<figure><img src="image_1.png" /></figure>

# Expanding the Sample: The Relationship between the Black Hole Mass of BCGs and the Total Mass of Galaxy Clusters

Frederika Phipps<fnref n="1" />,<fnref n="2" />, Ákos Bogdán<fnref n="2" />, Lorenzo Lovisari<fnref n="2" />, Orsolya E. Kovács<fnref n="2" />,<fnref n="3" />,<fnref n="4" />, Marta Volonteri<fnref n="5" />, and Yohan Dubois<fnref n="5" />

## Abstract

Supermassive black holes (BHs) residing in brightest cluster galaxies (BCGs) are overly massive when considering the local relationships between the BH mass and stellar bulge mass or velocity dispersion. Because of the location of these BHs within the cluster, large-scale cluster processes may aid the growth of BHs in BCGs. In this work, we study a sample of 71 galaxy clusters to explore the relationship between the BH mass, stellar bulge mass of the BCG, and the total gravitating mass of the host clusters. Because of difficulties in obtaining dynamically measured BH masses in distant galaxies, we use the Fundamental Plane relationship of BHs to infer their masses. We utilize X-ray observations taken by *Chandra* to measure the temperature of the intracluster medium, which is a proxy for the total mass of the cluster. We analyze the $M_{BH-kT}$ and $M_{BH-M_{bulge}}$ relationships and establish the best-fitting power laws: $\log_{10}(M_{BH}/10^9 M_{\odot}) = -0.35+2.08 \log_{10}(kT/1 \text{ keV})$ and $\log_{10}(M_{BH}/10^9 M_{\odot}) = -1.09+1.92 \log_{10}(M_{bulge}/10^{11} M_{\odot})$. Both relations are comparable with that established earlier for a sample of brightest group/cluster galaxies with dynamically measured BH masses. Although both the $M_{BH-kT}$ and the $M_{BH-M_{bulge}}$ relationships exhibit large intrinsic scatter, on the basis of Monte Carlo simulations, we conclude that dominant fraction of the scatter originates from the Fundamental Plane relationship. We split the sample into cool core and noncool core resembling clusters but do not find statistically significant differences in the $M_{BH-kT}$ relation. We speculate that the overly massive BHs in BCGs may be due to frequent mergers and cool gas inflows onto the cluster center.

**Key words:** galaxies: clusters: general – galaxies: clusters: intracluster medium – galaxies: elliptical and lenticular, cD – galaxies: evolution – X-rays: galaxies: clusters

## 1. Introduction

Throughout the evolution of galaxies, they undergo diverse physical processes that produce the observed galaxy populations and result in various relations between different galaxy properties (Croton et al. 2006; Faber et al. 2007). Black holes (BHs) are believed to have a profound effect on the evolution of their host galaxy because of their energetic feedback (Richstone et al. 1998; Cattaneo et al. 2009; Cheung et al. 2016). By precisely measuring the mass of BHs, relationships between the BH mass and the properties of the host galaxies have been established, the most well-known being the correlation between central stellar velocity dispersion ($\sigma$) and BH mass (e.g., Ferrarese & Merritt 2000; Gebhardt et al. 2000; McConnell & Ma 2013; DeGraf et al. 2015), as well as the relationship between stellar bulge mass and BH mass (e.g., Magorrian et al. 1998; Häring & Rix 2004; Kormendy & Ho 2013). These results have contributed to the development of the current theoretical paradigm, in which the BH and the host galaxies coevolve and regulate each other's growth (Fabian 1999; King 2003; Di Matteo et al. 2005; Hopkins et al. 2006; Shankar et al. 2006; Somerville et al. 2008).

The study of BHs and their host galaxies has been extensive, despite the difficulties in measuring the BH masses accurately. However, large-scale structures, in which most galaxies are embedded (Sepp & Gramann 2014), could influence the evolution of BHs. Most galaxy groups/clusters contain a unique type of elliptical galaxy, known as the brightest group/cluster galaxies (BGGs/BCGs), at their center (Crawford et al. 1999;

Bernardi et al. 2007). Typically, BGGs/BCGs are both the most massive and luminous galaxies in the group/cluster. As these galaxies are located at the bottom of the potential well for these large-scale structures, it is feasible that the BHs of BCGs undergo a different evolution than BHs residing in field or satellite galaxies. Studies of BHs in BCGs pointed out that many of these BHs are overmassive in comparison to the stellar bulge mass or velocity dispersion of the BCG (McConnell et al. 2011, 2012; Mezcua et al. 2018). This hints that the large-scale potential of clusters may aid the growth of these BHs.

There is a vast difference between the scales of BHs and galaxy groups/clusters. While the sphere of influence of BHs is ~100 pc for BHs with masses ~$10^9 M_{\odot}$, galaxy groups/clusters extend to megaparsec scales. Therefore, it is appealing to probe whether the growth of BHs in BCGs may be influenced by the large-scale structures. From theoretical considerations, a correlation between BH mass and cluster halo temperature is expected (Gaspari & Sadowski 2017). This relation had been previously quantified by Mittal et al. (2009), who used the luminosity of the BCG as a proxy for the BH mass. More recently, Bogdán et al. (2018) investigated using a sample of 17 galaxy groups/clusters with dynamically measured BHs in their BGGs/BCGs. By analyzing XMM-Newton X-ray observations, they found a tight correlation between the BH and total mass of the group/cluster, which was traced via the gas temperature of the intracluster medium (ICM). This relation had an intrinsic scatter in the x- and y-axes of 0.22 and 0.38. They concluded that the $M_{BH-kT}$ relation is

<footnote n="1">¹School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK</footnote>

<footnote n="2">²Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA</footnote>

<footnote n="3">³Konkoly Observatory, MTA CSFK, Konkoly Thege M. út 15-17, H-1121 Budapest, Hungary</footnote>

<footnote n="4">⁴Eötvös University, Department of Astronomy, Pf. 32, H-1518 Budapest, Hungary</footnote>

<footnote n="5">⁵Institut d'Astrophysique de Paris, Sorbonne Universités, UPMC Univ Paris 6 et CNRS, UMR 7095, 98 bis bd Arago, F-75014 Paris, France

Received 2018 May 25; revised 2019 March 11; accepted 2019 March 14; published 2019 April 24</footnote>
tighter and has less scatter than the $M_{BH}-M_{bulge}$ relation (which had scatter of 0.35 and 0.68 in each axis), hinting that the BH mass of BGGs/BCGs may be determined by physical processes that are governed by the properties of the large-scale potential. However, the sample of Bogdán et al. (2018) was relatively small, as it was limited by the available dynamical BH mass measurements in BGGs/BCGS. While this means their masses are more accurate, this also meant that they were unable to investigate these relationships for a notable sample of massive clusters. To further probe the findings of Bogdán et al. (2018), it is necessary to extend the sample of galaxy groups and clusters, especially including massive systems.

As measuring the mass of BHs using dynamical methods is challenging for less massive and/or distant BHs, we must rely on tracers. In this work, we utilize the Fundamental Plane relationship, which defines a plane between BH mass and its luminosity in the X-ray and radio (Merloni et al. 2003; Falcke et al. 2004; Körding et al. 2006; Gültekin et al. 2009). The Fundamental Plane relationship for BCGs was investigated by Hlavacek-Larrondo et al. (2012) for a sample of 18 BCGs. This work was extended by Mezcua et al. (2018), who measured the X-ray and radio properties for a large sample of BHs in BCGs using Chandra and Very Long Baseline Interferometry (VLBI) data. Here, we use their results to infer both BH masses and bulge masses of the BCGs. While they investigated the Fundamental Plane relationship of both Merloni et al. (2003) and Plotkin et al. (2012), in this work we opt to use the “standard” relation of Merloni et al. (2003) as this relation covers the largest range of radio luminosities and BH masses.

To derive the total mass of the galaxy clusters, we utilize Chandra X-ray observations. Specifically, the temperature of the ICM is a good proxy for the cluster's total mass (Horner et al. 1999; Ettori et al. 2013). Not only is it a good proxy, but through using this method we maintain a straightforward comparison with the work of Bogdán et al. (2018). We then use the temperature measured from these X-ray observations to study the BH mass—cluster temperature relationship.

The structure of the paper is as follows: in Section 2 we discuss the sample selection process and the observations used to obtain them. Section 3 describes the analysis of the Chandra data. Results are presented in Section 4 and in Section 5 we discuss the implications of these results. We summarize our results in Section 6. In this paper, we take the Hubble constant, $H_0$, to be 70 km s⁻¹ Mpc⁻¹ and $\Omega_M = 0.3$ and $\Omega_\Lambda = 0.7$. All error bars represent $1\sigma$ uncertainties, unless otherwise mentioned.

## 2. Sample

To study a larger set of galaxy clusters with indirect BH mass measurements, we rely on Mezcua et al. (2018), who investigated the Fundamental Plane relationship for BCGs. Mezcua et al. (2018) drew their sample from Hogan et al. (2015), who examined the radio properties of X-ray selected BCGs from a parent sample of three ROSAT X-ray catalogs. This sample includes galaxy clusters with redshifts in the range of $0.006 < z < 0.29$. By utilizing Chandra X-ray and VLBI radio observations, Mezcua et al. (2018) probed the Fundamental Plane relationship. They inferred the BH masses of the galaxies using the K-band luminosities of the BCGs. They concluded that the BHs were overly massive compared to the galaxy's stellar mass. However, they did not expand their analysis to include the effects of the cluster. In this work, we

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Redshift distribution for two cluster samples. The hatched (blue) distribution is the clusters in the sample studied by Bogdán et al. (2018). The empty (red) distribution shows the clusters studied in this work. Note that the galaxy clusters investigated in this work have significantly higher redshifts than those in Bogdán et al. (2018).</figcaption></figure>

utilize the Fundamental Plane relationship from Merloni et al. (2003), together with the X-ray and radio luminosities from Mezcua et al. (2018), in order to calculate the BH masses.

In Figure 1, we show the redshift distribution for our sample, as well as that for the sample studied by Bogdán et al. (2018). Given the larger redshift range of the present sample, we can explore a larger volume; hence, more massive clusters can be studied. However, while dynamically measured values for BH mass are preferable in constraining accurate relationships, they are limited to nearby and massive BHs. Therefore, without the use of tracers, we would not be able to significantly increase the sample size and explore higher redshift and, hence, more massive clusters.

## 3. Data Analysis

### 3.1. Chandra Data

#### 3.1.1. Data Reduction

To study the ICM of the galaxy clusters, we utilized Chandra X-ray observations. All the systems have been observed with Chandra—as it was also used by Mezcua et al. (2018). Although XMM-Newton data are available for some sources in our sample, in order to avoid calibration issues (e.g., see Schellenberger et al. 2015), we opted to use only Chandra data. The data analysis was performed using CIAO software version 4.9 and CALDB 4.7.6 (Fruscione et al. 2006).

