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\shortauthors{Suto et al.}
\shorttitle{HSE mass correction}
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\begin{document}
\title{Correction of Hydrostatic Mass of X-ray Clusters from Cosmological Simulations}
\author{
Daichi Suto\altaffilmark{1},
Tetsu Kitayama\altaffilmark{2},
Shin Sasaki\altaffilmark{3},\\
Yasushi Suto\altaffilmark{1,4,5},
Renyue Cen\altaffilmark{5}
and Klaus Dolag\altaffilmark{6,7}
}
\email{daichi@utap.phys.s.u-tokyo.ac.jp}
\altaffiltext{1}{Department of Physics, The University of Tokyo,
Tokyo 113-0033, Japan}
\altaffiltext{2}{Department of Physics, Toho University, Funabashi,
Chiba 274-8510, Japan}
\altaffiltext{3}{Department of Physics, Tokyo Metropolitan University,
Hachioji, Tokyo 192-0397, Japan}
\altaffiltext{4}{Research Center for the Early Universe, School of Science, The University of Tokyo, Tokyo 113-0033, Japan}
\altaffiltext{5}{Princeton University Observatory, Princeton, NJ 08544}
\altaffiltext{6}{Max-Planck Institut f\"ur Astrophysik, Karl-Schwarzschild Str. 1, D-85741 Garching, Germany}
\altaffiltext{7}{University Observatory M\"unchen, Scheinerstr. 1, 81679, M\"unchen, Germany
}
\begin{abstract}
We present a method to correct the hydrostatic mass to better estimate the total mass of galaxy clusters. We analyze six simulated clusters and find that the difference between the total mass and the hydrostatic mass is correlated with temperature of gas at large radii with a physical reason. Using this correlation, we define a correction mass term. In the three-dimensional space where no observational biases exist, the correction term plus the hydrostatic mass reproduces the total mass within a 10 percent at $r\gtrsim r_{500}$ where the hydrostatic mass alone underestimates the total mass by up to 30 \%. We also discuss the applicability of the method for X-ray observations by making mock X-ray observables from the simulation data and deprojecting the gas density and temperature from them. We show that the difference between the hysrostatic and total masses becomes larger (up to 40 \%) due to the deprojection process. As a result, the hydrostatic mass (estimated from X-ray observables) with the correction term can reproduce the total mass within 20 \%.
We also show that the main cause of the misestimate of mass in the deprojection process is the spherical assumption of the density and temperature profiles.
\end{abstract}
\keywords{cosmology: theory -- galaxies: clusters: general -- methods:
numerical -- X-rays: galaxies: clusters}
\section{INTRODUCTION}
Clusters of galaxies are the largest gravitationally bound systems in the universe and one of the most important cosmological applications of them is determination of cosmological parameters\citep[][for a recent review]{allen11}. They are primarily observed by the strong X-ray emission from the intractluster medium (ICM). More precise measurements of the ICM and a larger statistical sample of clusters than ever will be brought about by the coming X-ray satellites ASTRO-H and eROSITA/SRG. Combined with the Sunyaev-Zel’dovich (SZ) effect currently measured by the Atacama Cosmology Telescope (ACT) and South Pole Telescope (SPT), and/or the gravitational lensing effect by the Subaru Hyper Supreme Cam (HSC), galaxy clusters will offer a variety of new insights into the universe.
In view of the ongoing structure formation, the hydrostatic assumption is just an approximation and it will be invalid to some extent especially in the outer region of clusters, where the cluster is less dynamically relaxed. In fact, recent studies \citep{mahdavi08,zhang08,zhang10} have shown that the hydrostatic mass is systematically smaller (by $\sim$ 10 -- 20\% at $\sim r_{500}$) than the mass estimated by the gravitational lensing analysis, which is free from the assumption on the dynamical state of the gas. (Note that the lensing mass can also be biased by other factors.)
Clusters of galaxies are the largest gravitationally bound systems in the universe and offer a variety of cosmological and astrophysical applications \citep[][for a recent review]{allen11}. They are primarily observed by the strong X-ray emission from the intractluster medium (ICM). Recently, X-ray observations of the ICM have extended their observation area out to the virial radius of clusters, where the dynamical state of the ICM will provide information on still ongoing structure formation processes. More precise measurements of the ICM and a larger statistical sample of clusters than ever will be brought about by the coming X-ray satellites ASTRO-H and eROSITA/SRG. Combined with the Sunyaev-Zel’dovich (SZ) effect currently measured by the Atacama Cosmology Telescope (ACT) and South Pole Telescope (SPT), or the gravitational lensing effect by the Subaru Hyper Supreme Cam (HSC), galaxy clusters will offer a new insight into our universe. Thus it is a good time now to revisit biases in cluster observations.
