A GOOD MASS PROXY FOR GALAXY CLUSTERS WITH XMM-NEWTON

The spectral analysis is carried out by using XSPEC version 12.6.0. We fit the de-projected spectra with the absorbed Mekal model:

$$\begin{equation} {\rm Model} = {\rm Wabs}(N_{\rm H})\times {\rm Mekal}(T,z,A,{\rm norm}), \end{equation} \tag{ 2 }$$

in which Wabs is a photoelectric absorption model (Morrison & McCammon 1983), Mekal is a plasma emission model (Mewe et al. 1985; Kaastra 1992). The temperature T, metallicity A, and norm (emission measure) are free parameters. The redshift z are fixed as in Table 1. For the majority of observations, the absorption N_H are fixed parameters, in a few case (i.e., A478, EXO 0422, Hercules), N_H are not unique, we left them as a free parameter. After obtaining the temperature of each shell, we can fit the radial de-projected temperature profile by the following equation (Xue et al. 2004):

$$\begin{equation} {T(\rm r)} = {T_{\rm 0}}+\frac{A}{r/r_{\rm 0}}\exp \left(-\frac{(\ln r-\ln r_{\rm 0})^{2}}{\omega }\right), \end{equation} \tag{ 3 }$$

where T₀, A, r₀, and ω are free parameters.

For calculating the de-projected electron density profile, we divide the cluster into several annular regions (>14, depending on count rate of the cluster) centered on the emission peak. Then we use the de-projecting technique (see Jia et al. 2004, 2006 for detailed calculation) to calculate the photon counts in each shell. Since the de-projected temperature and abundance profiles are known, we can estimate the normalization "norm(i)" and its error in each shell. Then we can derive the de-projected electron density n_e of each region from Equation (4).

$$\begin{equation} {\rm norm}(i) = \frac{10^{-14}}{4\pi [D(1+z)]^{2}}\int n_{\rm e}n_{\rm H}dV, \end{equation} \tag{ 4 }$$

where D is the angular size distance to the source in cm. To obtain an acceptable fit for all clusters in this sample, we adopt a double-β model to fit the electron density profile (Chen et al. 2003):

$$\begin{equation} n_{\rm e}(r) = n_{\rm 01}\left[1+\left(\frac{r}{r_{c1}}\right)^{2}\right]^{-\frac{3}{2}\beta _{1}}+ n_{\rm 02}\left[1+\left(\frac{r}{r_{c2}}\right)^{2}\right]^{-\frac{3}{2}\beta _{2}}, \end{equation} \tag{ 5 }$$

where n₀₁ and n₀₂ are electron number density parameters, β₁ and β₂ are the slope parameters, and r_c1 and r_c2 are the core radii of the inner and outer components, respectively.

The primary parameters of all 39 galaxy clusters are given in Table 1.

Once we have obtained the de-projected radial profiles of electron density n_e(r) and temperature T(r), together with the assumptions of hydrostatic equilibrium and spherical symmetry, the gravitational mass of cluster within radius r can be determined as (Fabricant et al. 1980):

$$\begin{equation} M(< r) = -\frac{k_{\rm B}Tr^{2}}{G\mu m_{\rm p}}\left[\frac{d(\ln n_{\rm e})}{dr} + \frac{d(\ln T)}{dr}\right], \end{equation} \tag{ 6 }$$

where the mean molecular weight μ is assumed to 0.62. k_B, G, and m_p are the Boltzmann constant, the gravitational constant, and the proton mass, respectively. The mass of hot gas is calculated as

$$\begin{equation} M_{\rm g}(< r) = 4 \pi \mu _{\rm e} m_{\rm p}\int n_{\rm e}(r)r^{2}dr, \end{equation} \tag{ 7 }$$

where μ_e is the mean molecular weight of the electrons. We calculate the total mass and gas mass within r₅₀₀, in which the mean gravitational mass density is equal to 500 times the critical density at the cluster redshift. The total masses, M₅₀₀, and gas masses, M_g, within r₅₀₀ for all the clusters are listed in Table 2.