As the first step of the analysis, we reprocessed all Chandra observations using the `chandra_repro` tool. Because we aim to study the diffuse emission, we must identify and remove bright point sources. To this end, we utilized the `wavdetect` task, which correlates the data with a Mexican Hat wavelet function of different scales and generates a list of point sources. The applied scales were the square root series of two from $\sqrt{2}$ to 8.0. The point sources found by `wavdetect` were then masked from the analysis of the diffuse X-ray emission.

Once the point sources were excluded, we filtered the time periods that had high background due to flares. For each observation, we extracted light curves in the 2.3–7.3 keV energy range and binned them with a time interval of 200 s. Light curves were extracted within this energy range, as
Chandra is most sensitive to flares in this band (Hickox & Markevitch 2006). We applied the deflare tool with the parameter nsigma = 2. Hence, we removed all time periods that were ≥2σ outliers from the mean, which resulted in exposures that were typically 15%–25% shorter than the original exposures (Table 1). Although we could be less conservative by applying a 3σ clipping instead, we prefer to use a 2σ clip. We use this clipping because it is more effective in excluding soft proton flares and, therefore, more effective in avoiding bias in the spectral fit procedure.

To account for the sky and instrumental background components, we constructed the blank-sky background for each observation. We used the blanksky tool to create the background event files. While the spectrum of the background remains invariable, its normalization exhibits variations. To account for this, we used the count rates in the 10–12 keV energy range to rescale the blank-sky background files.

### 3.1.2. Measuring the Temperature of the ICM

To accurately measure the ICM temperature, we first identified the peak of the X-ray emission, which is considered to be the center of the cluster. To find the X-ray peak, we smoothed the 0.7–2 keV band images with a Gaussian with a kernel size of 3. We searched for the maximum on this smoothed image, which defined the peak of the emission. We note that the center of the cluster may be slightly offset from the BCG (Hudson et al. 2010). Several clusters in our sample exhibit double-peaked profiles, as these may be undergoing mergers (see Section 5). In these cases, the second peak was masked from the observations.

After we identified the center of each cluster, we determined the R$_{500}$ radius. As the first step, we computed the signal-to-noise ratio in the 0.7–2 keV band using concentric annuli. The first spectra were extracted within a radius where the signal-to-noise ratio peaked. After extracting the spectrum, we determined the initial temperature and an initial M$_{500}$<fnref n="6" /> using the kT–M$_{500}$ relation by Lovisari et al. (2015):

$$ \log_{10}\left(\frac{M_{500}}{5 \times 10^{13} h^{-1} M_{\odot}}\right) = 1.71 \log_{10}\left(\frac{kT}{2 \text{ keV}}\right) + 0.20, \quad (1) $$

where h is the reduced Hubble constant. This M$_{500}$ value was then used to calculate the R$_{500}$ radius.

Once the initial R$_{500}$ radius was retrieved, we followed the iterative process outlined in Lovisari et al. (2017) to measure the gas temperature of the ICM. Specifically, we extracted the source and background spectra from the (0.15–0.75)R$_{500}$ region, in order to exclude the central regions where there could be an extreme temperature gradient. These spectra were fit using XSpec (Arnaud 1996). From the fit we obtained a new temperature and, hence, M$_{500}$ value using the kT–M$_{500}$ relation in Lovisari et al. (2015). This new M$_{500}$ and the inferred R$_{500}$ defined the new extraction annulus. We continued to iterate via this process until the temperature remained invariant within 5%. Although Lovisari et al. (2017) used the M–Y$_{x}$ relation, in this work we rely on the kT–M$_{500}$ relation for consistency with Bogdán et al. (2018). For most of the clusters, the extraction region lay within the field of view of the analyzed Chandra observations. However, for a small sample of nearby clusters, only a relatively small fraction of the R$_{500}$ values were

included in the field of view. For these systems, we utilized multiple observations to increase the coverage of the clusters. In the Appendix, we show all the Chandra images of the clusters in the 0.7–2.0 keV band, as well as the annulus from which the final spectra was extracted as defined by the R$_{500}$ value.

Spectra along with the corresponding response files for the source and background data were extracted using the specextract and dmextract tools, respectively. The background spectra were renormalized using the count rate ratios observed in the 10–12 keV band, and the renormalized background spectrum was subtracted from the source spectrum. The background-subtracted source spectrum was grouped by count number with each bin requiring a minimum of 15 counts. This final spectrum was used to fit a model and determine the ICM temperature of the galaxy cluster.

We used the spectral analysis software, XSpec, to fit a model to the data (Arnaud 1996). Fitting was performed in the 0.7–5 keV energy band. We fit the emission with an apec model that describes collisionally ionized thermal plasma. We allowed the abundance, temperature, and normalization to vary. For the abundances, we used the table of Anders & Grevesse (1989) In addition, we included photoelectric absorption, whose value was the weighted average from the Leiden/ Argentine/Bonn (LAB) survey (Kalberla et al. 2005). This procedure resulted in acceptable fits for most clusters in our sample. However, for two clusters, A478 and J0352.9+1941, the column densities provided by the LAB survey were underestimated. We obtained significant residuals at <1 keV for the spectra of these clusters. The LAB survey underestimates column densities, and this underestimation becomes strongly significant for high values (excess of 10$^{21}$), such as for these clusters. Therefore, for these two systems, we allowed the column density to vary during the fitting process. Allowing this parameter to vary results in an acceptable fit.

In Table 2 we list the obtained R$_{500}$ and M$_{500}$ values, as well as the best-fit ICM temperatures.

### 3.2. CC and NCC Clusters

Given our larger sample size, we aim to explore whether the M$_{BH}$–kT relationship exhibits variations for different types of clusters. Specifically, we split our sample into cool core (CC) and noncool core (NCC) clusters. Because of their dynamics, CC clusters generally have greater gas inflow to the center (Hudson et al. 2010). This process may increase the growth rate of the BH residing in the BCG. This, in turn, could lead to differences in the M$_{BH}$–kT relation for CC and NCC clusters.

There are several parameters that can be used to distinguish between these two types of cluster, and the debate is ongoing as to which parameter is the most accurate. Therefore, we used four parameters to split the galaxy cluster samples into CC and NCC clusters to examine whether there is a difference in the M$_{BH}$–kT relationship. These parameters are central electron density, central entropy, cuspiness, and surface brightness density (Hudson et al. 2010). Instead of using a value from literature, the median value of each of the parameters was used to split our sample. We split our clusters in this way because, if we were to use a value from the literature, then we would need our data to be calibrated perfectly with the original work. As this is difficult, by using the median value instead of classifying the clusters definitively as a CC (NCC), we simply make a distinction between more relaxed (disturbed) systems. Therefore,

<footnote n="6">⁶ M$_{500}$ is the mass contained within the radius where the density of the cluster is 500 times the critical density of the universe.</footnote>
<table><caption>**Table 1**

