THE ENTIRE VIRIAL RADIUS OF THE FOSSIL CLUSTER RXJ 1159 + 5531. II. DARK MATTER AND BARYON FRACTION

For the ICM entropy profile we employ a power law with two breaks plus a constant,

$$\begin{eqnarray}&&S(r)={s}_{0}+{s}_{1}f(r),\end{eqnarray} \tag{ 3 }$$

where s₀ represents a constant entropy floor, and ${s}_{1}=S({r}_{{\rm{ref}}})-{s}_{0}$ for some reference radius ${r}_{{\rm{ref}}}$ (taken to be 10 kpc). The dimensionless function f (r) is

$$\begin{eqnarray*}&&f(r)=\left\{\begin{array}{l}{\left(\displaystyle \frac{r}{{r}_{{\rm{ref}}}}\right)}^{{\alpha }_{1}}\qquad \qquad r\leqslant {r}_{{\rm{b,1}}}\\ {f}_{1}{\left(\displaystyle \frac{r}{{r}_{{\rm{ref}}}}\right)}^{{\alpha }_{2}}\qquad \qquad {r}_{{\rm{b,1}}}\lt r\leqslant {r}_{{\rm{b,2}}}\\ {f}_{2}{\left(\displaystyle \frac{r}{{r}_{{\rm{ref}}}}\right)}^{{\alpha }_{3}}\qquad \qquad r\gt {r}_{{\rm{b,2}}}\end{array}\right.\ \end{eqnarray*}$$

where ${r}_{{\rm{b,1}}}$ and ${r}_{{\rm{b,2}}}$ are the two break radii, and the coefficients f₁ and f₂ are given by

$$\begin{eqnarray*}&&{f}_{n}={f}_{n-1}{\left(\displaystyle \frac{{r}_{{\rm{b}},n}}{{r}_{{\rm{ref}}}}\right)}^{{\alpha }_{n}-{\alpha }_{n+1}},\end{eqnarray*}$$

with ${f}_{0}\equiv 1.$ To enforce convective stability (Section 3) we require ${\alpha }_{1},{\alpha }_{2},{\alpha }_{3}\geqslant 0.$ Hence, this model has seven free parameters: s₀, s₁, ${r}_{{\rm{b,1}}}$, ${r}_{{\rm{b,2}}}$, ${\alpha }_{1}$, ${\alpha }_{2}$, and ${\alpha }_{3}$.

We represent the stellar mass of the cluster using the K-band light profile of the BCG (2MASX J11595215+5532053) from the Two Micron All-Sky Survey (2MASS) as listed in the Extended Source Catalog (Jarrett et al. 2000); i.e., an n = 4 Sérsic model (i.e., de Vaucouleurs) with ${r}_{e}=9.83\,{\rm{kpc}}$ and ${L}_{K}=1.03\times {10}^{12}\,{L}_{\odot }$. Additional (poorly constrained) stellar mass contributions from non-central galaxies and intracluster light (ICL) are treated as a systematic error in Section 6.2.

We consider the following models for the distribution of DM.

• 1.  
  NFW. As it is the current standard both for modeling observations and simulated clusters, we use the NFW profile (Navarro et al. 1997) for our fiducial DM model. It has two free parameters, a concentration ${c}_{{\rm{\Delta }}}$, and mass ${M}_{{\rm{\Delta }}}$, evaluated with respect to an overdensity Δ times the critical density of the universe.
• 2.  
  Einasto. The Einasto profile (Einasto 1965) is now recognized as a more accurate representation of the profiles of DM halos. We implement the mass profile following Merritt et al. (2006) but using the approximation for d_n given by Retana-Montenegro et al. (2012). As with the NFW profile, we express the parameters of the Einasto model in terms of a virial concentration, ${c}_{{\rm{\Delta }}}$, and mass, ${M}_{{\rm{\Delta }}}$. We fix $n=5\,(\alpha =0.2)$ appropriate for cluster halos (e.g., Dutton & Macciò 2014).
• 3.  
  CORELOG. To provide a strong contrast to the NFW and Einasto models, we investigate a model having a constant density core and with a density that approaches ${r}^{-2}$ at large radius. For consistency, we also express the free parameters of this model in terms of a virial concentration, ${c}_{{\rm{\Delta }}}$, and mass, ${M}_{{\rm{\Delta }}};$ e.g., see Section 2.1.2 of Buote & Humphrey (2012b) for more details.

All ${c}_{{\rm{\Delta }}}$, ${M}_{{\rm{\Delta }}}$, and ${r}_{{\rm{\Delta }}}$ values are evaluated at the redshift of RXJ 1159+5531 (i.e., z = 0.081). Below we quote concentration values for the total mass profile (i.e., stars+gas+DM) as, ${c}_{{\rm{\Delta }}}^{{\rm{tot}}}\equiv {r}_{{\rm{\Delta }}}^{{\rm{tot}}}/{r}_{s}^{{\rm{DM}}}$, where ${r}_{s}^{{\rm{DM}}}$ is the DM scale radius and ${r}_{{\rm{\Delta }}}^{{\rm{tot}}}$ is the virial radius of the total mass profile. We determine ${r}_{{\rm{\Delta }}}^{{\rm{tot}}}$ iteratively starting with the DM virial radius, adding in the baryon components, recomputing the virial radius, and stopping when the change in virial radius is less than a desired tolerance.

We fitted the model to the data using the "nested sampling" Bayesian Monte Carlo procedure implemented in the MultiNest code v2.18 (Feroz et al. 2009), and we adopted flat priors on the logarithms of all the free parameters. We used a ${\chi }^{2}$ likelihood function, where the ${\chi }^{2}$ consists of the temperature and projected ${\rho }_{{\rm{gas}}}^{2}$ data points, the model values, and the statistical weights. The weights are the variances of the data points obtained from the spectral fitting in Paper 1. We quote two "best" values for each parameter: (1) "Best Fit," which is the expectation value of the parameter in the derived posterior probability distribution, and (2) "Max Like," which is the parameter value that gives the maximum likelihood (found during the nested sampling). Finally, unless stated otherwise, all errors quoted are $1\sigma$, representing the standard deviation of the parameter computed in the posterior probability distribution. All models shown in the figures have been evaluated using the "Max Like" values.

In Figures 1 and 2 we display the projected ${\rho }_{{\rm{gas}}}^{2}$ ($\propto {{\rm{\Sigma }}}_{\nu }/{{\rm{\Lambda }}}_{\nu }(T,Z)$, where ${{\rm{\Sigma }}}_{\nu }$ is the surface brightness) and temperature data along with the best-fitting "fiducial" model (and residuals). The fiducial model consists of an entropy profile with two breaks, the n = 4 sersic model for the stellar mass of the BCG, and the NFW model for the DM. Inspection of the figures reveals that, overall, the fit is good. Most of the fit residuals are within $\approx 1\sigma$ of the model values, and the most deviant points lie within $\approx 2\sigma$ of the model values. Since for our Bayesian analysis we cannot easily formally assess the goodness-of-fit as in a frequentist approach, we have also fitted models to the data using a standard frequentist ${\chi }^{2}$ analysis. For our fiducial model we obtain a minimum ${\chi }^{2}=39.9$ for 39 degrees of freedom, which is formally acceptable from the frequentist perspective. (For reference, if the BCG component is omitted, ${\chi }^{2}=59.6;$ i.e., the data strongly require it.)

