citet{Chatzopoulos_14}. This lifts the assumptions of spherical symmetry and add rotation.\n\\newline\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\noindent Tsallis' thermostatistics is today a new paradigm for\nstatistics, with applications\n to several scientific disciplines \\cite{gellmann,1988,PP1997,FMP,TMP}.\n Notwithstanding its\n manifold applications, some details of the basic thermostatistical formalism\n remain unexplored. This is why {\\it analytical} results are to be\n welcome, specially if they are, as the ones to be here\n investigated, of a very general nature.\n We will provide this type of results for discrete\n probability distributions of fixed variance that maximize Tsallis' entropy.\n\n\\noindent Given a discrete probability distribution (DPD) $p= \\{\np_{k} \\}$, its Tsallis' information measure (or entropic form) is defined as\n\\begin{equation}\nH_{q} ( p ) =\\frac{1}{q-1} ( 1-\\sum_{k=-\\infty}^{+\\infty}\np_{k}^{q} ) . \\label{dino} \\end{equation} It is a classical result that\nas $q \\rightarrow1,$ Tsallis entropy reduces to\nShannon entropy\n\\[\nH_{1} ( p ) =-\\sum_{k=-\\infty}^{+\\infty}p_{k}\\log p_{k}.\n\\]\nWithout loss of generality, we will here consider only centered\nrandom variables of fixed variance.\n\n\\vskip 2mm \\noindent The aim of this paper is to provide accurate\nestimates of i) the parameters of the DPD's and ii) their behavior.\n The maximizers of Tsallis' information measure under variance\nconstraint in the {\\it continuous, multivariate case} have been\ndiscussed in \\cite{next2003}. \\vskip 3mm \\noindent Consider a quantum\nsystem whose eigenstates are characterized by a set of quantum\nnumbers\n that we collectively denote with an integer, running index $k$ (Cf. Eq.\n (\\ref{dino})), that will of course also label the\neigensolutions ($\\vert \\psi_k \\rangle,\\,\\,\\,\\,\\,\\epsilon_k$) of\nthe pertinent time-independent Schr\\oe dinger equation\n\n\\begin{equation} \\label{0q1} \\mathcal{H}\\, \\vert \\psi_k \\rangle =\n\\epsilon_k\\,\\vert \\psi_k \\rangle,\n \\end{equation} with $\\mathcal{H}$ the Hamiltonian. Let\n $p_k \\equiv \\vert \\psi_k \\vert^2$ be the probability of finding\n our system in the state $\\vert \\psi_k \\rangle $. The mixed state\n\n \\begin{equation} \\label{0q2} \\rho = \\sum_k\\,p_k\\, \\vert \\psi_k \\rangle \\langle \\psi_k \\vert, \\end{equation}\ncommutes with the Hamiltonian by construction and represents thus\na bona fide possible stationary state of the system. \\noindent If we now\nfind a physical quantity $Z$ whose mean value is proportional to\nthe variance, we can interpret $\\rho$ as the mixed state that\nmaximizes Tsallis measure subject to the a-priori known\nexpectation value of such a physical quantity. We will show below\nthat, in these circumstances, \\fbox{ {\\it universal expressions}\ncan be given for the $p_k$'s, and thus for $\\rho$}.\n\n\n\\noindent We discuss possible applications in the forthcoming section.\nAfterwards, after having introduced some definitions and\nnotations, we characterize the discrete Tsallis maximizers for\nfixed variance in both the $q<1$ and the $q>1$ cases, and discuss\nthermal stability questions. We pass then to\n analyze extensions to the multivariate cases.\n For the sake of completeness, some of the proofs of\nassertions referenced to in what follows are given in Annex.\n\n\\section{Possible physical applications}\n\n\n\n\\noindent Several physical models can be adapted to the scenario\ndescribed above (Cf. Eqs. (\\ref{0q1}) and (\\ref{0q2})) The\nweights $p_k$ in (\\ref{0q2})) will be of the Tsallis-power law\nform. With these weights, $\\rho$ maximizes Tsallis' entropy\nsubject to the constraint of a constant variance, which introduces\na Lagrange multiplier that we will call $\\beta$.\n\n\\noindent Given a physical quantity $Z$ whose mean value is\nproportional to the variance, $\\rho$ is that mixed state which\nmaximizes Tsallis measure subject to the a-priori known $\\langle Z\n\\rangle-$value. In the examples below $Z$ is the system's energy\n$E$, but many other possibilities can be imagined. Thus, $\\rho$\nwill be the state maximizing Tsallis' $H_q$ for a fixed value of\nthe expectation value $U =\\langle E\\rangle=\nTr\\,[\\mathcal{H}\\,\\rho]$ of the Hamiltonian. As a consequence, the\nmultiplier $\\beta$ can be thought of as an ``inverse temperature\"\n$T$, that is, set $\\beta=1\/(k_BT)$, with $k_B$ the Boltzmann\nconstant. This is so because we are at liberty of imagining that\n$U$ is kept constant because it is in contact with a heat\nreservoir \\cite{reif}. Of course, this is not necessarily the\ncase. $\\rho$ exists by itself and is a legitimate stationary mixed\nstate of our system. But we can think of $\\beta$ as either a\n``real\" or an ``equivalent\" inverse temperature.\n\n \\noindent Consider, for example, a system for which the energy spectrum\nconsists of a denumerable set of $N$ ($N$ possibly infinite)\n energy levels labeled by a {\\it quantum} number $k$ with\n $p_{-k}=p_{k},$\nso that all levels exhibit a degeneracy\n\n\\begin{equation} g_k=2\\,\\,\\, for \\,\\,\\,k \\ne 0;\\,\\,\\,k > 0;\\,\\,\\,and\n\\,\\,\\,g_0=1, \\end{equation} i.e., the sums in (\\ref{dino}) run from $0$ up to\n$\\infty$ and each summand is multiplied by $g_k$. Within the\npresent framework, $U$ \\footnote{Mean values linear in the PD are\nemployed here. They are quite legitimate and were used by Tsallis\nin his seminal 1988 paper \\cite{1988}. The Legendre structure of\nthermodynamics is definitely respected using them in conjunction\nwith Tsallis' entropy \\cite{PP1997}. Recently, Ferri, Martinez and\nPlastino have shown \\cite{FMP} that PD's obtained in this manner\ncan be easily translated into PD's constructed via MaxEnt using\n$q-$expectation values evaluated \\`a la Tsallis-Mendes-Plastino\n\\cite{TMP}, showing thereby that there is a one-to-one\ncorrespondence between the two types of PD.}\n\n\\begin{equation} U= \\langle E \\rangle = \\sum_{k=0}^{+\\infty} g_k p_{k} E_k,\\end{equation}\n becomes numerically equal to the DPD's variance, which by definition is fixed and assumedly known a priori.\n As just stated, we may think,\n if we wish, that our system is in contact with a heat reservoir,\n which fixes the mean energy, and that\n the associated Lagrange multiplier $\\beta$\n can be assimilated to an inverse temperature $T$.\n Among many examples of such a scenario we mention here:\n\n\n\n\n\n\\begin{itemize}\n\\item\n the planar rotor \\cite{dictio}, where $k$ is the magnetic quantum number\ncorresponding to the azimuthal angle usually denoted by $\\phi$.\nThe level-energies $E_k$ are proportional to $k^2$ and \\begin{equation} \\label{1} E_k = C_E\nk^2;\\,\\,\\,C_E\\,\\,\\, {\\rm has \\,\\,\\,dimension\\,\\,\\, of\\,\\,\\,\nenergy.}\\end{equation} We have $C_E=\\hbar^2\/2M_I$, with $M_I$ the system's moment of inertia \\cite{Wyllie}.\n For simplicity's sake we take here $C_E=1$, but\nretain, of course, its energy-units.\n\\item the three-dimensional rigid rotator \\cite{dictio}, although $k$ now means the quantum number $L$ associated\nto orbital angular momentum (for large $k$, the spectrum of (\\ref{1})\nlooks like that of the 3D rigid rotor) and, for all $k \\ge 0$, $g_k=1$.\n\\item a vibrating string of length $2l$ with fixed ends\n\\cite{mandl}, whose energy eigenvalues are $E_k \\propto\n[k\\pi\/2l]^2$. \\item the case of a particle (of mass $m$) in a box subject to periodic boundary conditions\n(at $(-L,\\,\\,+L)$) whose energy values are $\\epsilon_k=[2\\pi^2\\hbar^2 k^2\/(mL^2)]$ \\cite{reif}.\n\\end{itemize}\n\n\n\n\n\n\n\n\\section{Definitions and Notations}\n\n\\noindent In what follows, $m$ is a positive real number and $n$ a positive\ninteger. Often, $m$ will take the special form of an odd integer\n$m=2n+1.$\nThe solutions to the problem of maximization of Tsallis\nentropy ``under variance\nconstraint\" (here equivalent to ``for fixed mean energy\") will be called discrete Tsallis-probability laws (DTPL's) in\nthis manuscript. Since (see preceding Section) fixed mean energy can be thought of as implying contact with a heat reservoir at the fixed temperature $T=1\/(k_B \\beta)$ we will call the pertinent Lagrange multipliers $\\beta$ inverse temperatures in what follows. DTPL's are given by the following two theorems.\n\n\\begin{theorem}\n\\noindent If $\\frac{1}{3}1}\n\\end{equation}\nwhere $f_{q} ( T ) $ is the partition function, given by\n\\begin{equation}\nf_{q} ( T ) =\\sum_{k=0}^{+\\infty}g_k\\, ( 1+\\frac{k^{2}}\n{k_BT} ) ^{\\frac{1}{q-1}}, \\label{faqle1}\n\\end{equation}\nand $\\beta=1\/k_BT$ is a real positive Lagrangian multiplier such\nthat\n\\[\nU \\equiv \\sigma^{2}=\\sum_{k=0}^{+\\infty}g_k k^{2}p_{k}.\n\\]\nIn the limit $T \\rightarrow \\infty$, our probability distribution\nconverges to the\n classical discrete Gaussian distribution \\cite{Zb}.\n\\end{theorem}\n\n\\noindent Notice the existence of a lower bound $q=1\/3$. Smaller $q$\nvalues are unphysical because for them the mean energy of the\nmodel (for the PD (\\ref{discreteq>1})) becomes infinite.