Analyzed Chandra Observations in This Work</caption><thead><tr><th>Name</th><th>Obsid</th><th>Instrument</th><th>Redshift</th><th>t<sub>total</sub> (ks)</th><th>t<sub>clean</sub> (ks)</th><th>Date</th></tr></thead><tbody><tr><td>A1204</td><td>2205</td><td>ACIS-I</td><td>0.171</td><td>23.6</td><td>20.3</td><td>2001 Jun 1</td></tr><tr><td>A1367</td><td>514</td><td>ACIS-S</td><td>0.022</td><td>40.5</td><td>31.1</td><td>2000 Feb 26</td></tr><tr><td></td><td>17201</td><td>ACIS-I</td><td>0.022</td><td>61.3</td><td>48.5</td><td>2016 Jan 31</td></tr><tr><td>A1446</td><td>4975</td><td>ACIS-S</td><td>0.103</td><td>58.4</td><td>49.3</td><td>2004 Sep 29</td></tr><tr><td>A1644</td><td>7922</td><td>ACIS-I</td><td>0.048</td><td>51.5</td><td>40.1</td><td>2007 May 24</td></tr><tr><td>A1664</td><td>7901</td><td>ACIS-S</td><td>0.128</td><td>36.6</td><td>30.0</td><td>2006 Dec 4</td></tr><tr><td>A168</td><td>3203</td><td>ACIS-I</td><td>0.044</td><td>40.6</td><td>31.3</td><td>2002 Feb 5</td></tr><tr><td>A1763</td><td>3591</td><td>ACIS-I</td><td>0.228</td><td>19.6</td><td>17.0</td><td>2003 Oct 28</td></tr><tr><td>A1795</td><td>493</td><td>ACIS-S</td><td>0.063</td><td>19.6</td><td>14.4</td><td>2000 Mar 21</td></tr><tr><td>A1930</td><td>11733</td><td>ACIS-S</td><td>0.132</td><td>34.5</td><td>25.9</td><td>2010 Sep 9</td></tr><tr><td>A2009</td><td>10438</td><td>ACIS-I</td><td>0.153</td><td>19.9</td><td>8.88</td><td>2008 Dec 4</td></tr><tr><td>A2029</td><td>4977</td><td>ACIS-S</td><td>0.078</td><td>77.9</td><td>57.0</td><td>2004 Jan 8</td></tr><tr><td>A2033</td><td>15167</td><td>ACIS-I</td><td>0.078</td><td>8.97</td><td>8.59</td><td>2013 May 22</td></tr><tr><td>A2052</td><td>5807</td><td>ACIS-S</td><td>0.036</td><td>127.0</td><td>101.0</td><td>2006 Mar 24</td></tr><tr><td></td><td>10478</td><td>ACIS-S</td><td>0.036</td><td>119.1</td><td>101.5</td><td>2009 May 25</td></tr><tr><td>A2063</td><td>4187</td><td>ACIS-I</td><td>0.034</td><td>8.8</td><td>1.4</td><td>2003 Apr 20</td></tr><tr><td>A2110</td><td>15160</td><td>ACIS-I</td><td>0.098</td><td>7.98</td><td>7.00</td><td>2013 Aug 29</td></tr><tr><td>A2199</td><td>10748</td><td>ACIS-I</td><td>0.031</td><td>40.6</td><td>36.2</td><td>2009 Nov 19</td></tr><tr><td>A2204</td><td>7940</td><td>ACIS-I</td><td>0.151</td><td>77.1</td><td>63.7</td><td>2007 Jun 6</td></tr><tr><td>A2355</td><td>15097</td><td>ACIS-I</td><td>0.231</td><td>19.8</td><td>14.7</td><td>2013 Nov 5</td></tr><tr><td>A2390</td><td>4193</td><td>ACIS-S</td><td>0.233</td><td>95.1</td><td>69.0</td><td>2003 Sep 11</td></tr><tr><td>A2415</td><td>12272</td><td>ACIS-I</td><td>0.057</td><td>9.92</td><td>7.90</td><td>2010 Sep 24</td></tr><tr><td>A2597</td><td>7329</td><td>ACIS-S</td><td>0.083</td><td>60.1</td><td>43.6</td><td>2006 May 4</td></tr><tr><td>A262</td><td>7921</td><td>ACIS-S</td><td>0.017</td><td>111.0</td><td>92.1</td><td>2006 Nov 20</td></tr><tr><td></td><td>2215</td><td>ACIS-S</td><td>0.017</td><td>28.7</td><td>24.4</td><td>2001 Aug 3</td></tr><tr><td>A2626</td><td>16136</td><td>ACIS-S</td><td>0.055</td><td>111.0</td><td>86.3</td><td>2013 Oct 20</td></tr><tr><td>A2634</td><td>4816</td><td>ACIS-S</td><td>0.030</td><td>49.5</td><td>39.7</td><td>2004 Aug 31</td></tr><tr><td>A2665</td><td>12280</td><td>ACIS-I</td><td>0.057</td><td>9.92</td><td>5.92</td><td>2011 Jan 17</td></tr><tr><td>A2667</td><td>2214</td><td>ACIS-S</td><td>0.235</td><td>9.65</td><td>9.05</td><td>2001 Jun 19</td></tr><tr><td>A3017</td><td>15110</td><td>ACIS-I</td><td>0.220</td><td>14.9</td><td>12.0</td><td>2013 May 1</td></tr><tr><td>A3526</td><td>16223</td><td>ACIS-S</td><td>0.010</td><td>179.0</td><td>145.0</td><td>2014 May 26</td></tr><tr><td>A3528S</td><td>8268</td><td>ACIS-I</td><td>0.057</td><td>8.08</td><td>8.07</td><td>2007 Mar 20</td></tr><tr><td>A3581</td><td>12884</td><td>ACIS-S</td><td>0.022</td><td>84.5</td><td>68.7</td><td>2011 Jan 3</td></tr><tr><td></td><td>1650</td><td>ACIS-S</td><td>0.022</td><td>7.2</td><td>7.0</td><td>2001 Jun 7</td></tr><tr><td>A3695</td><td>12274</td><td>ACIS-I</td><td>0.089</td><td>9.87</td><td>9.37</td><td>2010 Aug 17</td></tr><tr><td>A4059</td><td>5785</td><td>ACIS-S</td><td>0.049</td><td>92.1</td><td>83.3</td><td>2005 Jan 26</td></tr><tr><td></td><td>897</td><td>ACIS-S</td><td>0.049</td><td>40.7</td><td>6.5</td><td>2000 Sep 24</td></tr><tr><td>A478</td><td>1669</td><td>ACIS-S</td><td>0.086</td><td>42.4</td><td>31.2</td><td>2001 Jan 27</td></tr><tr><td>A496</td><td>4976</td><td>ACIS-S</td><td>0.033</td><td>75.1</td><td>36.5</td><td>2004 Jul 22</td></tr><tr><td>AS1101</td><td>11758</td><td>ACIS-I</td><td>0.056</td><td>97.7</td><td>79.6</td><td>2009 Aug 24</td></tr><tr><td>AS780</td><td>9428</td><td>ACIS-S</td><td>0.234</td><td>39.6</td><td>34.6</td><td>2008 Jun 16</td></tr><tr><td>AS851</td><td>11753</td><td>ACIS-I</td><td>0.010</td><td>72.6</td><td>64.7</td><td>2009 Aug 19</td></tr><tr><td>Hercules</td><td>5796</td><td>ACIS-S</td><td>0.155</td><td>47.5</td><td>43.2</td><td>2005 May 9</td></tr><tr><td>Hydra</td><td>4970</td><td>ACIS-S</td><td>0.055</td><td>98.8</td><td>86.7</td><td>2004 Oct 22</td></tr><tr><td>RXJ0058.9+2657</td><td>6830</td><td>ACIS-I</td><td>0.048</td><td>94.4</td><td>78.2</td><td>2006 Sep 2</td></tr><tr><td>RXJ0107.4+3227</td><td>2147</td><td>ACIS-S</td><td>0.018</td><td>44.4</td><td>33.2</td><td>2000 Nov 6</td></tr><tr><td>RXJ0123.6+3315</td><td>2882</td><td>ACIS-I</td><td>0.017</td><td>43.6</td><td>33.9</td><td>2002 Jan 8</td></tr><tr><td>RXJ0341.3+1524</td><td>4182</td><td>ACIS-I</td><td>0.029</td><td>23.5</td><td>22.9</td><td>2003 Mar 11</td></tr><tr><td>RXJ0352.9+1941</td><td>10466</td><td>ACIS-S</td><td>0.109</td><td>27.2</td><td>24.3</td><td>2008 Dec 18</td></tr><tr><td>RXJ0439.0+0520</td><td>9369</td><td>ACIS-I</td><td>0.245</td><td>19.9</td><td>18.7</td><td>2007 Nov 12</td></tr><tr><td>RXJ0751.3+5012</td><td>15170</td><td>ACIS-I</td><td>0.024</td><td>97.7</td><td>74.7</td><td>2013 May 14</td></tr><tr><td>RXJ0819.6+6336</td><td>2199</td><td>ACIS-S</td><td>0.119</td><td>14.9</td><td>11.3</td><td>2000 Oct 19</td></tr><tr><td>RXJ1050.4-1250</td><td>3243</td><td>ACIS-S</td><td>0.015</td><td>29.5</td><td>22.6</td><td>2002 Nov 5</td></tr><tr><td>RXJ1304.3-3031</td><td>4998</td><td>ACIS-I</td><td>0.010</td><td>15.0</td><td>13.6</td><td>2004 Feb 15</td></tr><tr><td>RXJ1315.4-1623</td><td>9399</td><td>ACIS-S</td><td>0.009</td><td>82.7</td><td>66.9</td><td>2008 Mar 7</td></tr><tr><td></td><td>17196</td><td>ACIS-S</td><td>0.009</td><td>88.9</td><td>80.9</td><td>2015 May 11</td></tr><tr><td>RXJ1320.1+3308</td><td>6941</td><td>ACIS-S</td><td>0.038</td><td>38.6</td><td>31.2</td><td>2005 Nov 1</td></tr><tr><td>RXJ1501.1+0141</td><td>12952</td><td>ACIS-S</td><td>0.007</td><td>143.0</td><td>126.0</td><td>2011 Apr 5</td></tr><tr><td></td><td>9517</td><td>ACIS-S</td><td>0.007</td><td>98.8</td><td>82.0</td><td>2008 Jun 5</td></tr><tr><td>RXJ1504.1-0248</td><td>5793</td><td>ACIS-I</td><td>0.217</td><td>39.2</td><td>30.7</td><td>2005 Mar 20</td></tr><tr><td>RXJ1506.4+0136</td><td>7923</td><td>ACIS-I</td><td>0.006</td><td>90.0</td><td>76.4</td><td>2007 Jun 12</td></tr><tr><td>RXJ1522.0+0741</td><td>900</td><td>ACIS-I</td><td>0.045</td><td>57.3</td><td>53.2</td><td>2000 Apr 3</td></tr><tr><td>RXJ1524.2-3154</td><td>9401</td><td>ACIS-S</td><td>0.102</td><td>40.9</td><td>29.4</td><td>2008 Jan 7</td></tr></tbody></table>
<table><caption>**Table 1**
(Continued)</caption><thead><tr><th>Name</th><th>Obsid</th><th>Instrument</th><th>Redshift</th><th>t<sub>total</sub><br>(ks)</th><th>t<sub>clean</sub><br>(ks)</th><th>Date</th></tr></thead><tbody><tr><td>RXJ1539.5-8335</td><td>8266</td><td>ACIS-I</td><td>0.076</td><td>7.99</td><td>7.64</td><td>2007 Jun 24</td></tr><tr><td>RXJ1558.4-1410</td><td>9402</td><td>ACIS-S</td><td>0.097</td><td>40.1</td><td>31.4</td><td>2008 Apr 9</td></tr><tr><td>RXJ1604.9+2356</td><td>9423</td><td>ACIS-S</td><td>0.032</td><td>74.5</td><td>56.9</td><td>2008 May 18</td></tr><tr><td>RXJ1715.3+5725</td><td>4194</td><td>ACIS-I</td><td>0.028</td><td>47.3</td><td>25.9</td><td>2003 Sep 17</td></tr><tr><td>RXJ1720.1+2638</td><td>4361</td><td>ACIS-I</td><td>0.161</td><td>25.7</td><td>15.6</td><td>2002 Aug 19</td></tr><tr><td>RXJ1750.2+3504</td><td>12252</td><td>ACIS-I</td><td>0.171</td><td>19.8</td><td>16.8</td><td>2010 Oct 15</td></tr><tr><td>RXJ1844.1+4533</td><td>5295</td><td>ACIS-I</td><td>0.092</td><td>30.7</td><td>25.1</td><td>2004 Jan 29</td></tr><tr><td>RXJ2129.6+0005</td><td>9370</td><td>ACIS-I</td><td>0.235</td><td>29.5</td><td>27.0</td><td>2009 Apr 3</td></tr><tr><td>Z1665</td><td>15161</td><td>ACIS-I</td><td>0.031</td><td>9.95</td><td>7.11</td><td>2013 Feb 27</td></tr><tr><td>Z235</td><td>11735</td><td>ACIS-S</td><td>0.083</td><td>19.8</td><td>13.3</td><td>2009 Sep 6</td></tr><tr><td>Z3146</td><td>9371</td><td>ACIS-I</td><td>0.291</td><td>40.2</td><td>28.0</td><td>2008 Jan 18</td></tr><tr><td>Z7160</td><td>4192</td><td>ACIS-I</td><td>0.258</td><td>91.9</td><td>79.3</td><td>2003 Sep 5</td></tr><tr><td>Z808</td><td>12253</td><td>ACIS-I</td><td>0.169</td><td>18.8</td><td>14.3</td><td>2010 Oct 6</td></tr><tr><td>Z8193</td><td>14988</td><td>ACIS-S</td><td>0.175</td><td>18.2</td><td>11.8</td><td>2013 Oct 7</td></tr><tr><td>Z8276</td><td>11708</td><td>ACIS-S</td><td>0.075</td><td>45.4</td><td>38.9</td><td>2009 Nov 26</td></tr></tbody></table>

from now on, we will refer to the two populations as more relaxed, e.g., clusters showing CC properties, and more disturbed (clusters showing NCC properties).