The total mass of clusters is the most important quantity in most cosmological applications. Although various ways to estimate the mass (X-ray observations of ICM, gravitational lensing, the SZ effect etc.) have been developed, each of them is subject to its own biases. In X-ray observations, the mass is estimated from gas density and temperature profiles under the assumption of hydrostatic equilibrium (HSE). The validity of HSE is, however, not yet clearly known. In a recent paper \citep[][hereafter Paper I]{suto13}, we have analyzed a simulated cluster and shown that the mass estimated under the HSE assumption (HSE mass) intrinsically underestimate the true mass by $\sim$ 15 \% on average and $\sim$ 30 \% at most, which is consistent with previous studies \citep{fang09,lau09} and that the violation of HSE can be mainly attributed to acceleration of the gas, which was neglected in the previous studies. The large discrepancy between the total and HSE masses indicates that we need to correct the HSE mass somehow in order to better estimate the true mass.
In addition, the above level of violation of HSE is the one when the three-dimensional density and temperature are used. In real observations, density and temperature have to be deprojected from two-dimensional observables. Hence the violation can be enhanced (or weakened) due to the deprojection effect. \cite{rasia12} have performed mock observations of 20 simulated clusters along three orthogonal axes and shown that the HSE mass is lower than the true mass by 25 -- 35 \%. This result is a composition of the intrinsic violation of HSE, the deprojection effect and performance limitations of observational instruments. The purpose of this paper is to develop a correction method of the hydrostatic mass to estimate the total mass with high accuracy. To do so, we analyze six simulated clusters; one extracted from an AMR simulation and five from an SPH simulation. First we develop a correction method in three dimensional space (no observational biases are considered). Next we discuss the applicability of the method for X-ray observations using mock X-ray observables constructed by the simulation data. We also quantitatively evaluate observational biases in density, temperature and mass estimates.
The rest of this paper is organized as follows. Section 2 explains the simulations we analyze in this paper. The correction method of the HSE mass when there are no observational biases is constructed in Section 3. We evaluate in Section 4 the validity of the method in X-ray observations by making X-ray observables from the simulation data and estimating the density and temperature from them. In Appendix, observational biases are quantitatively evaluated.
\section{SIMULATIONS}
We use in this paper clusters extracted from cosmological simulations. We analyze one cluster from an adaptive mesh refinement (AMR) hydrodynamical simulation performed by \cite{cen12} and 5 clusters from a smoothed particle hydrodynamical (SPH) simulation
performed by \cite{dolag09}.
The AMR simulation in this study has a higher resolution than the SPH simulation at the cost of the number of realization: there is only a single cluster in the simulation while the SPH simulation contains many structures (a list of all the clusters is found in \cite{dolag09}). Another major difference between the two simulations is that the AMR simulation does not include the AGN feedback while the SPH simulation does. The consistency check between the two simulation methods is an important task in the area of cosmological simulations although we put less emphasis on it in this paper. As it turns out in the following sections, the cluster from the AMR simulation exhibits no significantly different behavior from the ones from the SPH simulation at least in the quantities at the radial range we focus on.
Before presenting results of our analysis, we briefly summarize the main features of
these two simulations in the following.
\subsection{AMR Hydrodynamical Simulation}
The AMR simulation in this study uses an Eulerian adaptive mesh refinement code, Enzo \citep{bryan99, bryannorman99, oshea04, joung09}. We refer readers to \cite{cen12} for more detail. First the simulation was carried out with a low resolution in a periodic box of 120 $h^{-1}$ Mpc on a side. A region centered on a cluster with a mass of $\sim 2\times 10^{14}h^{-1}M_\odot$ was selected and resimulated with a higher resolution. The size of the refined region is $21 \times 24 \times 20(h^{-1}$ Mpc)$^3$, where the mean interparticle separation and mass of dark matter particles are 117 $h^{-1}$ kpc and $1.07 \times 10^8 h^{-1} M_\odot$, respectively.
The simulation includes star formation, radiative cooling and feedback processes. Stellar particles are created based on \cite{cen92}. Their mass is $\sim 10^6 M_\odot$. The cooling and feedback processes include a metagalactic UV background \citep{haardt96}, shielding of UV radiation by neutral hydrogen \citep{cen05}, metallicity-dependent radiative cooling \citep{cen95} and a supernova feedback model\cite{cen05}. The feedback due to AGN is not considered, which may cause higher abundance of baryons than the reality in the innermost region of the cluster. The cosmological parameters in this simulation are $(\Omega_b, \Omega_m, \Omega_\Lambda, h, n_s, \sigma_8)=$(0.046, 0.28, 0.72, 0.70, 0.96, 0.82), following the WMAP7-normalized $\Lambda$CDM model \citep{komatsu11}.
\subsection{SPH Simulation}
The SPH simulations used in this study were performed by K. Dolag using the TreePM/SPH
code GADGET-2 \citep{springel01, springel05}. We refer readers to \cite{dolag06, dolag09} for more detail. First, 10 regions containing massive halos were extracted from a lower resolution DM-only simulation performed by \cite{yoshida01}. Then these regions were resimulated with higher resolution including baryon physics. The simulations include radiative cooling, heating due to a uniform redshift-dependent UV background according to \cite{haardt96}. Star formation and feedback processes are also included based on \cite{springel03}. The cosmological parameters in the simulations are based on a flat $\Lambda$CDM model with $(\Omega_m, h, f_{\rm bar}, \sigma_8)=$(0.3, 0.7, 0.13, 0.9).