\nAdditionally (see below) the system turns out to be thermodynamically\nunstable because the specific heat becomes negative. In this case where $q<1$, we define\nthe real positive parameter $m$ that we need below by\n\\begin{equation}\nm=\\frac{1+q}{1-q}.\n\\label{mforq<1}\n\\end{equation}\n\n\n\n\n\n\\begin{theorem}\n\\noindent If $q>1,$ the discrete probability distribution $p$ with zero mean\nand mean energy $U$ (equivalently, variance\n$\\sigma^{2}$) that maximizes Tsallis entropy is defined as\n\\begin{eqnarray} p_{k}&=&\\Pr \\{ X=k \\} = \\Bigg\\{\n\\begin{array}\n[c]{ll}\nf_{q}^{-1} ( T )\n ( 1-\\frac{k^{2}}{k_BT} ) ^{\\frac{1}{q-1}} & \\forall\nk\\in [0, \\lfloor k_BT \\rfloor -1 ] \\\\\n0 & \\text{otherwise}\n\\end{array}\n\\label{discreteq<1}\n\\end{eqnarray}\nwhere $f_q ( T ) $ is the partition function\n\\begin{equation}\nf_{q} ( T ) =\\sum_{k=0}^{ \\lfloor \\beta \\rfloor\n-1}g_k ( 1-\\frac{k^{2}}{k_BT} ) ^{\\frac{1}{q-1}},\n\\label{faqge1}\n\\end{equation}\n$ \\lfloor k_BT \\rfloor $ denotes the integer part of\n$k_BT$ whose inverse, namely, $\\beta$, is a real positive\nLagrangian multiplier such that\n\\[\nU=\\sigma^{2}=\\sum_{k=0}^{+\\infty}g_k\\, k^{2}p_{k}.\n\\]\n\\end{theorem}\nIn the present case where $q>1$, we define $m$ as a real positive parameter related to $q$ by\n\\begin{equation}\nm=\\frac{q+1}{q-1},\n\\label{mforq>1}\n\\end{equation}\nto be of use below. Note that this definition differs from definition (\\ref{mforq<1}) in the case $q<1$.\n\n\\noindent We do not give the proofs of theorems (1) and (2) here. That\nlaws defined by (\\ref{faqle1}) and (\\ref{faqge1}) are the\nsolutions of the Tsallis entropy maximization problem can be\nobtained by extending the results of the continuous case as\npresented in \\cite{next2003}.\n {\\it The cases i) $q<1$ and ii) $q>1$ differ essentially by the\nfact that the latter has a finite support.} As far as we know,\nthis is a novel way, in the $q-$literature, of dealing with the celebrated Tsallis-cut-off \\cite{1988}. \\vskip 3mm\n\n\\noindent We remark on the fact that the forms (\\ref{discreteq>1}) and\n(\\ref{discreteq<1}) coincide with the distribution-form given by\nTsallis in his pioneer 1988 paper \\cite{1988} (see also\n\\cite{TMP}), namely, \\begin{equation} \\label{tsallis} p_k= Z_q^{-1} [1-(q-1)\n\\beta^* k^2]^{1\/(q-1)},\\end{equation} that, for $q \\rightarrow 1$, tends to\nthe discrete Gaussian \\cite{1988} \\begin{equation} Z_1^{-1}\n\\,\\exp{[-\\beta^*k^2]}. \\label{gauss}\\end{equation} In the former case (Eq.\n(\\ref{discreteq>1})) we have $q<1$, while in the later (Eq.\n(\\ref{discreteq<1})) $q>1$. Thus, in Eq. (\\ref{discreteq>1}) we\nhave $\\beta= (1-q) \\beta^*$, while for (\\ref{discreteq<1}) $\\beta=\n(q-1) \\beta^*$. Of course, according to (\\ref{discreteq>1}),\n(\\ref{discreteq<1}), and (\\ref{gauss}), for $q \\rightarrow 1$ we\nhave $\\beta^* \\rightarrow \\beta$.\n\\section{Approximate treatments for $q<1$}\n\n\\subsection{The two regimes}\n\n\\noindent Let us recall that, for $q<1$, $m =(1+q)\/(1-q)$. Thus, $m$ is large if\n$q \\rightarrow 1$ (Boltzmann limit (BL)), while a small positive value of $m$ entails\n$q \\rightarrow -1$. Careful inspection of the partition\nfunction $f_{q} ( T ) $ in the cases $m=3\n \\leftrightarrow q=1\/2$, $m=5 \\leftrightarrow q=2\/3$,\n and $m=7 \\leftrightarrow q=3\/4$ indicates that two regimes should be distinguished:\n\n\n\\begin{itemize}\n\\item the first regime, called \\textquotedblright power law\\textquotedblright\n\\ regime, corresponds to the case $T \\ll1$ (low temperatures)\n\n\\item the second regime called the \\textquotedblright\nStudent-t\\textquotedblright\\, regime, corresponds to the case $T \\gg1$\n(high temperatures)\n\\end{itemize}\n\n\\noindent Anticipating the results\nprovided by the following theorems, we note that the cases\n\\textquotedblright small $T$\\textquotedblright\/\\textquotedblright\nlarge $T$\\textquotedblright\\ can be translated as joint range\nvalues of parameters $U$ and $m \\equiv (1+q)\/(1-q):$\ntypically, the large $T$ case corresponds to jointly large values\nof $U$ and $m,$ as expected, while the small case corresponds to\njointly small values of these parameters. This is illustrated on\nthe curve below (Fig. 1), where the large $T$ property is\ntranslated as $T\\geq100,$ and the small $T$ property as\n$T\\leq0.01.$ Note that the left bound $n=1\/2$ corresponds to\n$m=2n+1=2,$ i.e., $q=1\/3$, and thus to an (unphysical) infinite\n$U$. For economy's sake we set herefrom\n\n\\begin{equation} \\label{defa} a=\\sqrt{k_BT}. \\end{equation}\n\n\\subsection{The power law regime}\n\nIn the power-law regime ($T\\ll1$), a detailed characterization of\nthe distribution can be obtained for all real values of $m$, as\nshown in the following theorem.\n\n\\begin{theorem}\n$\\label{powerlawregime}$\\label{power_law} Assuming that\n$T\\ll1\\,\\,\\,(a \\ll 1)$ and with $m$ defined by (\\ref{mforq<1}), the following approximations hold:\n\n\\begin{enumerate}\n\\item for the partition function\n\\[\nf_{q} ( T ) \\simeq1+2a^{m+1}\\zeta ( m+1 ),\n\\]\n\n\n\\item for the probability law\n\\begin{equation}\np_{k}\\simeq \\bigg\\{\n\\begin{array}\n[c]{ll}\n\\frac{a^{m+1}}{1+2a^{m+1}\\zeta ( m+1 ) }k^{-m-1} & \\text{if }\nk\\neq0\\\\\n\\frac{1}{1+2a^{m+1}\\zeta ( m+1 ) } & \\text{if }k=0\n\\end{array}\n, \\label{pk_smalla}\n\\end{equation}\nand corresponds thus to a discrete power-law. It tends to a Gaussian for $q \\rightarrow 1$.\n\n\\item for the mean energy\n\\begin{equation} \\label{mono2} U\\simeq2a^{m+1}\\zeta ( m-1 ). \\end{equation}\n\n\\item If, moreover, $m=2n+1$ with $n\\in\\mathbb{N}$, $n\\geq1$, the characteristic\nfunction can be approximated as\n\\[\n\\phi_{m} ( u ) \\simeq\\frac{1}{ (\n1+2a^{2n+2}\\zeta ( 2n+2 ) ) } (\n1+2a^{2n+2}\\frac{ ( -1 ) ^{n} ( 2\\pi )\n^{2n+2}}{ ( 2n+2 ) !}B_{2n+2} ( \\frac{u}{2\\pi\n} ) ),\n\\]\nwhere $B_{n} ( u ) $ denotes the Bernoulli polynomial of order $n.$\n\\end{enumerate}\n\n\\begin{proof}\nthe proof is given in Annex 1.\n\\end{proof}\n\\end{theorem}\nThe specific heat $C=dU\/dT$ is proportional to $(m+1)$. This is\n{\\it positive}, and thus the system {\\it stable}, for all\n$\\frac{1}{3}\\frac{1}{3}$, the\nfirst values of $p_{k}.$\n\\begin{gather*}\n\\begin{tabular}\n[c]{|c|c|c|c|c|}\\hline\n& $k=0$ & $k=\\pm1$ & $k=\\pm2$ & $k=\\pm3$\\\\\\hline\n$q=\\frac{1}{2}$ & $0.999\\,78$ & $9.\\,\\allowbreak997\\,8\\times10^{-5}$ &\n$\\allowbreak6.\\,\\allowbreak248\\,6\\times10^{-6}$ & $\\allowbreak1.\\,\\allowbreak\n234\\,3\\times10^{-6}$\\\\\\hline\n$q=\\frac{3}{5}$ & $0.999\\,98$ & $9.\\,\\allowbreak999\\,8\\times10^{-6}$ &\n$\\allowbreak3.\\,\\allowbreak124\\,9\\times10^{-7}$ & $\\allowbreak4.\\,\\allowbreak\n115\\,1\\times10^{-8}$\\\\\\hline\n$q=\\frac{2}{3}$ & $\\simeq1.\\,\\allowbreak000\\,00$ & $1.\\,\\allowbreak\n000\\,00\\times10^{-6}$ & $\\allowbreak1.\\,\\allowbreak562\\,5\\times10^{-8}$ &\n$\\allowbreak1.\\,\\allowbreak371\\,7\\times10^{-9}$\\\\\\hline\n$q=\\frac{5}{7}$ & $\\simeq\\allowbreak1.\\,\\allowbreak000\\,00$ & $1.\\,\\allowbreak\n000\\,00\\times10^{-7}$ & $\\allowbreak7.\\,\\allowbreak812\\,5\\times10^{-10}$ &\n$4.\\,\\allowbreak572\\,5\\times10^{-11}$\\\\\\hline\n\\end{tabular}\n\\\\\n\\text{Table.1: first values of }p_{k}\\text{ for several values of }q \\text{ and } a=0.1\n\\end{gather*}\n\n\n\\subsection{The Student-t regime}\n\nIn the case of the Student-t regime (high temperatures), our main\nresult writes as follows:\n\n\\begin{theorem}\n$\\label{student-t}\\forall\\varepsilon>0,$ $\\exists a_{0}>0$ and\n$\\exists\nm_{0}>0$ such that if $a \\geq a_{0}$ and $2 1\/3$. In other words, the system is stable in the\ninterval $]1\/3,1].$ We remark also that, as a consequence of\n(\\ref{mono2}) and (\\ref{mono1}), the mean energy $U$ is an\nincreasing function of $a$ (and thus of $T$), as it\nshould.\n\\begin{remark}\nA remarkable result here is that expression (\\ref{pk_largea}) is\nexactly the sampled version - with the same partition function - of the continuous maximizer of the\nTsallis entropy whose expression is recalled here \\cite{TsallisStudent}:\n\n\\[\nf(x)=\n\\frac{\\Gamma ( \\frac{m+1}{2} )}{\\Gamma ( \\frac {m}{2} ) \\Gamma (\\frac{1}{2}) \\sqrt{U(m-2)}}\n( 1+\\frac{x^2 }{U(m-2)})\n^{-\\frac{m+1}{2}} \\,\\, \\forall x \\in \\mathbb{R}\n\\]\n\nThis entails that, at very high $T$, one can approximate sums over\ndiscrete energy levels by an integral. In order to understand the\nsituation, remember that the results holds for $m\\gg1$, which\nentails $q \\sim 1$, i.e., the Boltzmann limit. In this limit,\nreplacing sums by integrals is a commonplace text-book procedure.