To calculate these parameters, we constructed surface brightness profiles in the 0.7–2 keV band for each cluster. For each cluster, we created 38 concentric annuli with a minimum of 10 pixels and a maximum of 200 pixels. To account for the background, we used the blank-sky images in the same energy range and with the same annuli. The obtained background-subtracted surface brightness profiles were fit with a double-beta model (Hudson et al. 2010):

$$ \Sigma = A_{0,1} \left[ 1 + \left( \frac{r}{r_{c,1}} \right)^2 \right]^{-3\beta_1+0.5} + A_{0,2} \left[ 1 + \left( \frac{r}{r_{c,2}} \right)^2 \right]^{-3\beta_2+0.5}, \quad (2) $$

where $A_{0,1}$ and $A_{0,2}$ are the amplitudes, and $r_{c,1}$ and $r_{c,2}$ are the core radii. The values for amplitude, core radii, and $\beta_{1,2}$ were fit for each cluster. On the basis of these, we computed the four parameters to define the relaxed and disturbed cluster subsamples. The obtained distribution of these subsamples are shown in Figure 2.

### 3.2.1. Central Electron Density

To calculate the central electron density ($n_{e,0}$) from the double-beta model, we use

$$ n_{e,0} = \left[ n_{e,1}^2 \left( 1 + \left( \frac{r}{r_{c,1}} \right)^2 \right)^{-3\beta_1} + n_{e,2}^2 \left( 1 + \left( \frac{r}{r_{c,2}} \right)^2 \right)^{-3\beta_2} \right]^{\frac{1}{2}}, \quad (3) $$

where $n_{e,1}$ and $n_{e,2}$ are the central electron densities for each component in the double-beta model. The calculation of these quantities depends upon $n_0$, i.e., the central density:

$$ n_0 = \left( \frac{10^{14} 4 \pi D_A D_L \zeta N}{EI} \right)^{0.5}, \quad (4) $$

where N is the normalization obtained from the best-fit *apec* model. To calculate the value of $n_0$, we used 0.048R$_{500}$ as the

central extraction region (Hudson et al. 2010). The electron densities $n_{e,1}$ and $n_{e,2}$ for the double-beta model were calculated following Hudson et al. (2010).

The value of $n_{e,0}$ could be used to classify clusters as either disturbed or relaxed. The median which we took as a “splitting” value was $n_{e,0} = 0.040 \text{ cm}^{-3}$. Clusters with a density less than this value were classified as disturbed systems, as a hotter central temperature indicates a lower central density, which is characteristic of a NCC system.

### 3.2.2. Central Entropy

To split the clusters according to their central entropy ($K_0$), we compute this parameter using

$$ K_0 = k_B T_0 n_{e,0}^{-\frac{2}{3}}, \quad (5) $$

where $T_0$ is the central temperature, which is computed for the 0.048R$_{500}$ region, thereby maintaining consistency with the normalization in Equation (4). The “splitting” value for this parameter is $K_0 = 26.59 \text{ keV cm}^2$. If $K_0$ was less than this value, then the cluster would be classified as a relaxed cluster, as the gas at the center is cooler and, hence, less perturbed.

### 3.2.3. Cuspiness

The cuspiness parameter ($\alpha$) was suggested as a parameter for identifying CC clusters at large redshifts (Vikhlinin et al. 2007) and is defined as

$$ \alpha = - \frac{d \log(n_e(r))}{d \log(r)}, \quad (6) $$

when $r = 0.04R_{500}$. As our density function is based on the double-beta model, we can recast Equation (6) as the following:

$$ \alpha = 3r^2 \frac{\sum_{12} \mathrm{Li}_2 \beta_1 r_{c,1}^{-2} b_1' + \mathrm{Li}_1 \beta_2 r_{c,2}^{-2} b_2'}{\sum_{12} \mathrm{Li}_2 b_1 + \mathrm{Li}_1 b_2}, \quad (7) $$