The cooling time t_cool is a timescale during which the hot gas loses all of its thermal energy. We calculate the cooling time of the gas as

$$\begin{equation} t_{{\rm cool}}=\frac{5}{2}\frac{n_{e}+n_{i}}{n_{e}}\frac{kT}{n_{{\rm H}}\Lambda (A,T)}, \end{equation} \tag{ 8 }$$

where Λ(A, T) is the cooling function of the gas and we fix the metallicity at A = 0.3 Z_☉. n_e, n_H, and n_i are the number densities of the electrons, hydrogen, and ions, respectively. For the almost fully ionized plasma in clusters, n_e = 1.2n_H and n_i = 1.1n_H. The central cooling time t_c is derived from the central electron density n_e0 and the central temperature T₀. n_e0 is the electron density at r = 0.004r₅₀₀ (Hudson et al. 2010) and T₀ is the average temperature within 0–0.05r₅₀₀, where 0.05r₅₀₀ is the maximum value of the innermost annulus for the whole sample.

We derive the global temperature, $T_{(0.2\hbox{--}0.5)r_{\rm 500}}$, by the volume average of the radial temperature profile limited to the radial range of (0.2–0.5)r₅₀₀, as listed in Table 1. The temperatures within 0.2r₅₀₀ tend to show peculiarities linked to the cluster dynamical history, which are mainly affected by the cool cores (Smith et al. 2005). The upper boundary of 0.5r₅₀₀ is limited by the quality of the spectral data. Thus, using temperature within (0.2–0.5)r₅₀₀, we can minimize the scatter in the X-ray scaling relations and reach a better agreement between the X-ray scaling relations for the CCCs and NCCCs. Figure 3 shows the temperature profiles for the CCC sub-sample and the NCCC sub-sample, respectively. There is a special cluster (Hercules), which has a rapid fluctuation in the temperature profile for the CCC sub-sample, and the unique distribution is also confirmed by the Chandra data (Cavagnolo et al. 2009; C. K. Li et al., in preparation).

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In the following, we investigate the relationships between several parameters. We perform the relations with logarithmic values of the parameters in the form

$$\begin{equation} \log _{10}(Y) = A + B \cdot \log _{10}(X), \end{equation} \tag{ 9 }$$

where X and Y represent the variables, A and B are the two free parameters to be estimated. We firstly create the histogram of residuals from the best fitting relation in the log space, and then, the raw scatter can be obtained by a Gaussian fitting to the histogram. The intrinsic scatter is calculated as (Morandi et al. 2007)

$$\begin{eqnarray} &&S = \left[ \sum _{j} \left(\left(\log _{10} (Y_{j}) -A -B \log _{10} (X_{j}) \right)^2\right.\right.\nonumber \\ &&\left.\left. -\epsilon _{\log _{10} (Y_{j})}^2\right) / (N-2) \right]^{1/2}, \end{eqnarray} \tag{ 10 }$$

where $\epsilon _{\log _{10} (Y_j)} = \epsilon _{Y_{j}}/(Y_j \ln 10)$, with $\epsilon _{Y_j}$ being the statistical error of the measurement Y_j, and N is the total number of data.

It is well known that different regression methods may give different slopes even at the same population level (Akritas & Bershady 1996). Therefore, it is important to choose the most suitable method for different data. In M₅₀₀–T relation, we compare four kinds of BCES regression methods (BCES(Y|X), BCES(X|Y), BCES Bisector, BCES Orthogonal), all of which take into account both intrinsic scatter and the presence of errors on both variables (Akritas & Bershady 1996). The best power-law fits given by these four methods are shown in Table 3. The slopes from these four methods are consistent within 1σ errors. In order to compare with previous results, we use the BCES Bisector method. The results of power-law fits in the BCES Bisector method for all the relations discussed in Section 6 are summarized in Table 4.