In this paper, we use 5 regions named g1, g72, g914, g3344 and g1542 and analysis the most massive structure (labeled ``a'' after the name of the region) in each region. For the analysis in the following sections, we use a cubic region of 2 $r_{500}$\footnote{The radius $r_\Delta$ is defined so that the mean density inside $r_\Delta$ is $\Delta$ times the critical density of the universe.} on a side centered on the center of gravity of the cluster. We reform the data of particles in the region into 256$^3$ mesh data of gas density, temperature, velocity and acceleration for each cluster.
\begin{table}
\begin{center}
\begin{tabular}{cccccccccc}
\toprule
& $r_{2500}$& $M_{2500}$& $T_{2500}$& $r_{500}$& $M_{500}$& $T_{500}$& $r_{200}$& $M_{200}$& $T_{200}$\\
& [kpc] & [$10^{14}M_\odot$] & [keV] & [kpc] & [$10^{14}M_\odot$] & [keV] & [kpc] & [$10^{14}M_\odot] $& [keV]\\
\midrule
g1a & 534& 6.29& 9.53& 1206& 14.6& 5.74& 2026& 19.8& 3.38\\
g72a & 496& 5.05& 6.89& 1111& 11.4& 4.88& 1704& 15.6& 3.66\\
g914a & 304& 0.71& 1.54& 505& 1.07& 1.18& 755& 1.43& 0.88\\
g3344a & 304& 0.72& 1.53& 510& 11.1& 1.16& 749& 1.40& 0.87\\
g1542a & 300& 0.65& 1.43& 485& 0.95& 1.10& 728& 1.28& 0.79\\
AMR & 321& 1.07& 2.35& 643& 2.20& 1.54& 971& 3.04& 1.03\\
\bottomrule
\end{tabular}
\end{center}
\caption{The list of simulated clusters in this study and ther properties. The mass (in $10^{14} M_\odot$)and temperature (in keV) at $r_{2500}$, $r_{500}$ and $r_{200}$ (in $h^{-1}$ kpc) are shown.}
\label{t1}
\end{table}
We show in Figures \ref{fig1} and \ref{fig2} two-dimensional maps of the X-ray surface brightness and spectroscopic-like temperature (defined in section 4).
\section{HYDROSTATIC MASS CORRECTION}
In this section, we construct a correction method for the hydrostatic mass. We hereafter use $\bs{x}=(r,\theta,\varphi)$ to denote the spherical coordinates in the three-dimensional space, while $\bs{X}=(R,\Theta)$ denotes the circular coordinates in the two-dimensional space. For a three-dimensional quantity $a(\bs{x})$, its radial profile is written as $\ol{a}(r)$, while the radial profile of a two-dimensional quantity $A(\bs{X})$ is written as $\ol{A}(R)$
Throughout this paper, when we make a radial profile of a three-dimensional quantity, we follow the following procedure. First we divide the radial range of 0.2 $r_{500}$ -- $2\ r_{500}$ into 20 logarithmically equal intervals, and the polar (0 -- $\pi$) and azimuthal angles (0 -- $2\pi$) into 6 and 12 linearly equal intervals, respectively. For each divided element, we average the values of grids within the element. Finally we average the values of 6 $\times$ 12 elements of the same radial bin. Similarly, when we make a radial profile of a two-dimensional quantity, we divide the radial range of 0.2 $r_{500}$ -- $2\ r_{500}$ into 20 logarithmically equal intervals, and the polar angle (0 -- $2\pi$) into 12 linearly equal intervals. We have checked that changing the number of bins has no significant effect on the results of this paper.
To construct a method for correcting the hydrostatic mass, we give careful consideration to the nature of the hydrostatic mass. In the cosmological simulations we use, the gas follows the Euler equation:
\begin{equation}
\frac{D\bs{v}}{Dt}=-\frac{1}{\rhog}\nabla p-\nabla\phi,
\end{equation}
where $\bs{v}$, $\rhog$ and $p$ are the velocity, density and pressure of gas, and $\phi$ is the gravitational potential due to the total density field. Combining this equation with the Poisson equation $\lap\phi=4\pi G\rho_{\rm tot}$ and using the Gauss theorem, one can derive the following:
\begin{equation}
\Mtot=\Mth+\Mnth,
\end{equation}
\begin{equation}
\Mtot=\int d^3x\ \rhot,
\end{equation}
\begin{equation}
\Mth=-\frac{1}{4\pi G}\int d^2\bs{S}\cdot\frac{1}{\rhog}\nabla p
\end{equation}
and
\begin{equation}
\Mnth=-\frac{1}{4\pi G}\int d^2\bs{S}\cdot\frac{D\bs{v}}{Dt}=-\frac{1}{4\pi G}\int d^2\bs{S}\cdot\left[\pd{\bs{v}}{t}+(\bs{v}\cdot\nabla)\bs{v}\right].