where the core radii and the $\beta_{1,2}$ are the values found from fitting the double-beta model (Equation (2)) to the surface brightness profile. The values $\mathrm{Li}_i$ are the line emission measure for model i and $\sigma_{12}$ is the ratio of the central surface brightness
<table><caption>Table 2
Characteristics of the Analyzed Galaxy Clusters, Their BCGs, and BHs</caption><thead><tr><th>Name<br>(1)</th><th>N<sub>H</sub><br>(10<sup>20</sup> cm<sup>-2</sup>)<br>(2)</th><th>log<sub>10</sub>(M<sub>BH</sub>)<br>(M<sub>☉</sub>)<br>(3)</th><th>log<sub>10</sub>(M<sub>bulge</sub>)<br>(M<sub>☉</sub>)<br>(4)</th><th>kT<br>(keV)<br>(5)</th><th>M<sub>500</sub><br>(10<sup>14</sup>M<sub>☉</sub>)<br>(6)</th><th>R<sub>500</sub><br>(Mpc)<br>(7)</th><th>R<sub>frac</sub><br>(8)</th></tr></thead><tbody><tr><td>A1204</td><td>1.38</td><td>9.01 ± 0.62</td><td>11.77 ± 0.02</td><td>2.51 ± 0.31</td><td>1.67 ± 0.18</td><td>0.716 ± 0.016</td><td>1.00</td></tr><tr><td>A1367<sup>a</sup></td><td>2.39</td><td>&lt;11.19</td><td>11.40 ± 0.01</td><td>2.40 ± 0.07</td><td>2.11 ± 0.04</td><td>0.887 ± 0.007</td><td>0.46</td></tr><tr><td>A1446</td><td>1.5</td><td>10 ± 0.08</td><td>11.83 ± 0.02</td><td>3.17 ± 0.25</td><td>2.49 ± 0.17</td><td>0.867 ± 0.012</td><td>0.68</td></tr><tr><td>A1644<sup>a</sup></td><td>5.19</td><td>&lt;10.78</td><td>12.13 ± 0.04</td><td>2.90 ± 0.11</td><td>2.14 ± 0.07</td><td>0.0867 ± 0.006</td><td>0.73</td></tr><tr><td>A1664<sup>a</sup></td><td>8.72</td><td>&lt;10.58</td><td>11.76 ± 0.02</td><td>4.54 ± 0.42</td><td>4.60 ± 0.36</td><td>1.040 ± 0.017</td><td>0.73</td></tr><tr><td>A168<sup>a</sup></td><td>3.25</td><td>&lt;9.14</td><td>11.76 ± 0.02</td><td>1.69 ± 0.07</td><td>0.85 ± 0.03</td><td>0.640 ± 0.005</td><td>0.58</td></tr><tr><td>A1763</td><td>0.92</td><td>9.94 ± 0.11</td><td>12.37 ± 0.05</td><td>5.82 ± 0.66</td><td>7.04 ± 0.68</td><td>1.101 ± 0.022</td><td>1.00</td></tr><tr><td>A1795<sup>a</sup></td><td>1.17</td><td>&lt;10.83</td><td>12.05 ± 0.03</td><td>6.07 ± 0.35</td><td>7.56 ± 0.37</td><td>1.302 ± 0.014</td><td>0.69</td></tr><tr><td>A1930</td><td>1.13</td><td>9.79 ± 0.16</td><td>12.25 ± 0.05</td><td>3.34 ± 0.43</td><td>2.72 ± 0.3</td><td>0.870 ± 0.020</td><td>0.89</td></tr><tr><td>A2009</td><td>3.27</td><td>9.74 ± 0.08</td><td>12.17 ± 0.04</td><td>3.89 ± 0.55</td><td>3.53 ± 0.43</td><td>0.931 ± 0.024</td><td>1.00</td></tr><tr><td>A2029<sup>a</sup></td><td>3.15</td><td>&lt;9.38</td><td>11.90 ± 0.03</td><td>8.76 ± 0.05</td><td>10.1 ± 7.7</td><td>1.415 ± 0.361</td><td>0.71</td></tr><tr><td>A2033</td><td>2.94</td><td>9.68 ± 0.15</td><td>12.17 ± 0.04</td><td>2.12 ± 0.23</td><td>1.25 ± 0.12</td><td>0.705 ± 0.014</td><td>0.96</td></tr><tr><td>A2052</td><td>2.85</td><td>11.23 ± 0.12</td><td>11.93 ± 0.03</td><td>3.01 ± 0.05</td><td>2.57 ± 0.04</td><td>0.936 ± 0.005</td><td>0.68</td></tr><tr><td>A2063<sup>a</sup></td><td>3.04</td><td>&lt;9.03</td><td>11.76 ± 0.02</td><td>3.39 ± 0.50</td><td>2.90 ± 0.44</td><td>0.974 ± 0.049</td><td>0.61</td></tr><tr><td>A2110</td><td>2.39</td><td>8.88 ± 0.16</td><td>11.91 ± 0.03</td><td>2.63 ± 0.38</td><td>1.52 ± 0.21</td><td>0.738 ± 0.021</td><td>1.00</td></tr><tr><td>A2199<sup>a</sup></td><td>0.89</td><td>&lt;9.72</td><td>11.86 ± 0.02</td><td>3.25 ± 0.09</td><td>2.59 ± 0.06</td><td>0.940 ± 0.005</td><td>0.38</td></tr><tr><td>A2204<sup>a</sup></td><td>5.69</td><td>&lt;9.77</td><td>12.15 ± 0.04</td><td>5.97 ± 0.25</td><td>7.34 ± 0.26</td><td>1.191 ± 0.009</td><td>1.00</td></tr><tr><td>A2355</td><td>4.75</td><td>10.29 ± 0.26</td><td>12.21 ± 0.05</td><td>6.17 ± 1.03</td><td>7.77 ± 1.11</td><td>1.135 ± 0.034</td><td>1.00</td></tr><tr><td>A2390<sup>a</sup></td><td>6.89</td><td>&lt;12.05</td><td>12.33 ± 0.06</td><td>11.62 ± 0.97</td><td>22.9 ± 1.64</td><td>1.626 ± 0.024</td><td>0.89</td></tr><tr><td>A2415</td><td>4.79</td><td>11.13 ± 0.01</td><td>11.44 ± 0.02</td><td>2.18 ± 0.55</td><td>1.31 ± 0.28</td><td>0.731 ± 0.033</td><td>0.71</td></tr><tr><td>A2597<sup>a</sup></td><td>2.49</td><td>&lt;10.65</td><td>11.57 ± 0.02</td><td>3.42 ± 0.14</td><td>2.83 ± 0.10</td><td>0.921 ± 0.007</td><td>0.54</td></tr><tr><td>A262</td><td>5.46</td><td>9.01 ± 0.08</td><td>11.63 ± 0.02</td><td>2.33 ± 0.08</td><td>2.06 ± 0.06</td><td>0.885 ± 0.008</td><td>0.34</td></tr><tr><td>A2626</td><td>4.33</td><td>9.71 ± 0.01</td><td>11.98 ± 0.03</td><td>3.20 ± 0.14</td><td>2.53 ± 0.10</td><td>0.911 ± 0.007</td><td>0.75</td></tr><tr><td>A2634</td><td>5.06</td><td>10.88 ± 0.15</td><td>11.77 ± 0.02</td><td>5.89 ± 0.32</td><td>7.19 ± 0.33</td><td>1.322 ± 0.013</td><td>0.50</td></tr><tr><td>A2665</td><td>6.04</td><td>8.58 ± 0.15</td><td>11.99 ± 0.03</td><td>2.68 ± 1.79</td><td>1.87 ± 1.07</td><td>0.822 ± 0.099</td><td>0.66</td></tr><tr><td>A2667</td><td>1.65</td><td>10.35 ± 0.27</td><td>11.97 ± 0.04</td><td>7.05 ± 1.09</td><td>9.77 ± 1.29</td><td>1.222 ± 0.034</td><td>1.00</td></tr><tr><td>A3017</td><td>2.09</td><td>9.76 ± 0.17</td><td>11.73 ± 0.02</td><td>4.71 ± 0.84</td><td>4.90 ± 0.75</td><td>0.983 ± 0.031</td><td>1.00</td></tr><tr><td>A3526<sup>a</sup></td><td>8.1</td><td>&lt;10.55</td><td>11.77 ± 0.02</td><td>2.26 ± 0.01</td><td>1.40 ± 0.01</td><td>0.781 ± 0.001</td><td>0.82</td></tr><tr><td>A3528S<sup>a</sup></td><td>6.13</td><td>&lt;8.77</td><td>12.11 ± 0.04</td><td>2.06 ± 0.24</td><td>1.19 ± 0.12</td><td>0.707 ± 0.015</td><td>0.88</td></tr><tr><td>A3581</td><td>4.25</td><td>10.41 ± 0.05</td><td>11.48 ± 0.01</td><td>1.62 ± 0.05</td><td>1.63 ± 0.03</td><td>0.814 ± 0.005</td><td>0.65</td></tr><tr><td>A3695</td><td>3.7</td><td>10.37 ± 0.08</td><td>11.90 ± 0.03</td><td>2.80 ± 0.29</td><td>2.01 ± 0.18</td><td>0.817 ± 0.015</td><td>1.00</td></tr><tr><td>A4059<sup>a</sup></td><td>1.1</td><td>&lt;9.38</td><td>12.20 ± 0.04</td><td>5.96 ± 0.25</td><td>5.76 ± 0.34</td><td>1.208 ± 0.024</td><td>0.61</td></tr><tr><td>A478<sup>a</sup></td><td>26.8</td><td>&lt;9.90</td><td>11.93 ± 0.03</td><td>7.82 ± 0.26</td><td>8.48 ± 0.93</td><td>1.327 ± 0.028</td><td>0.71</td></tr><tr><td>A496<sup>a</sup></td><td>4.8</td><td>&lt;10.77</td><td>12.00 ± 0.03</td><td>7.51 ± 0.43</td><td>10.9 ± 0.53</td><td>1.513 ± 0.016</td><td>0.58</td></tr><tr><td>AS1101<sup>a</sup></td><td>1.83</td><td>&lt;9.42</td><td>11.92 ± 0.03</td><td>1.37 ± 0.01</td><td>0.59 ± 0.01</td><td>0.560 ± 0.001</td><td>0.85</td></tr><tr><td>AS780</td><td>7.72</td><td>11.14 ± 0.01</td><td>12.25 ± 0.05</td><td>5.57 ± 0.68</td><td>6.52 ± 0.68</td><td>1.068 ± 0.023</td><td>1.00</td></tr><tr><td>AS851</td><td>4.96</td><td>9.76 ± 0.08</td><td>11.61 ± 0.02</td><td>0.73 ± 0.01</td><td>0.20 ± 0.01</td><td>0.412 ± 0.001</td><td>0.77</td></tr><tr><td>Hercules<sup>a</sup></td><td>6.33</td><td>&lt;10.15</td><td>12.04 ± 0.03</td><td>3.25 ± 0.23</td><td>2.60 ± 0.16</td><td>0.840 ± 0.010</td><td>1.00</td></tr><tr><td>Hydra</td><td>4.84</td><td>10.99 ± 0.62</td><td>11.78 ± 0.02</td><td>4.13 ± 0.09</td><td>3.92 ± 0.07</td><td>1.054 ± 0.004</td><td>0.69</td></tr><tr><td>RXJ0058.9+2657</td><td>5.73</td><td>10.69 ± 0.08</td><td>11.91 ± 0.03</td><td>1.33 ± 0.07</td><td>0.57 ± 0.03</td><td>0.556 ± 0.006</td><td>0.79</td></tr><tr><td>RXJ0107.4+3227</td><td>5.41</td><td>9.88 ± 0.08</td><td>11.65 ± 0.02</td><td>6.07 ± 0.76</td><td>7.56 ± 0.81</td><td>1.361 ± 0.031</td><td>0.33</td></tr><tr><td>RXJ0123.6+3315</td><td>5.23</td><td>8.28 ± 0.01</td><td>11.69 ± 0.02</td><td>1.32 ± 0.03</td><td>0.56 ± 0.01</td><td>0.570 ± 0.002</td><td>0.26</td></tr><tr><td>RXJ0341.3+1524</td><td>16.17</td><td>9.53 ± 0.08</td><td>11.19 ± 0.02</td><td>1.72 ± 0.02</td><td>0.87 ± 0.01</td><td>0.656 ± 0.002</td><td>0.45</td></tr><tr><td>RXJ0352.9+1941</td><td>26.83</td><td>8.97 ± 0.06</td><td>11.85 ± 0.03</td><td>2.19 ± 0.15</td><td>1.96 ± 0.86</td><td>0.798 ± 0.019</td><td>0.75</td></tr><tr><td>RXJ0439.0+0520</td><td>10.3</td><td>11.95 ± 0.12</td><td>12.10 ± 0.04</td><td>3.19 ± 0.43</td><td>2.51 ± 0.29</td><td>0.770 ± 0.019</td><td>1.00</td></tr><tr><td>RXJ0751.3+5012<sup>a</sup></td><td>5.09</td><td>&lt;9.21</td><td>11.33 ± 0.01</td><td>0.99 ± 0.07</td><td>0.34 ± 0.02</td><td>0.482 ± 0.006</td><td>0.40</td></tr><tr><td>RXJ0819.6+6336</td><td>4.16</td><td>8.47 ± 0.12</td><td>12.12 ± 0.04</td><td>3.55 ± 0.68</td><td>3.01 ± 0.50</td><td>0.911 ± 0.032</td><td>0.75</td></tr><tr><td>RXJ1050.4-1250</td><td>4.5</td><td>7.95 ± 0.09</td><td>11.39 ± 0.01</td><td>0.96 ± 0.06</td><td>0.33 ± 0.02</td><td>0.478 ± 0.005</td><td>0.53</td></tr><tr><td>RXJ1304.3-3031<sup>a</sup></td><td>6.01</td><td>&lt;9.32</td><td>11.58 ± 0.02</td><td>0.95 ± 0.03</td><td>0.32 ± 0.01</td><td>0.477 ± 0.003</td><td>0.91</td></tr><tr><td>RXJ1315.4-1623</td><td>4.94</td><td>9.26 ± 0.08</td><td>11.44 ± 0.01</td><td>1.20 ± 0.03</td><td>1.43 ± 0.01</td><td>0.789 ± 0.003</td><td>0.30</td></tr><tr><td>RXJ1320.1+3308</td><td>1.05</td><td>9.32 ± 0.12</td><td>11.46 ± 0.01</td><td>1.04 ± 0.04</td><td>0.37 ± 0.01</td><td>0.487 ± 0.003</td><td>0.71</td></tr><tr><td>RXJ1501.1+0141</td><td>4.25</td><td>8.06 ± 0.08</td><td>11.21 ± 0.02</td><td>0.59 ± 0.09</td><td>1.22 ± 0.02</td><td>0.750 ± 0.004</td><td>0.23</td></tr><tr><td>RXJ1504.1-0248<sup>a</sup></td><td>6.1</td><td>&lt;10.69</td><td>11.93 ± 0.03</td><td>6.99 ± 0.57</td><td>9.63 ± 0.67</td><td>1.233 ± 0.018</td><td>1.00</td></tr><tr><td>RXJ1506.4+0136</td><td>4.24</td><td>8.97 ± 0.23</td><td>11.31 ± 0.01</td><td>0.89 ± 0.01</td><td>0.28 ± 0.01</td><td>0.460 ± 0.001</td><td>0.47</td></tr><tr><td>RXJ1522.0+0741<sup>a</sup></td><td>3.05</td><td>&lt;8.91</td><td>11.63 ± 0.02</td><td>2.96 ± 0.10</td><td>2.22 ± 0.06</td><td>0.880 ± 0.005</td><td>0.65</td></tr><tr><td>RXJ1524.2-3154<sup>a</sup></td><td>8.44</td><td>&lt;10.55</td><td>11.87 ± 0.03</td><td>3.00 ± 0.21</td><td>2.26 ± 0.13</td><td>0.841 ± 0.010</td><td>0.70</td></tr><tr><td>RXJ1539.5-8335</td><td>7.68</td><td>10.19 ± 0.15</td><td>12.10 ± 0.04</td><td>1.76 ± 0.28</td><td>0.91 ± 0.1</td><td>0.635 ± 0.018</td><td>0.67</td></tr><tr><td>RXJ1558.4-1410</td><td>11.47</td><td>12.23 ± 0.15</td><td>12.19 ± 0.04</td><td>3.84 ± 0.22</td><td>3.45 ± 0.17</td><td>0.972 ± 0.010</td><td>0.64</td></tr><tr><td>RXJ1604.9+2356</td><td>4.99</td><td>10.77 ± 0.06</td><td>11.85 ± 0.02</td><td>2.27 ± 0.28</td><td>1.41 ± 0.15</td><td>0.766 ± 0.017</td><td>0.53</td></tr><tr><td>RXJ1715.3+5725</td><td>2.6</td><td>9.96 ± 0.55</td><td>11.75 ± 0.02</td><td>1.37 ± 0.02</td><td>0.59 ± 0.01</td><td>0.575 ± 0.001</td><td>0.39</td></tr><tr><td>RXJ1720.1+2638<sup>a</sup></td><td>3.89</td><td>&lt;9.15</td><td>12.13 ± 0.04</td><td>5.39 ± 0.46</td><td>6.17 ± 0.45</td><td>1.115 ± 0.017</td><td>1.00</td></tr><tr><td>RXJ1750.2+3504</td><td>3.12</td><td>11.08 ± 0.15</td><td>12.17 ± 0.04</td><td>3.00 ± 0.49</td><td>2.27 ± 0.31</td><td>0.791 ± 0.023</td><td>1.00</td></tr></tbody></table>
<table><caption>**Table 2**
(Continued)</caption><thead><tr><th>Name<br>(1)</th><th>N<sub>H</sub><br>(10<sup>20</sup> cm<sup>-2</sup>)<br>(2)</th><th>log<sub>10</sub>(M<sub>BH</sub>)<br>(M<sub>☉</sub>)<br>(3)</th><th>log<sub>10</sub>(M<sub>bulge</sub>)<br>(M<sub>☉</sub>)<br>(4)</th><th>kT<br>(keV)<br>(5)</th><th>M<sub>500</sub><br>(10<sup>14</sup>M<sub>☉</sub>)<br>(6)</th><th>R<sub>500</sub><br>(Mpc)<br>(7)</th><th>R<sub>frac</sub><br>(8)</th></tr></thead><tbody><tr><td>RXJ1844.1+4533</td><td>6.32</td><td>10.72 ± 0.01</td><td>11.97 ± 0.03</td><td>1.72 ± 0.11</td><td>0.88 ± 0.01</td><td>0.618 ± 0.007</td><td>1.00</td></tr><tr><td>RXJ2129.6+0005<sup>a</sup></td><td>4.16</td><td>&lt;9.76</td><td>12.16 ± 0.04</td><td>4.30 ± 0.36</td><td>4.19 ± 0.3</td><td>0.921 ± 0.014</td><td>1.00</td></tr><tr><td>Z1665<sup>a</sup></td><td>2.74</td><td>&lt;9.09</td><td>11.65 ± 0.02</td><td>1.48 ± 0.26</td><td>0.68 ± 0.1</td><td>0.601 ± 0.019</td><td>0.56</td></tr><tr><td>Z235<sup>a</sup></td><td>3.91</td><td>&lt;10.65</td><td>11.95 ± 0.03</td><td>3.45 ± 0.35</td><td>2.87 ± 0.25</td><td>0.926 ± 0.017</td><td>0.55</td></tr><tr><td>Z3146<sup>a</sup></td><td>2.93</td><td>&lt;8.73</td><td>12.19 ± 0.06</td><td>5.59 ± 0.55</td><td>6.57 ± 0.55</td><td>1.024 ± 0.018</td><td>1.00</td></tr><tr><td>Z7160<sup>a</sup></td><td>3.22</td><td>&lt;9.03</td><td>12.41 ± 0.06</td><td>3.60 ± 0.26</td><td>3.09 ± 0.19</td><td>0.817 ± 0.011</td><td>1.00</td></tr><tr><td>Z808<sup>a</sup></td><td>7.55</td><td>&lt;9.06</td><td>11.95 ± 0.03</td><td>3.03 ± 0.58</td><td>2.30 ± 0.38</td><td>0.797 ± 0.027</td><td>1.00</td></tr><tr><td>Z8193<sup>a</sup></td><td>2.31</td><td>&lt;11.33</td><td>12.28 ± 0.05</td><td>4.47 ± 0.71</td><td>4.47 ± 0.61</td><td>0.989 ± 0.028</td><td>0.72</td></tr><tr><td>Z8276</td><td>3.66</td><td>10.53 ± 0.54</td><td>11.89 ± 0.03</td><td>3.79 ± 0.23</td><td>3.38 ± 0.17</td><td>0.985 ± 0.011</td><td>0.94</td></tr></tbody></table>