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Below we shall often refer to the "Best Fit" virial radii of the fiducial model: ${r}_{2500}=271\,{\rm{kpc}}$, ${r}_{500}=575\,{\rm{kpc}}$, ${r}_{200}=862\,{\rm{kpc}}$, and ${r}_{{\rm{vir}}}={r}_{108}=1.12$ Mpc.

The results for the entropy profile using the fiducial hydrostatic model are displayed in Figure 3, and the parameter constraints are listed in Table 1. In the figure we have plotted the "scaled" entropy in units of the characteristic entropy, ${S}_{500}=260.6$ keV cm² (see Equation (3) of Pratt et al. 2010). The entropy profile has a small, but significant floor at the center and then rises more steeply than the $\sim {r}^{1.1}$ profile out to the first break radius ($\approx 36$ kpc). The profile is then shallower than the baseline model out to the second break (≈700 kpc), after which the slope is uncertain, but consistent with the baseline model out to the largest radii investigated $(\sim {r}_{{\rm{108}}})$. As noted in Paper 1, at no radius does the scaled entropy fall below the baseline model, consistent with a simple feedback explanation. (See Paper 1 for more detailed discussion of how the entropy profile compares to theoretical models.)

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Table 1.  Pressure and Entropy

	${P}_{{\rm{ref}}}$	s0	s1	${r}_{{\rm{b,1}}}$	${r}_{{\rm{b,2}}}$	${\alpha }_{1}$	${\alpha }_{2}$	${\alpha }_{3}$
	(10−2 keV cm−3)	(keV cm2)	(keV cm2)	(kpc)	(kpc)
Best Fit	5.30 ± 0.25	4.48 ± 0.78	10.9 ± 1.3	35.6 ± 5.1	708 ± 277	1.81 ± 0.18	0.57 ± 0.06	1.01 ± 0.44
(Max Like)	$(5.49)$	$(4.51)$	$(10.7)$	$(42.4)$	$(180)$	$(1.80)$	$(0.36)$	$(0.74)$

Note. Constraints on the free parameters associated with the pressure and entropy for the fiducial hydrostatic model. ${P}_{{\rm{ref}}}$ is the normalization of the total thermal ICM pressure profile at the reference radius 10 kpc. The other parameters refer to the double broken power-law entropy model (Equation (3)). In the top row we list the marginalized "best-fitting" values and $1\sigma$ errors; i.e., the mean values and standard deviations of the parameters of the Bayesian posterior. In the second row we give the "maximum likelihood" parameter values that maximize the likelihood function.

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In Figure 3 following Pratt et al. (2010) we also show the result of rescaling the entropy profile by ${({f}_{{\rm{gas}}}/{f}_{{\rm{b}},{\rm{U}}})}^{2/3}$, where ${f}_{{\rm{gas}}}$ is the gas fraction as a function of radius and ${f}_{{\rm{b}},{\rm{U}}}=0.155$ is the baryon fraction of the universe. The overall very good agreement of this rescaled entropy profile with the baseline model suggests that the feedback has primarily served to spatially redistribute the gas rather than raise its temperature.

We note that the second break in the entropy profile is not required at high significance; i.e., the ratio of the Bayesian evidence for the two-break and one-break model is 1.9.

The results for the pressure profile using the fiducial hydrostatic model are displayed in Figure 3, and the constraints for the reference pressure are listed in Table 1. In the figure we have plotted the "scaled" pressure in units of the characteristic pressure, ${P}_{500}=5.9\times {10}^{-4}$ keV cm⁻³ (see Equation (5) of Arnaud et al. 2010), and compared it to the "universal" pressure profile of Arnaud et al. (2010). (Note that we quote results for the total gas pressure rather than the electron gas pressure and have accounted for this also in the definition of P₅₀₀.) In most of the region where the universal profile is expected to be valid, the pressure profile of RXJ 1159+5531 agrees within the 20% scatter in the pressure profiles of the clusters studied by Arnaud et al. (2010), reaching maximum deviations of 50%–60% at the endpoints; i.e., RXJ 1159+5531 has a pressure profile similar to the other clusters.

In Figure 4 we show the profiles for the total mass and different mass components for the fiducial hydrostatic model, while in Table 2 we list the results for ${M}_{\star }/{L}_{K}$ and the NFW parameters, concentration and mass, evaluated for several overdensities. Of all the mass components, the stellar mass has the weakest constraints $(\sim \pm 18 \% )$, while the gas mass is the best constrained (e.g., $\sim \pm 4 \%$ at ${r}_{{\rm{108}}}$.), considering only the statistical errors.

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Table 2.  Stellar and Total Mass

	${M}_{\star }/{L}_{K}$	c2500	M2500	c500	M500	c200	M200	${c}_{{\rm{vir}}}$	${M}_{{\rm{vir}}}$
	(${M}_{\odot }{L}_{\odot }^{-1}$)		(${10}^{13}{M}_{\odot }$)		(${10}^{13}{M}_{\odot }$)		(${10}^{13}{M}_{\odot }$)		(${10}^{13}{M}_{\odot }$)
Best Fit	0.61 ± 0.11	2.6 ± 0.4	3.1 ± 0.1	5.6 ± 0.7	5.9 ± 0.4	8.4 ± 1.0	7.9 ± 0.6	10.9 ± 1.3	9.4 ± 0.7
(Max Like)	$(0.53)$	$(3.0)$	$(3.1)$	$(6.3)$	$(5.6)$	$(9.4)$	$(7.4)$	$(12.2)$	$(8.8)$
Spherical	⋯	⋯	⋯	${}_{-0.32}^{+0.13}$	${}_{-0.04}^{+0.15}$	⋯	⋯	⋯	⋯
Einasto	−0.05	−0.2	−0.0	−0.4	0.1	−0.6	0.1	−0.9	0.1
CORELOG	−0.05	10.0	−0.4	23.1	0.5	37.5	2.4	51.7	4.7
1 Break	0.02	−0.1	0.0	−0.2	0.1	−0.2	0.2	−0.3	0.2
${{\rm{\Lambda }}}_{\nu }(T,Z)$	−0.04	0.2	−0.1	0.5	−0.4	0.7	−0.5	0.9	−0.6
Response	0.02	−0.2	0.0	−0.4	0.1	−0.6	0.3	−0.8	0.3
Proj. Limit	${}_{-0.00}^{+0.00}$	${}_{-0.0}^{+0.0}$	${}_{-0.0}^{+0.0}$	${}_{-0.0}^{+0.0}$	${}_{-0.0}^{+0.0}$	${}_{-0.0}^{+0.1}$	${}_{-0.0}^{+0.0}$	${}_{-0.0}^{+0.1}$	${}_{-0.0}^{+0.0}$
SWCX	0.00	−0.0	0.0	−0.0	0.0	−0.1	0.1	−0.1	0.1
Distance	${}_{-0.02}^{+0.03}$	${}_{-0.1}^{+0.1}$	${}_{-0.0}^{+0.0}$	${}_{-0.2}^{+0.1}$	${}_{-0.1}^{+0.1}$	${}_{-0.3}^{+0.2}$	${}_{-0.1}^{+0.1}$	${}_{-0.3}^{+0.2}$	${}_{-0.2}^{+0.2}$
${N}_{{\rm{H}}}$	−0.02	0.1	−0.0	0.2	−0.1	0.3	−0.2	0.4	−0.3
Fix ${Z}_{{\rm{Fe}}}({r}_{{\rm{out}}})$	${}_{-0.03}^{+0.05}$	${}_{-0.3}^{+0.2}$	${}_{-0.1}^{+0.2}$	${}_{-0.5}^{+0.3}$	${}_{-0.2}^{+0.5}$	${}_{-0.8}^{+0.5}$	${}_{-0.3}^{+0.7}$	${}_{-1.1}^{+0.6}$	${}_{-0.4}^{+0.8}$
Solar Abun.	−0.05	0.3	−0.1	0.5	−0.3	0.7	−0.5	1.0	−0.6
PSF	0.03	−0.1	0.1	−0.3	0.2	−0.4	0.3	−0.5	0.4
FI–BI	0.02	−0.1	−0.0	−0.1	0.0	−0.2	0.1	−0.2	0.1
NXB	${}_{-0.00}^{+0.01}$	−0.0	${}_{-0.0}^{+0.1}$	−0.0	${}_{-0.0}^{+0.1}$	−0.1	0.1	−0.1	${}_{-0.0}^{+0.1}$
CXB	−0.01	0.0	0.0	0.1	−0.0	0.1	−0.0	0.1	−0.1
CXBSLOPE	${}_{-0.04}^{+0.06}$	${}_{-0.4}^{+0.2}$	${}_{-0.1}^{+0.3}$	${}_{-0.7}^{+0.4}$	${}_{-0.4}^{+0.8}$	${}_{-1.0}^{+0.7}$	${}_{-0.5}^{+1.1}$	${}_{-1.4}^{+0.9}$	${}_{-0.6}^{+1.2}$