\end{equation}
In this paper, we take $d\bs{S}$ to be spherical surfaces. In this case, $\Mtot$ represents the total mass within the sphere. In Paper I, we have shown that the contribution from the convection term to the total mass is relatively small and that the Eulerian acceleration contributes up to 30 \% of the total mass at $r\gtrsim r_{500}$.
The left panel of Figure \ref{fig3} illustrates the integrands of $\Mth$ and $\Mnth$ of a simulated cluster, which shows there is little change in the integrand of $\Mnth$ compared to that of $\Mth$ . In addition, $\Mth$ scales as $r$ while $\Mnth$ scales as $r^2$ . Therefore, if the amplitude of the acceleration remains unchanged, its contribution to the mass estimate is greater at larger radii.
Based on the above facts, we define the correction mass term by $\Mcor=$ constant $\times r^2$ . If we determine the constant for each cluster from observables, we can correct the hydrostatic mass to better estimate the true mass. To this end, we check if the hydrostatic mass has statistical properties. The hydrostatic mass, however, has a bumpy profile which strongly differs from one cluster to another (see the red dotted line of Figure \ref{fig3}). To smooth the profile of $\Mth$, we replace the density and temperature profiles with the following parametric formulae:
\begin{equation}
\wh{n}(r)=n_0\frac{(r/r_c)^{-\alpha/2}}{(1+r^2/r_c^2)^{3\beta/2-\alpha/4}},
\label{nfit}
\end{equation}
\begin{equation}
\wh{T}(r)=T_0\frac{(r/r_t)^{-a}}{(1+(r/r_t)^b)^{c/b}}.
\label{tfit}
\end{equation}
The number of parameters is nine in total. These are simplified versions of the models used in an X-ray observation \citep{vikhlinin06}. Although these profiles are used in the range about 0.1 $r_{500}$ -- $1\ r_{500}$ in \cite{vikhlinin06}, they reproduce the density and temperature profiles of our simulated clusters in the range 0.2 $r_{500}$ -- $2\ r_{500}$ with less than 5 \% accuracy.
We define sets of parameters as $\nu=(n_0, r_c, \alpha, \beta)$, $\tau=(T_0, r_t, a, b, c)$ and $\lambda=(\nu, \tau)$. Since the parameters here are obtained from $\ol{n}$ and $\ol{T}$, we label the parameters $\nu_n$, $\tau_T$ and $\lambda_{n,T}$ to distinguish them from the ones in the following sections.
Using the above parametric profiles, one can construct the spherically symmetric hydrostatic mass:
\begin{equation}
\hMth(\lambda_{n,T};r)=-\frac{rk_B\wh{T}(\tau_T)}{\mu m_pG}\left[\frac{d\ln\wh{n}(\nu_n)}{d\ln r}+\frac{d\ln\wh{T}(\tau_T)}{d\ln r}\right],
\end{equation}
where $k_B$ , $\mu$ and $m_p$ are the Boltzmann constant, the mean molecular weight and the proton mass, respectively. The blue line in Figure \ref{fig3} demonstrates that $\hMth$ has no bumps due to local pressure gradient so that $\hMth$ is more suitable for a statistical discussion than $\Mth$ . In addition, as long as density and temperature profiles like Eqs. (\ref{nfit}) and (\ref{tfit}) are assumed in observations, the resulting hydrostatic mass is $\hMth$ , not $\Mth$ even if any other observational bias could be removed. Hence we use $\Mtot-\hMth$ , not $\Mtot-\Mth$.
As discussed before, the main cause of violation of hydrostatic equilibrium is the acceleration of gas especially at large radii. The acceleration is correlated with the total gravitation which is correlated with temperature in the process of virialization. Therefore it is expected that $\Mtot-\hMth$ is correlated with temperature at large radii. In fact, Figure \ref{fig4} shows the correlation between the parametrized temperature $\wh{T}$ and $\Mtot-\hMth$ at $r_{200}$, $r_{500}$ and $r_{2500}$. Note that $\Mtot-\hMth$ of the cluster g1a at $r_{2500}$ and of the AMR cluster at $r_{500}$ is negative and not plotted in the figure.
At $r_{200}$, $\Mtot-\hMth$ is positive for all the clusters since the effect of acceleration becomes larger at large radii. The correlation $r_{200}$ is approximated by the following line:
\begin{equation}
\log_{10}(\Mtot(r_{200})-\hMth(r_{200}))=1.71\log_{10}\wh{T}(r_{200})+13.5.
\label{cor}
\end{equation}
The number of simulated clusters is six so the above formula is statistically weak. Its physical basis is, however, solid as stated above.
From the above results, we redefine the correction mass term $\Mcor$ by
\begin{equation}
\Mcor=\left(\frac{\wh{T}_{200}}{\text{keV}}\right)^{1.71}\left(\frac{r}{r_{200}}\right)^2\times10^{13.5}\ M_\odot.