Note. Columns are as follows: (1) name of the galaxy cluster; (2) line-of-sight column density to the cluster (Kalberla et al. 2005); (3) BH mass obtained from the Fundamental Plane relation (Merloni et al. 2003); (4) stellar bulge mass of the BCG calculated using K-band luminosity and the mass-to-light ratio of 0.85 (Bell et al. 2003); (5) best-fit temperature of the ICM; (6) $M_{500}$ mass inferred from the best-fit temperature in column (5) using the kT-M500 relation of Lovisari et al. (2015); (7) $R_{500}$ radius of the cluster; and (8) the fraction of the $R_{500}$ radius included in the Chandra field of view. The errors associated with the $M_{500}$ mass and $R_{500}$ radius were computed from the temperature uncertainties.

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Distributions of relaxed and disturbed clusters based on the four parameters from Hudson et al. (2010); (a) central electron density, $n_{e,0}$, (b) central entropy, $K_0$, (c) cuspiness, $\alpha$, and (d) central surface brightness, $C_{sb}$. The relaxed clusters, which have properties similar to those of CCs, are shown in blue hatch. The disturbed clusters, which can be likened to NCC clusters, are shown in red. The black lines indicate the median for each parameter where the clusters split into the two subsamples.</figcaption></figure>

<footnote>a Sources where the BH mass is an upper limit.</footnote>
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 3.** Scaling relations investigated in this paper: (a) BH mass—cluster temperature relation, (b) BH mass—bulge mass relation, (c) BH mass—cluster mass relation, and (d) bulge mass—cluster temperature relation. The red squares are from this work, with downward arrows indicating upper limits in BH mass. The blue circles are for the BGGs/BCGs studied in Bogdán et al. (2018). The best-fitting lines for the data are shown with the solid line and were all calculated with BCES_REGRESS code. For the fitting procedure, the upper limits were included. The dotted-dashed line in the relation is from Bogdán et al. (2018), the dotted line is from a joint fit based on the sample of this work and Bogdán et al. (2018) data, and in panel (b) the dashed line is the $M_{BH}-M_{bulge}$ relation from McConnell & Ma (2013).</figcaption></figure>

of model 1 to model 2. Finally, $b_i, b'_i$ are defined as

$$b_i = \left( 1 + \left( \frac{r}{r_{c,i}} \right)^2 \right)^{-3\beta_i} \quad (8)$$

and

$$b'_i = \left( 1 + \left( \frac{r}{r_{c,i}} \right)^2 \right)^{-3\beta_i - 1}, \quad (9)$$

where $i = 1, 2$. For further details we refer to Hudson et al. (2010).

The median “splitting” value for this parameter is $\alpha = 1.05$. If $\alpha > 1.05$, then the cluster is classified as disturbed.

### 3.2.4. Surface Brightness Density

The final parameter we used to identify clusters as either CC or NCC was the surface brightness density, $C_{sb}$. This parameter was first used by Santos et al. (2008) as a way to measure the excess of brightness at the core of a cluster. The

$C_{sb}$ is defined as

$$C_{sb} = \frac{\sum(r < 40 \text{ kpc})}{\sum(r < 400 \text{ kpc})}. \quad (10)$$

In other words, the $C_{sb}$ is the ratio of the integrated surface brightness within a radius 40 kpc to that within a radius of 400 kpc. For this parameter, a median “splitting” value of $C_{sb} = 0.66$ is used where clusters with a $C_{sb}$ greater than this value are classified as relaxed clusters, as CC clusters are bright at the center (Fabian et al. 1984).

We note that the median values we use to split the samples into relaxed and disturbed clusters differ from the literature, but this is to be expected because of differences between our samples and the data used.

## 4. Results

### 4.1. Correlations

In the top left and right panels of Figure 3, we depict the observed relation between the BH mass and the cluster temperature and stellar bulge mass, respectively. Note that the BH masses are inferred from the Fundamental Plane relation (Merloni et al. 2003). The errors in the BH masses
<table><caption>**Table 3**
Best-fit Parameters on Linear Regression for Our Data</caption><thead><tr><th>Relation</th><th>α</th><th>β</th><th>σ<sup>x</sup><sub>intron</sub></th><th>σ<sup>y</sup><sub>intron</sub></th><th>r<sup>a</sup></th><th>ρ<sup>b</sup></th></tr></thead><tbody><tr><td>M<sub>BH</sub>–kT</td><td>−0.35 ± 0.21</td><td>2.08 ± 0.43</td><td>0.35</td><td>0.91</td><td>0.26</td><td>0.24</td></tr><tr><td>M<sub>BH</sub>–M<sub>bulge</sub></td><td>−1.09 ± 0.39</td><td>1.92 ± 0.45</td><td>0.49</td><td>0.96</td><td>0.26</td><td>0.24</td></tr><tr><td>M<sub>BH</sub>–M<sub>500</sub></td><td>−1.08 ± 0.25</td><td>1.24 ± 0.19</td><td>0.69</td><td>0.99</td><td>0.18</td><td>0.20</td></tr><tr><td>M<sub>Bulge</sub>–kT</td><td>0.39 ± 0.05</td><td>1.06 ± 0.08</td><td>0.23</td><td>0.22</td><td>0.66</td><td>0.62</td></tr></tbody></table>

**Notes.**

<sup>a</sup> Pearson correlation coefficient.

<sup>b</sup> Spearman correlation coefficient.

were calculated using the uncertainties obtained for the X-ray and radio luminosities of the BHs. To infer the BH mass from the Fundamental Plane, it is necessary to detect the BHs both in radio and X-ray wavelengths. However, in the sample of Mezcua et al. (2018), several BHs remain undetected in the X-ray band. Therefore, for these BHs, we compute upper limits on the BH mass. To derive the stellar mass of the BCGs, we rely on Mezcua et al. (2018), who provide the K-band absolute magnitude of the galaxies. We convert the K-band luminosities to stellar mass using the K-band mass-to-light ratio of 0.85, which is typical for massive elliptical galaxies (Bell et al. 2003). On the plots, we add the data points from Bogdán et al. (2018), who used nearby galaxy groups/clusters with dynamically measured BHs. In the bottom left and right panels of Figure 3, we show the relations between the inferred BH mass against cluster mass inferred from the kT–M<sub>500</sub> relation (Lovisari et al. 2015), as well as the relation between bulge mass and best-fit cluster temperature, respectively. We note that, for six clusters in our sample, the fraction of the R<sub>500</sub> radius covered by the field of view of Chandra is R<sub>frac</sub> < 0.4. After examining these clusters, we concluded that the BHs associated with their BCGs are not preferentially overmassive or undermassive relative to other systems. As such, these data points do not introduce a bias in our results.

