Chandra X-Ray Galaxy Clusters at z < 1.4 z < 1.4 z ... - Springer Link

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galaxy clusters, namely, α decreases with increasing baryonic mass content. ... 2 α. [(r/r−2)α − 1],. (3) where ρ−2 is the density at r−2 and r−2 is the radius at.
c Pleiades Publishing, Ltd., 2014. ISSN 1063-7729, Astronomy Reports, 2014, Vol. 58, No. 9, pp. 587–610.  c Yu.V. Babyk, A. Del Popolo, I.B. Vavilova, 2014, published in Astronomicheskii Zhurnal, 2014, Vol. 91, No. 9, pp. 679–704. Original Russian Text 

Chandra X-Ray Galaxy Clusters at z < 1.4 1.4: Constraints on the Inner Slope of the Density Profiles Yu. V. Babyk1* , A. Del Popolo2, 3 , and I. B. Vavilova1 1

Main Astronomical Observatory, National Academy of Sciences of Ukraine, Kyiv, Ukraine 2 Universita` di Catania, Catania, Italy 3 Universidade de Sao ˜ Paulo, Sao ˜ Paulo, Brazil Received October 10, 2013; in final form, March 17, 2014

Abstract—A reconstruction of the total mass (the fraction of dark matter, intercluster gas, and the brightest galaxy of the cluster) of 128 X-ray galaxy clusters at redshifts 0.01–1.4 based on Chandra observations is presented. The total mass M200 and the baryonic mass Mb have been measured for all the sample objects, as well as the concentration parameter c200 , which characterizes the size of the dark a /(1 + z)b matter halo. The existence of a tight correlation between c200 and M200 is confirmed, c ∝ Mvir with a = −0.56 ± 0.15 and b = 0.80 ± 0.25 (95% confidence level), in good agreement with the predictions of numerical simulations and previous observations. Fitting the inner dark-matter density slope α with a generalized NFW model yields α = 1.10 ± 0.48 at the 2σ confidence level, combining the results for the entire sample, for which the model gives a good description of the data. There is also a tight correlation between the inner slope of the dark-matter density profile α and the baryonic mass content Mb for massive galaxy clusters, namely, α decreases with increasing baryonic mass content. A simple power-law model is used to fit the α − Mb distributions, yielding the break point for the inner slope of the dark-matter density profile b = 1.72 ± 0.37 (68% confidence level). DOI: 10.1134/S1063772914090017

1. INTRODUCTION

The NFW profile is given by ρ(r) =

The problem of the formation of dark matter (DM) halos has a long history going back to the seminal papers [1, 2], who studied the formation of the DM density profile in the framework of the collapse of a spherical perturbation in an expanding background (see [3, 4] for reviews, [5–7] for changes to halo formation due to dark energy, and [8] for limits of the DM scenario). Through this approach and several subsequent analytical models, such as [9–11], it was found that the DM density profiles can be described by power laws in the entire radius range. More recent semi-analytical models (e.g., [12–22]) have shown that these profiles are not power laws. The same conclusion was indicated by N -body simulations (e.g., [23, 24] (Navarro–Frenk–White, hereafter NFW), and [25–31]), which found that the spherically averaged density profiles of the DM halos were similar, regardless of the halo mass or cosmological model [32– 35]. *

ρc Δc ρ0 = , (r/rs )(1 + r/rs )2 (r/rs )(1 + r/rs )2 (1)

where ρc is the critical density of the Universe at the cluster redshift z, and Δc is the virial overdensity. The scale radius rs is connected to the virial radius rvir through the concentration parameter c = rvir /rs .1 NFW showed that the concentration parameter was larger for halos with smaller masses, since these formed before higher-mass halos, when the universe was denser. The concentration–mass relation has been considered in many studies. Several simple models for explaining the concentration-mass relation were proposed in [28, 37]. The model of [28] used two parameters: K, determining the initial concentration parameter of a collapsing halo, and F , the ratio of the initial collapsing mass and the final virial mass 1

E-mail: [email protected]

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The value of rvir in a spherical model for a collapse is analytically defined as rvir = 178 [for a Cold Dark Matter (CDM) model], approximated as r180 , while the density contrast corresponding to rvir = r101 [36] in the ΛCDM cosmology. The values of the virial radius required to reduce observations are equal, and usually taken to be r500 or r200 [27].

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of the halo at the current epoch. A two-parameter power-law model was used in [38, 39] and the more recent studies [37, 40]. The slope of the logarithmic density on small scales, also known as the inner slope, is given by  d log ρ  α=− = 1. (2) d log r  r→0

r −1

in the inner part, and This profile diverges as ρ ∝ behaves as ρ ∝ r −3 at large radii. An even steeper inner slope was obtained in [41] (ρ ∝ r −1.5 ). More recent simulations (e.g., [29, 31, 42, 43]) have shown that the DM density profiles are better fitted by an Einasto profile, which becomes shallower towards the halo center. An Einasto profile is a rolling power-law, first introduced to describe the distribution of stars in the Milky Way. It has the form 2 (3) ln(ρ/ρ−2 ) = − [(r/r−2 )α − 1], α where ρ−2 is the density at r−2 and r−2 is the radius at which the logarithmic slope of the density is isothermal (i.e., –2), analogous to rs in the NFW profile. Unfortunately, N -body simulations do not always yield results that correspond well to observations. Observations of the inner parts of the DM density profiles of dwarf galaxies and low-surface-brightness (LSB) dwarf galaxies are characterized by a core-like structure [44–52]. A similar problem is encountered in studies of the inner parts of the DM density profiles of clusters of galaxies, which have been investigated through both X-ray observations and strong and weak lensing. Computations of the masses of X-ray galaxy clusters usually assume that the intra-cluster medium (ICM) is in hydrostatic equilibrium and the cluster is approximately spherical. In this case,   d ln ρgas d ln T kT r − − , (4) M (r) = μmp G d ln r d ln r where k is the Boltzmann constant, T is the gas temperature, μ is the mean molecular weight of the gas, and mp is the proton mass. The gas density, ρgas , is usually described by the β model [53], which is based on the assumption that the galaxy mass density distribution is in good agreement with a King profile, and the gas is in hydrostatic equilibrium:  −3β/2 , (5) ρgas (r) = ρ0,gas (r) 1 + (r/rc,gas )2 where β is the beta parameter, rc,gas is the core radius of the gas, and ρ0,gas is the gas density at the center of the galaxy. The model’s parameters are determined by analyzing the X-ray surface brightness profile obtained from X-ray images. The total mass M (r) can

be obtained when ρgas and T (r) are known from Xray observations. An exact mass determination (in the NFW model) gives a consistent description of the mass distribution in relaxed clusters [54–56]. X-ray temperature measurements provide some information on the structures of galaxy clusters on scales of 50–500 kpc [36]. On smaller radii, temperature measurements are limited by the instrumental resolution and the presence of substructure in the cluster [57]. X-ray measurements are also limited by the presence of “cooling flows” and violation of the assumption of hydrostatic equilibrium [58]. For this reason, calculation of the inner slopes of DM density profiles based on X-ray data can yield discrepant values (e.g., 0.6 [59], 1.2 [60], 1.9 [61]). Another technique used to study the DM distribution in clusters is gravitational lensing. Weak lensing of background galaxies is used to reconstruct the mass distribution in the outer parts of clusters [62]. The resolution that can be achieved with modern telescopes is able to constrain profiles within 100 kpc. Strong lensing can be used to study the DM distribution in the inner parts of clusters, and typically is sensitive to the projected mass distribution inside 100−200 kpc (or 10−20 kpc at the limit of capabilities [63–66]). Discrepant results have sometimes been obtained when using the lensing method. Smith et al. [67] found α > 1 at 1% of rvir by analyzing the tangential and radial arcs of A383. A much smaller α value was obtained for the same cluster in [68, 69] using lensing and taking into account the stellar kinematics of the central region. Tyson et al. [70] found α = 0.57 ± 0.024 for CL 0024+1654, while Kneib et al. [71] found that a NFW profile fits the profile at radii from 0.1 to several times rvir . Sand et al. [72] found a profile for the central nuclear region with α = 0.35 for MS 2137.3-2353, while it was concluded in [63, 73] that the precise value of the slope depends on the mass-to-light ratio of the Brightest Cluster Galaxy (BCG). Thus, N -body simulations are not always in agreement with the inner slopes of the DM density profiles of dwarf galaxies, LSB dwarf galaxies, and clusters of galaxies, where the observations may even disagree for the same cluster [63, 64], [74, 75], [57], [36]. There could be several reasons for such discrepancies, such as (a) different approaches to calculating the slope, which sometimes corresponds to the dark matter and sometimes to the total mass; (b) use of observational techniques with different/limited radius ranges; (c) failure to take into account the stellar mass of the BCG. Obtaining firmer constraints on the central part of the DM density profiles requires the use of combined methods [71, 76], such as weak + strong ASTRONOMY REPORTS Vol. 58

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lensing [77], X-ray data + weak lensing [78], or applying better constraints on the central part of the DM density profiles using the stellar kinematics of the central galaxy (in the region of 1−200 kpc). In some studies of the clusters MS 2137-23, A383, A963, RXJ1133, MACS 1206, and A1201 [68, 72, 79], the DM halo was considered separately from the baryonic and stellar mass of the BCG. The derived DM density profiles were flatter than α < 1, except for the profile of RXJ1133. Newman et al. [69] found α < 1 for A383 (95% confidence level). In their detailed analysis of the DM and baryonic distributions in A611, Newman et al. [76] found α < 0.3 (68% confidence level). Thus, some galaxy clusters have inner DM density profile slopes shallower that those obtained in N body simulations, in better agreement with the profiles of dwarf galaxies and LSB dwarf galaxies [44, 46–50]. Note that, similarly to the results of Sand et al. [68, 72, 79] for the clusters refered to above, NGC 2976, NGC 4605, NGC 5949, NGC 5963, and NGC 6689 have inner slopes varying from very flat to cuspy (with a break); using a sample from “The HI Nearby Galaxy Survey” (THINGS), de Blok et al. [80] found that the best fits to the rotation curves, and then the inner slopes α of their DM density profiles, depend on their mass. An important factor for understanding this problem is the fundamental role of baryonic matter in determining the structural form of the density profiles; various processes can lead to the formation of a flatter inner DM density profile, such as the transfer of energy from stellar baryons to dark matter, stimulating heating of the DM and lowering the central density of DM [15, 21, 81–95]. Similar to the study of Schmidt and Allen [57] (hereafter SA07), we used Chandra X-ray data to study the properties of 128 dynamically relaxed galaxy clusters. We selected a sample with redshifts from 0.01 to 1.4, with the aim of determining their total and gas mass profiles and analyzing the measured distributions of c200 , M200 , and the baryonic mass content. We determine the baryon content of each cluster by subtracting the DM mass from the total mass. We emphasize that, when fitting the DM density profile using a generalized NFW model, ρ0 , (6) ρ(r) = α (r/rs ) (1 + r/rs )3−α the slope α is correlated with the baryonic mass of the cluster, in agreement with a number of studies [13, 96–101]. We assumed H0 = 73 km s−1 Mpc−1 and Ωm = 0.27. Section 2 describes our sample of X-ray galaxy clusters based on Chandra observations obtained to determine the baryonic and total mass profiles, as well as the techniques used. Section 3 presents our ASTRONOMY REPORTS

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main results, which are discussed in Section 4. Our conclusions are formulated in Section 5. 2. METHOD