\label{mcor}
\end{equation}
Figure \ref{fig5} shows that $\hMth+\Mcor$ reproduces $\Mtot$ within 10 \% at $r\gtrsim r_{500}$ for all clusters. We are not so serious about the discrepancy between $\Mtot$ and $\hMth+\Mcor$ at small radii because cosmological applications mostly require the mass at $r_{500}$ or the virial radius. Also, physical properties in the central region of simulated clusters often exhibit different behavior from reality probably due to an inadequate understanding of high energy baryon processes. Hence there is little point in discussing the discrepancy at small radii of the simulated clusters. Note that, at $r_{500}$, $\hMth$ alone reproduces $\Mtot$ so the correction is not needed.
\section{APPLICATION TO X-RAY OBSSERVATIONS}
We have constructed the method for correcting the hydrostatic mass in the previous section. In this section, we test the applicability of the method to X-ray observations. In X-ray observations, the gas density and temperature are estimated from two-dimensional observables. The deprojection process can introduce additional biases on the density and temperature estimate. If the density and temperature are misestimated, the difference between the total and the hydrostatic mass can be larger than the intrinsic one. In addition, the correction mass is also affected by the temperature misestimate and the correction may not work.
To evaluate the applicability of the correction method, we make two-dimensional observables using the simulation data and estimate the gas density and temperature by deprojecting the observables. In order to focus on the effect of deprojection, we do not consider the performance of any specific observational instruments.
In X-ray observations, the primary observables are X-ray surface brightness $I_X$ and spectroscopic temperature $T_{\rm spec}$. The surface brightness is defined by the integral of energy loss rate along a line of sight:
\begin{equation}
I_X(\bs{X})=\frac{1}{4\pi(1+z)^4}\int dl\ n^2(\bs{x})\Lambda(T(\bs{x})).
\end{equation}
In our analysis, the line of sight is set to the $z$-axis of the simulation box. The emissivity $\Lambda(T)$ is calculated using the spectral fitting software {\tt SPEX}\footnote{http://www.sron.nl/spex} in the X-ray energy band (0.5 – 10 keV) under the assumption of the collisional ionization equilibrium. The redshift is arbitrarily set to 0.05, where 100 kpc roughly corresponds to 2 arcmin. Note that the value of redshift affects only the amplitude of $\Ix$ which is irrelevant to the conclusion of this paper.
Instead of $T_{\rm spec}$, for simplicity, we use the spectroscopic-like temperature $\Tsl$ introduced by \cite{mazzotta04}:
\begin{equation}
\Tsl(\bs{X})=\frac{\int dl\ n^2(\bs{x})T^{1/4}(\bs{x})}{\int dl\ n^2(\bs{x})T^{-3/4}(\bs{x})}.
\end{equation}
\cite{mazzotta04} have shown that $\Tsl$ reproduces $T_{\rm spec}$ within a few percent for temperatures higher than a few keV for simulated clusters.
The deprojection process is as follows. First we make the two-dimensional data of $\Ix$ and $\Tsl$ and calculate their radial profiles. We assume that the gas density and temperature are of the form (\ref{nfit}) and (\ref{tfit}), respectively, and determine such parameters that best reproduce the radial profiles $\lIx$ and $\lTsl$ by fitting.
When fitting, we define the error of the value of $\lIx$ and $\lTsl$ as the variance in each annulus. The parameters here are labeled as $\nu_{\Ix,\Tsl}$, $\tau_{\Ix,\Tsl}$ and $\lambda_{\Ix,\Tsl}$.
Figure \ref{fig6} demonstrates the deprojection process for the cluster g1a. The density is overestimated and the temperature is underestimated especially at large radii. In Appendix A, we show that the misestimates stem mainly from the assumption of the spherical symmetry of the density and temperature profiles.
\begin{equation}
\hMth(\lambda_{\Ix,\Tsl};r)=-\frac{rk_B\wh{T}(\tau_{\Ix,\Tsl})}{\mu m_pG}\left[\frac{d\ln\wh{n}(\nu_{\Ix,\Tsl})}{d\ln r}+\frac{d\ln\wh{T}(\tau_{\Ix,\Tsl})}{d\ln r}\right],
\end{equation}
Using these profiles, the hydrostatic mass is calculated. Since the density is overestimated at large radii, its gradient is underestimated. Coupled with the underestimate of temperature, the hydrostatic mass is underestimated especially at large radii. The dotted lines of Figure \ref{fig7} shows the hydrostatic mass underestimates the total mass by up to 30 -- 40 \% at $r\gtrsim r_{500}$.
The correction mass is obtained by Eq. (\ref{mcor}) with the deprojected temperature $\wh{T}$ at $r_{200}$. Since the temperature is underestimated, the correction mass is also underestimated. As a result, the correction is inadequate compared to that in Section 3. However, the underestimate of mass is alleviated (up to 20 \%). Figure \ref{fig7} also shows the total mass can be estimated within 10 \% at around $r_{500}$, .
\section{CONCLUSION}
We have constructed the correction method of the hydrostatic mass to better estimate the total mass of galaxy clusters by analyzing six clusters from the cosmological simulations. We showed that the difference between the total mass and hydrostatic mass (violation of hydrostatic equilibrium) is correlated with the temperature of gas. Using this correlation, we defined the correction mass term (Eq. (\ref{mcor})) and showed that $\Mth+\Mcor$ can reproduce the total mass within 10 \% when no observational biases exist.