Table 4. Summary of the Fits to the Mass Scaling Relations

Y	X	Number of Clusters	B	A	Comments
$\frac{M_{\rm 500}}{M_{\odot }} \; E(z)$	$\frac{T}{\rm keV}$	39	1.94 ± 0.17	13.12 ± 0.10	ALL
		25	1.85 ± 0.22	13.17 ± 0.12	CCCs
		14	2.03 ± 0.26	13.08 ± 0.15	NCCCs
$\frac{M_{\rm 500}}{M_{\odot }}\;E(z)$	$\frac{M_{\rm g}}{M_{\odot }} \; E(z)$	39	0.79 ± 0.07	3.67 ± 0.95	ALL
		25	0.72 ± 0.09	4.66 ± 1.17	CCCs
		14	0.89 ± 0.11	2.36 ± 1.41	NCCCs
$\frac{L_{\rm bol}}{\rm erg\;s^{-1}}\;E(z)^{-1}$	$\frac{M_{\rm 500}}{M_{\odot }} \; E(z)$	39	1.73 ± 0.16	19.68 ± 2.26	ALL
		25	2.03 ± 0.22	15.35 ± 3.19	CCCS
		14	1.38 ± 0.17	24.58 ± 2.33	NCCCs
$\frac{M_{\rm 500}}{M_{\odot }}\;E(z)$	$\frac{Y_{\rm X}}{M_{ \odot }\;{\rm keV}} \; E(z)$	39	0.57 ± 0.04	6.29 ± 0.58	ALL
		25	0.52 ± 0.06	5.47 ± 0.80	CCCs
		14	0.63 ± 0.06	6.95 ± 0.76	NCCCs

Notes. The relations are given in the form: log₁₀(Y) = A + B · log₁₀(X). For each relation, we list the normalization parameter A and the slop B. In addition to the results for all the sample, we also list the results both for the CCC sub-sample and the NCCC sub-sample.

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Figure 4 shows the M₅₀₀–T relation for our sample. Considering the whole sample, our best-fit slope 1.94 ± 0.17 is higher than the self-similar predictive value, 1.5. The slope is consistent with the result in Mantz et al. (2010), 2.08 ± 0.08 for a 238 galaxy cluster sample observed by Chandra or ROSAT. Reichert et al. (2011) derived a slope of 1.76  ±  0.08 from the 14 literature samples. Our slope is also consistent with the result of Sanderson et al. (2003), 1.84  ±  0.01 for the M₂₀₀–T relation using ASCA temperature profiles. Previous studies have suggested that the slopes of the M₂₀₀–T relation for the high mass and low mass parts may be different (Finoguenov et al. 2001; Dos Santos & Dorë 2002). The cross-over temperature between the two parts is typically ∼3.0 keV (Finoguenov et al. 2001). Sanderson et al. (2003) investigated a sample, which included 66 clusters with a broad range of temperature (0.5–15 keV), and found no obvious break in the M₂₀₀–T relation at ∼3.0 keV. We also obtain a similar slope of 1.99  ±  0.30 for a sub-sample with T > 3.0 keV from our sample. Thus, our slope is steeper than the self-similar prediction, and it does not vary with the temperature (or mass) range of the sample.

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Fixing the slope to 2.08, we find that the normalization of our relation is lower by ∼28% than that in Mantz et al. (2010). With the slope fixed to 1.76, the normalization for our sample is lower than that in Reichert et al. (2011) by ∼27%.

Moreover, the slopes and the normalizations of the M₅₀₀–T relation for the CCCs and NCCCs are consistent within errors as listed in Table 4. This indicates that no evident influence of cool core is found on this relation. This agrees with the result obtained by Chen et al. (2007), who considered a 88 cluster sample based on ASCA and ROSAT observations.

The right panel of Figure 4 shows a histogram of the log space residuals from the best fitting M₅₀₀–T relation for the whole sample. The residuals are corresponding to the value of vertical distances to the best fitting line (Pratt et al. 2009). The raw scatter of the M₅₀₀–T relation is well described by a Gaussian fit in log space with σ_M(T) = 0.18 dex. The intrinsic scatter in M₅₀₀ around the M₅₀₀–T relation is 0.14 dex.

Figure 5 shows the M₅₀₀–M_g relation for our sample and the best-fit gives a slope of 0.79 ± 0.07. The slope is in agreement with 0.81 ± 0.07 derived from Chandra data (Kravtsov et al. 2006; Nagai et al. 2007) and in marginal agreement with 0.91 ± 0.08 presented by Zhang et al. (2008). The slope is shallower than the self-similar prediction ($M_{\rm 500}\propto {f_{\rm g}^{-1}}M_{\rm g}$). This may be due to the trend of observed f_g with the total mass. Giodini et al. (2009) investigated a sample of 41 clusters, which spanned the total mass range 1.5 × 10¹³ M_☉–1.1 × 10¹⁵ M_☉, and found f_g∝M^0.21. Taking this trend into account, the slope of the M₅₀₀–M_g relation is about 0.79, which is in good agreement with our result. The normalization of our M₅₀₀–M_g relation is lower by ∼24% using a fixed slope of 0.91, and higher by 9% with the slope fixed to 0.81 for the Chandra data. The slopes of the M₅₀₀–M_g for the CCCs and NCCCs are consistent within errors. The influence of cool core is not found on this relation.