2.1. Sample of Chandra X-Ray Galaxy Clusters at z < 1.4 The main criterion used to compile our sample of galaxy clusters was the X-ray flux. The X-ray flux should be as low as possible in order to form a representative sample, but must be fairly high to enable more exact measurement of the total masses of the clusters. The X-ray flux used in [102, 103] was fX = 7.0 × 10−12 erg s−1 cm−2 at energies 0.5–7.0 keV. We selected the clusters having the largest X-ray fluxes. We also applied a second criterion: that the clusters cover a wide range of redshifts. The sample cluster with the highest redshift is ISCS J1438+3414, with z ∼ 1.41, while most of the clusters have redshifts in the range 0.01–0.3, so that 0.01 ≤ z ≤ 1.41. We added 14 MACS clusters and 8 RCS clusters to our sample, as some of the most distant clusters observed by Chandra. As in SA07, all the sample clusters have mass-weighted Xray temperatures within the radius r2500 of kT2500 ≤ 5 keV. To reduce systematic errors and enable the most accurate tests of the predictions of the CDM model, we considered only highly relaxed clusters. All other clusters in the sample have regular X-ray morphologies, indicative of a relaxed state, enabling reliable determination of the total mass profile using the hydrostatic equilibrium equation. We did not include observations with exposure times shorter then 5 ks. This helped to decrease the uncertainties in the spectral modeling and the mass calculations. The results are presented in Table 1. As was already noted, we increased the number of clusters in our sample, compared to the sample of SA07. A comparison of our tables with those of SA07 indicates a number of clusters in common (e.g., A1795, A2029, A478, A1413, A2204, A383, A963, A1835, A611, A2537, MACS J0242.6-2132, MACS J1427.62521, MACS J2229.8-2756, MACS J1391.8-2635, MACS J1115.8+0129, MACS J1720.3+3536, MACS J1311.0-0311, MACS J1423.8+2404). Note that the coordinates in Table 1 are those from the NED database, which were, in turn, taken from [104]. Since these coordinates were determined in [104] using the Palomar telescope, there exist some differences between the NED optical coordinates and the corresponding peaks of the X-ray emission of the intracluster gas. These differences are not critical, and are essentially insignificant, but we have used the central coordinates from the maximum of the X-ray emission.

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Table 1. Summary of the Chandra observations. The columns list the (1) cluster name, (2) cluster redshift, (3) observation ID, (4) net exposure time, (5) detector used, (6) column density, and (7)–(8) central coordinates taken from NED Cluster 1 A85 A119 A133 A168 A209 A262 A383 A399 A401 A478 A496 A521 A539 A576 A644 A697 A754 A773 A907 A963 A1060 A1068 A1201 A1240 A1361 A1413 A1446 A1569 A1644 A1650 A1651 A1664 A1689 A1736 A1795 A1835 A1914 A1995 A2029 A2034 A2052

z 2 0.055 0.044 0.056 0.045 0.206 0.016 0.187 0.072 0.074 0.088 0.033 0.250 0.028 0.039 0.070 0.280 0.054 0.217 0.150 0.206 0.012 0.138 0.169 0.159 0.117 0.140 0.104 0.074 0.047 0.084 0.085 0.128 0.183 0.046 0.063 0.253 0.171 0.318 0.077 0.113 0.035

ObsID 3 904 7918 2203 3203 522 2215 524 3230 518 1669 931 430 5808 3289 2211 4217 577 533 3205 903 2220 1652 4216 4961 2200 537 4975 6100 2206 5823 4185 1648 540 4186 493 495 542 906 891 2204 890

texp , ks 4 38.91 45.63 35.91 41.12 10.09 29.12 10.09 49.28 18.24 42.94 19.16 39.60 24.60 39.10 30.08 19.70 44.77 10.50 47.70 36.76 32.32 27.17 40.17 52.03 16.96 9.34 59.12 41.75 18.96 40.13 9.77 9.90 10.45 15.12 19.88 19.77 8.17 58.23 20.07 54.70 37.23

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nH , 1020 cm−2 6 3.26 3.44 1.55 3.48 1.71 5.36 3.84 11.50 10.30 14.80 4.45 6.05 12.90 5.66 6.44 3.41 4.45 1.44 5.46 1.40 4.79 1.39 1.66 1.98 2.23 2.05 1.41 2.22 4.97 1.57 1.88 8.91 1.81 5.38 1.20 2.30 0.93 1.45 3.07 1.59 2.78

RA(J2000) 7 00h 41m 37.8s 00 56 21.4 01 02 39.0 01 15 09.8 01 31 53.0 01 52 50.4 02 48 06.9 02 57 56.4 02 58 56.9 04 13 20.7 04 33 37.1 04 54 10.1 05 16 35.1 07 21 24.1 08 17 26.4 08 42 53.3 09 08 50.1 09 17 59.4 09 58 21.1 10 17 13.9 10 36 51.3 10 40 47.1 11 13 01.1 11 23 32.1 11 43 45.1 11 55 18.9 12 01 51.5 12 36 18.7 12 57 14.8 12 58 46.2 12 59 22.9 13 03 41.8 13 11 29.5 13 26 52.1 13 49 00.5 14 01 02.0 14 26 03.0 14 52 50.4 15 10 56.0 15 10 13.1 15 16 45.5

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DEC(J2000) 8 –09◦ 20 33 –01 15 47 –21 57 15 +00 14 51 –13 36 42 +36 08 46 –03 29 32 +13 00 59 +13 34 56 +10 28 35 –13 14 46 –10 15 11 +06 27 14 +55 44 20 –07 35 21 +36 20 12 –09 38 12 +51 42 23 –11 03 22 +39 01 31 –27 31 35 +39 57 19 +13 25 40 +43 06 32 +46 21 21 +23 24 31 +58 01 18 +16 35 30 –17 21 13 –01 45 11 –04 11 10 –24 13 06 –01 20 17 –27 06 33 +26 35 07 +02 51 32 +37 49 32 +58 02 48 +05 44 41 +33 31 41 +07 00 01 No. 9

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Table 1. (Contd.) Cluster 1 A2063 A2065 A2124 A2142 A2147 A2163 A2199 A2204 A2218 A2219 A2244 A2256 A2319 A2390 A2462 A2537 A2589 A2597 A2634 A2657 A2667 A2670 A2717 A2744 A3112 A3158 A3266 A3376 A3391 A3395 A3526 A3558 A3562 A3571 A3667 A3827 A3921 A4038 A4059 AWM4 CLJ 1226.9+3332 Coma (A1656) IIIZw 54

z 2 0.035 0.073 0.065 0.091 0.035 0.203 0.030 0.152 0.175 0.225 0.097 0.058 0.056 0.228 0.073 0.295 0.041 0.085 0.031 0.040 0.230 0.076 0.049 0.308 0.075 0.060 0.059 0.045 0.051 0.050 0.011 0.048 0.049 0.039 0.055 0.098 0.093 0.030 0.047 0.032 0.890 0.023 0.0311

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ObsID 3 6263 3182 3238 1228 3211 545 497 499 553 896 4179 519 3231 500 4159 4962 3210 922 4816 4941 2214 4959 6973 2212 6972 3712 899 3202 4943 4944 504 1646 4167 4203 889 7920 4973 4992 897 9423 3180 10672 4182

texp , ks 4 17.04 50.09 19.61 12.26 18.12 9.57 19.72 10.20 5.96 42.84 57.72 15.00 14.62 9.96 39.74 36.67 13.86 39.86 50.16 16.36 9.77 40.14 47.65 25.14 30.16 31.35 30.14 44.85 18.69 22.17 32.12 14.61 19.54 34.44 50.96 46.17 29.76 33.97 41.19 75.46 32.12 28.91 23.76

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nH , 1020 cm−2 6 2.98 2.94 1.68 4.25 3.40 12.10 0.87 5.66 3.30 1.75 2.10 4.11 7.85 6.94 3.11 4.50 4.15 2.50 5.02 5.86 1.64 2.91 1.12 1.60 2.72 1.62 1.71 4.71 5.58 6.13 8.07 3.89 3.83 3.91 4.95 2.12 2.82 1.54 1.10 5.03 1.37 0.93 15.50

RA(J2000) 7 15 23 01.8 15 22 42.6 15 44 59.3 15 58 16.1 16 02 17.2 16 15 34.1 16 28 38.5 16 32 45.7 16 35 54.0 16 40 21.1 17 02 44.0 17 03 43.5 19 20 45.3 21 53 34.6 22 39 05.2 23 08 16.4 23 24 00.5 23 25 18.0 23 38 18.4 23 44 51.0 23 51 47.1 23 54 13.7 00 02 59.4 00 14 19.5 03 17 52.4 03 42 39.6 04 31 11.9 06 00 43.6 06 26 15.4 06 27 31.1 12 48 51.8 13 27 54.8 13 33 31.8 13 47 28.9 20 12 30.1 22 01 49.1 22 49 38.6 23 47 37.0 23 56 40.7 16 04 57.0 12 26 58.0 12 59 48.7 03 41 17.6

DEC(J2000) 8 +08 38 22 +27 43 21 +36 03 40 +27 13 29 +15 53 43 –06 07 26 +39 33 06 +05 34 43 +66 13 00 +46 41 16 +34 02 48 +78 43 03 +43 57 43 +17 40 11 –17 21 22 –02 10 44 +16 49 29 –12 06 30 +27 01 37 +09 08 40 –26 00 18 –10 25 08 –36 02 06 –30 23 19 –44 14 35 –53 37 50 –61 24 23 –40 03 00 –53 40 52 –54 23 58 –41 18 21 –31 29 32 –31 40 23 –32 51 57 –56 49 00 –59 57 15 –64 23 15 –28 07 42 –34 40 18 +23 55 14 +33 32 54 +27 58 50 +15 23 44

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Table 1. (Contd.) Cluster 1 ISCS J1438+3414 MACS J0011.7–1523 MACS J0159.8–0849 MACS J0242.6–2132 MACS J0429.6–0253 MACS J0647.7+7015 MACS J0744.8+3927 MACS J1115.8+0129 MACS J1311.0–0311 MACS J1423.8+2404 MACS J1427.6–2521 MACS J1720.3+3536 MACS J1931.8–2635 MACS J2129.4–0741 MACS J2229.8–2756 MKW 3s MKW 4 MKW 8 PKS 0745–19 RXC J0043.4–2037 RXC J0232.2–4420 RXC J0307.0–2840 RXC J0516.7–5430 RXC J0547.6–3152 RXC J0605.8–3518 RXC J1131.9–1955 RXC J2014.8–2430 RXC J2129.6+0005 RXC J2337.6+0016 Zw CL 1215 RCS J0224–0002 RCS J0439–2904 RCS J1107–0523 RCS J1419.2+5326 RCS J1620+2929 RCS J2156+0123 RCS J2318+0034 RCS J2319+0038 RX J0439.0+0520 RX J0848.7+4456 RX J0849+4452 RX J0910+5422 RX J1113.1–2615 RX J1221.4+4918

z 2 1.41 0.379 0.405 0.314 0.399 0.59 0.697 0.355 0.49 0.545 0.318 0.391 0.352 0.589 0.324 0.045 0.020 0.027 0.103 0.292 0.283 0.253 0.294 0.148 0.141 0.307 0.161 0.235 0.273 0.075 0.778 0.951 0.735 0.64 0.87 0.335 0.78 0.904 0.208 0.570 1.26 1.106 0.730 0.700