\n\\end{remark}\n\n\n\\section{The $q>1$ instance}\n\n\\subsection{A general result and its consequences}\n\nRecall that in the case $q>1$, $m=(q+1)\/(q-1)$. The equivalent of theorem (\\ref{student-t}) writes as\nfollows:\n\n\\begin{theorem}\n\\label{Student-r}$\\forall\\varepsilon>0,$ $\\exists a_{0}>0$ and\n$\\exists\nm_{0}>0$ such that if $a\\geq a_{0}$ and $m\\geq m_{0}$ then\n\\begin{equation}\n\\left \\vert \\sum_{k=- ( a-1 ) }^{ a-1} ( 1-\\frac{k^{2}}{ a^{2}\n} ) ^{\\frac{m-1}{2}}-\\frac{ a\\sqrt{\\pi}\\Gamma ( \\frac{m+1}\n{2} ) }{\\Gamma ( \\frac{m}{2}+1 ) } \\right \\vert \\leq\n\\varepsilon. \\label{approx}\n\\end{equation}\n\n\n\\begin{proof}\nThe proof of this result is given in Annex 3. It essentially\nfollows the steps of the proof of theorem (\\ref{student-t}).\n\\end{proof}\n\\end{theorem}\n\n\\noindent We depict in Fig. 3 the area (north-east), delimited in the plane $ ( a,m )$, that contains\n values of $a$ and $m$ for which approximation (\\ref{approx})\nholds, with $\\varepsilon=10^{-10}$.\n\n\\begin{corollary}\nFor $ a\\gg1\\,\\,\\,(T\\gg 1)$ the mean energy $U$ verifies\n\\begin{equation}\nU \\simeq\\frac{ a^{2}}{m+2} \\label{varapprox}\n\\end{equation}\n\n\n\\begin{proof}\nDenoting $n=\\frac{m-1}{2}$ and $f_{n}(a)$ for $f_{q}(a)$, the normalization constant verifies\nthe following\nrecurrence\n\\begin{align*}\nf_{n+1} ( a ) & =\\sum_{k=-a}^{+a} ( 1-\\frac{k^{2}}{a^{2}\n} ) ^{n+1}=\\sum_{k=-a}^{+a} (\n1-\\frac{k^{2}}{a^{2}} )\n^{n}-\\frac{1}{a^{2}}\\sum_{k=-a}^{+a}k^{2} (\n1-\\frac{k^{2}}{a^{2}} )\n^{n}\\\\\n& =f_{n} ( a ) -\\frac{U}{a^{2}}f_{n+1} ( a )\n\\end{align*}\nso that\n\\[\nU =a^{2} (\n1-\\frac{f_{n+1} ( a ) }{f_{n} ( a ) } ) .\n\\]\nMoreover, using the result of theorem (\\ref{Student-r}), we have\n\\[\nf_{n} ( a ) \\simeq\\frac{\\Gamma (\n\\frac{m}{2}+1 )\n}{a\\sqrt{\\pi}\\Gamma ( \\frac{m+1}{2} ) }\n\\]\nwe obtain after some algebra\n\\begin{equation} \\label{ener} U\n\\simeq\\frac{a^{2}}{m+2}. \\end{equation}\n\n\\end{proof}\n\\end{corollary}\nWe see from the form of (\\ref{ener}) that the specific heat $C$ is\nproportional to $1\/(m+2)$. Thus, the is system stable for all $q\n>1$. Also, since $a^2=k_BT$, we see that the mean energy $U$ is an\nincreasing function of $T$, as it should.\n\n\\subsection{Convergence to a sampled Student-r law}\n\nAs a consequence of theorem (\\ref{Student-r}), the Tsallis\nmaximizers can be approximated as follows, for $a$ large enough\n($T$ large enough).\n\n\\begin{theorem}\nIf $a\\gg1$ then the following approximation holds\n\\begin{equation}\np_{k}\\simeq\\frac{\\Gamma ( \\frac{m}{2}+1 )\n}{\\sqrt{\\pi}\\Gamma ( \\frac{m+1}{2} ) \\sqrt{U}\n\\sqrt{m+2}} ( 1-\\frac{k^{2}}{U ( m+2 ) } ) ^{\\frac{m-1}{2}}. \\label{sampled}\n\\end{equation}\n\n\n\\begin{proof}\n>From (\\ref{varapprox}), we have\n\\[\na^{2}\\simeq ( m+2 )U.\n\\]\nand the result follows from Theorem (\\ref{Student-r}).\n\\end{proof}\n\\end{theorem}\n\n\\noindent {\\rm Note that here}\n\\begin{equation} \\label{r2} (m+2)\\, U= k_B \\,T = \\beta^{-1}, \\end{equation} {\\rm a result that will be needed below.}\n\\noindent We remark the similarity to the corresponding $q<1$ probability\nlaw.\n\n\\noindent Eq. (\\ref{sampled}) corresponds to the sampled version of a\ncontinuous Student-r distribution -which maximizes Tsallis entropy\nfor fixed mean energy \\cite{TsallisStudent} - with the same partition function.\n\n\\subsection{Detailed results}\n\nIn the case where $m$ is an odd integer, more detailed results\ncan be obtained concerning the behavior of this probability law.\n\n\\begin{theorem}\n\\label{integercase}For $a\\gg1,$ if $m=2n+1$ is an odd integer with\n$n\\geq2$,\nthen\n\\begin{equation}\nf_{m} ( a ) =\\frac{ 2^{2n+1} ( n ! ) ^{2}}{(2n+1)!}a+o ( a^{-2} ) \\label{fapprox}\n\\end{equation}\n\n\nand $\\forall a\\geq1,$ and if $m=3$\n\\[\nf_{3} ( a ) =a\\frac{4}{3}-\\frac{a^{-1}}{3}.\n\\]\nMoreover, the mean energy $U$ verifies\n\\begin{equation}\nU =\\frac{a^{2}}{m+2}+o ( a^{-1} ).\n\\label{sigmaapprox}\n\\end{equation}\n\n\n\\begin{proof}\nthe proof is given in Annex 4.\n\\end{proof}\n\\end{theorem}\nThis quasi linearity of $f_m (a ) $ in $a$ for high\ntemperatures is illustrated in Fig. 4, while the quadratic\nbehavior of $U$ vs. $a$ is depicted in Fig. 5.\n\n\n\n\\section{Convergence to the discrete normal distribution}\n\nWe show now that, in the case where parameter $m$ grows to infinity, both discrete Tsallis distributions corresponding to either $q<1$ or $q>1$ converge to the discrete normal distribution.\n\n\\begin{proposition}\nIf $a\\gg1$ and if the mean energy $U$ is fixed to a constant\nvalue, then, as $m$ grows to infinity (i.e.\n$q \\rightarrow 1^{\\pm}$), both discrete Tsallis probability distributions\n(\\ref{discreteq>1}) and (\\ref{discreteq<1}) converge to the maximizer of Shannon's discrete entropy,\ni.e., the discrete normal distribution.\n\n\\begin{proof}\nin the case $q<1,$ the result follows by taking $m \\rightarrow +\\infty$ in\n(\\ref{pk_largea}) with\n\\[\n\\lim_{m \\rightarrow +\\infty}\\frac{\\Gamma ( \\frac{m+1}{2} ) }\n{\\Gamma ( \\frac{m}{2} ) \\Gamma ( \\frac{1}{2} )\n\\sqrt{U(m-2})}=\\frac{1}{\\sqrt{2U\\pi}}\n\\]\nand (Cf. (\\ref{r1}))\n\\[\n\\lim_{m \\rightarrow +\\infty} ( 1+\\frac{k^{2}}{U (\nm-2 ) } ) ^{-\\frac{m+1}{2}}=\\exp (\n-\\frac{k^{2}}{2U} ) .\n\\]\nIn the case $q>1,$ the result follows similarly by taking $m \\rightarrow\n+\\infty$ in (\\ref{sampled}) with\n\\[\n\\lim_{m \\rightarrow +\\infty}\\frac{\\Gamma ( \\frac{m}{2}+1 ) }\n{\\sqrt{\\pi}\\Gamma ( \\frac{m+1}{2} ) \\sqrt{m+2}}=\\frac{1}{\\sqrt\n{2\\pi}}\n\\]\nand (Cf. (\\ref{r2}))\n\\[\n\\lim_{m \\rightarrow +\\infty} ( 1-\\frac{k^{2}}{U ( m+2 ) } )\n^{\\frac{m-1}{2}}=\\exp ( -\\frac{k^{2}}{2U} ).\n\\]\n\n\\end{proof}\n\\end{proposition}\n\\noindent\n\nThis result shows that, for high enough temperatures, and if $m \\rightarrow \\infty$ (or equivalently $q \\rightarrow \\pm 1$) the distribution that maximizes Tsallis'\nentropy converges to the Boltzmann one, both for $q<1$ and $q>1$.\n Of course, for fixed $T \\gg 1$ and $q \\rightarrow +\\infty$, we obtain the uniform distribution, whereas for $T \\ll 1$ and $q \\rightarrow +\\infty$ we obtain the deterministic case $p_{k} = \\delta_{k}$.\n\n\\section{Extension to the multivariate case}\n\nIn the last part of this communication\n we give a heuristic discussion of\n the multivariate discrete Tsallis laws with $q<1.$\n\n\\begin{theorem}\n\\label{multivariate}\nif $\\frac{d}{d+2}1.$\nMost of the ensuing results are given by approximations, but\nnumerical simulations show that they can be regarded as very\naccurate ones.\n\n\\noindent The discrete probability distributions $p_k$ define, for any Hamiltonian $\\mathcal{H}$ whose eigenvalue equation reads $\\mathcal{H}\\vert \\psi_k\\rangle =\\epsilon_k \\vert \\psi_k\\rangle$ mixed states of the form \\begin{equation} \\label{qqq2} \\rho = \\sum_k\\,p_k\\, \\vert \\psi_k \\rangle \\langle \\psi_k \\vert, \\end{equation}\nthat commute with the Hamiltonian by construction and represent thus bona fide stationary states of the system defined by $\\mathcal{H}$. These weights $p_k$ are of the Tsallis-power law form. With these weights, $\\rho$ maximizes Tsallis' entropy subject to variance constraint, which introduces a Lagrange multiplier $\\beta$. In this work we have considered situations for which the system's mean energy is proportional to the variance.\n\n\n\\noindent Our present consideration allows one to nitidly appreciate the\neffects of non-extensivity, as, varying $q$, one passes from\nunstable to stable systems and even to unphysical situations of\ninfinite energy.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nMagnetic properties of transition metals (TM) are generally determined by the $3d$ valence electrons. Resonant soft X-ray scattering at $L_{2,3}$ absorption edges of TM involves $2p$-to-$3d$ transitions, thus being an element-selective probe with possibility to distinguish magnetic signal from different elements in multicomponent magnets \\cite{fink2013resonant}. Moreover, the spatial coherence of the polarized soft X-ray beams provided by modern synchrotron radiation sources and free-electron lasers give vast opportunities for the lensless imaging using coherent diffraction imaging \\cite{turner2011x,flewett2012method,ukleev2018coherent}, ptychography \\cite{tripathi2011dichroic,shi2016soft}, and holographic techniques \\cite{eisebitt2004lensless,duckworth2011magnetic}. Both coherent resonant soft X-ray scattering (RSXS) imaging and holography allow to perform real-space imaging of the magnetization distribution in thin samples with various environments, such as in high magnetic fields or at low temperatures. Flexibility of the environment and enhanced robustness of these methods against the specimen displacements are significant advantage of the lensless techniques compared to scanning transmission magnetic X-ray microscopy (STXM) \\cite{fischer2015x}, although the possibility of cryogenic STXM imaging also has been recently demonstrated \\cite{simmendinger2018transmission}. Coherent diffraction allows the solution of classical crystallographic inverse problem of phase retrieval by using the iterative reconstruction algorithms applied to the resonant diffraction intensities \\cite{miao1999extending}. X-ray magnetic holography is based on the utilization of the interference between magnetic scattering from the object under investigation and reference wave generated by charge scattering from the prepared source. Imaging experiments using holographic approaches can be realized in a few different ways: Fourier transform holography (FTH) is based on a reference wave scattered from one or multiple small ($30-150$\\,nm) pinholes placed near the sample aperture \\cite{eisebitt2004lensless}. Alternatively, holography with extended reference by autocorrelation linear differential operation (HERALDO) \\cite{guizar2007holography} can be performed. In contrast to FTH, HERALDO technique implies the scattering from extended reference object, such as a narrow slit or a sharp corner, which allows to improve the contrast of the real-space image without compromising the resolution \\cite{zhu2010high}. Moreover, fabrication of the extended reference is less challenging than the array of the reference pinholes \\cite{zhu2010high,buttner2013automatable}. Previously, FTH and HERALDO techniques were successfully applied for imaging of the element-specific magnetic domain patterns in thin films and multilayers with perpendicular magnetic anisotropy \\cite{stickler2010soft,camarero2011exploring,weder2017multi}. Both FTH and HERALDO with references milled at an oblique angle into the masks also allow the imaging at a tilted angle, which is relevant for the spintronic devices with an in-plane magnetic anisotropy, such as spin-valves \\cite{duckworth2013holographic}, and magnetic nanoelements \\cite{tieg2010imaging,parra2016holographic,bukin2016time}. Thus the coherent soft X-ray scattering and imaging are powerful tools to study the spin ordering in multicomponent magnetic compounds with element selectivity.\n \nRecently, several bulk materials that exhibit non-trivial topological spin textures and contain two or more magnetic elements have been discovered: doped B20-type alloys \\cite{adams2010skyrmion, shibata2013towards}, Co-Zn-Mn compounds with $\\beta$-Mn structure \\cite{tokunaga2015new}, molybdenum nitrides \\cite{li2016emergence} and Heusler alloys \\cite{meshcheriakova2014large,phatak2016nanoscale,nayak2017magnetic}. Competition between the magnetic interactions in non-centrosymmetric compounds results in the complex phase diagram. The interplay between exchange interaction, antisymmetric Dzyaloshinskii-Moriya interaction (DMI), and magnetocrystalline anisotropy may cause incommensurate spin phases such as helical, conical and Bloch-type skyrmion lattice states \\cite{bak1980theory,rossler2006spontaneous}.\nThe typical size of a magnetic skyrmion varies in a range from a few to a few hundreds nm which makes them promising candidates for future spintronic applications such as skyrmion racetrack memory and logic devices \\cite{fert2017magnetic}. The skyrmions can be manipulated by current pulses with ultra-low current densities \\cite{iwasaki2013universal}, electric \\cite{white2014electric,okamura2016transition} and microwave fields \\cite{onose2012observation,okamura2013microwave,wang2015driving}, and temperature gradients \\cite{everschor2012rotating,kong2013dynamics,mochizuki2014thermally}. In the past decade skyrmion textures have been extensively studied by means of small-angle neutron scattering (SANS) \\cite{adams2011long,seki2012formation,moskvin2013complex} and Lorentz transmission electron microscopy (LTEM) \\cite{yu2010real,yu2011near}. Also, several groups reported on the resonant X-ray diffraction \\cite{langner2014coupled,zhang2016multidomain}, small-angle scattering \\cite{yamasaki2015dynamical,okamura2017directional,okamura2017emergence,zhang2017room} studies of Bloch-type skyrmions in the chiral magnets and imaging of N\\'eel-type skyrmions stabilized by interfacial DMI \\cite{buttner2015dynamics,blanco2015nanoscale,Woo2016,Woo2017}. Since X-ray magnetic circular dichroism (XMCD) is sensitive to the component of the magnetization parallel to the incident X-ray beam, transmission soft X-ray imaging is a complementary method to LTEM, which is sensitive to the in-plane magnetic flux inside the sample \\cite{chapman1984investigation}. Hence for the N\\'eel-type skyrmions, where the curl of magnetization lies in the sample plane and produce no contrast for LTEM without tilting the sample \\cite{jiang2016mobile,pollard2017observation}, soft X-ray imaging, has been successfully employed for the room-temperature N\\'eel-type skyrmion-hosting thin films \\cite{buttner2015dynamics,blanco2015nanoscale,Woo2016,Woo2017}, but not yet for the thin plates of polar magnets, that order magnetically at cryogenic temperatures \\cite{kezsmarki2015neel,kurumaji2017neel}.\n\nRoom-temperature magnetic ordering of the chiral $\\beta$-Mn-type Co-Zn-Mn alloy \\cite{tokunaga2015new} makes these materials promising for applications. The $\\beta$-Mn-type compound Co$_8$Zn$_8$Mn$_4$ exhibits a transition from the paramagnetic state to a helical or Bloch-type skyrmion lattice state at $T_c \\approx 300$\\,K with a magnetic modulation period of 125\\,nm \\cite{karube2016robust}, and undergoes a spin glass transition\nat $T_g\\approx8$\\,K, probably due to freezing of Mn spins \\cite{karube2018sciadv}. Frustration at the Mn site ultimately results in increment of a spin-glass transition temperature ($T_g\\approx30$\\,K) in the compound with higher Mn concentration Co$_7$Zn$_7$Mn$_6$. Moreover, a low-temperature frustration-induced equilibrium skyrmion phase has been recently found in the latter \\cite{karube2018sciadv}. In the present work we utilized the polarization-dependent soft X-ray magnetic spectroscopy, coherent RSXS, and HERALDO techniques to perform an element-selective study of the magnetic interactions and long-range ordering in Co$_8$Zn$_8$Mn$_4$ compound. The coherence of the synchrotron radiation allowed us to successfully combine small-angle scattering in transmission geometry with coherent diffraction imaging and employ the small-angle scattering patterns for the real-space reconstruction of the local magnetization distribution via iterative phase retrieval algorithm. The coherent diffraction imaging results were compared to the real-space reconstruction results provided by HERALDO. \n\n\\section{Experimental}\n\nX-ray spectroscopy, scattering and imaging experiments were performed at the variable-polarization soft X-ray beamline BL-16A of the Photon Factory (KEK, Japan) \\cite{amemiya2010k} and BL29 BOREAS of the ALBA synchrotron radiation laboratory (Cerdanyola del Vall\\'es, Spain) \\cite{barla2016design}.\n\nExperimental geometry of soft X-ray absorption (XAS) and XMCD experiments are shown in Fig.\\ref{Fig1}a. A bulk polycrystalline Co$_8$Zn$_8$Mn$_4$ specimen was obtained from an ingot grown by Bridgman method as described in Ref.\\cite{karube2016robust}. The sample was polished to remove the potentially oxidized surface layer prior to the experiment. Sample was placed to the vacuum chamber with pressure of $10^{-9}$\\,Torr equipped with a 5\\,T superconducting magnet. XAS and XMCD signals were measured with energy resolution of 0.1\\,eV using the surface-sensitive total electron yield (TEY) method near Co and Mn $L_{2,3}$ absorption edges with right and left circularly polarized (RCP and LCP) X-rays. \n\n\\begin{figure*}\n\\includegraphics[width=16cm]{Fig1.png}\n\\caption{(Color online) Sketch of the (a) XMCD, (b) RSXS and (c) HERALDO experiments. (d) SEM image of the RSXS sample aperture. (e) Thin plate of the Co$_8$Zn$_8$Mn$_4$ fixed onto the membrane. (f) SEM image of the sample aperture and the reference slit drilled in the 1-$\\mu$m-thick gold plate for HERALDO experiment. (g) Co$_8$Zn$_8$Mn$_4$ thin plate fixed onto gold plate by tungsten contact. The strong white\/black contrast in the top\/bottom parts of the panels (d--g) corresponds to the Au wires sputtered onto the membrane \\cite{okamura2017directional}, that is irrelevant for this study.}\n\\label{Fig1}\n\\end{figure*}\n\nRSXS and HERALDO experiments were performed in the transmission geometry (Figs. \\ref{Fig1}b,c). Commercial silicon nitride membranes from Silson Ltd (Southam, UK) were processed for soft X-ray experiments. The front side of each membrane was coated with $\\approx 4$\\,$\\mu$m-thick layer of gold to absorb the incoming X-ray beam. Further treatment of the membranes and thin plate fabrication were carried out by using focused ion beam (FIB) setup Hitachi NB5000 equipped with scanning electron microscope (SEM). Since the attenuation length for soft X-rays in Co$_8$Zn$_8$Mn$_4$ alloy is of about 100\\,nm, the scattering and imaging experiments were carried out on thin plates of Co$_8$Zn$_8$Mn$_4$ with thickness of 200\\,nm and 150\\,nm, respectively. Two FIB-thinned plates of a Co$_8$Zn$_8$Mn$_4$ containing (001) plane were cut from the bulk single crystal. For the RSXS experiment a thin plate was attached directly to the Si$_3$N$_4$ membrane (Figs.