We also investigated the biases in the mass estimate which stem from the deprojection of the two-dimensional observables (Appendix A). Due to the assumption of spherical symmetry of the density and temperature, the density is overestimated by $\sim$ 40 \% and the temperature is underestimated by $\sim$ 20 \% at around $r_{200}$. These lead to the underestimate of the mass by $\sim$ 40 \% at around $r_{200}$ (the hydrostatic mass originally underestimates the total mass by $\sim$ 30 \% there).
The underestimate of the temperature also leads the underestimate of the correction term. As a result, the hydrostatic mass plus the correction term reproduce the total mass within $\sim$ 20 \% at around $r_{200}$ where the hydrostatic mass underestimates the total mass by up to 40 \% without the correction. At around $r_{500}$, the total mass can be reproduced within 10 \% by using the correction term. Note that, however, there is another bias in determination of $r_{200}$ (Appendix B).
The correlation (\ref{cor}) is constructed from six simulated clusters. Hence its statistical significance is not strong, although there is a physical reason for it. There may be a cluster which does not follow the correlation (\ref{cor}) especially if it is unrelaxed. So the validity of the correlation (\ref{cor}) should be confirmed with more simulated clusters of various dynamical states.
To better estimate the total mass, the density and temperature should be estimated accurately. Especially, the misestimate of temperature directly leads to the misestimate of mass since $\Mth\sim T\times$ gradients of $n$ and $T$. The temperature misestimate is mainly due to the assumption of spherical symmetry of temperature (Appendix A), so the inhomogeneity of the observables may be able to be used to correct the misestimate. The variance of $\Tsl$ is, however, of order $10^{-2}$ and it is difficult to extract the information on the inhomogeneity of $T$. (The variance of $\Ix$ is of order 10, but stems mainly from the inhomogeneity of $n$.)
We again stress that the effect of acceleration (violation of hydrostatic equilibrium) is inevitable at large radii. The mass at large radii should be carefully estimated while recent observations with Suzaku and Chandra have extended their observable region to around the virial radius. In this paper, the spectroscopic-like temperature is used instead of the spectroscopic temperature. Also, noises due to observational instruments is neglected. The observational biases may need an evaluation specialized for each observational instrument, which will be discussed elsewhere.
\section*{ACKNOWLEDGMENTS}
Y.S. gratefully acknowledges support from the Global Scholars Program of Princeton University. This work is supported in part by the Grants-in-Aid for Scientific Research by the Japan Society for the Promotion of Science (JSPS) (21740139, 20340041, 24340035). Computing resources were in part provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center. The work of R.C. is supported in part by grants NNX11AI23G.
\appendix
\section{BIASES IN THE DEPROJECTION PROCESSES}
In section 4, the density estimated from the surface brightness and spectroscopic-like temperature is overestimated, while the temperature is underestimated especially at large radii. In this appendix, we identify the cause of these misestimates.
\subsection{Effect of the Simultaneous Fitting}
Since we ignore in this paper noises of any observational instruments, the remaining possible causes are the simultaneous fitting of the density and temperature, and the spherical assumption for the density and temperature profiles. To evaluate the first possibility, we fit one of the density and temperature with the other assumed to be known.
First we assume that the temperature is known. In this case we can use the emission measure $EM$ defined by
\begin{equation}
EM(\bs{X})=\int dl\ n^2(\bs{x})
\end{equation}
instead of $\Ix$. We calculate $EM$ from the simulation data and determine such parameters $\nu$ for the density profiles that best reproduce $\ol{EM}$. We label the set of parameters as $\nu_{EM}$.
In order to fit the temperature independently, we calculate the spectroscopic-like temperature with the density set to $\wh{n}(\nu_n)$ (obtained in section 2):
\begin{equation}
\Tsl^*(\bs{X})=\frac{\int dl\ \wh{n}^2(\nu_n)T^{1/4}(\bs{x})}{\int dl\ \wh{n}^2(\nu_n)T^{-3/4}(\bs{x})}.
\end{equation}
The asterisk is written in distinction from $\Tsl$ of the original definition. Similar to the other quantities, we determine such parameters $\tau$ for the density profiles that best reproduce $\ol{\Tsl^*}$. We label the set of parameters as $\tau_{\Tsl^*}$.
Using the parameters $\lambda_{EM,\Tsl^*}=(\nu_{EM},\tau_{\Tsl^*})$, we calculate the hydrostatic mass:
\begin{equation}
\hMth(\lambda_{EM,\Tsl^*};r)=-\frac{rk_B\wh{T}(\tau_{\Tsl^*})}{\mu m_pG}\left[\frac{d\ln\wh{n}(\nu_{EM})}{d\ln r}+\frac{d\ln\wh{T}(\tau_{\Tsl^*})}{d\ln r}\right].