ObsID 3 10461 3261 3265 3266 3271 3196 536 3275 3258 4195 3279 3280 3282 3199 3286 900 3234 4942 508 9409 4993 9414 9331 9419 12899 3276 11757 552 3248 4184 4987 3577 5825 3240 3241 674 4938 5750 527 927 945 2227 915 1662

texp , ks 4 150.00 22.00 18.00 12.00 24.00 18.85 19.69 16.00 15.00 113.40 17.00 21.00 14.00 17.69 17.00 58.03 30.36 23.45 28.33 20.18 23.71 19.16 9.64 20.04 5.06 14.10 20.18 10.09 9.31 12.22 90.15 77.17 50.12 10.03 37.13 45.76 51.12 21.18 9.71 126.74 128.45 84.20 105.95 80.13

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nH , 1020 cm−2 6 2.01 2.05 2.08 2.69 5.25 5.63 5.68 4.39 1.88 2.38 6.18 3.37 9.11 4.84 1.34 3.04 1.90 2.80 42.70 1.54 2.42 1.36 6.80 2.02 4.47 4.46 7.77 4.29 3.82 1.74 2.92 2.63 4.24 1.18 2.72 2.56 4.13 4.16 10.50 2.66 2.50 2.35 5.47 1.45

RA(J2000) 7 14 33 22.4 00 11 42.9 01 59 48.0 02 42 36.0 04 29 36.0 06 47 45.9 07 44 51.8 11 15 53.3 13 11 00.0 14 23 48.3 14 27 39.4 17 20 15.5 19 31 48.0 21 29 26.0 22 29 48.0 15 21 51.9 12 03 57.7 14 40 43.1 07 47 31.32 00 43 23.1 02 32 18.7 03 07 01.1 05 16 35.2 05 47 34.2 06 05 52.4 11 32 00.7 20 14 49.7 21 29 37.9 23 37 39.7 12 17 40.60 02 24 00.0 04 39 38.0 11 07 22.80 14 19 12.0 16 20 09.40 15 47 34.2 23 18 30.67 23 19 53.00 04 39 02.2 08 48 47.2 08 53 43.6 09 10 45.36 11 13 05.2 12 21 24.5

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DEC(J2000) 8 +22 18 35.4 –15 23 22 –08 49 00 –21 32 00 –02 53 00 +70 15 03 +39 27 33 +01 29 47 –03 11 00 +24 04 47 –25 21 02 +35 36 21 –26 35 00 –07 41 28 –27 56 00 +07 42 31 +01 53 18 +03 27 11 –19 17 40.0 –20 37 35 –44 20 41 –28 40 30 –54 16 37 –31 53 01 –35 18 22 –19 53 34 –24 30 30 +00 05 39 +00 17 37 +03 39 45.0 –00 02 00 –29 04 55 –05 23 49.0 +53 26 00 +29 29 26.0 +26 38 29.0 +00 34 03.0 +00 38 00.0 +05 20 43 +44 56 17 +35 45 53.8 +54 22 07.3 –26 15 26 +49 18 13 No. 9

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2.2. Data Reduction The preliminary processing of the Chandra data, which were obtained using the Advanced CCD Imaging Spectrometer (ACIS), included: (a) the Chandra pipeline processing of the list of observations; (b) re-processing with the CIAO software package; (c) refinement of the rejection of cosmic ray events using the VFAINT mode; (d) cleaning of the data as recommended by the Chandra X-ray Centre (to remove periods of anomalously high background). The data reduction was performed using the Chandra CIAO v.4.2, including CalDB 4.5.1 for the mapping and calibration. The main data processing steps were carried out using the methods discussed in [105–110], namely: (a) removing point sources in the images using the wavdetect routine; (b) determining the X-ray peaks in the images after removal of the point sources; (c) centering the X-ray peaks with rings with different widths with the same number of X-ray photons in each;2 and (d) extracting the annular spectra. The average of the outermost radius is about 1 Mpc. We generated ARF and RMF files using the mkarf and mkrmf command in the CIAO package. The spectra were grouped to have at least 50 counts per spectral bin using the grppha command. Background spectra for galaxy clusters with 0.01 < z < 0.3 were created using blank-field data sets available from the Chandra X-ray center. The background spectra for more distant clusters were created using spectra for regions near the X-ray source with the same size, without any point sources. We usually used observations carried out on the ACIS chips 0, 1, 2, 3, and 7, since these chips are most accurately calibrated, although ACIS chips 4, 5, and 6 were also used. All the background spectra were cleaned in the same way as was done for the main spectra for the clusters. We used the DSDEPROJ routine method [111, 112] to process the annular spectra, in order to determine the X-ray temperature and other parameters. The DSDEPROJ deprojection routine assumes only spherical geometry, and enables the solution of some model-dependent problems. DSDEPROJ produces a set of “deprojected spectra,” which can then be fitted using a suitable spectral model in the Xspec spectral analysis package. Our use of this procedure showed that this method does not generate oscillating temperature profiles for multi-temperature clusters, and produces stable solutions for elongated clusters 2

We minimized systematic errors in the temperature profiles by requiring that the number of photons in each annulus be at least 5 × 103 and that the number of annuli be at least 10. ASTRONOMY REPORTS

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and for clusters with breaks in their temperature or density.3 The spectral fitting in Xspec (version 12.6) [113] was carried out using WABS*MEKAL model4 [41, 115–117]. The WABS(nH × MEKAL(T , Z, z, normMEKAL (EM))) model includes five parameters: the temperature, column density, metal abundance, redshift, and normalization parameter normMEKAL . The metal abundance was taken to be Z = 0.3 Z . The energy range 0.5–7.0 keV was used during the fitting. The errors were determined through the Xspec routine error. For each annulus, we determined the deprojected temperature distributions, kT , and the Emission  Measure, EM ∼ ne nH dV , which is proportional to the electron and hydrogen densities, ne and nH . The EM profiles were calculated in the band 0.5–7.0 keV, which provides the optimal ratio of the cluster and background flux in the Chandra data. Note that, for the clusters in our sample with the lowest redshifts, the statistical accuracy of the EM at large radii is limited primarily by the Chandra field of view.

2.3. Mass Profile of the Galaxy Clusters To calculate the total mass profiles of the galaxy clusters, we assumed that the clusters were spherically symmetrical and in hydrostatic equilibrium. We determined the total mass of the clusters using two independent methods. In the first, we parameterized the cluster total-mass profiles (luminous + dark matter) using an NFW model (slope α = 1, Method 1). The density profiles for the intergalactic gas were also calculated at this stage, analogous to SA07. In Method 2, we took the total mass profile to be composed of three components: the DM halo (fitted with a generalized NFW profile), the gaseous component (using an improvement of the beta model described in the Appendix), and the luminous component of the central galaxy (approximated with a Jaffe profile [118]). For the DM density profile, we used the NFW model with the DM density profile given by Eq. (1) and the mass profile within a specified radius given by  r r

3 , (7) − M (< r) = 4πρ0 rs ln 1 + rs rs + r where ρ0 and rs were defined in Section 1. 3

The DSDEPROJ source code is available at wwwxray.ast.cam.ac.uk/papers/dsdeproj. 4 MEKAL is a model describing the emission from a hot diffuse plasma (ICM), and WABS models the photo-electric absorption (taken from [114]).

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2.4. Contribution of the Diffuse X-ray Gas to the Cluster Mass Schmidt and Allen [57] obtained their gas profiles using Method 1, and did not include a functional form for the gas profile. In Method 2, it is necessary to apply a model for the gas density profile in order to approximate the contribution of the gas component to the total mass, and thereby derive the DM profile. The widely used beta model [53] given by Eq. (5) of Section 1 does not determine the gas distribution very accurately. Namely, it overestimates the baryon density at large radii in models with flat profiles at the cluster center, and yields biases in the gas distribution and gas fraction in clusters. According to SA07, the beta model has some influence on the DM profile, since the gas fraction is 10% of the total mass. We propose here an improved model for the gas distribution, which gives better results than the beta model (see the Appendix). 2.4.1. Stellar component. The studied clusters have a single, optically dominant BCG near their centers. This makes it possible to determine the mass of the stars in the central galaxy using the Jaffe model [118] with the same parameters as in SA07: ρ0,J , (8) ρJ (r) = (r/rc )2 (1 + r/rc )2 where ρ0,J is the central density and rc the core radius. The total-mass profile was obtained by adding the stellar mass to the generalized NFW potential of the dark matter and the mass of emitting gas calculated using the improved beta model. Since the model for the central galaxy had little effect on the results, we limited our modeling of the stellar mass in the cluster to the stellar component of the central galaxy (similarly to SA07), since the influence of the stellar mass of the other galaxies is even less than for the central one. Thus, the two parameters of the model, ρJ and rc , were fixed as follows: Re = 0.76rc = 25 kpc and ρ0,J = MJ /(4πrc3 ), where MJ = 1.14 × 1012 M is the estimated stellar mass of the BCG of MS21372353 [57, 72]. Recall that the effective radius Re is the radius containing half of the luminosity. The stellar mass of all the clusters was assumed to be the same. As was shown by SA07, using a different model for the stars of the central galaxy (for example, [119]) does not affect the results. 2.4.2. Best-fitting parameters. All the computations were made using the χ2 criterion  Vobs − Vmodel 2 2 , (9) χ = σobs N

where Vobs and Vmodel are the observed and modeled values. We determined the best-fitting parameter values for each mass model. The regions with residual

substructure (in the innermost regions) produced by various effects [102, 120, 121] were down-weighted in the mass analysis, similarly to SA07. In particular, we added 30% uncertainties in quadrature to the binned temperatures in regions with residual substructures. In other words, we discarded the corresponding central points in the fit of the NFW profile. 2.4.3. Mass profile scaling. Simple analytical and numerical simulations of structure formation predict that galaxy clusters should be self-similar, for both the DM component and the hot ICM. Scaling laws relate each physical property to the total mass M and redshift of the cluster [24, 122–126]. The DM density profiles are approximately universal on large mass scales [23, 127]. We scaled the mass profiles with the parameters R200 and M200 , where R200 is the radius within which the mean halo density is 200 times the critical density, and M200 is the total mass of the galaxy cluster inside a sphere with radius R200 . We defined the total mass within a given overdensity Δ as: 4 3 , (10) MΔ = πΔρc RΔ 3 3 = c r is the radius within where (see Section 1) RΔ Δ s which the mean cluster overdensity is Δ times ρc , and the relation with the concentration cΔ and the scale radius rs holds by definition for a NFW mass profile. We assumed Δ = 200 in our computations. An important aim of our study was to determine the influence of the baryonic content in a galaxy cluster on the slope of the density profile. We calculated the fraction Mb /M200 for each of the sample clusters, by subtracting the DM mass from the (2) total mass in each cluster. Taking the R200 values (2) for the clusters whose total-mass profiles M200 were modeled with our Method 2 (Table 3)—i.e., using a three-component model (stellar matter M∗ , diffuse gas Mgas , and dark matter MDM )—we can obtain the total mass for each cluster as: (2)

M200 = M∗ + Mgas + MDM = Mb + MDM . (11) Similarly, taking the R2001 values for the clusters modeled with Method 1 (Table 2)—i.e., using a twocomponent model (stellar matter and dark matter)— yields a total mass for each cluster (1)

(12)

M200 = M∗ + MDM . The baryonic mass is then given by (2)

(1)

Mb = M200 − M200 + M∗ = Mgas + M∗ .