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The right panel in Figure 5 shows a histogram of the log space residuals from the M₅₀₀–M_g relation for the whole sample. This raw scatter is well described by a Gaussian fit in log space with $\sigma _{ M(M_{g})}$ = 0.14 dex. The intrinsic scatter in M₅₀₀ around the M₅₀₀–M_g relation is 0.14 dex.

The L_bol–M₅₀₀ relation is very important for the application to cosmological cluster surveys. Figure 6 shows the L_bol–M₅₀₀ relation for our sample and the best fit gives a slope of 1.73 ± 0.16. Our slope is higher than the self-similar expected value, 1.33, and agrees with the results obtained by Reiprich & Böhringer (2002) (1.80 ± 0.08) and Ettori et al. (2004) (1.88 ± 0.42), in both of which the same core-uncorrected L_bol are used as ours. Our slope is lower than the core-corrected result of Zhang et al. (2008, 2.36 ± 0.07). The slope of our relation is consistent with the result of Morandi et al. (2007, 2.00 ± 0.28), in which the L_bol was obtained by excluding the r < 100 kpc region.

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The slope for the NCCCs (1.38 ± 0.17) is consistent with the self-similar expected value, while the slope (2.03 ± 0.22) for the CCCs is higher. There is a significant normalization discrepancy between CCCs and NCCCs. These may be due to the high luminosity of the cool core in CCCs.

The right panel in Figure 6 shows a histogram of the log space residuals from the L_bol–M₅₀₀ relation for the whole sample. This raw scatter in M₅₀₀ is well described by a Gaussian fit in log space with $\sigma _{M(L_{{\rm bol}})}$ = 0.27 dex. The intrinsic scatter in M₅₀₀ around the L_bol–M₅₀₀ relation is 0.15 dex.

We present the M₅₀₀–Y_X relation for this sample in Figure 7, in which the best-fit gives a slope of 0.57 ± 0.04 being consistent with 0.6 expected by the self-similar model. Our slope is consistent with the result of 0.62 ± 0.06 in Zhang et al. (2008), who used the same temperature, $T_{\rm (0.2\hbox{--}0.5)r_{500}}$, as ours. The slope is also in good agreement with the value 0.57 ± 0.03 in Vikhlinin et al. (2009), in which a extended temperature was used in the radial range (0.15–1)r₅₀₀ for 10 relaxed Chandra clusters. Moreover, our result is supported by the simulation (0.57  ±  0.01) (Nagai et al. 2007). Thus, the slope of M₅₀₀–Y_X relation is stable and not sensitive to the temperature definition.

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The right panel in Figure 7 shows a histogram of the log space residuals from the M₅₀₀–Y_X relation for all clusters. This raw scatter in M₅₀₀ is well described by a Gaussian fit in log space with $\sigma _{M(Y_{\rm X})}$ = 0.13, which is the smallest among the relation between the observables and the total mass. We also obtain a small intrinsic scatter of 0.13 dex. Fixing the slope to 0.62, we find that the normalization of our relation is lower than that in Zhang et al. (2008) by ∼19%. We also fix the slope to 0.57, and find that the normalization of our slope is similar with the simulation value by ∼2%.

The slope for the relation of the NCCCs agrees with that for the CCCs within errors as listed in Table 4. Both slopes are close to the self-similar prediction (0.6). Taking the errors into account, the normalization for the NCCCs is also consistent with that for the CCCs.

In this section, we compare the four mass scaling relations (L_bol–M₅₀₀, M₅₀₀–T, M₅₀₀–M_g, and M₅₀₀–Y_X), to find out which relation is the best to estimate cluster mass. The comparison of these relations are listed in Table 5.