\\,\\ref{Fig1}d,e). In case of the RSXS sample the aperture with a diameter of 4.5\\,$\\mu$m and asymmetric shape, which provides a better convergence of the phase retrieval algorithm \\cite{fienup1986phase} was drilled in the gold coating (Fig. \\ref{Fig1}d). \n\nIn the case of the HERALDO sample we fabricated the sample aperture by a different fabrication approach: a large hole with a diameter of 6\\,$\\mu m$ was drilled in the gold-coated membrane, and covered by 1\\,$\\mu$m-thick gold plate fabricated by FIB from a bulk specimen. Then, a circular sample aperture with a diameter of 700\\,nm and a reference slit with a length of 1\\,$\\mu$m and width 40\\,nm were milled in the Au plate (Fig. \\ref{Fig1}f). The slit length and distance from the aperture were chosen according to the separation conditions preventing the overlapping of sample autocorrelation and sample-reference cross-correlation at the reconstruction \\cite{guizar2010differentially}. \n\nTo prevent the specimen damage by Ga$^+$ ions in the aperture and reference milling process the thin plate was fixed to the membrane by means of a tungsten contact after the mask treatment (Fig. \\ref{Fig1}e). For both RSXS and HERALDO the thin plates were attached to the corresponding membranes by the single tungsten contacts (Fig. \\ref{Fig1}g) to avoid the possible strain \\cite{shibata2015large,okamura2017emergence}.\n\nThe RSXS setup at Photon Factory, Japan was equipped with a high-vacuum chamber with a background pressure of $10^{-8}$\\,Torr \\cite{yamasaki2015dynamical}. The scattered intensity was collected by an in-vacuum charge-coupled device (CCD) area X-ray detector of $2048\\,\\times\\,2048$ pixels (Princeton Instruments, Trenton, New Jersey, USA). The RSXS endstation MARES was used at ALBA synchrotron \\cite{barla2016design}. Resonant diffraction intensity was collected by a custom-designed CCD detector of $2148\\,\\times\\,2052$ pixels (XCAM Co, ltd, UK).\nSince small-angle scattering intensity is distributed near the transmitted direct beam, a tungsten beamstop was introduced to protect the detector for RSXS experiment, while in case of HERALDO the smaller aperture size allowed us to measure the holograms without using any beamstop.\nThe magnetic field was applied parallel to the incident X-ray beam and perpendicular to the thin plate. A He-flow-type cryostat was used to control the sample temperature in range from $\\sim 15$\\,K to $\\sim 320$\\,K, as measured by the Si diode thermometers attached next to the sample holder and cryostat head \\cite{yamasaki2013diffractometer}. A radiation shield was wrapped around the sample holder to reduce the heating of the specimen from the warm environment.\n\n\\section{Results and discussions}\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{Fig2.png}\n\\caption{(Color online) (a) XAS spectra of the bulk Co$_8$Zn$_8$Mn$_4$ near Co and Mn $L_{2,3}$ absorption edges: measured points are shown as the solid lines; calculated spectra are shown as the dashed lines. (b) XMCD signal. The measurements were performed at temperature $T=130$\\,K at applied magnetic field $B=0.5$\\,T.}\n\\label{Fig2}\n\\end{figure}\n\n\\begin{figure*}\n\\includegraphics[width=17cm]{Fig3.png}\n\\caption{(Color online) (a) Schematic phase diagram of Co$_8$Zn$_8$Mn$_4$ and the procedure of the resonant soft x-ray scattering (RSXS) measurements. The sample is field-cooled (FC) in the applied magnetic field $B = 70$\\,mT from room temperature down to 25\\,K (indicated by blue arrow) and then warmed back to 300\\,K in the same field (indicated by red arrow). Coherent RSXS speckle patterns measured for Co$_8$Zn$_8$Mn$_4$ sample at $E=779$\\,eV corresponding to $L_3$ absorption edge of Co at different temperatures (b) $T=25$\\,K, (c) $T=120$\\,K, (d) $T=150$\\,K and applied field $B=70$\\,mT. White scale bar corresponds to 0.05\\,nm$^{-1}$.}\n\\label{Fig3}\n\\end{figure*}\n\nThe XAS signals averaged between RCP and LCP spectra near the $L_{2,3}$ edges of Mn and Co at $T=130$\\,K are shown in Fig. \\ref{Fig2}a. Surprisingly, despite the metallic nature of the Co$_8$Zn$_8$Mn$_4$ alloy, the Mn absorption shows a multiplet structure at the $L_3$ and $L_2$ edges. Well-resolved peaks at 1.3 and 3.5\\,eV above the absorption maxima at $E=641.0$\\,eV and a doublet structure at the $L_2$ edge, which splits into two maximums at $E=651.7$\\,eV and $E=653.2$\\,eV are clearly observable. Meanwhile, the Co $L_3$ and $L_2$ peak shapes at $E=780.0$\\,eV and $E=794.1$\\,eV are similar to the broad spectrum of metallic Co \\cite{van1991strong,schwickert1998x,dhesi2001anisotropic}. This suggests that the fine structures of Mn $2p \\rightarrow 3d$ transition should result from the localization of Mn $3d$ electrons rather than the oxidation of surface \\cite{miyamoto2003soft,grabis2005element,gilbert2003multiple}: otherwise, the multiplet structure of Co $L_{2,3}$ edges should be either observed \\cite{regan2001chemical,magnuson2002electronic}. To qualitatively illustrate the features of the measured XAS, simulations were performed using the small cluster approach in the Xclaim software tool \\cite{fernandez2015xclaim}. Mn $3d^5$ was assumed as an initial state and Mn $2p^5$ $3d^6$ was the final configuration. The simulated XAS lines were broadened with a Lorentz function with the full width $\\Gamma$ at half maximum (FWHM) of 1\\,eV. The spectrum for Co was calculated using initial $3d^8$ and final $2p^5$ $3d^9$ configurations, which reproduces the measured data despite some broadening of the latter. The observed fine structure of the XAS spectrum of Mn also nicely corresponds to the one calculated from the multiplet effects (Fig. \\ref{Fig2}a).\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{Fig4.png}\n\\caption{(Color online) Temperature dependence of the modulation vector of skyrmion lattice $q_{Sk}$. Inset shows magnetic field dependence of $q_{Sk}$ measured at $T=25$\\,K and $T=50$\\,K.}\n\\label{Fig4}\n\\end{figure}\n\nXMCD signal measured in a magnetic field of $B=0.5$\\,T well above the saturation is shown in Fig. \\ref{Fig2}b for both elements. Despite the magnitude of the XMCD measured at $L_{2,3}$ edges of Mn is about \\sfrac{1}{5} times smaller than XMCD measured at Co $L_{2,3}$ edges, it is clear that the signs of the dichroic signals are the same. XMCD signal at Mn $L_2$ edge is notably suppressed, indicating quenching of the orbital moment.\nThe sum rule analysis \\cite{o1994orbital} allows to estimate the orbital to spin moment ratio for Mn and Co as $\\mu_l$(Mn)\/$\\mu_s$(Mn)=0.03 and $\\mu_l$(Co)\/$\\mu_s$(Co)=0.0025.\nMagnetic field dependence of the element-selective XMCD signals measured at 135\\,K can be found in the Supplementary materials \\cite{supplementary}. The signs and magnitudes of the XMCD signals indicate a ferromagnetic coupling of Co and Mn moments and a partial cancellation of Mn magnetization. This is in a good agreement with the magnetization measurements, which have shown the reduction of magnetization and critical temperature with increment of Mn concentration in $\\beta$-Mn-type Co-Zn-Mn compounds \\cite{tokunaga2015new}. One can assume that while the Co-Co and Co-Mn couplings are ferromagnetic, the Mn-Mn interaction should be antiferromagnetic. This scenario is also suggested by recent neutron diffraction study of the Co-Zn-Mn alloys in a wide composition range accompanied by the density functional theory (DFT) calculations \\cite{bocarsly2019deciphering}. This is in contrast to the case of Co-Mn alloys where Mn moments tend to align antiparralel to the host Co magnetization \\cite{nakai1978magnetic,menshikov1985magnetic,wildes1992polarized}. On the other hand, the parent $\\beta$-Mn compound shows strongly antiferromagnetic nearest-neighbor correlations in the $12d$ Mn sublattice, while strong ferromagnetic correlations between next-nearest-neighbors were found \\cite{paddison2013emergent}. Further, recent DFT calculations of the $\\beta-$Mn-type Co-Zn-Mn suggested larger localization of the Mn atoms at $12d$ site than at $8c$, being consistent with our observations \\cite{bocarsly2019deciphering}. For further quantitative discussions on the XAS and XMCD features, such as origin of multiplet spectrum of Mn, and determination of spin and orbital contributions, additional measurements of different Co-Zn-Mn concentrations and spectra calculations from \\textit{ab initio} theory are highly desired.