\end{equation}
We compare $\hMth(\lambda_{EM,\Tsl^*})$ with $\hMth(\lambda_{\Ix,\Tsl})$. The difference between the two represents the effect of the simultaneous fitting of the density and temperature. Figure \ref{fig8} shows that $\hMth(\lambda_{EM,\Tsl^*})$ is close to $\hMth(\lambda_{\Ix,\Tsl})$ and the effect of the simultaneous fitting is small.
\subsection{Effect of the Spherical Assumption}
We next evaluate the effect of the assumption of the spherical symmetry of the radial profiles of the density and temperature. To this end, we define the following quantities:
\begin{equation}
\hIx(\lambda_{n,T};R)=\frac{1}{4\pi(1+z)^4}\int dl\ \wh{n}^2(\nu_n)\Lambda(\wh{T}(\tau_T))
\end{equation}
and
\begin{equation}
\hTsl(\lambda_{n,T};R)=\frac{\int dl\ \wh{n}^2(\nu_n)\wh{T}^{1/4}(\tau_T)}{\int dl\ \wh{n}^2(\nu_n)\wh{T}^{-3/4}(\tau_T)}.
\end{equation}
These are the observables when the three-dimensional density field and temperature field are spherically symmetric. If we fit the density and temperature to $\hIx(\lambda_{n,T})$ and $\hTsl(\lambda_{n,T})$, the obtained density and temperature profiles reproduce $\wh{n}(\nu_n)$ and $\wh{T}(\tau_T)$ which resemble the true density and temperature profiles $\ol{n}$ and $\ol{T}$.
The left panel of Figure \ref{fig9} shows that $\hIx$ is smaller than $\lIx$. The density profile which best reproduces the true one is obtained from $\hIx$, but the observable is $\lIx$, not $\hIx$. Therefore, the density profile estimated from $\lIx$ is larger than the true one, as shown in the right panel of Figure \ref{fig9}. The difference between $\lIx$ and $\hIx$ is mainly due to the fact that the observables depend on density in the form of $n^2$.
On the other hand, the $\hTsl$ is higher than $\lTsl$ because $\Tsl$ is weighted by $n^2T^{-3/4}$ and more affected by lower temperatures (see Figure \ref{fig10}). Hence the temperature estimated from $\lTsl$ underestimates the true one.
\section{BIAS IN DETERMINATION OF $r_\Delta$}
The correction mass $\Mcor$ (Eq. (\ref{mcor})) is calculated with reference to $r_{200}$, but $r_{200}$ must be observationally estimated. The correction does not work if $r_{200}$ is misestimated. In this appendix, we test the validity of estimates of $r_{500}$ and $r_{200}$.
In X-ray observations, $r_\Delta$ of a cluster is usually determined by using an existing $r$-$T$ relation or its hydrostatic mass profile (unless it is obtained by gravitational lensing observations). To test the first method, we use the following relations constructed by \cite{arnaud05}:
\begin{equation}
h(z)r_{500}=1104\left(\frac{T_X}{5\ \text{keV}}\right)\ \text{kpc},
\end{equation}
and
\begin{equation}
h(z)r_{200}=1674\left(\frac{T_X}{5\ \text{keV}}\right)\ \text{kpc},
\end{equation}
where $T_X$ is the spectroscopic temperature of the $0.1\ r_{200}$ -- $0.5\ r_{200}$ region. We calculate the average spectroscopic-like temperature $\Tsl$ in the same region and substitute the above relations. Figure \ref{fig11} shows that $r_{500}$ and $r_{200}$ are underestimated for the clusters we use. In other words, our clusters do not follow the above $r$-$T$ relations. Since the fiducial radius is underestimated by $\sim$ 20 \%, the correction mass $\Mcor$ is overestimated by $\sim$ 40 \%.
Another way to determine $r_\Delta$ is to assume the hydrostatic mass reproduces the total mass. By comparing the hydrostatic mass profile and $4\pi \rho_cr^3/3$ ($\rho_c$ is the cosmic critical density), $r_\Delta$ can be determined. This method reflects the validity of the hydrostatic equilibrium assumption.
We use the $\hMth(\lambda_{\Ix,\Tsl})$ and determine $r_{500}$ and $r_{200}$. Figure \ref{fig11} shows that $r_500$ is well reproduced while $r_{200}$ is underestimated up to $\sim$ 20 \%. This is because the hydrostatic mass more deviates the total mass at larger radii.