The slope of the L_bol–M₅₀₀ relation (1.73 ± 0.16) is steeper than the self-similar expectation (1.33) significantly. It may be due to that the L_bol is dominated by cluster center and thus particularly susceptible to nongravitational processes, while the self-similarity is roughly preserved in the outer region. For the M₅₀₀–T relation, we perform the fittings for both the hot clusters (T > 3.0 keV) and the whole sample, and find that the slopes are steeper than the self-similar value and do not vary with the cluster temperature range. The slope of the M₅₀₀–M_g relation is shallower than the self-similar prediction, moreover, the gas fraction seems to depend on the M₅₀₀ rather than a constant. The M₅₀₀–Y_X relation agrees with the self-similarity and is consistent with the simulations (Kravtsov et al. 2005; Nagai et al. 2007).

Both intrinsic scatters and raw scatters show that the M₅₀₀–Y_X relation is the tightest one compared with the M₅₀₀–T, M₅₀₀–M_g, and L_bol–M₅₀₀ relations. Numerical simulations also indicate that the Y_X is a low-scatter mass proxy (with only ≈5%–8% intrinsic scatter; Nagai et al. 2007; Kravtsov et al. 2006), which is smaller than any other mass proxies even in the presence of significant dynamical activity.

The L_bol–M₅₀₀ relation has the largest discrepancy in the slopes between CCCs (2.03 ± 0.22) and NCCCs (1.38 ± 0.17), indicating a significant influence of the cool core on this relation. The slope for the NCCCs is consistent with the self-similar expected, while the slope for the CCCs is steeper (Table 3). In contrast, the slopes of the other relations display much smaller differences between CCCs and NCCCs. That indicates the cool core has little influence on the M₅₀₀–T, M₅₀₀–M_g, and M₅₀₀–Y_X relations.

Many large cluster samples have been used to investigate the scaling relations. We compare recent results in Table 6. Both simulations (Nagai et al. 2007; Fabjan et al. 2011) and observations (Arnaud et al. 2007; Vikhlinin et al. 2009; C. K. Li et al., in preparation) show that the relation of M₅₀₀–Y_X is in good agreement with the self-similar expected value, 0.6, and does not change with cluster sample, while the other relations are quite different between samples. Compared to the other relations, the M₅₀₀–Y_X relation is the most insensitive to variations in the physical processes included in the simulation (Poole et al. 2007; Rasia et al. 2011; Fabjan et al. 2011).

Table 6. Recent Results on the Slopes of the Scaling Relations

Relation	Slope	Comments	Reference
Lbol–M500	1.63 ± 0.08	115 clusters, z = 0.1–1.3, core excised Lbol, r < 0.15 r500	Maughan07
L0.1–2.4–M500	1.82 ± 0.13	106 clusters, ROSAT/ASCA	Chen07
Lbol–M500	1.71 ± 0.46	24 clusters, Chandra, core excised Lbol, r < 100 kpc	Morandi07
Lbol–M500	2.33 ± 0.70	37 clusters, XMM, core corrected Lbol, r < 0.2 r500	Zhang08
L0.1–2.4–M500	1.76 ± 0.13	31 clusters, XMM	Arnaud10
Lbol–M500	1.51 ± 0.09	14 literature samples	Reichert11
M500–T	1.54 ± 0.06	106 clusters, ROSAT/ASCA	Chen07
M500–T	1.71 ± 0.09	10 clusters, $T_{0.1\hbox{--}0.5\,r_{200}}$, XMM	Arnaud07
M500–T	1.74 ± 0.09	70 clusters, z = 0.18–1.24	O'Hara07
M500–T	1.65 ± 0.26	$T_{0.2\hbox{--}0.5\,r_{500}}$	Zhang08
M500–T	1.53 ± 0.08	17 clusters, $T_{0.15\hbox{--}1\,r_{500}}$, Chandra	Vikhlinin09
M500–T	2.04 ± 0.04	238 clusters, Chandra/ROSAT	Mantz10
M500–T	1.76 ± 0.08	14 literature samples	Reichert11
M500–Mg	0.80 ± 0.04	10 clusters, XMM	Arnaud07
M500–Mg	0.81 ± 0.07	simulation	Nagai07
M500–Mg	0.91 ± 0.08	37 clusters, XMM	Zhang08
M500–Y	0.57 ± 0.01	simulation	Nagai07
M500–Y	0.59 ± 0.01	simulation	Fabjan11
M500–Y	0.56 ± 0.03	10 clusters, XMM	Arnaud07
M500–Y	0.57 ± 0.03	17 clusters, Chandra	Vikhlinin09
M500–Y	0.62 ± 0.06	37 clusters, XMM	Zhang08

Notes. The references are from top to bottom: Maughan (2007), Chen et al. (2007), Morandi et al. (2007), Zhang et al. (2008), Arnaud et al. (2010), Reichert et al. (2011), Arnaud et al. (2007), O'Hara et al. (2007), Vikhlinin et al. (2009), Mantz et al. (2010), Nagai et al. (2007), Fabjan et al. (2011).