To compute the best-fit relations, we used the BCES_REGRESS code (Akritas & Bershady 1996). This linear regression code uses bivariate correlated errors and an intrinsic scatter fitting method. It is advantageous to use this over standard linear regression fitting as it allows for the errors in both the x and y measurements to be taken into account. We performed the fits in log–log space and used the bisector method. In order to perform a fitting that included the upper limited sources, we adopted the following method. For the nondetections (shown with arrows in Figure 3), we calculated a lower limit on the BH mass and then assumed that the true BH mass could be represented by a random uniform distribution between the upper and lower limits. The lower limits were calculated using the M<sub>BH</sub>–M<sub>bulge</sub> relation of McConnell & Ma (2013). We take the lower limits to be 3σ below the predicted value from this relation. In order to implement the BCES_REGRESS code, all variables need an error. Therefore, for the nondetections for the BH masses, we assume that the errors are 25% of the range between the measured upper limit and the calculated lower limit. We tested different percentages for the errors but found no significant difference in the result. For the detected BH mass points, we assumed a Gaussian distribution between the error range centered on the measured value. With these two random distributions for the detections and nondetections, we implemented the fitting code via a Monte Carlo simulation and repeated 10<sup>4</sup> times. The values quoted below and

in Table 3 are the mean from all these realizations. We obtained the following relationships:

$$ \log_{10}\left(\frac{M_{\mathrm{BH}}}{10^{9} M_{\odot}}\right) = -0.35 + 2.08 \log_{10}\left(\frac{kT}{1 \mathrm{keV}}\right), \quad (11) $$

$$ \log_{10}\left(\frac{M_{\mathrm{BH}}}{10^{9} M_{\odot}}\right) = -1.09 + 1.92 \log_{10}\left(\frac{M_{\mathrm{bulge}}}{10^{11} M_{\odot}}\right), \quad (12) $$

$$ \log_{10}\left(\frac{M_{\mathrm{BH}}}{10^{9} M_{\odot}}\right) = -1.08 + 1.45 \log_{10}\left(\frac{M_{\mathrm{500}}}{10^{13} M_{\odot}}\right), \quad (13) $$

$$ \log_{10}\left(\frac{M_{\mathrm{Bulge}}}{10^{11} M_{\odot}}\right) = 0.39 + 1.06 \log_{10}\left(\frac{kT}{1 \mathrm{keV}}\right). \quad (14) $$

The intrinsic scatter in both the x and y planes were calculated following Lovisari et al. (2015):

$$ \begin{aligned} \sigma_{\mathrm{intron}}^{y} &= \sqrt{(\sigma_{\mathrm{tot}}^{y})^{2} - (\sigma_{\mathrm{stat}}^{y})^{2} - (a^{2}(\sigma_{\mathrm{stat}}^{x})^{2})} \\ \sigma_{\mathrm{intron}}^{x} &= \sqrt{(\sigma_{\mathrm{tot}}^{x})^{2} - (\sigma_{\mathrm{stat}}^{x})^{2} - (a^{-2}(\sigma_{\mathrm{stat}}^{y})^{2})}, \end{aligned} \quad (15) $$

where σ<sup>x,y</sup><sub>tot</sub> and σ<sup>x,y</sup><sub>stat</sub> represent the total and statistical scatter along x and y, and the value, a, is the gradient calculated from the BCES_REGRESS code. The best-fit relations, the scatter, and the Pearson and Spearman correlation coefficients are tabulated in Table 3.

The best-fit M<sub>BH</sub>–kT relation is somewhat steeper and exhibits larger scatter than the relation found in Bogdán et al. (2018). We note that the bulge masses in this work and in that of Bogdán et al. (2018) were calculated using different methods. To make the M<sub>BH</sub>–M<sub>bulge</sub> relations comparable, we recomputed the bulge masses for the sample of Bogdán et al. (2018) using the method applied in this work. The M<sub>BH</sub>–M<sub>bulge</sub> relations are identical within 1σ uncertainties. The intrinsic scatter in the M<sub>BH</sub>–kT and M<sub>BH</sub>–M<sub>bulge</sub> relations are σ<sup>y</sup><sub>intron</sub> = 0.91 and 0.96, respectively, which exceeds those found in Bogdán et al. (2018) and is comparable with the scatter in the Fundamental Plane relation of 0.88 found by Merloni et al. (2003).

In the bottom left panel of Figure 3 we plot the BH mass against the M<sub>500</sub> mass of each cluster. To convert the cluster temperature to M<sub>500</sub>, we used the kT–M<sub>500</sub> relation of Lovisari et al. (2015) (Equation (1)). The M<sub>BH</sub>–M<sub>500</sub> relationship (Equation (13)) is consistent to within 1σ with that found by Bogdán et al. (2018). In the bottom right panel of Figure 3, we show the relation between the stellar bulge mass and the cluster temperature. While this relation exhibits smaller scatter (see Table 3), the best-fit relation and the correlation coefficients are comparable to that obtained in Bogdán et al. (2018).
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 4.** Panel (a) shows the relationship between BH mass and cluster temperature. The red squares are the results from this work, and downward arrows indicate upper limits. The light gray circles—or cloud of data—are the simulated data points from the Monte Carlo program. The red solid line is the relation from Equation (11) calculated using BCES_REGRESS code. The black lines either side the red one is the $1\sigma$ contours from the line of best fit for the simulated data. Panel (b) shows the distribution in $\sigma_{\text{intron}}^y$ for the 1000 simulations.</figcaption></figure>

<table><caption>**Table 4**
Best-fit Parameters on Linear Regression for Joint Fitting</caption><thead><tr><th>Relation</th><th>α</th><th>β</th><th>σ<sub>intron</sub><sup>x</sup></th><th>σ<sub>intron</sub><sup>y</sup></th><th>r<sup>a</sup></th><th>ρ<sup>b</sup></th></tr></thead><tbody><tr><td>M<sub>BH</sub>–kT</td><td>−0.21 ± 0.10</td><td>1.98 ± 0.24</td><td>0.35</td><td>0.84</td><td>0.37</td><td>0.35</td></tr><tr><td>M<sub>BH</sub>–M<sub>bulge</sub></td><td>−1.07 ± 0.26</td><td>1.94 ± 0.32</td><td>0.53</td><td>0.96</td><td>0.31</td><td>0.28</td></tr><tr><td>M<sub>BH</sub>–M<sub>500</sub></td><td>−0.82 ± 0.12</td><td>1.16 ± 0.10</td><td>0.72</td><td>0.92</td><td>0.31</td><td>0.31</td></tr><tr><td>M<sub>Bulge</sub>–kT</td><td>0.48 ± 0.04</td><td>0.93 ± 0.07</td><td>0.23</td><td>0.25</td><td>0.70</td><td>0.69</td></tr></tbody></table>

**Note.**

<sup>a</sup> Pearson correlation coefficient.

<sup>b</sup> Spearman correlation coefficient.

## 4.2. Joint Fit Correlations

The sample investigated in this work explores the high-mass end of the relations, while Bogdán et al. (2018) studied the low-mass end of the scaling relations. Therefore, here, we combine the two samples and perform joint fitting to establish relations, which are constrained across a broad range of galaxy groups and clusters with temperatures of $kT = 0.4–12$ keV. We find the best-fitting equations to be

$$ \log_{10}\left(\frac{M_{\text{BH}}}{10^9 M_{\odot}}\right) = -0.21 + 1.98 \log_{10}\left(\frac{kT}{1 \text{ keV}}\right), \quad (16) $$

$$ \log_{10}\left(\frac{M_{\text{BH}}}{10^9 M_{\odot}}\right) = -1.07 + 1.94 \log_{10}\left(\frac{M_{\text{bulge}}}{10^{11} M_{\odot}}\right), \quad (17) $$

$$ \log_{10}\left(\frac{M_{\text{BH}}}{10^9 M_{\odot}}\right) = -0.82 + 1.16 \log_{10}\left(\frac{M_{500}}{10^{13} M_{\odot}}\right), \quad (18) $$

$$ \log_{10}\left(\frac{M_{\text{Bulge}}}{10^{11} M_{\odot}}\right) = 0.48 + 0.92 \log_{10}\left(\frac{kT}{1 \text{ keV}}\right). \quad (19) $$

The intrinsic scatter in both axes was also calculated for the joint sample using Equation (15), as well as the Pearson and Spearman correlation coefficients. All results are summarized in Table 4. The joint fit relations are visualized in Figure 3 as the dotted green line. From Table 4, it is clear that the $M_{BH}–kT$

relation is slightly tighter than the $M_{BH}–M_{bulge}$ relation. This is likely due to the relation for the joint fit being driven by the dynamically measured BHs. The scatter in both the $M_{BH}–kT$ and the $M_{BH}–M_{bulge}$ relation has been reduced; however, the scatter arising from the Fundamental Plane relation likely plays a notable role.

## 4.3. Monte Carlo Simulations

Given that the scatter in the $M_{BH}–kT$ and the Fundamental Plane relation are comparable, we investigated whether the intrinsic scatter of the Fundamental Plane relation may be responsible for the observed scatter in the $M_{BH}–kT$ relation. Therefore, we performed Monte Carlo simulations.

We assumed that the relation in Equation (11) was perfect and used it to determine the masses of BHs using a random distribution of $10^4$ temperatures within the temperature range of our data. A Gaussian distribution was created using a scatter of 0.88 that characterizes the Fundamental Plane relationship of Merloni et al. (2003), which was convolved with the BH masses produced to give a randomly scattered sample of simulated data. We then calculated the best-fit relation using BCES_REGRESS code and the intrinsic scatter in the y-direction ($\sigma_{\text{intron}}^y$) for the simulated data. We repeated this $10^3$ times and derived the gradient, y-intercept, and $\sigma_{\text{intron}}^y$ for each realization.
<figure><img src="image_10.png" /><figcaption>**Figure 5.** Plot of BH mass against cluster temperature for each of the four parameters used to distinguish between relaxed and disturbed systems; (a) central electron density, (b) central entropy, (c) cuspiness, and (d) concentrated surface brightness. The blue circles and red squares indicate relaxed and disturbed clusters, respectively, where points represented with arrows are upper limits. Darker colors indicate sources that were identified as either CC-like or NCC-like by more than three parameters. The solid line represents the best-fit $M_{BH}-kT$ relation obtained for the full joint sample (Equation (16)).</figcaption></figure>

The left panel of Figure 4 shows one representative realization of the simulation along with the observed data for the $M_{BH}-kT$ relationship. All the observed points lie within the range set by the simulated data, hinting that at least part of the scatter in our data may be due to the Fundamental Plane relation.

In the right panel of Figure 4 we show the distribution of $\sigma_{\text{intron}}^y$ for the simulations. We find that 68% of the simulations have a scatter in the range of $\sigma_{\text{intron}}^y = 0.87-0.89$, which is similar to the scatter of the Fundamental Plane relation. We find that the observed scatter of $\sigma_{\text{intron}}^y = 0.91$ is observed in none of the realizations. It is possible that the dominant fraction of the scatter is introduced because of the Fundamental Plane relation that was used to infer the BH masses, but further factors may also contribute to the scatter. These are further discussed in Section 5.

### 4.4. Investigating CC and NCC Clusters

In Section 3.2, we used the median of each morphological parameter to split the sample into the most relaxed (CC-like) and most disturbed (NCC-like) systems.