Notes. Best-fitting values, maximum likelihood values, and $1\sigma$ errors (see notes to Table 1) for the stellar mass-to-light ratio $({M}_{\star }/{L}_{K})$, concentration, and enclosed mass corresponding to the total mass profile (stars+gas+DM) of the fiducial hydrostatic model computed for radii corresponding to several different overdensities. (Note: "vir" refers to Δ = 108.) We also provide a detailed systematic error budget as explained in Section 6. For most of the entries we list the values using the same precision. If the particular error has a smaller value, it appears as a zero; e.g., "0.0" or "−0.0." Briefly, the systematic test names refer to tests of, ("Spherical," Section 6.1): the assumption of spherical symmetry; ("Einasto," "CORELOG," Section 6.4): using Einasto or CORELOG instead of NFW for the DM profile; ("1 Break," Section 6.3): only using one break radius for the entropy model; ("${{\rm{\Lambda }}}_{\nu }(T,Z)$," Section 6.8): different treatments for deprojecting the plasma emissivity; ("Response," Section 6.8): response weighting; ("Proj. Limit"), Section 6.8): outer radius of model cluster; ("SWCX," Section 6.6): Solar Wind Charge Exchange emission; ("Distance," Section 6.8): assumed distance; ("${N}_{{\rm{H}}}$," 6.7): assumed Galactic hydrogen column density; ("Fix ${Z}_{{\rm{Fe}}}({r}_{{\rm{out}}})$," Section 6.5)): effect of fixing the metal abundance in the outer Suzaku aperture to either $0.1{Z}_{\odot }$ or $0.3{Z}_{\odot }$; ("Solar Abun.," Section 6.5): using different solar abundance tables; ("PSF," Section 6.7): the sensitivity to the PSF mixing procedure for the Suzaku data; ("FI–BI," Section 6.7): the flux calibration of the different Suzaku CCDs; ("NXB," Section 6.6): the sensitivity to the adopted model of the non-X-ray background for the Suzaku data; ("CXB," Section 6.6): fixing the normalization of the CXB power law to the average value determined from surveys; and ("CXBSLOPE," Section 6.6): the sensitivity to the assumed slope in the CXB power la.

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Most of the possible systematic errors we consider in Section 6, and listed in Table 2, are not significant since they induce parameter changes of the same size or smaller than the $1\sigma$ statistical error. The most significant changes result from aspects of the background modeling (Section 6.6), the treatment of the plasma emissivity (Section 6.8), and how the metal abundances are treated (Section 6.5), particularly in the outermost apertures; i.e., the CXBSLOPE, ${{\rm{\Lambda }}}_{\nu }(T,Z)$, Solar Abun, and Fix ${Z}_{{\rm{Fe}}}({r}_{{\rm{out}}})$ rows in Table 2. It is, however, reassuring that even these changes generally lead to parameter changes not much larger than $1\sigma$. (See Section 6 for a more detailed discussion of the systematic error budget.)

The value we obtain for the K-band stellar mass-to-light ratio (${M}_{\star }/{L}_{K}\approx 0.6{M}_{\odot }/{L}_{\odot }$) of the BCG agrees very well with our previous determinations (Gastaldello et al. 2007; Humphrey et al. 2012a) and also is consistent with the value expected from stellar population synthesis models. Using the published relationship between stellar mass-to-light ratio and color from Zibetti et al. (2009), we obtain ${M}_{\star }/{L}_{K}\approx 0.55{M}_{\odot }/{L}_{\odot }$ for $g-i=1.41$, where the g and i magnitudes are taken from the "Model" entries in the NASA/IPAC Extragalactic Database (NED) referring to the Sloan Digital Sky Survey Data Release⁵ 6. This good agreement should be considered only a mild consistency check, since (1) there is significant scatter depending on which bands are used for the color (we used g − i as favored by Zibetti et al. 2009), and (2) the relationship between color and ${M}_{\star }/{L}_{K}$ depends on the assumed stellar initial mass function (IMF). If instead we use the relationship between stellar mass-to-light ratio and color of Bell et al. (2003), who employ a Salpeter-like IMF, we obtain ${M}_{\star }/{L}_{K}\approx 0.94{M}_{\odot }/{L}_{\odot }$, about $3\sigma$ above our measured value. While there is considerable scatter depending on the color used, the X-ray analysis favors the lower ${M}_{\star }/{L}_{K}$ obtained from Zibetti et al. (2009) who adopt a Milky Way IMF (Chabrier 2003). Several previous studies have found that massive early-type galaxies instead favor a Salpeter IMF (e.g., Conroy & van Dokkum 2012; Newman et al. 2013a; Dutton & Treu 2014), although the more recent study by, e.g., Smith et al. (2015) and most of our previous X-ray studies (e.g., Humphrey et al. 2009, 2012b) favor a Milky Way IMF.