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In summary, the M₅₀₀–Y_X relation is the stablest one among these four scaling relations, which is in agreement with the self-similar model and does not changes with the cluster sample. In addition, this relation has the smallest scatter in M₅₀₀ and does not affected by the cool core. Thus, the parameter Y_X is the best proxy to estimate cluster mass.

For X-ray flux-limited cluster samples, more luminous clusters will be selected from a survey volume and this will induce the Malmquist bias. Many works have pointed out that the observed L_X–M scaling relation may be significantly affected by Malmquist bias if the scatter in luminosity for fixed mass is large (Stanek et al. 2006; Vikhlinin et al. 2009; Pratt et al. 2009). We can give the mean bias in ln L for given mass (Vikhlinin et al. 2009):

$$\begin{eqnarray} {\rm Bias}(\ln L|\ln L_0)&=&\langle \ln L\rangle - \ln L_0 \nonumber \\ &=& \frac{\int _{-\infty }^{\infty }(\ln L - \ln L_0)\,p(\ln L)\,V(\ln L)\,d\ln L}{\int _{-\infty }^{\infty } p(\ln L)\, V(\ln L)\, d\ln L} \nonumber \\ \end{eqnarray} \tag{ 11 }$$

where ln L₀ is the mean L for given mass, and p(ln L) has a log-normal distribution (p(ln L)∝ − (ln L − ln L₀)²/(2σ²)), which characterizes the scatter of L in the L_X–M relation. For the low redshift and flux-limited sample, the evolution on the L_X–M relation can be neglected within the survey's effective redshift depth (Vikhlinin et al. 2009), and the survey volume is a power law function of the object luminosity (V(L)∝L^3/2 in Euclidean space). Thus, Equation (11) can be worked out analytically with Bias(ln L|ln L₀) = 3/2σ². We also perform a Gaussian fitting for the L_bol–M₅₀₀ relation and find σ = 0.65, the bias is 0.63. Thus, the Malmquist bias leads to a 87% overestimation in the normalization of the observed L_bol–M₅₀₀ relation in our sample, which is much smaller than the factor of two bias advocated by Stanek et al. (2006) but larger than the value 26% in Vikhlinin et al. (2009).

The total cluster mass may be affected by some factors. First, the assumption of spherical symmetry is not satisfied in some clusters, thus, the masses of these clusters may be overestimated (or underestimated). To estimate the uncertainty due to spherical symmetry, Landry et al. (2013) found a ±6% systematic uncertainty in the total mass of the most disturbed clusters, A520, at r₅₀₀. All the clusters in our sample are regular and round in projected image, but in case of merging along the line-of-sight, the inaccuracy may be large due to the assumption of spherical symmetry. Second, all the clusters in our sample are nearby, at redshift z < 0.1. The temperature profiles to r₅₀₀ are unavailable for some clusters. In this case, we use the extended temperature profile to calculate the cluster mass, which will introduce some errors. We select an NCCC (A3391) and a CCC (A1650), whose temperatures at r₅₀₀ are available, to test the uncertainties due to the possibility of mis-extrapolating the temperature profile. We fit the temperature profiles without the outmost observed data, and then obtain their cluster masses again. Compared with the previous masses, the biases in the cluster mass due to the mis-extrapolating temperature profiles for A1650 and A3391 are 17% and 4%, respectively. Third, the hydrostatic equilibrium may be destroyed and non-thermal pressure components become more significant at large radius (Nagai et al. 2007; Zhang et al. 2008; Vikhlinin et al. 2009). The subsonic turbulent motions of the ICM gas in relaxed clusters in the Nagai et al. (2007) sample seem to result in a ∼15% underestimate in the hydrostatic estimates of M₅₀₀.