(13)

As the result we obtain a relation between the obtained values of α, i.e., the fits to the dark matter component for a generalized NFW model, and the baryonic content Mb . ASTRONOMY REPORTS Vol. 58

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Table 2. Results of modeling the total mass profiles of the studied clusters of galaxies in the case of considering the three-component model cluster (dark matter halo, intercluster X-ray gas, the brightest galaxy clusters or cD) with an NFW model with slope α = 1. The columns list the (1) cluster name, (2) parameter rs , (3) concentration parameter c200 = R200 /rs , (4) radius within which the matter density is 200 times the critical density for the total and dark matter profiles R200 (in Mpc), (5) the total cluster mass at radius R200 , and (6) χ2 /dof. The confidence levels are 68% Cluster 1 A85 A119 A133 A168 A209 A262 A383 A399 A401 A478 A496 A521 A539 A576 A644 A697 A754 A773 A907 A963 A1060 A1068 A1201 A1240 A1361 A1413 A1446 A1569 A1644 A1650 A1651 A1664 A1689 A1736 A1795 A1835 A1914 A1995 A2029 A2034

rs , Mpc 2 0.76+0.15 −0.17 0.64+0.23 −0.22 0.29+0.15 −0.05 0.24+0.12 −0.10 0.65+0.18 −0.21 0.13+0.07 −0.06 0.46+0.16 −0.17 1.43+0.28 −0.36 0.96+0.36 −0.37 0.58+0.26 −0.21 0.17+0.09 −0.06 0.15+0.03 −0.04 0.18+0.05 −0.04 0.59+0.16 −0.17 1.33+0.43 −0.29 0.36+0.12 −0.13 3.08+0.39 −0.48 0.63+0.11 −0.13 0.27+0.06 −0.08 0.41+0.15 −0.13 0.19+0.07 −0.03 0.58+0.16 −0.19 0.25+0.11 −0.10 0.27+0.08 −0.10 0.28+0.04 −0.08 0.31+0.14 −0.10 0.23+0.05 −0.03 0.21+0.05 −0.04 0.15+0.04 −0.08 1.52+0.33 −0.29 0.57+0.16 −0.17 0.35+0.11 −0.14 0.27+0.05 −0.05 0.11+0.00 −0.01 0.53+0.20 −0.21 0.92+0.18 −0.27 3.62+0.48 −0.37 0.95+0.33 −0.28 0.42+0.17 −0.19 0.99+0.17 −0.15

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c200 3 3.55+0.26 −0.22 4.12+0.25 −0.23 8.11+0.25 −0.26 7.37+0.26 −0.27 3.11+0.77 −0.67 10.11+1.02 −0.98 3.44+0.47 −0.36 2.15+0.37 −0.33 3.19+0.38 −0.33 4.51+0.51 −0.48 11.26+0.83 −0.81 11.26+0.83 −0.78 10.37+2.04 −2.01 4.28+0.83 −0.81 2.19+0.25 −0.27 5.58+1.11 −1.22 1.02+0.27 −0.22 3.33+0.61 −0.53 6.25+1.01 −0.89 4.35+1.01 −0.99 10.27+2.45 −2.17 3.05+0.21 −0.22 6.27+0.83 −0.81 6.38+0.28 −0.15 7.28+0.83 −0.81 5.85+0.67 −0.44 9.16−0.67 −0.72 10.37+2.01 −2.04 15.35+3.89 −4.17 2.03+0.35 −0.37 4.29+0.74 −0.71 7.24+0.83 −0.81 8.35+0.91 −0.85 16.27+4.16 −4.13 4.61+0.55 −0.88 2.66+0.35 −0.44 1.04+0.36 −0.33 3.17+0.38 −0.33 6.99+0.41 −0.43 2.46+0.59 −0.62

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R200 , Mpc 4 2.71+0.65 −0.73 2.66+0.13 −0.15 2.33+0.19 −0.12 1.74+0.45 −0.56 2.01+0.54 −0.67 1.33+0.12 −0.09 1.57+0.21 −0.17 3.07+1.37 −1.36 3.07+1.21 −0.99 2.60+0.41 −0.55 1.93+0.48 −0.35 1.67+0.16 −0.12 1.88+0.33 −0.18 2.55+0.56 −0.73 2.92+0.77 −0.81 2.04+0.18 −0.22 3.15+1.26 −1.15 2.11+0.26 −0.35 1.67+0.15 −0.21 1.77+0.14 −0.18 1.99+0.36 −0.34 1.77+0.14 −0.06 1.56+0.51 −0.48 1.73+0.66 −0.58 2.01+0.84 −0.81 1.83+0.66 −0.57 2.11+0.48 −0.43 2.22+0.84 −0.81 2.25+0.39 −0.36 3.10+0.72 −0.73 2.45+0.46 −0.37 2.55+0.72 −0.77 2.23+0.03 −0.07 1.85+0.42 −0.37 2.46+0.36 −0.54 2.45+0.31 −0.24 3.77+0.76 −0.67 3.01+0.43 −0.42 2.94+0.96 −0.82 2.45+0.51 −0.50

M200 , 1014 M 5 25.68+4.89 −3.18 24.06+4.16 −2.18 16.34+3.55 −2.17 6.74+2.03 −1.78 12.05+2.34 −2.11 2.93+0.73 −0.67 5.63+2.04 −1.19 37.90+3.94 −4.11 37.97+4.16 −2.81 23.35+2.62 −2.17 9.10+1.19 −1.27 7.21+1.16 −1.13 8.38+1.46 −1.77 21.11+2.16 −1.19 32.55+3.51 −2.99 13.56+1.27 −1.14 40.30+3.71 −3.16 14.09+1.72 −1.28 6.54+0.73 −0.72 8.22+1.24 −0.91 9.81+1.32 −1.16 7.71+0.71 −0.78 5.43+0.59 −0.66 7.34+0.94 −0.81 11.07+1.34 −1.18 8.53+1.18 −0.92 12.66+1.26 −1.15 14.35+1.34 −1.11 14.60+1.47 −1.17 39.44+3.74 −2.78 19.48+2.45 −2.17 22.84+2.81 −2.19 16.09+2.81 −1.73 8.11+1.15 −1.18 19.34+2.18 −2.16 22.86+2.86 −3.11 76.85+5.81 −5.28 45.32+5.16 −3.18 33.43+3.71 −4.12 19.98+2.54 −1.37

χ2 /dof 6 12.31/8 11.57/9 31.37/8 22.11/8 11.03/9 18.35/10 52.02/9 42.47/8 32.47/9 10.84/11 10.92/10 15.29/14 33.32/8 12.03/17 19.05/16 45.32/8 16.38/12 17.77/11 41.38/13 25.83/11 23.16/9 12.04/14 11.22/10 12.11/13 13.00/17 31.99/9 31.37/9 15.12/12 48.11/8 18.22/14 18.28/11 15.66/14 41.54/12 44.33/8 16.33/11 52.77/8 13.28/8 12.22/8 14.20/9 14.38/9

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Table 2. (Contd.) Cluster 1 A2052 A2063 A2065 A2124 A2142 A2147 A2163 A2199 A2204 A2218 A2219 A2244 A2256 A2319 A2390 A2462 A2537 A2589 A2597 A2634 A2657 A2667 A2670 A2717 A2744 A3112 A3158 A3266 A3376 A3391 A3395 A3526 A3558 A3562 A3571 A3667 A3827 A3921 A4038 A4059 AWM4 CLJ 1 226.9+3332 Coma (A1656) IIIZw 54

rs , Mpc 2 0.19+0.03 −0.04 0.27+0.14 −0.10 1.31+0.21 0.18 0.19+0.04 −0.02 0.54+0.13 −0.10 0.20+0.02 −0.03 2.07+0.37 −0.35 0.34+0.09 −0.10 0.90+0.11 −0.07 0.26+0.01 −0.03 0.84+0.15 −0.17 1.24+0.15 −0.17 2.87+0.44 −0.37 2.57+0.32 −0.19 1.26+0.14 −0.16 0.19+0.03 −0.06 0.44+0.12 −0.14 0.32+0.09 −0.06 0.24+0.05 −0.03 0.18+0.05 −0.03 0.36+0.11 −0.10 1.04+0.23 −0.22 0.44+0.13 −0.16 0.35+0.05 −0.04 .30 1.380−0.27 +0.04 0.26−0.03 1.24+0.13 −0.14 0.35+0.05 −0.05 0.37+0.03 −0.04 0.26+0.05 −0.03 0.79+0.11 −0.10 0.18+0.04 −0.03 0.30+0.04 −0.07 0.27+0.05 −0.06 0.32+0.06 −0.03 0.29+0.06 −0.04 0.87+0.09 −0.13 0.53+0.08 −0.07 0.20+0.02 −0.01 0.22+0.02 −0.00 0.24+0.03 −0.02 1.75+0.13 −0.11 2.35+0.43 −0.55 0.17+0.01 −0.02

c200 3 10.33+1.39 −1.33 7.36+0.38 −0.33 2.38+0.64 −0.56 11.36+2.84 −2.81 5.27+0.83 −0.79 10.46+3.81 −4.27 2.04+0.37 −0.33 6.27+0.25 −0.26 2.83+0.32 −0.44 6.33+2.34 −1.55 3.44+0.73 −0.73 2.43+0.84 −0.81 1.16+0.37 −0.27 1.28+0.27 −0.22 2.11+0.52 −0.62 13.23+0.72 −0.67 4.88+0.24 −0.24 6.27+0.75 −0.72 8.28+1.27 −1.26 11.38+2.17 −2.16 5.49+0.83 −0.81 2.25+0.17 −0.15 5.33+1.11 −1.09 6.12+0.84 −0.81 1.72+0.15 −0.17 9.36+0.73 −0.73 2.19+0.80 −0.81 8.36+0.83 −0.81 7.38+1.01 −1.03 9.26+1.14 −1.36 3.74+0.35 −0.37 10.43+0.81 −0.83 8.37+0.78 −0.83 8.26+0.25 −0.27 8.92+1.98 −2.10 9.36+1.25 −1.36 4.45+0.67 −0.77 5.27+0.73 −0.71 9.38+1.03 −1.01 9.36+2.02 −2.01 7.21+1.02 −1.03 2.04+0.32 −0.36 1.37+0.25 −0.27 11.25+1.33 −1.38

R200 , Mpc 4 1.93+0.55 −0.61 2.01+0.45 −0.37 3.11+0.46 −0.37 2.16+0.63 −0.61 2.82+0.59 −0.61 2.05+0.14 −0.17 4.22+0.68 −0.66 2.14+0.28 −0.19 2.55+0.71 −0.66 1.65+0.16 −0.12 2.89+0.78 −0.91 3.01+0.51 −0.39 3.33+0.62 −0.58 3.29+1.12 −1.29 2.65+0.61 −0.54 2.51+0.27 −0.22 2.15+0.63 −0.55 1.99+0.63 −0.51 2.01+0.57 −0.73 2.11+0.05 −0.11 1.98+0.15 −0.11 2.35+0.45 −0.36 2.33+1.03 −1.02 2.16+0.79 −0.81 2.38+0.36 −0.31 2.43+0.31 −0.26 2.71+0.38 −0.62 2.95+0.52 −0.77 2.75+0.43 −0.62 2.45+0.84 −0.35 2.97+0.96 −0.84 1.88+0.25 −0.27 2.55+0.38 −0.36 2.22+0.15 −0.17 2.88+0.46 −0.37 2.75+0.61 −0.52 3.89+0.16 −0.43 2.81+0.33 −0.26 1.87+0.25 −0.17 2.03+0.28 −0.34 1.73+0.27 −0.22 3.57+0.51 −0.52 3.22+0.59 −0.62 1.91+0.51 −0.43