Since fossil clusters like RXJ 1159+5531 are thought to be highly evolved, early forming systems, it is interesting to examine whether they possess large concentrations for their mass compared to the general population. According to the results of Dutton & Macciò (2014), the mean value of c₂₀₀ for a "relaxed" DM halo with ${M}_{200}=7.9\times {10}^{13}\,{M}_{\odot }$ in the Planck cosmology is 5.3, which is a little more than $3\sigma$ lower than the value we measure for RXJ 1159+5531 $(8.4\pm 1.0)$. In addition, with respect to the intrinsic scatter of the theoretical relation, the c₂₀₀ value is a $\sim 2\sigma$ outlier. Hence, with respect to the fiducial hydrostatic model with an NFW DM halo, RXJ 1159+5531 appears to possess a significantly above-average concentration, consistent with forming earlier than the average halo population. For the Einasto DM halo, our measured ${c}_{200}=7.8\pm 1.0$ is slightly more than $2\sigma$ above the mean predicted value of 5.6 and a little above the $1\sigma$ intrinsic scatter (${c}_{200}=7.5$) using the results of Dutton & Macciò (2014). The reduced significance for the Einasto case arises from a combination of our smaller measured c₂₀₀ and also larger theoretical intrinsic scatter for the Einasto profile compared to NFW.

In the right panel of Figure 4 we compare the results for the different DM models. As is readily apparent from visual inspection, the gas mass profiles are virtually identical for all three cases; i.e., the inferred gas mass profile is robust to the assumed DM model. Over most of the radii investigated, the total mass profile is also quite insensitive to the assumed DM model, though CORELOG leads to a more massive halo than either NFW or Einasto. (Note in particular the "pinching" of the total mass profile near 10–20 kpc, about 1–2 R_e, where the DM crosses over the stellar mass.) The largest deviations appear at the very largest radii (≳r₁₀₈). The fitted stellar mass profile is very similar for all three DM models and yields consistent ${M}_{\star }/{L}_{K}$ values in each case.

All the DM models produce fits of similar quality in terms of the magnitude of their fractional residuals. When we perform a frequentist ${\chi }^{2}$ fit we obtain minimum ${\chi }^{2}$ values of 39.4 for Einasto, and 40.1 for CORELOG compared to 39.9 for NFW; i.e., the fits are statistically indistinguishable from the frequentist perspective. Moreover, from the Bayesian analysis we can use the ratio of evidences to compare the Einasto $(\,{\mathrm{log}}_{10}\,Z=-53.00)$ and NFW $(\,{\mathrm{log}}_{10}\,Z=-53.81)$ models since they have essentially the same free parameters and priors: We obtain an evidence ratio of 2.2 in favor of the Einasto model, which is not very significant. It is not straightforward to compare the evidences of the CORELOG and NFW models because they have different prior volumes (and the models are not "nested"). Nevertheless, the frequentist minimum ${\chi }^{2}$ values clearly show that, despite the high-quality X-ray data covering the entire virial radius in projection, the data do not statistically disfavor the CORELOG model.

For completeness, we have also examined allowing the Einasto index n to be a free parameter. Since, as just noted, the data are unable to distinguish clearly between the NFW, Einasto $(n=5)$, and CORELOG profiles (each of which have two free parameters), and these models already provide formally acceptable fits, it follows that adding another free parameter does not improve the fit very much. Indeed, for the frequentist fit we find that ${\chi }^{2}$ is reduced by only 0.08 and gives a large $1\sigma$ error range on the index: $n={5.8}_{-2.0}^{+4.6}$ or $\alpha =1/n={0.17}_{-0.08}^{+0.09}$. Despite the large uncertainty, the best-fitting Einasto index matches well the value expected for a DM halo of the mass of RXJ 1159+5531 (Dutton & Macciò 2014).

The results for the baryon and gas fraction profiles using the fiducial hydrostatic model are displayed in Figure 5, and the parameter constraints are listed in Table 3 including the systematic error budget. Similar to the results for the total mass, most of the systematic errors are insignificant in the sense that the estimated parameter changes in the gas and baryon fraction are comparable to or less than the $1\sigma$ statistical error. Again, the most important changes occur for aspects of the background modeling (Section 6.6) and the treatment of the plasma emissivity (Section 6.8) and metal abundances (Section 6.5), particularly in the outermost apertures; i.e., the CXBSLOPE, ${{\rm{\Lambda }}}_{\nu }(T,Z)$, Fix ${Z}_{{\rm{Fe}}}({r}_{{\rm{out}}})$, and Solar Abun rows in Table 3. The CXBSLOPE and Fix ${Z}_{{\rm{Fe}}}(r)$ result in systematic errors almost $3\sigma$ in magnitude. (See Section 6 for a more detailed discussion of the systematic error budget.)

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Table 3.  Gas and Baryon Fraction

	fgas,2500	fb,2500	fgas,500	fb,500	fgas,200	fb,200	fgas,vir	fb,vir
	0.051 ± 0.001	0.072 ± 0.004	0.093 ± 0.003	0.104 ± 0.004	0.126 ± 0.007	0.134 ± 0.007	0.152 ± 0.010	0.159 ± 0.010
(Max Like)	$(0.052)$	$(0.070)$	$(0.093)$	$(0.103)$	$(0.127)$	$(0.134)$	$(0.155)$	$(0.161)$
${M}_{{\rm{stellar}}}^{{\rm{other}}}$	⋯	${}^{+0.017}$	⋯	${}^{+0.016}$	⋯	${}^{+0.016}$	⋯	${}^{+0.015}$
Spherical	⋯	⋯	${}_{-0.0009}^{+0.0006}$	⋯	⋯	⋯	⋯	⋯
Einasto	−0.001	0.000	−0.002	−0.001	−0.002	−0.001	−0.002	−0.001
CORELOG	0.003	0.002	−0.003	−0.001	−0.018	−0.015	−0.037	−0.035
1 Break	0.000	−0.000	−0.001	−0.001	−0.001	−0.001	−0.000	−0.000
${{\rm{\Lambda }}}_{\nu }(T,Z)$	0.000	0.001	0.004	0.004	0.010	0.010	0.015	0.016
Response	0.001	0.001	−0.001	−0.001	−0.003	−0.003	−0.004	−0.004
Proj. Limit	${}_{-0.000}^{+0.000}$	${}_{-0.000}^{+0.000}$	${}_{-0.000}^{+0.000}$	${}_{-0.000}^{+0.000}$	${}_{-0.000}^{+0.001}$	${}_{-0.000}^{+0.001}$	${}_{-0.000}^{+0.001}$	${}_{-0.000}^{+0.001}$
SWCX	−0.000	−0.000	−0.000	−0.000	−0.000	−0.000	−0.000	−0.000
Distance	${}_{-0.001}^{+0.002}$	${}_{-0.000}^{+0.001}$	${}_{-0.001}^{+0.002}$	${}_{-0.001}^{+0.002}$	${}_{-0.002}^{+0.003}$	${}_{-0.001}^{+0.003}$	${}_{-0.002}^{+0.003}$	${}_{-0.002}^{+0.002}$
${N}_{{\rm{H}}}$	−0.000	0.000	0.001	0.001	0.002	0.002	0.003	0.004
Fix ${Z}_{{\rm{Fe}}}({r}_{{\rm{out}}})$	0.000	${}_{-0.000}^{+0.001}$	${}_{-0.007}^{+0.003}$	${}_{-0.007}^{+0.003}$	${}_{-0.016}^{+0.006}$	${}_{-0.016}^{+0.006}$	${}_{-0.024}^{+0.009}$	${}_{-0.024}^{+0.009}$
Solar Abun.	−0.000	0.001	0.004	0.004	0.007	0.007	0.008	0.008
PSF	0.001	−0.000	−0.001	−0.002	−0.003	−0.003	−0.004	−0.004
FI–BI	0.002	0.001	0.003	0.003	0.005	0.005	0.007	0.007
NXB	${}_{-0.001}^{+0.001}$	${}_{-0.000}^{+0.001}$	${}_{-0.002}^{+0.002}$	${}_{-0.002}^{+0.001}$	${}_{-0.006}^{+0.001}$	${}_{-0.005}^{+0.001}$	${}_{-0.010}^{+0.000}$	${}_{-0.010}^{+0.000}$
CXB	−0.001	−0.000	−0.001	−0.001	−0.002	−0.002	−0.004	−0.004
CXBSLOPE	${}_{-0.001}^{+0.001}$	${}_{-0.001}^{+0.001}$	${}_{-0.009}^{+0.005}$	${}_{-0.008}^{+0.005}$	${}_{-0.018}^{+0.011}$	${}_{-0.018}^{+0.011}$	${}_{-0.027}^{+0.018}$	${}_{-0.027}^{+0.018}$