\n\nComplex scattering factor $f$ for resonant magnetic X-ray scattering can be described as\n\n\\begin{equation}\nf=(\\bm{s}\\cdot \\bm{s}') f_c + i (\\bm{s}\\times \\bm{s}')\\cdot \\bm{M}f_{m}^{1} + (\\bm{s}\\cdot\\bm{M})(\\bm{s}'\\cdot\\bm{M})f_{m}^{2},\n\\label{eq1}\n\\end{equation}\nwhere $\\bm{s}$ and $\\bm{s}'$ are polarizations of the incident and scattered X-rays, respectively; $f_c$ is the charge scattering factor; $\\bm{M}$ is the local magnetization; $f_{m}^1$ and $f_{m}^2$ are factors attributed to the magnetic scattering maximized at the resonant condition. The last term in Eq. (\\ref{eq1}) containing scattering factor $f_{m}^2$ is quadratic in $\\bm{M}$ and generally smaller than the other two terms \\cite{hannon1988x}. Therefore the scattering patterns $(|f|^2)$ measured at resonant conditions mainly consist of the squares of charge and magnetic scattering factors $|f_c|^2$ and $|f_{m}^1|^2$, and their interference ($|f_c^* f_{m}^1|$). For linear polarization the measured intensities are dominated by the pure charge and magnetic scattering terms, while those measured using circular polarization contain charge-magnetic cross term \\cite{hill1996resonant,eisebitt2003polarization}. Consequently, the intensity of magnetic scattering can be simply distinguished from the charge scattering by subtracting the diffraction pattern measured 1) at off-resonant condition and appropriately normalized or 2) in the field-induced polarized state of the sample or 3) above the critical temperature.\n\nRSXS experiments were carried out at Mn ($E=640.5$\\,eV) and Co ($E=779$\\,eV) $L_{3}$ edges, where the magnetic scattering intensity was maximized. Far-field RSXS patterns were acquired with an exposition time of 1\\,second (excluding readout time $\\sim 0.4$\\,s) and total 200\\,expositions were averaged. In case of the off-resonant conditions, the absorption was reduced and the measurement time was consequently reduced by a factor of two. RSXS was measured in the temperature range between $T=25$\\,K and $T=270$\\,K during a field \n-warming-after-field-cooling (FWAFC) in $B=70$\\,mT. The sample position was realigned after each temperature change to compensate the effect of the thermal contraction (expansion) of the sample manipulator with a buffer time between the measurements (in average $\\approx 40$\\,minutes) for stabilization. In the present experiment the linear polarization was used to minimize the influence of the charge-magnetic scattering interference term. In order to isolate the charge scattering contribution we have measured the scattering intensity at off-resonance ($E=645$\\,eV and $E=785$\\,eV) conditions. \n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=15cm]{Fig5.png}\n\\caption{(Color online) Differential scattering patterns between LCP and RCP X-rays of energies (a) $E=779$\\,eV and (b) $E=640.5$\\,eV the taken at $T=20$\\,K. The colorbar is given for both patterns in arbitrary units. Note that the strong linear patterns inclined upward right should be due to the incomplete subtraction of the charge scattering from the reference slit. (c) Fourier transform of (a) after applying linear differential filter and rotation of the image by 39$^\\circ$. (d), (e) Magnetic texture recorded at Co and Mn $L_3$ edges, respectively. The colorbar is given for both images in arbitrary units. The region outside the field of view is filled with the black background manually for clarity. (f) Fourier ring correlation analysis of the reconstructed images for Co and Mn. Orange line represents calculated half-bit threshold and the dashed line indicates irreversible cross-point between the FRC function and half-bit threshold.}\n\\label{Fig5}\n\\end{center}\n\\end{figure*}\n\nTypical coherent RSXS speckle patterns measured at $E=779$\\,eV at temperatures $T=25$\\,K, $T=120$\\,K, and $T=150$\\,K are shown in Fig.\\ref{Fig3}. The missing area in the left part of each panel is due to the beamstop shadow. Tuning the energy to the Mn $L_3$ edge results in the scaling of the scattering pattern due to the difference in the photons wavelength. Magnetic scattering intensity is weaker at Mn edge, which can be explained by the lower Mn concentration in this compound, stronger absorption, and magnetization reduction, as expected from XMCD experiment. At room temperature and $T=150$\\,K the small-angle scattering patterns demonstrate six-fold symmetry indicating single-domain triangular skyrmion lattice. As it has been already shown by SANS and LTEM experiments, the skyrmion lattice phase can be supercooled by a field cooling process down to the low temperatures \\cite{karube2016robust,nakajima2017skyrmion,morikawa2017deformation}. According to the previous LTEM observations on a Co$_8$Zn$_8$Mn$_4$ thin plate, hexagonal skyrmion lattice to amorphous state transition is reversible and accompanied by the elongation of individual skyrmions along one of the principal crystallographic axes, while the skyrmion density is conserved \\cite{morikawa2017deformation}.\n\nUpon the field cooling (or FWAFC), the RSXS transforms to a homogeneous ring-like pattern corresponding to the intermediate \"amorphous\" phase at the temperature $T\\approx100$\\,K. Below $T\\approx100$\\,K four wide peaks appear at the coherent diffraction speckle pattern. Moreover, below the transition temperature, the skyrmion lattice parameter gradually decreases from $a_{Sk}=112$\\,nm ($T=100$\\,K) to $a_{Sk}=76$\\,nm ($T=25$\\,K). Temperature dependence of the magnetic modulation $q_{Sk}$ is shown in Fig. \\ref{Fig4}. $q_{Sk}$-vector is also dependent on magnetic field similar to the bulk case \\cite{karube2016robust} (inset in Fig. \\ref{Fig4}). Elongation of the $q_{Sk}$ vector is, presumably, caused by increasing antiferromagnetic interaction between Mn ions superimposed on the helical order. Upon cooling from $T=100$\\,K to $T=25$\\,K a coherent RSXS speckle pattern with four-fold symmetry can be observed. According to the previous results, the hexagonal skyrmion lattice \\cite{karube2016robust} is recovered by applying stronger magnetic field of 300-500\\,mT at low temperature. Indeed, the magnitude of the $q_{Sk}$ vector tends to shrink upon increment of the magnetic field (inset in Fig. \\ref{Fig4}). However, in present experiment the magnetic scattering intensity arising from isolated skyrmions is hardly distinguishable in presence of the background even when the charge scattering is subtracted. Corresponding RSXS patterns can be found in the Supplementary materials \\cite{supplementary}.\n\nThe integrated speckle pattern intensities are demonstrating similar features for Co and Mn except the difference in signal-to-noise ratio, which is better at $E=779$\\,eV. Radially integrated azimuthal profiles of the scattering patterns measured using soft X-rays with energies $E=779$\\,eV and $E=640.5$\\,eV can be found in the Supplementary materials \\cite{supplementary}. \n\nFourier transform holography with extended reference allows to reconstruct the real-space image by single Fourier transform of the measured pattern multiplied by the linear differential operator in direction parallel to the reference slit. Circularly polarized soft X-rays with energies matching to Co and Mn $L_3$ edges were used for HERALDO experiment. Far-field diffraction patterns were collected with RCP and LCP to produce the interference pattern and reconstruct the magnetic texture. In this case information about the magnetic texture was simply encoded in the interference term $f_c^* f_{m}^1$ between charge scattering arising from the reference slit and magnetic scattering from the Co$_8$Zn$_8$Mn$_4$ sample. The difference between the holograms taken with the two opposite X-ray helicities provided the isolated interference term which could be inverted to the real-space image via single Fourier transform and linear differential operator \\cite{guizar2007holography}. HERALDO patterns were acquired with an exposition time of 6\\,s (excluding readout time $\\sim 0.4$\\,s), and total 200 expositions were averaged, resulting in a total acquisition time of 20 minutes for each hologram. The sample was cooled down to the temperature $T=20$\\,K while the magnetic field of $B=70$\\,mT was applied along $[001]$ direction parallel to incident soft X-ray beam propagation vector.\n\n\\begin{figure*}\n\\includegraphics[width=16cm]{Fig6.png}\n\\caption{(Color online) Real-space magnetic textures imaged at Co $L_3$ edge ($E=779$\\,eV) at temperatures (a) $T=20$\\,K, (b) $T=100$\\,K and (c) $T=120$\\,K and applied magnetic field $B=70$\\,mT.}\n\\label{Fig6}\n\\end{figure*}\n\nDifferences between the diffraction patterns taken with the two opposite X-ray polarizations at photon energies $E=779$\\,eV and $E=640.5$\\,eV are shown in Figs. \\ref{Fig5}a and b, respectively. The highest harmonics of the interference pattern can be found at $q_{max}=0.2$~nm$^{-1}$ which corresponds to the real-space resolution of 32\\,nm. The difference between Fourier transform images taken with RCP and LCP at Co $L_3$ edge after applying the differential filter in the slit direction is shown in Fig. \\ref{Fig5}c. Reconstructed real-space image shows the sample autocorrelation and two object-reference cross-correlations delivered by both ends of the slits, as well as their complex conjugates. Magnetic contrast is inverted between the reconstructed object and conjugate. By taking into account the widths of magnetic domains where the local magnetization points antiparallel (parallel) to the applied field $B=70$\\,mT at different temperatures (Fig. \\ref{Fig6}), we conclude that the regions with the negative (positive) values in Figs. \\ref{Fig5}d and e correspond to magnetization pointing antiparallel (parallel) to the field.