M200 , 1014 M 5 +1.26 9.12−1.72 +1.94 10.30−1.29 +5.82 39.43−4.18 +1.37 13.12−1.27 +2.47 29.87−3.17 +1.27 10.93−1.28 +5.92 111.18−7.38 +2.18 12.38−1.29 +3.16 23.35−2.17 +0.82 6.46−0.83 +4.92 36.49−3.39 +3.16 36.52−3.18 +5.37 47.78−5.29 +4.57 46.00−5.82 +3.15 28.21−2.39 +2.47 20.73−2.49 +2.16 16.12−2.19 +1.27 10.05−1.29 +1.27 10.76−1.22 +2.15 11.88−1.27 +2.17 9.89−1.28 +2.61 19.71−1.27 +1.25 16.62−1.62 +2.11 12.94−1.26 +2.14 22.17−2.26 +3.14 18.84−2.16 +3.17 25.80−2.72 +4.16 33.25−3.27 +3.72 26.61−2.91 +1.25 18.91−1.28 +4.27 33.67−4.12 +0.72 8.26−0.28 +3.27 21.27−2.16 +1.56 14.05−1.46 +3.75 30.41−3.84 +2.48 26.84−2.46 +5.26 78.90−4.27 +2.15 29.61−1.77 +1.18 8.26−0.92 +0.92 10.72−1.28 +0.85 6.55−0.37 143.19+15.26 −17.26 +4.72 41.94−3.92 +1.04 8.81−0.93

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χ2 /dof 6 11.32/9 11.92/8 11.20/9 11.47/9 23.29/11 15.28/10 16.75/10 11.17/11 15.28/9 81.10/9 25.18/8 41.19/8 12.11/8 11.22/13 13.02/15 15.05/16 21.66/11 12.48/10 21.12/9 13.47/9 11.55/9 41.28/12 23.21/9 31.83/9 21.36/9 34.27/9 31.22/9 31.21/8 11.29/9 21.02/9 12.07/9 12.66/10 21.33/11 21.17/11 13.36/15 13.22/13 16.02/17 16.33/13 15.32/11 17.37/11 18.22/10 14.45/10 16.44/13 18.66/12 No. 9

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Table 2. (Contd.) Cluster 1 ISCS J1438.1+3414 MACS J0011.7–1523 MACS J0159.8–0849 MACS J0242.6–2132 MACS J0429.6–0253 MACS J0647.7+7015 MACS J0744.8+3927 MACS J1115.8+0129 MACS J1311.0–0311 MACS J1423.8+2404 MACS J1427.6–2521 MACS J1720.3+3536 MACS J1931.8–2635 MACS J2129.4–0741 MACS J2229.8–2756 MKW3s MKW4 MKW8 PKS 0745–191 RXC J0043.4–2037 RXC J0232.2–4420 RXC J0307.0–2840 RXC J0516.7–5430 RXC J0547.6–3152 RXC J0605.8–3518 RXC J1131.9–1955 RXC J2014.8–2430 RXC J2129.6+0005 RXC J2337.6+0016 Zw CL 1215 RCS J0224–0002 RCS J0439–2904 RCS J1107–0523 RCS J1419.2+5326 RCS J1620+2929 RCS 2156+0123 RCS J2318+0034 RCS J2319+0038 RX J0439.0+0520 RX J0848.7+4456 RX J0849+4452 RX J0910+5422 RX J1113.1–2615 RX J1221.4+4918 ASTRONOMY REPORTS

rs , Mpc 2 6.81+0.55 −0.72 0.75+0.13 −0.23 0.54+0.12 −0.13 0.32+0.06 −0.04 0.65+0.13 −0.10 3.42+0.35 −0.36 3.44+0.39 −0.37 1.07+0.09 −0.11 0.60+0.07 −0.06 0.85+0.29 −0.22 0.30+0.17 −0.32 0.54+0.10 −0.11 0.65+0.05 −0.06 2.06+0.32 −0.27 0.35+0.06 −0.03 0.18+0.03 −0.03 0.06+0.01 −0.00 0.12+0.02 −0.01 0.45+0.06 −0.06 0.19+0.03 −0.01 1.19+0.13 −0.16 0.62+0.07 −0.06 0.83+0.09 −0.06 0.48+0.06 −0.07 0.40+0.02 −0.01 0.83+0.11 −0.02 0.50+0.07 −0.04 0.47+0.07 −0.06 0.35+0.06 −0.03 0.52+0.08 −0.06 0.56+0.09 −0.07 0.25+0.01 −0.03 0.40+0.03 −0.03 0.28+0.03 −0.05 0.31+0.05 −0.06 1.93+0.12 −0.24 1.83+0.13 −0.15 0.71+0.11 −0.09 0.24+0.05 −0.03 1.40+0.14 −0.17 1.89+0.27 −0.25 0.96+0.15 −0.09 0.38+0.06 −0.04 1.10+0.15 −0.10

c200 3 0.55+0.05 −0.08 4.01+0.23 −0.33 5.35+0.82 −0.72 7.88+1.64 −1.23 3.36+1.76 −1.65 1.00+0.12 −0.11 1.01+0.37 −0.16 2.23+0.53 −0.55 5.01+0.73 −0.35 3.33+0.65 −0.63 8.28+1.99 −1.77 5.25+0.61 −0.36 4.11+0.36 −0.24 1.70+0.38 −0.41 8.54+1.67 −1.36 11.37+2.18 −2.73 19.93+3.15 −3.17 15.28+3.14 −3.16 6.45+0.73 −0.61 8.01+3.89 −3.03 1.88+0.67 −0.66 3.22+0.88 −0.77 2.45+1.55 −0.77 4.14+0.57 −0.63 4.14+0.33 −0.23 2.55+1.24 −1.67 3.88+0.54 −0.67 3.77+0.38 −0.33 5.02+1.55 −1.77 5.62+0.42 −0.40 2.78+0.25 −0.25 4.12+0.25 −0.21 3.15+0.56 −0.45 6.24+0.73 −0.71 4.38+0.72 −0.71 1.23+0.41 −0.30 1.18+0.19 −0.17 3.27+0.56 −0.55 7.71+1.27 −1.77 0.82+0.05 −0.01 1.14+0.11 −0.10 2.64+0.25 −0.21 3.28+0.73 −0.71 2.39+0.37 −0.35

Vol. 58 No. 9 2014

R200 , Mpc 4 3.75+0.33 −0.38 3.01+0.30 −0.31 2.89+0.51 −0.52 2.55+0.46 −0.45 2.17+0.78 −0.84 3.42+0.45 −0.54 3.48+0.45 −0.47 2.39+0.64 −0.63 3.02+0.33 −0.35 2.83+0.17 −0.32 2.47+0.81 −0.84 2.83+0.68 −0.67 2.67+0.91 −0.92 3.51+0.71 −0.72 3.02+0.81 −0.84 2.11+0.12 −0.18 1.21+0.10 −0.04 1.81+0.33 −0.28 2.92+0.22 −0.18 1.58+0.17 −0.17 2.25+0.34 −0.43 2.01+0.26 −0.29 2.05+0.32 −0.25 1.99+0.14 −0.43 1.66+0.18 −0.15 2.11+0.34 −0.28 1.95+0.12 −0.17 1.77+0.22 −0.21 1.75+0.22 −0.18 2.95+0.56 −0.63 1.56+0.16 −0.17 1.04+0.14 −0.12 1.26+0.29 −0.27 1.76+0.28 −0.21 1.35+0.18 −0.17 2.38+0.28 −0.29 2.17+0.37 −0.31 2.34+0.48 −0.29 1.83+0.48 −0.36 1.15+0.15 −0.13 2.16+0.52 −0.51 2.53+0.63 −0.63 1.26+0.15 −0.11 2.64+0.31 −0.35

M200 , 1014 M 5 293.15+18.27 −36.84 +3.82 48.34−4.15 +4.84 44.00+4.44 +1.99 27.44−2.19 +1.82 18.50−1.63 89.51+10.14 −9.28 +8.82 106.48−6.18 +4.17 23.59−3.18 +4.84 55.11−5.58 +4.72 48.21−4.72 +2.36 25.04−2.26 +5.17 40.70−4.92 +3.91 32.78−4.19 96.66+10.26 −10.15 +4.92 46.06−5.17 +2.47 12.02−3.19 +1.84 2.22−2.10 +1.48 7.47−2.57 +3.17 33.52−3.17 +0.82 6.38−0.84 +2.16 18.25−1.82 +1.72 12.62−1.82 +1.54 13.96−1.22 +1.17 11.06−1.10 +0.75 6.37−0.72 +1.45 15.43−1.66 +1.48 10.53−1.47 +0.94 8.46−1.10 +1.01 8.50−0.89 +4.17 33.71−4.29 +1.11 10.51−1.28 +0.56 3.79−0.27 +0.74 5.27−0.45 +1.64 12.91−1.72 +1.02 7.57−1.04 +3.28 22.80−2.88 +4.11 28.37−3.18 +4.25 40.96−5.22 +1.75 9.11−2.27 +0.73 3.32−0.77 +0.66 4.67−0.47 +5.67 64.92−7.34 +0.73 5.30−0.48 +5.22 46.64−3.49

χ2 /dof 6 17.73/8 13.43/9 11.49/8 14.33/8 51.38/9 11.22/9 14.28/9 13.21/8 12.18/9 11.88/9 12.18/8 11.72/9 11.44/11 21.46/10 31.33/10 11.22/10 21.37/10 11.92/9 12.82/8 11.11/8 12.22/9 11.38/9 14.72/9 12.34/8 14.38/9 12.88/8 22.83/8 21.33/8 21.18/8 11.04/8 21.09/9 21.20/9 21.05/9 14.48/9 15.47/9 13.05/9 25.10/9 11.36/9 15.46/8 21.66/8 16.88/8 51.46/9 21.22/8 51.07/9

598

BABYK et al.

Table 3. Results for the DM profiles of the Chandra X-ray galaxy clusters for the three-component fits. The columns list the (1) cluster name; (2)–(6) parameters for the NFW fit: rs , c200 , mass of baryonic matter Mb , fraction of baryonic matter, and χ2 /dof; and (7)–(8) the parameters of the generalized NFW: the inner slope α and χ2 /dof. The confidence levels are 68% Generalized NFW model (α is free parameter)

NFW model (α = 1) Cluster rs , Mpc

c200

Mb , 1013 M

fMb /Mtot

χ2 /dof

α

χ2 /dof

2

3

4

5

6

7

8

A85

0.69+0.13 −0.20

3.15+0.23 −0.32

42.86+4.83 −6.46

0.166

10.11/8

+0.16 0.85−0.28

12.11/7

A119

0.54+0.13 −0.12 0.24+0.13 −0.06 0.18+0.07 −0.06 0.58+0.08 −0.11 0.11+0.03 −0.02 0.43+0.13 −0.16 1.22+0.38 −0.46 0.84+0.26 −0.27 0.51+0.16 −0.11 0.13+0.06 −0.06 0.11+0.05 −0.05 0.15+0.04 −0.05 0.51+0.13 −0.12 1.12+0.33 −0.19 0.20+0.10 −0.05 2.84+0.29 −0.38 0.55+0.09 −0.10 0.22+0.04 −0.05 0.35+0.10 −0.11 0.15+0.05 −0.02 0.53+0.11 −0.13 0.20+0.10 −0.06 0.21+0.08 −0.10 0.22+0.06 −0.06 0.24+0.10 −0.11 0.21+0.05 −0.03 0.19+0.05 −0.04 0.12+0.04 −0.08 1.44+0.33 −0.29 0.51+0.16 −0.17 0.29+0.11 −0.14 0.21+0.05 −0.05 0.10+0.00 −0.01