Note. Best-fitting values, maximum likelihood values, $1\sigma$ statistical errors, and estimated systematic errors (see notes to Table 1) for the gas and baryon fractions of the fiducial hydrostatic model computed for radii corresponding to several different overdensities. (Note: "vir" refers to Δ = 108.)

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For most radii ${f}_{{\rm{b}}}\lt {f}_{{\rm{b,U}}}$, where ${f}_{{\rm{b,U}}}=0.155$ is the mean baryon fraction of the universe as determined by Planck (Planck Collaboration et al. 2014). Near r₁₀₈, ${f}_{{\rm{b}}}$ is consistent with ${f}_{{\rm{b,U}}}$, with the hot ICM consisting of 96% of the total baryons. These results are virtually identical to those for the Einasto model, where ${f}_{{\rm{b}}}=0.157\pm 0.011$ at r₁₀₈, whereas CORELOG has a smaller value, ${f}_{{\rm{b}}}=0.121\pm 0.008$.

The preceding discussion considered the baryons contained in the hot ICM and stellar baryons associated only with the BCG as determined by the fitted ${M}_{\star }/{L}_{K}$. To estimate the stellar baryons from non-central cluster members and ICL we follow the procedure we adopted in Section 4.3 of Humphrey et al. (2012a). Since the contribution of these non-BCG stellar baryons is not measured directly, we list it as a systematic error in Table 3 (see Section 6.2). At the largest radii these baryons are expected to increase the baryon fraction by $\sim 10 \% ,$ which is not much larger than the statistical error at ${r}_{108}.$

It is now well established that the total mass profiles of massive elliptical galaxies have density profiles very close to $\rho \sim {r}^{-\alpha }$ with $\alpha \approx 2$ over a wide range in radius. This relation extends to higher masses with smaller α in the central regions of clusters (e.g., Humphrey & Buote 2010; Newman et al. 2013b; Courteau et al. 2014; Cappellari et al. 2015 and references therein). In particular, in Humphrey & Buote (2010) it was shown that the total mass profiles inferred from hydrostatic studies of hot gas in massive elliptical galaxies, groups, and clusters, are fairly well approximated by a single power law over 0.2–10 stellar half-light radii $({R}_{e})$ so that, approximately, $\alpha =2.31-0.54\mathrm{log}({R}_{e}/\mathrm{kpc}),$ a result consistent with that obtained from combination of stellar dynamics and strong gravitational lensing for massive elliptical galaxies (Auger et al. 2010).

In Figure 6 we display the radial logarithmic derivatives (i.e., slopes) of the total mass and total density profiles for the fiducial hydrostatic model. The slopes are indeed slowly varying for our models, ranging from $\approx 1.2\mbox{--}1.4$ for the mass to ≈−1.6 to −2.0 for the density over radii 0.2–10 R_e, representing a radial variation of ±20%. In Table 4 we quote the mass-weighted total density slope $\langle \alpha \rangle$ following Equation (2) of Dutton & Treu (2014),

$$\begin{eqnarray}&&\langle \alpha \rangle =3-\displaystyle \frac{d\mathrm{ln}M}{d\mathrm{ln}r},\alpha \equiv -\displaystyle \frac{d\mathrm{ln}\rho }{d\mathrm{ln}r},\end{eqnarray} \tag{ 4 }$$

where M is the total mass enclosed within radius r. Within 10R_e, $\langle \alpha \rangle =1.74\pm 0.04$, which is consistent with the value of $\alpha ={1.67}_{-0.10}^{+0.11}$ we obtained previously from a power-law fit to the mass profile (Humphrey & Buote 2010) and with $\alpha =1.77$ obtained using the $\alpha -{R}_{e}$ scaling relation.

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Table 4.  Mass-weighted Total Density Slope

Radius	Radius
(kpc)	(Re)	$\langle \alpha \rangle$
4.9	0.5	1.86 ± 0.07
9.8	1.0	1.80 ± 0.08
19.7	2.0	1.67 ± 0.07
49.2	5.0	1.62 ± 0.02
98.3	10.0	1.74 ± 0.04

Note. The mass-weighted slope is computed for the fiducial hydrostatic model with Equation (4).

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While the presence of DM is largely accepted by the astronomical community, it is worthwhile to examine interpretations of the observations that instead consider a modification of the gravitational force law. Here we consider the most widely investigated and successful modified gravity theory, MOND (Milgrom 1983), which nevertheless is unable to explain observations of galaxy clusters considering only the known baryonic matter (e.g., Sanders 1999; Pointecouteau & Silk 2005; Angus et al. 2008; Milgrom 2015). We investigate whether MOND can obviate the need for DM in RXJ 1159+5531 following the approach of Angus et al. (2008).

For an isolated spherical system the gravitational acceleration in Newton gravity, ${g}_{{\rm{N}}}={{GM}}_{{\rm{N}}}(\lt r)/{r}^{2}$, and MOND, ${g}_{{\rm{M}}}={{GM}}_{{\rm{M}}}(\lt r)/{r}^{2}$, have similar forms, where ${M}_{{\rm{N}}}(\lt r)$ and ${M}_{{\rm{M}}}(\lt r)$ are the respective enclosed masses within radius r. For some interpolating function, $\mu ({g}_{{\rm{N}}}/{a}_{0})$, the accelerations are related by ${g}_{{\rm{M}}}=\mu ({g}_{{\rm{N}}}/{a}_{0}){g}_{{\rm{N}}}$, so that ${M}_{{\rm{M}}}=\mu ({g}_{{\rm{N}}}/{a}_{0}){M}_{{\rm{N}}}$, and ${a}_{0}\approx 1.2\times {10}^{-8}$ cm s⁻² is the MOND acceleration constant. For $\mu (x)=x/(1+x)$, where $x={g}_{{\rm{N}}}/{a}_{0}$, we have

$$\begin{eqnarray}&&{M}_{{\rm{M}}}(\lt r)=\displaystyle \frac{{M}_{{\rm{N}}}(\lt r)}{1+{a}_{0}/{g}_{{\rm{N}}}(r)}.\end{eqnarray} \tag{ 5 }$$

In this way we easily compute the MONDian mass using the Newtonian mass we have derived previously; i.e., there is no different fitting required. Equation (5), being derived from the simple interpolating function μ, has the undesirable property that ${M}_{{\rm{M}}}(\lt r)$ reaches a maximum value at some radius and then decreases (for more discussion of this point see Angus et al. 2008). Hence, for the moment we focus on results roughly within the radius where the MONDian mass reaches a maximum value.