\n\nFigures \\ref{Fig5}d and e show magnification of the sample-reference cross-correlations taken from reconstructions of holograms taken for Co and Mn, respectively. Due to the difference of the wavelengths, the real-space image corresponding to magnetic texture of Mn ions was scaled by factor $E_1\/E_2=779\/640.5\\approx1.22$. The magnetic structures exhibited by magnetic elements Mn and Co are similar to each other within the resolution limit. Signal-to-noise ratio is worse in case of the measurement at Mn $L_3$ edge due to the higher absorption of soft X-rays at $E=640.5$\\,eV and smaller concentration of Mn atoms compared to Co. Different magnitude of the magnetic moment between Co and Mn is also a reason for this. Therefore, the real-space image reconstructed for Mn atoms is slightly blurred (Fig. \\ref{Fig5}e). Additionally the reconstruction resolution is estimated from the Fourier ring correlation (FRC) \\cite{banterle2013fourier}. By using FRC the spatial frequency dependence of the cross-correlation of two real-space magnetic textures obtained from the reconstructions of Co and Mn HERALDO patterns (Fig. \\ref{Fig5}f) is analyzed. The resolution was calculated at the point where the FRC curves irreversibly cross the half-bit threshold \\cite{van2005fourier}. The real-space resolution that was found according to this criteria is similar to the resolution of 32\\,nm determined from the highest interference harmonics observed at the Fourier-space pattern.\n\n\\begin{figure*}\n\\includegraphics[width=17cm]{Fig7.png}\n\\caption{(Color online) Reconstruction of coherent RSXS patterns at (a) $T=25$\\,K, (b) $T=55$\\,K and (c) $T=120$\\,K collected at Co L$_3$ edge. The colorbar for intensity ($z$-scale) is given in arbitrary units. (d) Magnification of the region highlighted by the blue square in (a) is shown for the the different temperatures.}\n\\label{Fig7}\n\\end{figure*}\n\nElongated skyrmions can be observed at the holograms taken both at $E=779$\\,eV and $E=640.5$\\,eV indicating similar magnetic texture of Co and Mn sub-lattices, which coincides with the XMCD and RSXS data. Magnetic holograms were recorded during a FWAFC procedure up to 120\\,K at which the magnetic scattering is highly reduced due to the thermal shrinkage of the magnetic moments. Corresponding evolution of the magnetic texture measured at Co absorption edge is shown in Fig. \\ref{Fig6}. The transformation of the elongated skyrmions to more conventional shape with corresponding expansion takes place at $T=120$\\,K (Fig. \\ref{Fig6}c). Unfortunately, due to a thermal shrinkage of magnetic moments the real-space reconstruction of magnetic texture at higher temperatures can be hardly distinguished from the background fluctuations.\n\n\\begin{figure*}\n\\includegraphics[width=17cm]{Fig8.png}\n\\caption{(Color online) Micromagnetic simulations of Co$_8$Zn$_8$Mn$_4$ $10\\times10$\\,$\\mu$m$^2$ thin plate: $z-$projection of the magnetic texture is shown for various exchange from stiffness parameters $A_{ex}=9.2\\cdot10^{-12}$\\,J\/m (a), $5.8\\cdot10^{-12}$\\,J\/m (b), $3.2\\cdot10^{-12}$\\,J\/m (c). Bottom panels show magnification of the rectangular regions highlighted by the corresponding red boxes.}\n\\label{Fig8}\n\\end{figure*}\n\nSince the sample area probed by HERALDO was limited by the aperture size, we additionally performed a real-space inversion off the measured RSXS patterns by the iterative phase retrieval. Square root of the measured magnetic scattering intensity $\\sqrt{I_{m}}$ shown in Fig. \\ref{Fig3} isolated from the charge scattering as described in Ref. \\cite{ukleev2018coherent}, was used as real part of the Fourier-space constraint. Fixed sample aperture size, shape and orientation were used as real-space support. Pixels, that were missing at the center of diffraction pattern due to the beamstop shadow and subtracted charge scattering, were substituted by the Fourier transform of the support. This approach is similar to the substitution used in the guided hybrid input-output (HIO) algorithm \\cite{chen2007application}. We used the implementation of HIO algorithm \\cite{fienup1982phase} with a feedback parameter $\\beta=0.9$ with assuming positivity and reality constraints for the real-space pattern. Phase-retrieval algorithm tended to stagnate to the local minima after 500 iterations. For each coherent scattering pattern the final real-space image was calculated from average of 500 algorithm trials with random initial phases. Reliability of the reconstructed real-space images was qualitatively and quantitatively examined by comparing the reconstructions performed for the data measured at Co and Mn $L_3$ edges, respectively. Indeed, based on the results of the XMCD and HERALDO experiments the same orientation of the local magnetic moments of Co and Mn atoms was expected.\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{Fig9.png}\n\\caption{(Color online) Measured dependence of $q_{Sk}$ over temperature (symbols) and calculated dependence of $q_{Sk}$ over the exchange stiffness ($A_{ex}$) parameter.}\n\\label{Fig9}\n\\end{figure}\n\n\\begin{figure*}\n\\includegraphics[width=17cm]{Fig10.png}\n\\caption{(Color online) Magnetic field evolution of the $z-$projection of magnetization derived from the micromagnetic simulation of Co$_8$Zn$_8$Mn$_4$ thin plate in magnetic field $B=150$\\,mT (a), $B=250$\\,mT (b), $B=300$\\,mT (c), $B=350$\\,mT (d), $B=400$\\,mT (e), $B=450$\\,mT (f) showing the hexagonal SkX restoration.}\n\\label{Fig10}\n\\end{figure*}\n\nTo quantitatively estimate the reliability of the reconstructions performed for Co and Mn we have introduced the two-dimensional function \n$$R(i,j)=1-\\frac{|\\psi_{E_1}^{*(i,j)} - \\psi_{E_2}^{*(i,j)}|} {|\\psi_{E_1}^{*(i,j)}|},$$\nwhere $\\psi_{E_1}^{*(i,j)}$ and $\\psi_{E_2}^{*(i,j)}$ are values of $(i,j)$ pixels of the reconstructed real-space patterns normalized by maximum value for corresponding image at Co($E_1$) and Mn($E_2$) $L_3$ edges. This normalization is needed to compensate the difference in total magnetic contrast for Mn and Co patterns and transmission coefficient for the photons with different energies. Figure \\ref{Fig7} shows the real-space magnetic patterns reconstructed from data taken at Co absorption edge at different temperatures. At $T=25$\\,K and $T=55$\\,K (Figs. \\ref{Fig7}a,b) an array of magnetic skyrmions elongated along one of the crystallographic axes can be observed. As the temperature increases, some of the elongated skyrmions transform to the conventional circular-shaped (Fig. \\ref{Fig7}c), denoted as amorphous state. Unfortunately, similarly to the HERALDO experiment, the magnetic scattering intensity gradually decreases upon warming due to the thermal shrinkage of the magnetic moments. As a result, the signal to noise ratio is insufficient to reliably reconstruct the real-space image of the single-domain triangular skyrmion lattice at $T>120$\\,K. In the Supplementary information \\cite{supplementary} we show a few examples of the reconstructions performed for the skyrmion lattice and multidomain helical phases at higher temperatures. However, the real-space images obtained for these conditions correspond to the local minima in the solution space and vary sufficiently for the different starting phases. Two-dimensional reliability maps $R(i,j)$ for successful reconstructions can be found in the Supplementary information \\cite{supplementary}. Here we just note that the averaged reliability value $R=\\frac{1}{N}\\sum R(i,j)$, where $N$ is the sample area in pixels, is not less than 89\\% for the datasets measured at 25\\,K.