3.83+0.35 −0.33 7.83+0.35 −0.46 7.03+0.17 −0.20 2.86+0.27 −0.37 9.75+1.11 −0.93 3.32+0.27 −0.32 2.01+0.47 −0.43 3.00+0.28 −0.23 4.04+0.41 −0.58 10.73+1.83 −1.81 10.77+1.43 −1.28 10.21+1.44 −1.21 4.03+0.33 −0.21 2.00+0.22 −0.21 5.21+0.61 −0.52 0.84+0.17 −0.12 3.04+0.41 −0.33 6.01+1.11 −0.80 4.13+0.51 −0.69 9.58+1.45 −1.17 2.75+0.51 −0.32 5.88+0.63 −0.61 6.11+0.18 −0.25 6.81+0.63 −0.51 5.85+0.67 −0.44 9.06−0.67 −0.72 10.37+2.01 −2.04 14.65+3.89 −4.17 1.83+0.35 −0.37 3.69+0.74 −0.71 6.64+0.83 −0.81 7.15+0.91 −0.85 15.87+4.16 −4.13

47.97+8.35 −9.12 35.02+3.87 −6.45 19.76+2.17 −3.17 10.47+1.85 −1.82 7.25+1.53 −1.67 11.03+3.11 −2.18 94.53+10.11 −9.16 +6.31 75.91−5.83 30.86+2.88 −3.17 10.86+1.51 −1.22 14.47+1.95 −2.17 15.06+1.25 −1.19 23.87+2.14 −2.11 44.62+3.17 −2.91 13.49+1.17 −1.19 61.80+6.17 −5.19 24.47+2.15 −1.98 13.12+1.13 −1.19 22.63+2.14 −2.19 24.28+2.16 −1.93 14.64+1.16 −1.19 9.79+1.15 −0.93 16.42+1.73 −1.67 21.56+2.15 −2.17 14.48+1.42 −1.18 28.20+2.16 −2.17 37.01+3.26 −3.17 33.95+4.12 −3.16 81.43+5.85 −4.81 37.81+3.75 −3.61 40.35+3.71 −3.78 39.52+4.17 −3.92 11.26+1.93 −2.18

0.199

21.46/9

12.57/8

0.214

11.55/8

0.293

11.37/8

0.087

24.01/9

0.246

13.64/10

0.195

17.55/9

0.249

15.89/8

0.199

24.05/9

0.132

15.44/11

0.119

42.46/10

0.200

10.42/14

0.179

19.79/8

0.113

16.11/17

0.137

10.37/16

0.099

10.99/8

0.153

15.53/12

0.173

11.73/11

0.200

27.00/13

0.275

23.75/11

0.247

27.47/9

0.189

10.89/14

0.180

10.80/10

0.223

20.23/13

0.194

10.94/17

0.169

12.69/9

0.222

27.22/9

0.257

24.57/12

0.232

22.32/8

0.206

21.06/14

0.194

11.94/11

0.176

11.76/14

0.245

12.45/12

0.138

11.38/8

+0.27 0.82−0.25 +0.08 0.69−0.08 +0.17 1.01−0.11 +0.16 1.72−0.13 +0.03 0.69−0.04 +0.22 1.58−0.22 +0.17 0.51−0.15 +0.29 0.86−0.37 +0.22 0.96−0.16 +0.05 0.82−0.07 +0.12 0.93−0.13 +0.21 1.08−0.22 +0.18 0.74−0.18 +0.11 1.03−0.06 +0.14 0.94−0.13 +0.12 1.08−0.09 +0.13 0.82−0.11 +0.13 1.12−0.12 +0.13 1.52−0.12 +0.08 0.89−0.11 +0.19 0.98−0.19 +0.10 1.72−0.10 +0.08 0.91−0.08 +0.29 0.80−0.28 +0.08 0.93−0.08 +0.73 1.34−0.72 +0.18 0.67−0.19 +0.07 0.75−0.07 +0.12 0.43−0.22 +0.22 0.91−0.08 +0.26 0.72−0.25 +0.18 1.10−0.16 +0.13 1.08−0.27

1

A133 A168 A209 A262 A383 A399 A401 A478 A496 A521 A539 A576 A644 A697 A754 A773 A907 A963 A1060 A1068 A1201 A1240 A1361 A1413 A1446 A1569 A1644 A1650 A1651 A1664 A1689 A1736

ASTRONOMY REPORTS Vol. 58

21.22/7 21.04/7 71.53/8 12.39/9 71.02/8 12.47/7 22.13/8 10.24/10 10.42/9 12.19/13 23.32/7 14.63/16 13.55/15 41.32/7 13.98/11 47.22/10 11.18/12 14.83/10 33.16/8 15.54/13 23.22/9 21.21/12 11.50/16 10.99/8 24.37/8 2.12/10 3.11/7 25.22/13 21.18/10 25.36/13 16.54/11 26.34/7 No. 9

2014

CHANDRA X-RAY GALAXY CLUSTERS

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Table 3. (Contd.) Generalized NFW model (α is free parameter)

NFW model (α = 1) Cluster rs , Mpc

c200

Mb , 1013 M

fMb /Mtot

χ2 /dof

α

χ2 /dof

1

2

3

4

5

6

7

8

A1795 A1835 A1914 A1995 A2029 A2034 A2052 A2063 A2065 A2124 A2142 A2147 A2163 A2199 A2204 A2218 A2219 A2244 A2256 A2319 A2390 A2462 A2537 A2589 A2597 A2634 A2657 A2667 A2670 A2717 A2744 A3112 A3158 A3266 A3376 A3391 A3395 A3526 A3558 A3562

0.43+0.20 −0.21 0.87+0.18 −0.27 3.05+0.48 −0.37 0.88+0.33 −0.28 0.37+0.17 −0.19 0.90+0.23 −0.22 0.13+0.03 −0.04 0.22+0.11 −0.05 1.22+0.21 0.18 0.11+0.04 −0.02 0.50+0.12 −0.11 0.16+0.02 −0.03 2.00+0.47 −0.45 0.24+0.09 −0.10 0.76+0.21 −0.17 0.21+0.05 −0.03 0.66+0.15 −0.17 1.12+0.25 −0.27 2.67+0.44 −0.17 2.49+0.22 −0.29 1.17+0.34 −0.26 0.13+0.03 −0.06 0.40+0.12 −0.11 0.31+0.09 −0.07 0.18+0.05 −0.03 0.15+0.05 −0.03 0.33+0.11 −0.10 1.00+0.23 −0.12 0.41+0.13 −0.16 0.30+0.05 −0.04 .30 1.320−0.37 +0.04 0.21−0.03 1.20+0.13 −0.24 0.30+0.05 −0.05 0.32+0.03 −0.04 0.21+0.05 −0.03 0.71+0.11 −0.20 0.10+0.04 −0.03 0.22+0.04 −0.07 0.22+0.05 −0.06

4.21+0.55 −0.88 2.36+0.35 −0.44 0.74+0.36 −0.33 2.77+0.38 −0.33 6.39+0.41 −0.43 2.37+0.49 −0.42 9.63+1.19 −1.13 6.89+0.48 −0.43 2.02+0.34 −0.46 10.76+1.84 −1.81 5.10+0.43 −0.49 10.06+1.81 −1.27 1.88+0.27 −0.23 5.98+0.45 −0.36 2.56+0.42 −0.54 6.13+1.14 −0.65 3.24+0.43 −0.53 2.23+0.34 −0.31 1.06+0.47 −0.37 1.08+0.37 −0.32 1.91+0.42 −0.32 12.63+0.42 −0.37 4.18+0.34 −0.44 5.27+0.45 −0.52 8.02+0.77 −0.86 10.68+1.17 −1.16 5.14+0.43 −0.51 2.21+0.47 −0.25 5.01+0.51 −0.59 5.88+0.54 −0.51 1.52+0.35 −0.57 9.16+0.43 −0.63 2.02+0.60 −0.61 8.10+0.53 −0.51 7.01+1.11 −1.13 9.06+0.54 −0.76 3.34+0.45 −0.47 10.13+1.11 −1.23 7.89+0.48 −0.43 7.96+0.55 −0.57

20.46+3.16 −2.19 41.93+5.16 −4.28 93.76+9.61 −10.27 56.22+6.17 −4.91 42.42+5.18 −4.19 21.22+1.93 −2.18 13.45+2.46 −1.28 20.06+2.47 −2.46 94.05+10.27 −11.28 20.67+2.15 −2.19 21.69+3.16 −1.28 18.09+2.52 −1.27 143.51+14.92 −15.37 24.25+2.38 −2.19 43.67+5.82 −5.19 8.95+2.15 −0.93 47.06+5.38 −4.19 35.20+4.16 −3.82 57.77+5.97 −6.92 71.45+8.14 −6.93 50.89+5.67 −4.86 28.33+3.14 −3.11 31.45+4.28 −3.19 18.43+3.15 −2.91 12.34+1.27 −1.28 22.11+2.63 −2.18 19.53+2.15 −2.72 37.59+4.72 −3.18 30.09+3.61 −3.18 25.13+2.34 −2.15 48.97+5.16 −4.72 13.61+2.17 −1.27 16.76+2.15 −1.27 42.05+5.42 −4.17 30.67+4.27 −3.27 30.61+4.28 −3.27 45.40+6.87 −5.28 18.24+2.36 −1.92 28.64+4.25 −2.37 33.07+4.27 −3.27

0.105 0.183 0.121 0.124 0.126 0.106 0.147 0.194 0.238 0.157 0.072 0.165 0.129 0.195 0.186 0.138 0.128 0.096 0.120 0.155 0.180 0.136 0.195 0.183 0.114 0.186 0.197 0.190 0.180 0.194 0.220 0.072 0.065 0.126 0.115 0.161 0.134 0.220 0.134 0.235

11.05/11 11.83/8 11.21/8 12.24/8 13.26/9 11.06/9 14.47/9 11.94/8 42.38/9 11.57/9 10.72/11 10.65/10 11.29/10 11.95/11 11.86/9 11.38/9 21.28/8 10.96/8 11.20/8 10.55/13 11.80/15 31.36/16 11.95/11 31.83/10 13.14/9 11.86/9 31.97/9 10.90/12 10.80/9 31.94/9 26.20/9 10.72/9 20.65/9 11.26/8 11.15/9 21.61/9 12.34/9 23.20/10 11.34/11 22.35/11

+0.45 1.04−0.28 +0.13 0.68−0.15 +0.18 0.43−0.18 +0.18 0.55−0.18 +0.37 0.91−0.17 +0.37 1.06−0.27 +0.27 1.62−0.22 +0.27 0.80−0.28 +0.29 0.41−0.27 +0.12 0.91−0.16 +0.12 0.93−0.17 +0.19 1.12−0.19 +0.14 0.51−0.15 +0.39 0.90−0.10 +0.36 0.85−0.37 +0.14 1.78−0.15 +0.37 0.83−0.33 +0.16 0.81−0.17 +0.11 1.12−0.27 +0.12 0.86−0.10 +0.12 0.94−0.11 +0.15 1.21−0.10 +0.13 1.18−0.11 +0.17 1.34−0.17 +0.11 0.92−0.18 +0.19 1.21−0.24 +0.13 1.18−0.15 +0.14 0.74−0.11 +0.12 0.74−0.12 +0.14 1.03−0.15 +0.12 0.70−0.13 +0.15 0.83−0.18 +0.14 0.95−0.12 +0.15 1.23−0.17 +0.35 1.63−0.17 +0.19 0.72−0.18 +0.14 0.82−0.13 +0.14 0.69−0.13 +0.18 0.76−0.28 +0.18 0.67−0.19