In Figure 7 for the fiducial hydrostatic model we compare the cumulative DM fractions $({M}_{{\rm{DM}}}/{M}_{{\rm{total}}})$ inferred from Newtonian gravity and MOND. As in the Newtonian case, MOND requires a dominant fraction of DM with increasing radius to match the X-ray data. At r = 100 kpc the DM fraction is 85.0% ± 2.5%; i.e., the mass discrepancy is a factor of 6.7. Even when considering the contribution of baryons from (rather uncertain) non-central baryons (Section 6.2), the DM fraction is reduced by only a small amount to $\approx 82 \%$.

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It is also interesting to view the performance of MOND from a different perspective. Solving the MOND equation for ${g}_{{\rm{N}}}$ with the same interpolating function yields

$$\begin{eqnarray}&&{g}_{{\rm{N}}}=\displaystyle \frac{{g}_{{\rm{M}}}}{2}\left(1+\sqrt{1+4\displaystyle \frac{{a}_{0}}{{g}_{{\rm{M}}}}}\,\,\right).\end{eqnarray} \tag{ 6 }$$

That is, given ${g}_{{\rm{M}}}$ evaluated using the baryonic mass profiles (i.e., stellar and gas) derived from the Newtonian analysis, this expression gives ${g}_{{\rm{N}}}$, and thus the Newtonian mass profile (with DM) that MOND would predict. In Figure 7 we compare this "MOND-predicted Newtonian mass profile" with the actual Newtonian total mass profile. The MOND profile underpredicts the Newtonian mass over most radii, with the largest deficit again occurring near r = 100 kpc such that the predicted mass is $\approx 0.30{M}_{{\rm{N}}}$. The predicted profile crosses over the Newtonian profile shortly before r₂₀₀ and then exceeds the Newtonian profile afterwards.

In sum, consistent with previous results for X-ray groups and clusters, MOND requires a large fraction of DM similar to Newtonian gravity to explain the X-ray data.

It is interesting to compare our results to those obtained by Humphrey et al. (2012a) who used only the Chandra data and the North Suzaku observation. Humphrey et al. (2012a) obtained best-fitting values, ${M}_{\star }/{L}_{K}=0.54$ (in solar units), ${c}_{{\rm{vir}}}=11.2$, ${M}_{{\rm{vir}}}=9.3\times {10}^{13}\,{M}_{\odot }$, and f_b,500 = 0.124 (includes non-BCG stellar baryons). All of these results agree within $1\sigma$ of the values obtained in our study (Tables 2 and 3).

The excellent agreement between the two studies is notable for several reasons. First, the addition of the Suzaku observations covering out to r₂₀₀ in the S, E, and W directions does not modify the results significantly, consistent with the results of Paper 1, indicating only small azimuthal variation in the ICM properties at large radius. Second, improved background modeling incorporating point sources resolved by offset Chandra observations—as well as improved calibration and data processing in the Chandra and Suzaku pipelines—does not change the results significantly. Finally, the consistent results between the studies provide a useful consistency check on the different implementations of the entropy-based hydrostatic modeling (e.g., treatment of self-gravity of gas mass; see Section 3) between the two studies using entirely different modeling software.

Buote & Humphrey (2012b) showed that assuming a cluster is spherical when in fact it is ellipsoidal does not typically introduce large errors into the quantities inferred from hydrostatic modeling of the ICM. For a large range of intrinsic flattenings, they computed orientation-averaged biases (mean values and scatter) of several derived quantities, including halo concentration, total mass, and gas fraction.

We use the "NFW-EMD" results from Table 1 apjaa2652t1 of Buote & Humphrey (2012b) to provide an estimate of the error arising from the assumption of spherical symmetry within r₅₀₀; i.e., the error from assuming that the cluster is spherical when in fact it is a flattened ellipsoid viewed at a random orientation to the line of sight. From a study of cosmological DM halos Schneider et al. (2012) find that for a halo having mass similar to RXJ 1159+5531 the typical intrinsic short-to-long axis ratio is $\approx 0.5.$ We adopted this value for the error estimates on the concentration, mass, and gas fraction within r₅₀₀ in Tables 2 and 3 ("Spherical"). In all cases the effect is insignificant.

While the BCG dominates the stellar mass in the central region of the cluster, smaller non-central galaxies and diffuse ICL contribute significant stellar mass at larger radius. Due to the greater uncertainty of the amounts and distributions of these non-central baryons, we treat their contribution as a systematic effect to the baryon fraction. To account for these additional stellar baryons, we follow the procedure described in Section 4.3 of Humphrey et al. (2012a). For the non-central galaxies we use the result of Vikhlinin et al. (1999) that these galaxies comprise $\sim 25 \%$ of the V-band stellar light. We assume that this result also applies in the K-band with the same ${M}_{\star }/{L}_{K}$ as the BCG. Since we do not have a precise observational constraint on the ICL, we use the result from the theoretical study by Purcell et al. (2007) that the ICL contains up to ∼2 times the stellar mass of the BCG and adopt this value to give a conservative reflection of the systematic error. We assume that both components of non-central baryons are spatially distributed as the DM in our models.

The contribution of the non-central baryons to the baryon fraction are listed in Table 3 (${M}_{{\rm{stellar}}}^{{\rm{other}}}$). These stellar baryons increase f_b,vir by ∼10% to 0.174, fully consistent with the value reported in Humphrey et al. (2012a) containing the contributions from both the BCG and non-central stellar baryons. This modified f_b,vir exceeds the cosmic value ${f}_{{\rm{b,U}}}=0.155$ by $2\sigma$ considering only the statistical error on our fiducial model, although the disagreement should be considered less significant given the uncertainties in the non-central stellar baryons.

We examined the effect of restricting the entropy broken power-law model to only a single break (ONEBREAK). The effect is everywhere insignificant.

The effects of using a DM profile different from NFW are indicated in the rows "Einasto" and "Corelog" in Tables 2 and 3. We have discussed the magnitudes of these systematic differences in Sections 5.4 and 5.5.

We considered how choices made in the measurement of the metal abundances from the spectral fitting affected the results. (We refer the reader to Paper 1 for details on the spectral analysis.) First, we examined the impact of using different solar reference abundances. Whereas our default analysis used the solar abundance table of Asplund et al. (2006), the effects of instead using the tables of Anders & Grevesse (1989) or Lodders (2003) are listed in the "Solar Abun" row in Tables 2 and 3. The differences do not exceed the $1\sigma$ statistical error.