\n\nFrom this aspect, the results of lensless imaging by iterative phase retrieval and HERALDO are consistent with each other. Both techniques provide a spatial resolution of 30\\,nm and sensitive to the signal-to-noise ratio. Data analysis in case of coherent diffraction imaging is more sophisticated compared to HERALDO, but the latter requires advanced sample fabrication. Further attempts to improve the nanofabrication routine are required to obtain better real-space resolution. \n\nThe experimental results for the temperature- and field-dependent transformation of the skyrmion lattice obtained by soft X-ray scattering and imaging agree well with the results of Landau-Lifshitz-Gilbert (LLG) simulations calculated with realistic parameters for Co$_8$Zn$_8$Mn$_4$ thin plate by using mumax$^3$ package \\cite{vansteenkiste2014design}. As hinted by present XMCD experiment and previous magnetization measurements \\cite{tokunaga2015new}, the effective ferromagnetic exchange interaction in Co$_8$Zn$_8$Mn$_4$ system may decrease with temperature due to antiferromagnetic correlations of Mn sub-lattice. Moreover, manifestation of the cubic anisotropy is considerable at low temperatures. For the simulation we used exchange stiffness $A_{ex} = 9.2$\\,pJ\/m and DMI constant $D=0.00053$\\,J\/m measured experimentally by microwave spin-wave spectroscopy \\cite{takagi2017spin}, and cubic anisotropy in the form of an effective field \\cite{vansteenkiste2014design}:\n\\begin{equation*}\n\\begin{aligned}\n\\mathbf{B}_{anis} &= - 2 K_c \/ M_{s} \\big(((\\mathbf{c}_{1} \\cdot \\mathbf{m})\\mathbf{c}_{1})((\\mathbf{c}_{2} \\cdot \\mathbf{m})^2 + (\\mathbf{c}_{3} \\cdot \\mathbf{m})^2)+ \\\\\n&((\\mathbf{c}_{2} \\cdot \\mathbf{m})\\mathbf{c}_{2})((\\mathbf{c}_{1} \\cdot \\mathbf{m})^2 + (\\mathbf{c}_{3} \\cdot \\mathbf{m})^2)+ \\\\\n&((\\mathbf{c}_{3} \\cdot \\mathbf{m})\\mathbf{c}_{3})((\\mathbf{c}_{1} \\cdot \\mathbf{m})^2 + (\\mathbf{c}_{2} \\cdot \\mathbf{m})^2)\\big),\n\\end{aligned}\n\\end{equation*}\nwhere $K_c = 5000$\\,J\/m$^3$ is the 1$^{st}$-order cubic anisotropy constant determined from electron spin resonance (ESR) experiment \\cite{kezmarki2018}; $\\mathbf{c}_{1}$, $\\mathbf{c}_{2}$ and $\\mathbf{c}_{3}$ is a set of mutually perpendicular unit vectors indicating the anisotropy directions (cubic axes), and $M_s=350$\\,kA\/m is the saturation magnetization \\cite{takagi2017spin}. A two-dimensional 200\\,nm-thick plate with area of $10\\times10$\\,$\\mu$m$^2$ with open boundary conditions and elementary cell size $5\\times5\\times200$\\,nm$^3$ was simulated. To follow the experimental field-cooling protocol, we started the simulation with random initial spin configuration in the applied field along $[001]$ direction. The field needed for robust SkX formation was set to $B=150$\\,mT. The discrepancy between the observed and calculated field values is, presumably, caused by demagnetization effects. The characteristic helical pitch $\\lambda=4\\pi A_{ex}\/D$ and skyrmion size are determined by the ratio of the exchange stiffness $A_{ex}$ and the Dzyaloshinskii constant $D$ \\cite{bak1980theory}. Therefore it is reasonable to assume two possible scenarios: 1) the temperature-dependent $q$-vector variation is caused by changing exchange integral due to the antiferromagnetic correlations in Mn sub-lattice towards lower temperatures; 2) the emergence of the $q$-vector elongation can be also caused by enhancement of antisymmetric DMI due to the local non-complanar structure of frustrated Mn. At the moment, the experimental data does not allow to unambiguously distinguish between variations of $A_{ex}$ and $D$. To mimic the first scenario, we reduced the exchange stiffness parameter $A_{ex}$ from the measured value $9.2$\\,pJ\/m to $3.2$\\,pJ\/m in a linear fashion, while the magnetic field $B=150$\\,mT remained constant. Relaxation time of 10 ns was introduced between each step for magnetic texture stabilization. Upon gradual decrease of the exchange interaction in the system, the skyrmion lattice exhibits deformation similar to the LTEM and soft X-ray experiments (Figs. \\ref{Fig7}a--c). Elongation of the skyrmions is manifested due to the overall reduction of the exchange interaction, while its directionality along the cubic axes is dictated by the cubic anisotropy. Notably, this deformation takes place via directional expansion of each topological vortex, but not by merging of the neighboring skyrmions -- in the latter case the total topological charge of the system would decrease, which is opposite to the topological charge conservation scenario reported earlier \\cite{morikawa2017deformation}. Radial averages of the two-dimensional fast Fourier transformation (FFT) patterns of out-of-plane projection of magnetization distribution was used to calculate the $q$-vector magnitude dependence for the resultant magnetic textures. The simulated variation in $q_{Sk}$ induced by change in the exchange stiffness parameter $A_{ex}$ is in the very good qualitative agreement with the temperature dependence (Fig. \\ref{Fig9}). The shoulder in Fig. \\ref{Fig9} at $A_{ex}$ between 5 -- 6\\,pJ\/m corresponds to the smooth transition from the disk-shaped to elongated skyrmions. Therefore we assume that the linear decrease of the exchange interaction in Co$_8$Zn$_8$Mn$_4$ takes place with decreasing temperature. In general, this consideration is consistent with the previous studies of the $\\beta-$Mn alloys, that have shown presence of antiferromagnetic correlation of moments in the $12d$ Mn sublattice \\cite{shiga1994polarized,nakamura1997strong,stewart2009magnetic,paddison2013emergent,bocarsly2019deciphering}. We plan to directly address to this question in our future studies. For example, recently developed spin-wave sensitive SANS technique allows to directly probe $A_{ex}$ of a bulk specimen as a function of temperature \\cite{grigoriev2015spin}.\n\nLLG simulation was either used to probe the magnetic field evolution of the elongated skyrmion texture. The \"field-cooled\" magnetic texture shown in Fig. \\ref{Fig10}c with $A_{ex}=3.2$\\,pJ\/m was used. Except the discrepancy between the field magnitude values, the result is consistent with present X-ray and previous LTEM experiments -- restoration of the hexagonal lattice of circular skyrmions from the deformed state by ramping the magnetic field was successfully reproduced (Figs. \\ref{Fig10}a--f).\n\n\\section{Conclusion}\n\nIn conclusion, by means of element-selective soft X-ray circular magnetic dichroism we have revealed the ferromagnetic arrangement of Co and Mn ions in a room-temperature skyrmion-hosting compound Co$_8$Zn$_8$Mn$_4$. Moreover, by using the coherent resonant small-angle soft X-ray scattering and holography with extended reference we can conclude that the topological magnetic texture is the same for both type of atoms in whole temperature range above $T_g$ that is reliable for real-space reconstruction. Our results are consistent with each other and with the previous neutron scattering and Lorentz microscopy experiments and shows the transition from hexagonal skyrmion crystal to elongated skyrmion state that is accompanied by deformation of the individual vortices. Micromagnetic simulation suggests that such transition is driven by decreasing exchange interaction in the system and effect of the cubic anisotropy. This effective decrease of the ratio of symmetric exchange interaction to antisymmetric Dzyaloshinskii-Moriya mimics low-temperature antiferromagnetic frustration of Mn sub-lattice. At lower temperature, antiferromagnetic correlations of Mn atoms is superimposed onto the long-range helical (skyrmion) modulation, resulting in shortening of the helical pitch and deformation of skyrmions. However, this effect is reversible and hexagonal skyrmion lattice from elongated skyrmion state can be restored by increasing magnetic field even when the exchange stiffness is reduced, as learned from the micromagnetic simulation and previous experiments \\cite{karube2016robust,morikawa2017deformation}.\n\nWe have demonstrated first to our knowledge lensless soft X-ray imaging of the magnetic texture at cryogenic temperatures and applied magnetic fields. HERALDO imaging was used with the circularly polarized soft X-rays, while coherent diffraction imaging was performed with a linearly polarized beam. Both methods did not require focusing X-ray optics to perform magnetic imaging with resolution of few tens of nanometers. Practically, the signal to noise ratio sufficient for the successful reconstruction was achieved only in the temperature range from $T=20$\\,K to $T=120$\\,K due to the overall decay of the intensity of the magnetic scattering and charge-magnetic interference with increasing temperature.\n\nSoft X-ray imaging methods allow to simultaneously obtain element-selective real-space information and will be useful for further investigations of non-trivial magnetic textures in thin plates of polar magnets, since N\\'eel-type skyrmions produce no contrast in Lorentz transmission electron microscopy.\n\n\\section*{Acknowledgments}\nThe authors wish to acknowledge P. Gargiani and BOREAS beamline staff for the technical assistance. We also thank T. Honda for providing the membranes. Soft X-ray scattering experiments were performed as a part of the proposals no.: 2015S2-007 (Photon Factory) and 2016081774 (ALBA Synchrotron Light Laboratory).\nThis research was supported in part by PRESTO Grant Number JPMJPR177A from Japan Science and Technology Agency (JST), \"Materials research by Information Integration\" Initiative (MI$^2$I) project of the Support Program for Starting Up Innovation Hub from JST, the Japan Society for the Promotion of Science through the Funding Program for World-Leading Innovative R\\&D on Science and Technology (FIRST Program), and JSPS KAKENHI Grant Number 16H05990. V.U. acknowledges funding from the SNF Sinergia CDSII5-171003 NanoSkyrmionics. M. V. acknowledges additional funding to the MARES endstation by grants MICINN ICTS-2009-02, FIS2013-45469-C4-3-R and FIS2016- 78591-C3-2-R (AEI\/FEDER, UE).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}