24.13/10 11.77/7 10.28/7 11.22/7 21.20/8 21.38/8 20.32/8 10.92/7 11.30/8 25.47/8 11.29/10 21.28/9 10.75/9 21.17/10 21.28/8 27.10/8 11.18/7 23.19/7 11.11/7 21.22/12 23.02/14 12.05/15 23.66/10 2.48/9 14.12/8 23.47/8 21.55/8 12.28/11 10.21/8 20.83/8 27.36/8 10.27/8 11.22/8 10.21/7 12.29/8 21.02/8 21.07/8 12.66/9 16.33/10 15.17/10

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Vol. 58 No. 9 2014

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Table 3. (Contd.) Generalized NFW (α is free parameter)

NFW model (α = 1) Cluster rs , Mpc

c200

Mb , 1013 M

fMb /Mtot

χ2 /dof

α

χ2 /dof

1

2

3

4

5

6

7

8

A3571 A3667 A3827 A3921 A4038 A4059 AWM4 CLJ 1226.9+3332 Coma (A1656) IIIZw 54 ISCS J1438.1+3414 MACS J0011.7–1523 MACS J0159.8–0849 MACS J0242.6–2132 MACS J0429.6–0253 MACS J0647.7+7015 MACS J0744.8+3927 MACS J1115.8+0129 MACS J1311.0–0311 MACS J1423.8+2404 MACS J1427.6–2521 MACS J1720.3+3536 MACS J1931.8–2635 MACS J2129.4–0741 MACS J2229.8–2756 MKW3s MKW4 MKW8 PKS 0745–191 RXC J0043.4–2037 RXC J0232.2–4420 RXC J0307.0–2840 RXC J0516.7–5430 RXC J0547.6–3152 RXC J0605.8–3518 RXC J1131.9–1955 RXC J2014.8–2430 RXC J2129.6+0005 RXC J2337.6+0016 Zw CL1215

0.25+0.06 −0.03 0.23+0.06 −0.04 0.81+0.09 −0.23 0.46+0.08 −0.17 0.15+0.02 −0.01 0.17+0.02 −0.00 0.15+0.03 −0.02 1.54+0.13 −0.11 2.27+0.33 −0.45 0.12+0.01 −0.02 6.41+0.35 −0.42 0.65+0.23 −0.13 0.44+0.11 −0.11 0.27+0.06 −0.04 0.55+0.23 −0.11 3.12+0.15 −0.16 3.14+0.19 −0.17 1.00+0.19 −0.12 0.50+0.07 −0.06 0.73+0.19 −0.21 0.26+0.07 −0.02 0.48+0.08 −0.10 0.57+0.15 −0.16 1.96+0.22 −0.37 0.27+0.06 −0.03 0.13+0.03 −0.03 0.05+0.01 −0.00 0.10+0.02 −0.01 0.38+0.03 −0.02 0.14+0.03 −0.01 1.27+0.12 −0.13 0.55+0.17 −0.16 0.74+0.19 −0.16 0.41+0.06 −0.07 0.34+0.02 −0.02 0.75+0.11 −0.13 0.46+0.07 −0.04 0.42+0.07 −0.06 0.29+0.06 −0.03 0.41+0.08 −0.16

8.52+0.98 −1.10 8.86+0.65 −0.66 4.15+0.47 −0.57 4.87+0.43 −0.31 8.78+1.03 −1.21 8.86+1.02 −1.01 6.61+1.32 −1.11 1.54+0.42 −0.34 1.26+0.35 −0.37 10.65+1.13 −1.28 0.43+0.05 −0.08 3.83+0.33 −0.23 4.85+0.22 −0.32 7.18+1.24 −1.13 2.96+1.36 −1.15 0.90+0.11 −0.21 0.81+0.27 −0.16 1.83+0.33 −0.25 4.51+0.23 −0.15 2.93+0.35 −0.23 7.58+0.99 −0.77 4.85+0.31 −0.46 3.61+0.26 −0.34 1.32+0.28 −0.21 7.64+0.67 −1.16 10.47+1.18 −1.73 18.93+1.15 −1.17 14.68+1.14 −1.16 5.75+0.53 −0.41 7.42+1.89 −1.03 1.58+0.47 −0.36 2.52+0.38 −0.37 2.35+0.55 −0.67 3.64+0.77 −0.53 3.74+0.23 −0.33 2.25+1.14 −0.67 3.58+0.34 −0.47 3.17+0.48 −0.34 4.72+0.55 −0.77 5.12+0.42 −0.51

39.35+4.46 −3.28 44.18+4.56 −4.74 70.79+8.36 −6.48 18.56+2.36 −2.74 14.98+2.10 −1.48 13.64+1.25 −1.28 15.61+1.47 −1.47 94.12+10.16 −9.27 76.82+7.62 −4.58 17.99+1.74 −2.15 316.22+112.92 −74.83 59.97+4.88 −4.28 52.57+5.18 −4.92 48.48+4.28 −4.83 31.31+3.24 −3.18 98.25+8.12 −7.28 97.81+9.16 −7.99 49.38+4.76 −5.17 68.15+7.15 −6.72 34.90+3.26 −3.55 34.75+3.17 −3.33 37.61+4.29 −4.83 42.24+7.11 −6.18 95.79+7.91 −6.19 35.64+4.82 −5.19 20.87+3.99 −4.18 7.69+1.94 −2.19 8.34+2.37 −1.87 30.04+2.84 −2.93 10.29+2.55 −3.19 16.51+1.48 −1.55 21.28+2.10 −2.11 17.59+2.10 −1.99 18.82+2.12 −2.18 13.84+1.38 −1.28 20.92+2.10 −1.28 18.27+1.58 −1.93 12.27+1.33 −1.26 13.75+1.44 −1.55 36.33+3.17 −2.38

0.129 0.164 0.089 0.062 0.180 0.127 0.238 0.065 0.183 0.204 0.107 0.124 0.119 0.176 0.169 0.109 0.091 0.209 0.123 0.072 0.138 0.092 0.129 0.099 0.077 0.173 0.346 0.111 0.089 0.161 0.090 0.128 0.126 0.150 0.117 0.115 0.123 0.115 0.131 0.118

11.29/15 11.64/13 10.89/17 20.62/13 11.80/11 21.27/11 25.38/10 10.65/10 11.83/13 21.04/12 11.07/8 11.24/9 11.19/8 12.76/8 11.69/9 10.09/9 30.91/9 21.09/8 11.23/9 30.72/9 14.38/8 40.92/9 13.29/11 20.99/10 10.77/10 11.73/10 33.46/10 11.11/9 10.89/8 11.61/8 20.90/9 11.68/9 11.26/9 11.70/8 24.17/9 31.35/8 10.73/8 10.45/8 10.61/8 11.08/8

0.91+0.17 −0.19 0.87+0.17 −0.16 0.53+0.24 −0.34 1.16+0.48 −0.29 0.77+0.12 −0.13 0.83+0.12 −0.15 1.52+0.25 −0.22 0.52+0.16 −0.11 0.63+0.16 −0.16 1.43+0.28 −0.27 0.35+0.14 −0.14 0.63+0.16 −0.18 0.72+0.24 −0.36 0.64+0.17 −0.17 0.93+0.17 −0.17 0.77+0.37 −0.27 0.65+0.19 −0.19 0.64+0.26 −0.25 0.54+0.23 −0.13 0.63+0.18 −0.18 0.84+0.33 −0.23 0.79+0.17 −0.17 0.77+0.38 −0.33 0.64+0.18 −0.18 0.84+0.19 −0.19 0.85+0.17 −0.19 1.75+0.28 −0.27 1.83+0.18 −0.17 1.88+0.18 −0.19 1.43+0.18 −0.18 1.31+0.34 −0.38 1.21+0.19 −0.19 1.34+0.17 −0.16 1.64+0.15 −0.15 1.54+0.19 −0.19 1.42+0.18 −0.18 1.64+0.19 −0.18 1.73+0.24 −0.22 1.67+0.19 −0.19 0.74+0.14 −0.13

24.36/14 26.22/12 26.02/16 32.33/12 24.32/10 20.37/10 12.22/9 11.45/9 14.24/12 10.16/11 27.73/7 11.33/8 11.19/7 21.33/7 12.38/8 20.12/8 24.28/8 22.21/7 11.18/8 42.88/8 11.18/7 21.72/8 11.44/10 21.46/9 21.33/9 21.22/9 12.37/9 10.92/8 10.82/7 21.11/7 21.22/8 21.38/8 21.72/8 12.34/7 12.38/8 12.88/7 11.83/7 11.33/7 11.18/9 11.14/7

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Table 3. (Contd.) Generalized NFW model (α is free parameter)

NFW model (α = 1) Cluster c200

2

3

4

0.51+0.09 −0.17 0.20+0.01 −0.03 0.34+0.03 −0.03 0.22+0.03 −0.05 0.31+0.15 −0.06 1.77+0.22 −0.34 1.65+0.23 −0.25 0.57+0.11 −0.09 0.17+0.05 −0.03 1.33+0.34 −0.27 1.73+0.37 −0.25 0.83+0.15 −0.19 0.26+0.06 −0.04 1.01+0.15 −0.11

2.48+0.35 −0.35 3.92+0.35 −0.23 2.75+0.36 −0.35 5.74+0.53 −0.31 3.68+0.42 −0.41 1.03+0.43 −0.33 1.08+0.29 −0.27 2.92+0.36 −0.35 7.31+0.27 −0.77 0.57+0.15 −0.11 0.74+0.21 −0.12 2.44+0.15 −0.11 2.98+0.23 −0.21 2.02+0.27 −0.25

22.45+0.19 −0.11 9.01+1.66 −2.18 12.65+1.47 −1.48 33.91+2.88 −3.17 14.15+1.47 −1.82 42.97+5.82 −5.28 51.45+5.27 −5.82 91.91+8.88 −9.45 11.43+2.16 −1.37 9.36+3.56 −2.19 67.87+6.48 −6.84 66.84+6.77 −7.55 12.72+1.49 −1.48 75.08+6.32 −4.29

1 RCS J0224–0002 RCS J0439–2904 RCS J1107–0523 RCS J1419.2+5326 RCS J1620+2929 RCS 2156+0123 RCS J2318+0034 RCS J2319+0038 RX J0439.0+0520 RX J0848.7+4456 RX J0849+4452 RX J0910+5422 RX J1113.1–2615 RX J1221.4+4918

Mb , 1013 M fMb /Mtot

rs , Mpc

Table 2 summarizes our results for the modeling of the total mass profiles of the Chandra galaxy clusters with an NFW model (α = 1), similarly to SA07. Note that SA07 used both an NFW model and an isothermal sphere model (ρ(r) ∝ r −2 ), and concluded that the latter gives a much worse fit for almost all the sample clusters. Therefore, we used only the NFW model in the current study. The columns of Table 2 give (1) the name of the galaxy cluster, (2)–(3) the parameters rs and c200 = R200 /rs , (4) R200 , (5) the total mass at R200 , and (6) the goodness of fit χ2 /dof. (dof denotes the number of degrees of freedom). The confidence levels given are 68%. The NFW model is a good fit for most clusters, as in SA07. For 110 of 128 clusters, the probability of obtaining the corresponding χ2 /dof value based on a χ2 distribution is 0.04 or better; for only 18 clusters was this probability below 0.002. Combining the results for all the galaxy clusters studied, the NFW model gave a total χ2 of 2603.48 for 1267 degrees of freedom. ASTRONOMY REPORTS