Second, since the spectra in the outermost apertures (i.e., at the virial radius) are the most background dominated and subject to systematic errors, we also examined the impact of fixing the metal abundance there at $0.1{Z}_{\odot }$ and $0.3{Z}_{\odot }$, bracketing the best-fitting value of $\sim 0.2{Z}_{\odot }$ (referred to as "Δabun" in Table 3 of Paper 1). As can be seen in the row "Fix ${Z}_{{\rm{Fe}}}({r}_{{\rm{out}}})$" in Tables 2 and 3, this effect leads to one of the two largest systematic errors as mentioned above in Sections 5.4 and 5.5.

We considered the impact of several choices made in the treatment of the background in the spectral fitting (see Paper 1). The effect of including a model for the solar wind charge exchange emission in the spectral analysis is listed in row "SWCX" in Tables 2 and 3. We find the effect to be insignificant in all cases.

To assess the sensitivity of the results to the particle background in the Suzaku observations, we artificially increased and decreased the estimated non-X–ray background component by ±5% and list the results in row "NXB" in Tables 2 and 3. In all instances the differences are insignificant for the stellar and total mass parameters. While this is also true at most radii for the gas and baryon fractions, at ${r}_{{\rm{vir}}}$ the differences are comparable to the $1\sigma$ error.

We also explored the sensitivity of our results to the extragalactic cosmic X-ray background (CXB) power-law component (see Section 5.1 of Paper 1). In Paper 1 by default we assumed a power-law component in our spectral fits with a slope fixed at ${\rm{\Gamma }}=1.41$ (e.g., De Luca & Molendi 2004) and normalization free to vary. If we fix the normalization to the value expected for the cosmic average (see Section 2.3 of Paper 1), we obtain the results listed in row "CXB" in Tables 2 and 3. It is reassuring that the differences are all negligible. If instead we keep the normalization free to vary but change the slopes used to ${\rm{\Gamma }}=1.3,1.5$, we obtain the results listed in row "CXBSLOPE" (corresponding to "CXB-Γ" in Table 3 of Paper 1). As already noted in Sections 5.4 and 5.5, this effect leads to one of the two largest systematic errors. The differences are comparable to the $1\sigma$ errors within r₂₅₀₀ and increase to 2–3σ at ${r}_{{\rm{vir}}}$, where the data are most dominated by the background.

We performed other tests associated with the spectral fitting that we summarize here. In all cases they did not produce significant parameter differences in Tables 2 and 3. (1) The effect of varying the adopted value of Galactic ${N}_{{\rm{H}}}$ (Dickey & Lockman 1990) by ±20% is shown in row "${N}_{{\rm{H}}}$" in Tables 2 and 3 (see Section 5.6 of Paper 1). (2) We varied the spectral mixing between extraction annuli by ±5% from our default case to assess the impact of small changes on how we account for the large, energy-dependent Suzaku PSF (see Section 5.3 of Paper 1). The results are given in row "PSF" in Tables 2 and 3. (3) By default we allowed the normalization of the ICM model for annuli on the Suzaku/XIS front-illuminated (FI) and back-illuminated (BI) chips to be varied separately to allow for any calibration differences. To assess the impact of this choice, we also performed the spectral fitting requiring the same normalizations for the FI and BI chips (Section 5.7 of Paper 1), and the results are given in the "FI–BI" row in Tables 2 and 3.

Here we describe a few remaining tests we performed associated with choices made in our hydrostatic modeling procedure. First, we varied the cluster redshift by ±5% in our hydrostatic models (it was similarly varied in the spectral fitting in Paper 1), and the results are listed in the "Distance" row of Tables 2 and 3. The differences are everywhere insignificant. Second, we explored the impact of changing the default bounding radius of the cluster model. Whereas the default employed is 2.5 Mpc, in the "Proj. Limit" row of Tables 2 and 3 we show the differences resulting from instead using either 2.0 Mpc or 3.0 Mpc for the bounding radius. In all cases the effect is negligible. Next we examined the sensitivity of the results to the "response weighting" (Section 3) by instead performing no such weighting. As indicated in the "Response" row of Tables 2 and 3 the differences are insignificant.

Finally, we examined how our choices regarding the modeling of the plasma emissivity affect the results. As discussed in Section 3, the hydrostatic models by default take the measured metal abundance profile in projection and assigns it to be the true three-dimensional profile, which is then used (along with the temperature) to compute the plasma emissivity ${{\rm{\Lambda }}}_{\nu }(T,Z)$. Our default procedure to do this assignment uses the Chandra data and only the N Suzaku observation. We investigated using instead each of the other three Suzaku pointings. For a more rigorous test, we also fitted a projected, emission-weighted parametric model to the projected iron abundance profile (Figure 13 of Paper 1). We employed a multi-component model consisting of two power laws mediated by an exponential (Equation (5) of Gastaldello et al. 2007) and a constant floor of $0.05{Z}_{\odot }$. The results of both of these tests for the plasma emissivity are listed in row "${{\rm{\Lambda }}}_{\nu }(T,Z)$" of Tables 2 and 3. The differences are among the largest, though in most cases are less than the $1\sigma$ error. In detail, both the metallicity model test and the cases where the E and S Suzaku pointings are used produce negligible results. The differences indicated in the tables are largest when the W Suzaku pointing is used to assign the metallicity profile.

Previously in Zappacosta et al. (2006) we made the observation that in massive galaxy clusters (i.e., with virial mass larger than a few ${10}^{14}\,{M}_{\odot }$) the total gravitating mass profile inferred from X-ray studies is itself generally well described by a single NFW profile without any need for a distinct component for stellar mass from the central BCG. On the other hand, individual massive elliptical galaxies (e.g., Humphrey et al. 2006, 2011, 2012b) and group-scale systems (e.g., Gastaldello et al. 2007; Zhang et al. 2007; Démoclès et al. 2010; Su et al. 2014) with virial masses less than ${10}^{14}\,{M}_{\odot }$ studied with X-rays usually (but do not always) require distinct stellar BCG and DM mass components.

The most massive clusters where X-ray observations clearly require distinct stellar BCG (with a reasonable stellar mass-to-light ratio) and DM (without an anomalously low concentration) mass components are RXJ 1159+5531 (${M}_{200}=7.9\pm 0.6\times {10}^{13}\,{M}_{\odot }$) and A262 (${M}_{200}=9.3\pm 0.8\times {10}^{13}\,{M}_{\odot }$, Gastaldello et al. 2007). Hence, from the perspective of X-ray studies, $\sim {10}^{14}\,{M}_{\odot }$ appears to represent a point of demarcation above which the total mass profile, rather than the DM profile, is represented by a single NFW component.

Since X-ray images of the central regions of cool core clusters typically display irregular features (e.g., cavities) believed to be associated with intermittent AGN feedback, it is tempting to speculate that the inability to detect a distinct stellar BCG component in massive cool core clusters reflects simply a strong violation of the approximation of hydrostatic equilibrium used to measure the mass profile. However, Newman and colleagues (Newman et al. 2013b) have used a combination of stellar dynamics and gravitational lensing to perform detailed studies of the radial mass profiles in several galaxy clusters. In their most recent study of 10 clusters (Newman et al. 2015), they also propose $\sim {10}^{14}\,{M}_{\odot }$ as the mass above which the total mass profile, rather than the DM, is well described by a single NFW component.