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\end{document}
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig1a.eps}\gap
\includegraphics[width=7cm]{fig1b.eps}\\
\includegraphics[width=7cm]{fig1c.eps}\gap
\includegraphics[width=7cm]{fig1d.eps}\\
\includegraphics[width=7cm]{fig1e.eps}\gap
\includegraphics[width=7cm]{fig1f.eps}
\end{center}
\caption{The surface brightness $\Ix$ of the six simulated clusters.}
\label{fig1}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig2a.eps}\gap
\includegraphics[width=7cm]{fig2b.eps}\\
\includegraphics[width=7cm]{fig2c.eps}\gap
\includegraphics[width=7cm]{fig2d.eps}\\
\includegraphics[width=7cm]{fig2e.eps}\gap
\includegraphics[width=7cm]{fig2f.eps}
\end{center}
\caption{The spectroscopic-like temperature $\Tsl$ of the six simulated clusters. Note that the temperature range of the clusters g1a and g72a is different from that of the others.}
\label{fig2}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig3a.eps}\gap
\includegraphics[width=7cm]{fig3b.eps}
\end{center}
\caption{The left panel shows the integrand of $\Mth$ (red solid) and $\Mnth$ (green dot-dashed) for the cluster g1a. Their sum is also shown in black dashed line for reference. Their corresponding mass term $\Mth$ and $\Mnth$ are shown in the right panel along with the total mass $\Mtot$. Note that the $\Mtot$ is calculated {\it not} by integrating the black line in the left panel, but by summing the density of all the components in the simulation. }
\label{fig3}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig4.eps}
\end{center}
\caption{The correlation between $\Mtot-\hMth$ and $\wh{T}$ at $r_{2500}$ (triangles), $r_{500}$ (circles) and $r_{200}$ (squres). The symbol for each cluster has its own color; g1a (black), g72a (red), g914a (green), g3344a (blue), g1542a (cyan) and AMR (magenta). The line shows $\log{10} (\Mtot(r_{200}-\hMth(r_{200}))=1.71\log_{10}\wh{T}(r_{200})+13.5$.}
\label{fig4}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig5.eps}
\end{center}
\caption{The hydrostatic mass compared to the total mass. The dotted lines represent the hydrostatic mass calculated using the parameters which reproduce the true density and temperature profiles. The solid lines show the hydrostatic mass plus the correction mass. The color of the line differs from cluster to cluster in the same manner as Figure \ref{fig4}.}
\label{fig5}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig6a.eps}\gap
\includegraphics[width=7cm]{fig6b.eps}\\
\includegraphics[width=7cm]{fig6c.eps}\gap
\includegraphics[width=7cm]{fig6d.eps}
\end{center}
\caption{A demonstration of deprojection of the density and temperature from the surface brightness $\Ix$ and spectroscopic-like $\Tsl$ temperature using the cluster g1a. We find the parameters in Eqs. (\ref{nfit}) and (\ref{tfit}) so that they reproduce the radial profiles of $\Ix$ and $\Tsl$ (crosses in the top panels) simultaneously. The best-fit profiles of $\Ix$ and $\Tsl$ are shown in blue in the top panels. Using the best-fit parameters, the density $\wh{n}$ and temperature $\wh{T}$ (blue solid lines in the bottom panels) are calculated and compared with the true profiles (black dashed lines in the bottom panels).}
\label{fig6}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig7.eps}
\end{center}
\caption{Same as Figure \ref{fig5}, but the parameters for $\hMth$ are obtained from the surface brightness $\Ix$ and the spectroscopic-like temperature $\Tsl$.}
\label{fig7}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig8.eps}
\end{center}
\caption{The hydrostatic mass with the parameters obtained from $\Ix$ and $\Tsl$ (blue solid), and $EM$ and $\Tsl^*$ (red dashed). The black dotted line shows the total mass for reference.}
\label{fig8}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig9a.eps}\gap
\includegraphics[width=7cm]{fig9b.eps}
\end{center}
\caption{The left panel shows the surface brightness $\hIx$ (blue solid in the left panel) constructed from the sphericalized density ($\wh{n}(\nu_n)$) and temperature ($\wh{T}(\tau_T)$)fields compared to the radial profile of the original surface brightness $\lIx$ (black dashed in the left panel). The density estimated from $\hIx$ reproduces the true radial profile of the density $\ol{n}$ (black dashed in the right panel), while the density $\wh{n}$ (blue solid in the right panel) estimated from $\lIx$ overestimates the true density.}
\label{fig9}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig10a.eps}\gap
\includegraphics[width=7cm]{fig10b.eps}
\end{center}
\caption{The left panel shows the spectroscopic-like temperature $\hTsl$ (blue solid in the left panel) constructed from the sphericalized density ($\wh{n}(\nu_n)$) and temperature ($\wh{T}(\tau_T)$)fields compared to the radial profile of the original spectroscopic-like temperature $\lTsl$ (black dashed in the left panel). The temperature estimated from $\hTsl$ reproduces the true radial profile of the temperature $\ol{T}$ (black dashed in the right panel), while the temperature $\wh{T}$ (blue solid in the right panel) estimated from $\lTsl$ underestimates the true temperature.}
\label{fig10}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig11a.eps}\gap
\includegraphics[width=7cm]{fig11b.eps}
\end{center}
\caption{The fiducial radii $r_{500}$ (left) and $r_{200}$ (right) estimated by using $r$-$T$ relation (circles) and by assuming the HSE mass is equal to the total mass (squares).}
\label{fig11}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{test.eps}\gap
\includegraphics[width=7cm]{test1.eps}\gap
\includegraphics[width=7cm]{test2.eps}
\end{center}
\end{figure}
\end{document}