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α

χ2 /dof

6

7

8

0.113

21.13/9

11.39/8

0.207

21.37/9

0.219

12.39/9

0.222

12.62/9

0.147

11.87/9

0.181

13.88/9

0.141

11.81/9

0.204

24.24/9

0.225

11.25/8

0.241

12.81/8

0.165

11.45/8

0.143

11.03/9

0.219

12.39/8

0.151

11.61/9

+0.15 1.02−0.14 +0.36 1.84−0.33 +0.54 1.53−0.53 +0.12 0.67−0.13 +0.14 1.04−0.15 +0.11 0.58−0.12 +0.15 0.64−0.14 +0.15 0.54−0.16 +0.29 1.04−0.38 +0.14 1.01−0.15 +0.17 0.55−0.16 +0.13 0.63−0.13 +0.33 1.04−0.37 +0.18 0.54−0.18

5

22.20/8 11.25/8 11.28/8 11.17/8 11.25/8 11.10/8 24.36/8 24.46/7 11.66/7 11.28/7 42.46/8 24.22/7 14.27/8

3.2. Dark Matter Profile

3. RESULTS

3.1. Total Mass Profile

χ2 /dof

As was already noted, we carried out our analysis using two approaches. In the first, we examined models in which the inner slope of the DM density profile was fixed at α = 1, as in the NFW model. In the second, we decomposed the total mass of the galaxy cluster into its dark matter, diffuse gas, and BCG components. Table 3 presents our results when the cluster mass distributions were separated into the dark matter halo (fitted with a generalized NFW model), the X-ray emitting halo, and the stellar component of the central galaxy. We also calculated the concentration parameter for each cluster, in order to obtain the massconcentration parameter: cvir ≡ c200 and Mvir ≡ M200 .5 Similarly to SA07, we defined the concentotal /r dark ).6 tration parameters as cvir = (rvir s The simplest analytic form that describes the mass-concentration relation is a power law  a M c0 . (14) c(z) = (1 + z)b 8 × 1014 h−1 M 5

Note that cvir and Mvir in SA07 are related through the equa3 tion Mvir = 43 πrvir Δc (z)ρc , with Δc = 178Ωm (z)0.45 [128]. 6 The virial radii and virial masses were obtained for the threecomponent total mass model.

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When b = 1, this reduces to the analogous relation in [38]. As was shown in SA07, the relation between the concentrations and total masses of CDM halos is not well represented by the formula proposed in [38] (i.e., Eq. (14) with b = 1). To improve the description of Chandra data, as in SA07 and in contrast to [28], we considered the evolution over a wide range of redshifts, with b = 1 in (14). The motivation for this is that galaxy clusters are more complex objects than was assumed in [28]. They contain X-ray emitting gas, the galactic matter, and dark matter, whose properties evolve with time. For the most massive galaxy clusters, the model of [28] is able to predict the evolution of cvir with z, as was shown by Zhao et al. [129]. Moreover, Chandra has obsreved several massive galaxy clusters, including some in our sample, providing additional arguments for evolutionary studies of the mass–concentration relation. The results of standard χ2 fits using the three free parameters c0 , a, and b are shown by the bold solid line (average value) and the thin solid lines (±2σ) in Fig. 1. We concluded that this model (b = 1) gives an improved description of our sample with χ2 = 288.7 for 125 degrees of freedom than the same model with b = 1. For 34 galaxy clusters and using the same model, SA07 obtained the best-fitting parameters c0 = 7.55 ± 0.90, a = −0.45 ± 0.12 and b = 0.71 ± 0.52 with χ2 = 41.5 for 29 degrees of freedom. We also compared our results to those of [9] and [40], also plotted in Fig. 1. These two models are in good agreement with our data, and with our mass– concentration relation. Using the best-fit parameters from our fit, we determined the intrinsic scatter in our sample to be ∼0.2, which is also in good agreement with simulations and the results of SA07 and [28].

3.3. Inner Slope of the DM Profiles We fitted the DM density profiles using a generalized NFW model to derive the inner slopes of the DM density profiles. The results are summarized in Table 3. The goodnesss of fit for 110 of the 128 galaxy clusters yield χ2 /dof values corresponding to probabilities of 0.03 or better. The value α = 1.1 ± 0.4 (95% confidence limits) was obtained by summing the χ2 data (as a function of α) for the 128 clusters. This result is consistent with the predictions of ΛCDM simulations. Similarly to SA07, to obtain a combined DM density profile, we summed the χ2 values as a function of α for the clusters for which the NFW model provides an adequate fit to the data. These results are shown in Fig. 2, where the best fit corresponds to α = 1.10+0.40 −0.48 (95% confidence limits).

Note that three separate components (DM, gas, and the BCG) were included in the fits. The slopes are in the range 0.4 < α < 1.8 and the density profiles for most clusters are compatible with a ΛCDM model, 0.7 < α < 1.5 [57]. Some clusters (∼32%) have flatter profiles than the cuspy DM profile predicted by ΛCDM simulations. A few clusters with slopes lower than 0.7 are usually characterized by higher Mb values, compared to clusters with higher α values. The correlation between the intercluster X-ray gas and the inner slope indicates that the baryonic component influences the DM distribution in the central region of the cluster, in agreement with other results, such as those of [130].

3.4. Inner Slope and Baryon Content We discussed in the Introduction that the DM profiles of some clusters are not fit adequately by the NFW model [68, 69, 72, 76, 79]. The role of baryons is among the most important processes that can lead to the formation of a flat inner DM profile (see the Introduction). The effect of baryons, and the central BCG in shaping the inner DM density profile was recently considered in [130]. It was shown that a higher baryon content can increase the slopes of the inner DM profiles (for example, as in A611 and A383, which have flatter slopes than the profiles in an NFW model), while a smaller baryon content can produce steeper inner DM profiles, consistent with the NFW model (e.g., MACS J1423.8+2404, RXJ1133). Motivated by these results, we studied the possible correlation between the inner slope of the DM density profile, α, and the baryon content of the studied X-ray galaxy clusters. As in SA07, our analysis was based on a three-component model (the total cluster mass was divided between the masses of the DM halo, Xray emitting cluster gas, and the BCG). We fitted the DM profile with a generalized NFW model; the results, including the α values, are given in Table 3. The mass Mb was calculated by summing the contributions from the diffuse intercluster gas and the baryons contained in the central BCG in each cluster. Figure 3 plots the inner slope, α, versus the baryon content at the radius R200 . This plot shows a tight correlation between the inner slope α and the baryon content: the higher the baryonic content, the flatter the inner profile. This is in agreement with studies of the role of baryons on the inner DM density profiles of clusters [90, 92, 101, 130]. We fitted the αgN F W –Mb relation, where gN F W denotes a generalized NFW model, using a simple power law (y(x) = kx−b ); this yielded the slope b = 1.72 ± 0.37 at the 68% confidence level. We calculated the internal systematic scatter that could be present in the α−Mb relation. Using the simple ASTRONOMY REPORTS Vol. 58

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c200 102 Best-fit model (SA07) 2σ confidence level Duffy et al. (2008) [9] Bhattacharya et al. (2013) [40]

101

100

10–1 0 10

101

102 M200,

1014

M

Fig. 1. The mass–concentration relation for the total mass (c200 −M200 ). Thick (thin) solid line: best-fitting model and 2σ CL. The dashed line represents the [9] model, and the dotted line the [40] model (see the text for details).

Δχ2 14 12 10 8 6 4 2 0

0.5

1.0 α

1.5

2.0

Fig. 2. The summed χ2 values as a function of α for the clusters, for which the NFW model provides a reasonable fit to the +0.40 data. The dashed line indicates the 2σ confidence levels. Overall best fit: α = 1.10−0.48 (95% confidence level).

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αg NFW z < 0.15 z > 0.15

1

0.4

0

101

102 Mb,

1013

M

Fig. 3. αgeneralized NFW −Mb relation for all sample of galaxy clusters. The data were obtained fitting the DM profiles of clusters with the generalized NFW model. The baryonic mass was obtained, as described in the text, by subtracting the DM component from the total mass. The solid line represents the power-low fit with slope b = 1.72 ± 0.37 (68% confidence level). The dashed lines correspond to the region enclosed by the mean ±1σ standard deviation.

power-law model with k and b as free parameters, we found the scatter to be Δ log(b) ∼ 0.2. The flattening of the slope could be due to the fact that the higher the baryo content, the higher the transfer of energy and angular momentum from the baryons to the DM, causing the DM to move to higher orbits, reducing the inner density [15, 16, 90–92, 94, 101, 130] and forming halos that are rounder than those seen in simulations of DM in clusters [131–133]. However, as was noted in the Introduction, slopes flatter than those obtained in the simulations, α < 0.7 are observed in some clusters [68, 69, 72, 76, 79], and in most dwarf galaxies and LSB dwarf galaxies. A few of our studied clusters have slopes α < 0.7. As was already noted, NFW N -body simulations predict a DM density profile with an inner slope ∝r −1 and an outer slope ∝ r −3 . More recently, it has been shown that the DM density profiles are fitted better by an Einasto profile, and become shallower towards the center of the halo (e.g., [31]). In any case, the lower values for the inner slope obtained in N -body simulations, α = 0.8 at 120 pc [43], are in agreement with the analytical studies and numerical simulations aimed at solving the Jeans equation [134,

135]. Inner slopes of the DM density profiles in the range 0.7 < α < 1.5 are in agreement with ΛCDM models (SA07). Given the firm evidence supporting ΛCDM models on large scales, the discrepancy in our results could be related to the fact that physical processes occuring in baryonic matter are of fundamental importance in the formation and evolution of clusters. We are comparing dissipationless systems generated by N -body simulations with real, dissipative structures, whose physics is different from the dissipationless physics typical for dark matter. We should not forget that the inner 10 kpc of clusters are dominated by baryonic matter [68, 69, 72, 76], whose presence strongly influence the distribution of DM. While baryons can steepen the inner slope of the DM density profile of clusters through the adiabatic contraction of the darm matter [136–138], heating of the dark matter due to dynamical friction with other cluster galaxies can counteract the effect of adiabatic contraction and flatten the inner DM profile [15, 16, 90]. Other mechanisms can also flatten the inner slope of the DM density profile. The spatial distributions ASTRONOMY REPORTS Vol. 58

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of the components of a galaxy cluster can change due to feedback processes [139], gas cooling, star formation, and dynamical interactions between the different mass components (e.g., the BCG galaxy and the ICM in relaxed clusters). As was already noted, adiabatic contraction steepens the density profile and may produce rounder halos. The influence of both active nuclei [139] and supernova outbursts [94, 140] can influence the smoothing of the inner DM density profiles in galaxies. Supermassive black holes also produce changes in the inner gas distribution [141], and gravitational interactions (galaxy–galaxy, rampressure stripping, etc.) also play a role in this very dense environment [141–143]. 4. DISCUSSION

4.1. Comparison with Other Estimates of the Inner Slopes of the DM Density Profiles We compared our numerical estimates for the sloeps of the inner DM density profiles with other corresponding data. As was already noted, we applied the approach proposed in SA07 to our sample of cluster galaxies observed with Chandra. Our sample is appreciably larger than the sample of SA07, while the inner slopes of the cluster DM density profiles are comparable. In addition, some clusters, such as A383, A611, A963, A1835, A2029, A2204, and A2589 were included in other studies. A611 was studied by Newman et al. [76], who combined weak lensing from multicolor Subaru imaging, strong lensing using Hubble Space Telescope data, and stellar velocity dispersions measured with the Keck Telescope applied to a sample of DM profiles from 3 kpc to 3.25 Mpc. The resulting values rs = 320+240 −110 kpc and are in agreement with the results of SA07, c = 5.1+1.7 −1.6 but α < 0.3 (