The consistent picture obtained by X-ray, stellar dynamics, and lensing studies provides strong evidence for the reality of this transition mass. This has important implications for models of cluster formation since $\sim {10}^{14}\,{M}_{\odot }$ appears to represent the mass scale where dissipative processes become important in the formation of the central regions of galaxy groups and clusters; i.e., above this mass the impact of dissipative processes on the total mass profile has been counteracted by late-time collisionless merging that re-establishes the NFW profile (e.g., Loeb & Peebles 2003; Laporte & White 2015).

As discussed in Section 5.6, the various mass components of RXJ 1159+5531 combine to produce a nearly power-law total mass profile with slowly varying logarithmic density slope ranging between $\alpha \approx -1.6$ to −2.0 within a radius ∼100 kpc ($\sim 10{R}_{e}$). This result is not strongly dependent on the assumed DM profile (NFW, Einasto, CORELOG). The slope α is consistent with that inferred from the scaling relation with R_e proposed by Humphrey & Buote (2010) and Auger et al. (2010).

This nearly power-law behavior in the total mass obeying scaling relations between α and R_e (and stellar density) for early-type galaxies is known as the "Bulge-Halo Conspiracy." Using empirically constrained ΛCDM models Dutton & Treu (2014) argue that a complex balance between feedback and baryonic cooling is required to explain this "conspiracy."

While some early X-ray studies indicated that fossil groups and clusters have unusually large NFW DM concentrations, essentially all of those measurements were inflated by neglecting to include the stellar BCG component in the mass modeling (e.g., Mamon & Lokas 2005). For the fossil cluster RX J1416.4 + 2315, Khosroshahi et al. (2006) accounted for the presence of the stellar BCG component and inferred a DM concentration of ${c}_{200}=11.2\pm 4.5$, which is about $2\sigma$ above the value of ∼4 expected for a relaxed halo with ${M}_{200}=3.1\times {10}^{14}\,{M}_{\odot }$ according to Dutton & Macciò (2014). The result we obtained for RXJ 1159+5531 in Section 5.4 is a $3\sigma$ discrepancy, providing even stronger evidence for an above-average concentration in a fossil cluster, suggestive of a halo that formed earlier than the general population. (We note that we have not included adiabatic contraction (e.g., Blumenthal et al. 1986; Gnedin et al. 2004) of the DM halo in our model, which would lead to an even larger concentration.) We caution, however, that as noted in Section 5.4, when the Einasto DM model is employed the significance of RXJ 1159+5531 as an outlier is somewhat reduced.

It is also worth emphasizing that from the perspective of relaxed, low-redshift X-ray clusters, the concentration of RXJ 1159+5531 is not very remarkable. The only study that has measured the X-ray concentration-mass relation for a sizable number of relaxed systems having masses both lower and higher than RXJ 1159+5531 is Buote et al. (2007), which also included the measurement of RXJ 1159+5531 by Gastaldello et al. (2007). Our value for RXJ 1159+5531 is only $\sim 1\sigma$ larger than the mean relation obtained by Buote et al. (2007). The higher normalization of the X-ray concentration-mass relation can be explained by including reasonable star formation and feedback in the cluster simulations (Rasia et al. 2013) ,which are not present in the DM-only simulations of Dutton & Macciò (2014).

Recently, Eckert et al. (2015) have used gas masses inferred from XMM-Newton and total masses from weak lensing to measure gas fractions at r₅₀₀ for a large cluster sample. They conclude that the gas fractions measured in this way are significantly smaller than those obtained from hydrostatic studies (e.g., Ettori 2015). By assuming that the weak-lensing masses are accurate, Eckert et al. (2015) infer a hydrostatic mass bias of ${0.72}_{-0.07}^{+0.08}$. When compared to numerical hydrodynamical simulations (Le Brun et al. 2014), the small gas fractions obtained by Eckert et al. (2015) favor an extreme feedback model in which a substantial amount of baryons are ejected from cluster cores.

From our hydrostatic analysis of RXJ 1159+5531 we measured within a radius r₅₀₀: ${M}_{500}=(5.9\pm 0.4)\times {10}^{13}\,{M}_{\odot }$ and ${f}_{{\rm{gas}}}=0.093\pm 0.003$ (Tables 2 and 3). For this value of M₅₀₀, the best-fitting relation for the gas fraction obtained by Eckert et al. (2015) gives ${f}_{{\rm{gas}}}=0.05$, implying a hydrostatic mass bias of $\approx 80 \%$. Since the wealth of evidence indicates that the ICM in RXJ 1159+5531 is very relaxed (see below in Section 7.4), including that the value of ${f}_{{\rm{gas}}}\approx 0.09$ we measure agrees much better with the cluster gas fractions produced in the simulations of Le Brun et al. (2014) considering plausible cooling and supernova feedback, we do not believe that such a large hydrostatic mass bias in RXJ 1159+5531 is supported by the present observations. Instead, we believe that these results for RXJ 1159+5531 support the suggestion by Eckert et al. (2015) that the weak lensing masses are biased high.

As we have noted previously in the related context of elliptical galaxies (see Section 8.2.2.1 of Buote & Humphrey 2012a), even without possessing direct, precise measurements of the hot gas kinematics, it is still possible to identify relaxed systems where the hydrostatic equilibrium approximation should be most accurate. RXJ 1159+5531 is a cool core cluster and displays a very regular X-ray image from the smallest scales probed by Chandra (with little or no evidence for AGN-induced disturbances) out to r₂₀₀. The Suzaku images mapping the radial region from $\sim {r}_{2500}-{r}_{200}$ with full azimuthal coverage display remarkably homogeneous ICM properties (Paper 1).

We are able to obtain a good representation of the X-ray data with hydrostatic models with reasonable values for the model parameters: (1) the stellar mass-to-light ratio is consistent with stellar population synthesis models (Section 5.4); (2) the value of the concentration, while statistically higher than the expected mean of the halo population, is within the $2\sigma$ cosmological intrinsic scatter (Section 5.4), and, at any rate, deviations from hydrostatic equilibrium due to additional non-thermal pressure support should tend to produce anomalously low, not high, concentrations by leading to smaller inferences of the virial mass and radius (e.g., see discussion in Section 6.2 of Buote et al. 2007); and (3) the baryon fraction is consistent with the cosmic value.

Although it is tempting to ascribe the very relaxed state of RXJ 1159+5531 to it being a fossil cluster, recent evidence suggests that the X-ray properties of fossil systems are not significantly distinguishable from the general cluster population (Girardi et al. 2014). The apparently highly relaxed ICM within r₂₀₀ implies that non-thermal pressure support is small throughout the cluster and, as such, constrains theories that predict a large amount of non-thermal pressure support from the magneto-thermal instability in the ICM outside of r₅₀₀ (e.g., Parrish et al. 2012).