In Figure 5 we plot the BH mass against cluster temperature for each of the four parameters to probe whether there is any distinction between the two types. We find that relaxed and disturbed clusters do not occupy different parts of the parameter space. This hints that BHs do not undergo drastically different evolution in the centers of these two categories of cluster. Therefore, although galaxy/cluster mergers and cooling flows could be contributing to the unexpected growth of the BH in the BCG, we cannot identify the most significant cause. To further investigate the relations, we make a distinction between clusters for which three or more parameters identified them as relaxed or disturbed. However, we do not find statistically significant differences between the two populations. We implemented the same fitting method as in 4.1 using the BCES_REGRESS code for each of the parameters. We found that for each parameter, the difference in the $M_{BH}-kT$ relation for the two subsamples was statistically insignificant; therefore, we still cannot justify the cause of the unexpected growth for clusters that resemble properties similar to CC and NCC systems.
# 5. Discussion

## 5.1. Tightness of the Scaling Relations

In this work, we investigated the tightness of the relations between the total mass of galaxy clusters (traced by the best-fit gas temperature), the stellar mass of BCGs, and the mass of the central BH.

We found that the best-fit $M_{BH}-kT$ relation is similar to that found by Bogdán et al. (2018), albeit the scatter is significantly larger (Table 3). While the dominant fraction of the scatter likely originates from the intrinsic scatter in the fundamental plane relation, part of the scatter in the relation may arise from uncertainties in measuring the X-ray and radio luminosity of the BH as well as measuring the cluster temperature. Specifically, resolving the nuclear X-ray source and deriving its luminosity is challenging in the center of luminous galaxy clusters (Mezcua et al. 2018). In a fraction of the BCG sample, the BHs remained undetected, which might be due to either the dormant nature of the BHs or the luminous ICM in which the BCG is embedded. Thus, the large observed scatter in the $M_{BH}-kT$ relation is likely due to the combination of the intrinsic scatter of the Fundamental Plane relation, measurement uncertainties in the X-ray luminosity of the BH and ICM temperature, and the intrinsic scatter in the relation.

The scatter in the $M_{BH}-M_{bulge}$ relation is comparable to that obtained for the $M_{BH}-kT$ relation. Because the scatter for both relations is dominated by the intrinsic scatter of the Fundamental Plane, our results do not contradict those of Bogdán et al. (2018), who found lower scatter and tighter correlation for the $M_{BH}-kT$ relation. This result is further supported by the joint fitting of this and the Bogdán et al. (2018) data sets. Again, the $M_{BH}-kT$ relation is tighter than the $M_{BH}-M_{bulge}$, but both exhibit similar scatter.

The discrepancies in the $M_{BH}-kT$ relation between our work and that of Bogdán et al. (2018) are likely due to the different populations of galaxy groups and clusters. The sample of Bogdán et al. (2018) was mainly dominated by galaxy groups, and their sample included only two systems with an ICM temperature greater than 2 keV. This implies that their relation in the high-mass end is not well constrained. In this work, we mainly study massive galaxy clusters. Indeed, this sample consists of 53 clusters with a temperature greater than 2 keV. Therefore, the nature of the $M_{BH}-kT$ relationship is dictated largely by the high-mass clusters, resulting in a steeper relation.

Our results suggest that the $M_{bulge}-kT$ is the tightest relationship (Table 3). The scatter and the tightness of this relation is comparable with that obtained in Bogdán et al. (2018). However, we note that both $M_{bulge}$ and $kT$ are directly measured quantities, unlike those where the $M_{BH}$ is inferred from the Fundamental Plane. Therefore, it is not surprising that there is less scatter observed in this relationship. From the joint fitting, we can draw a similar conclusion.

Hence, the observed scaling relations are not inconsistent with the results of Bogdán et al. (2018). However, to conclusively confirm that the $M_{BH}-kT$ relation is tighter than the $M_{BH}-M_{bulge}$ relation, more accurate BH mass measurements would be required.

## 5.2. Processes Aiding the Growth of BHs in BCGs

Our results demonstrate that for a given stellar bulge mass, BCGs have more massive BHs than satellite galaxies. This can be seen in panel (b) of Figure 3 where we have plotted BH

mass against BCG stellar mass. On this plot, we added the $M_{BH}-M_{bulge}$ relation of McConnell & Ma (2013) to emphasize how overly massive most of these particular BHs are. It is worth noting that the best-fit $M_{BH}-M_{bulge}$ relation presented in McConnell & Ma (2013) (see Figure 3) includes BCGs. Excluding BCGs would result in a shallower $M_{BH}-M_{bulge}$ relation, implying that the BHs studied in this work would be even stronger outliers. Therefore, we overview the physical processes that may aid the growth of these BHs.

A candidate process that could be contributing to the accelerated growth of BHs in BCGs is galaxy–galaxy mergers. Mergers play an important role in the growth of BCGs because BCGs grow a factor of 1.8 in mass between $z=1$ and the present-day universe (Burke & Collins 2013). At the centers of galaxy clusters, mergers are more frequent than in field or satellite galaxies. Clusters with similar total mass can exhibit different BCG merging histories. Within a 50 kpc radius, a BCG can have, on average, ~6.45 companions, which, over time, will infall and merge with the BCG (Burke & Collins 2013). A merger of a gas-rich galaxy could directly feed the BH and induce star formation; hence, the $M_{BH}-M_{bulge}$ relation may still hold. However, Kaviraj (2014) suggested that certain merger events weaken the coupling between stellar mass and BH growth, which would allow for a larger scatter in this relationship, especially for BCGs.

Structure formation simulations established that BHs with masses $>10^9 M_\odot$ predominantly grow through BH–BH mergers (Dubois et al. 2014). For central cluster galaxies, BH–BH mergers, in which one of the BHs does not get ejected from the center, occurs more frequently than in field or satellite galaxies (see Yoo & Miralda-Escudé 2004, and references therein). In addition, BCGs undergo many dry mergers. During these gas-poor mergers, the star formation is less likely to be induced, but the central BH will still grow because of the BH–BH merger (Volonteri & Ciotti 2013). These processes together could result in the higher BH masses of BCGs. Therefore, it is possible that mergers aid the growth of BHs in BCGs.

Cold gas flows that directly feed the BH could also play a role in aiding the growth of BHs. Cold gas flows may occur within a radius occupied by the BCG (Reisenegger et al. 1996), if the cooling time of the gas is shorter than the Hubble timescale. While subsonic cold flows may enhance star formation (see O’Dea et al. 2010, and references therein), many BCGs with cooling flows do not show signatures of active star formation. It is possible that the low-angular-momentum gas flows do not give rise to significant star formation, but support the rapid growth of BHs. Three clusters exhibit cooling flows in our sample—A1795, A2597, and Hydra—which cover a broad range of cluster temperatures with $kT = 1.72–6.07$ keV (O’Dea et al. 2004; Rafferty et al. 2006). The inferred mass of these BHs exceeds the expected value from the $M_{BH}-M_{bulge}$ relation of McConnell & Ma (2013), and these particular clusters are $3.5\sigma–4.0\sigma$ outliers from the local $M_{BH}-M_{bulge}$ scaling relation. In addition, Rafferty et al. (2006) examined the star formation rate and BH growth rate of these clusters and concluded that, for Hydra, the BH is growing faster than expected from the relation of Magorrian et al. (1998). We also examined the location of these BHs on the $M_{BH}-kT$ and $M_{BH}-M_{bulge}$ relations. We find that all three are within $1\sigma_{intron}^y$ of the $M_{BH}-kT$ relation. However, they are $2\sigma_{intron}^y$ above our best-fit $M_{BH}-M_{bulge}$ relation. Overall, these results are consistent with the findings of Rafferty et al. (2006).
Therefore, it is feasible that low-angular-momentum cold flows play a notable role in fueling the growth of BHs in BCGs. This is in slight contradiction with the results found in Section 4.4, where we observed that CC clusters do not host significantly larger BHs than NCC ones. However, our sample has only three clusters with distinct cooling flows; therefore, any dependence of the $M_{BH-kT}$ relation on CC properties may have been washed out because of the sample splitting process.

## 6. Conclusions

In this work, we investigated the relationship between cluster mass, BCG stellar mass, and BH mass. The main results can be summarized as follows:

1. We studied a sample of 71 galaxy clusters in the redshift range of $z = 0.006-0.29$. The BH mass of the BCGs was inferred from the Fundamental Plane relation, and the total mass of the galaxy clusters was traced through the temperature of the ICM.

2. We concluded that the BH mass of BCGs significantly exceeds that expected from the local scaling relations, implying that additional processes aid the growth of these BHs. We also derived scaling relations between the BH mass, BCG stellar mass, and galaxy cluster mass.

3. We found that the best-fit $M_{BH-kT}$ relation is steeper than that of Bogdán et al. (2018) and a larger scatter is present. Using Monte Carlo simulations, we determined that most of this scatter may originate from the intrinsic scatter of the Fundamental Plane relation.

4. We split the galaxy cluster sample using different criteria and explored whether CC and NCC clusters exhibit a different $M_{BH-kT}$ relation. However, we did not find a statistically significant difference, suggesting that the BHs in BCGs do not undergo different evolution on CC and NCC clusters.

5. We discussed the potential causes of the unexpected growth in these BHs, from BH-BH mergers up to cluster-cluster mergers and cooling flows. Some clusters in our sample exhibited these features. Hence, we compared their BH masses to the relations we found in Section 4.1, which showed that these processes may have some influence over the aided growth of BHs in BCGs. However, because of the small number of objects in our sample displaying either a cluster-cluster merger or a cooling flow, we cannot definitively say to what extent the effect is that these processes have. We conclude that, in order to test the effects of these processes, we will need to probe galaxy evolution simulations.

We thank Vinay Kashyap and the Statistical Consulting Service of the Harvard Statistics Department for helpful discussion about the fitting procedures. This research has made use of *Chandra* data provided by the *Chandra* X-ray Center (CXC). The publication makes use of software provided by the CXC in the application package CIAO. In this work, the NASA/IPAC Extragalactic Database (NED) has been used. Á.B. acknowledges support from the Smithsonian Institute. L.L. acknowledges support from the *Chandra* X-ray Center through NASA contract No. NNX17AD83G. M.V. acknowledges funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013 grant agreement No. 614199, project BLACK).

## Appendix

Here, we display images of each cluster as seen in the X-ray, as well as displaying the annulus within which the final spectra were extracted (Figure 6).
<figure><img src="image_11.png" /><figcaption>**Figure 6.** *Chandra* X-ray images of the galaxy clusters studied in this work. Overplotted are the annuli that were used to characterize the galaxy clusters.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Figure 6. (Continued.)</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Figure 6. (Continued.)</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Figure 6. (Continued.)</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Figure 6. (Continued.)</figcaption></figure>
ORCID iDs

Marta Volonteri https://orcid.org/0000-0002-3216-1322

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