#  The relation between the cool-core radius and the host galaxy clusters:
thermodynamic properties and cluster mass

FanLam Ng Academia Sinica Institute of Astronomy and Astrophysics (ASIAA), No.
1, Section 4, Roosevelt Road, Taipei 10617, Taiwan  Yuanpei College, Peking
University, Yi He Yuan Road 5, Hai Dian District, Beijing 100871, China
Shutaro Ueda Academia Sinica Institute of Astronomy and Astrophysics (ASIAA),
No. 1, Section 4, Roosevelt Road, Taipei 10617, Taiwan  Institute of
Astronomy, National Tsing Hua University, Hsinchu 30013, Taiwan

###### Abstract

We present a detailed study of cool-core systems in a sample of four galaxy
clusters (RXCJ1504.1-0248, A3112, A4059, and A478) using archival X-ray data
from the Chandra X-ray Observatory. Cool cores are frequently observed at the
centers of galaxy clusters and are considered to be formed by radiative
cooling of the intracluster medium (ICM). Cool cores are characterized by a
significant drop in the ICM temperature toward the cluster center. We extract
and analyze X-ray spectra of the ICM to measure the radial profiles of the ICM
thermodynamic properties including temperature, density, pressure, entropy,
and radiative cooling time. We define the cool-core radius as the turnover
radius in the ICM temperature profile and investigate the relation between the
cool-core radius and the properties of the host galaxy clusters. In our
sample, we observe that the radiative cooling time of the ICM at the cool-core
radius exceeds 10 Gyr, with RXCJ1504.1-0248 exhibiting a radiative cooling
time of $32^{+5}_{-11}$ Gyr at its cool-core radius. These results indicate
that not only radiative cooling but also additional mechanisms such as gas
sloshing may play an important role in determining the size of cool cores.
Additionally, we find that the best-fit relation between the cool-core radius
and the cluster mass ($M_{500}$) is consistent with a linear relation. Our
findings suggest that cool cores are linked to the evolution of their host
galaxy clusters.

Intracluster medium (858), Galaxy clusters (584), X-ray astronomy (1810)

††facilities: CXO††software:  astropy (Astropy Collaboration et al., 2013,
2018), CIAO (Fruscione et al., 2006), XSPEC (Arnaud, 1996)

##  1 Introduction

Galaxy clusters contain a large amount of diffuse, hot X-ray emitting gas
known as the intracluster medium (ICM, $T\sim 10^{7-8}$ K), which is trapped
and thermalized in the deep gravitational potential well dominated by dark
matter. Because of the high temperature of the ICM, the ICM emits X-ray
radiation through thermal bremsstrahlung. Since the X-ray emissivity of the
ICM is proportional to the square of the electron density, the ICM loses its
thermal energy by X-ray radiation faster in the central region of galaxy
clusters than in the outskirts.

The radiative cooling time of the ICM at the centers of galaxy clusters
exhibiting strong X-ray surface brightness peaks is much shorter than the
inferred age of galaxy clusters1117.7 Gyr ($z\sim 1$) is often adopted as the
age of low-$z$ galaxy clusters. (e.g., Peterson & Fabian, 2006, for a review).
Thus, it was expected that runaway cooling occurs and triggers massive star
formation in the centers of galaxy clusters (Fabian, 1994). However, this
expectation is inconsistent with observational evidence. Neither such expected
massive star formation nor a large amount of cooled ICM that serves as fuel
for star formation is found (e.g., Tamura et al., 2001; Peterson et al., 2001;
O’Dea et al., 2008; McDonald et al., 2011, 2018). This inconsistency indicates
that runaway cooling must be suppressed, meaning the presence of heating
sources.

Although a heating source is required, X-ray observations have revealed that
the temperature of the ICM drops toward the cluster center. The ICM
temperature at the center is measured at a few keV, corresponding to $30-50$ %
of the peak value of the ICM temperature profile (e.g., Vikhlinin et al.,
2005; Simionescu et al., 2011). Such a region, consisting of cooler ICM (or
cooling ICM), is referred to as a cool core (Molendi & Pizzolato, 2001). The
presence of cool cores indicates that not only the cooling of the ICM is still
dominant in cool cores but also the heating is balanced with the cooling of
the ICM at least within a specific region. Therefore, exploring cool-core
systems in galaxy clusters is important to understand not only the the origin
of cool cores but also the evolution of cool cores under conditions of the
balance between cooling and heating.

Cool cores are characterized by a significant drop in the ICM temperature
toward the cluster center (Molendi & Pizzolato, 2001). However, various
alternative observables of the ICM have been used to identify cool cores:
central electron density (Hudson et al., 2010; Barnes et al., 2018), central
radiative cooling time (e.g., Bauer et al., 2005; Hudson et al., 2010; Wang et
al., 2023), central entropy profile (e.g., Bauer et al., 2005; Hudson et al.,
2010), classical mass deposition rate (e.g., Chen et al., 2007), and X-ray
luminosity ratio (e.g., Sayers et al., 2013; Shitanishi et al., 2018). These
observables have also been used to distinguish between cool-core and non-cool-
core clusters. For instance, Su et al. (2020) used the central radiative
cooling time of the ICM as an indicator to identify cool-core clusters from a
sample of galaxy clusters obtained from cosmological simulations such as
IllustrisTNG. Additionally, Barnes et al. (2018) investigated the cool-core
faction in a sample of galaxy clusters from IllustrisTNG using six parameters:
central electron density, radiative cooling time, entropy profile, X-ray
concentration parameters within a certain or fiducial radius, and cuspiness
parameter of X-ray morphology. Laganá et al. (2019) also investigated the
optimal parameters for identifying cool cores using some observables including
the cuspiness of the gas density profile, central gas density, and properties
of the brightest cluster galaxies (BCG).

Hudson et al. (2010) presented an in-depth study of the ICM properties in the
central regions of galaxy clusters listed in a catalog of the extended HIghest
X-ray FLUx Galaxy Cluster Sample (HIFLUGCS: Reiprich & Böhringer, 2002). They
applied 16 cool-core diagnostics parameters to their sample to find the most
appropriate parameter for characterizing cool-core clusters. They found that
the radiative cooling time at the center and cuspiness serve as the most
effective indicators for low-$z$ and high-$z$ cool-core clusters,
respectively. In addition, they introduced two categories of cool cores:
strong and weak cool cores, based on the radiative cooling time at the center.
Strong cool cores are defined as those with a radiative cooling time at the
center shorter than 1 Gyr, while weak cool cores are identified when a
radiative cooling time ($t_{\rm cool}$) at the center falls within the range
of 1 Gyr $<t_{\rm cool}<$7.7 Gyr.

A universal form in the ICM temperature profile has been reported (e.g., Allen
et al., 2001; Vikhlinin et al., 2005; Sanderson et al., 2006). Allen et al.
(2001) analyzed the ICM temperature profiles scaled by $r_{2500}$222We adopt
$M_{\Delta}$ as the mass enclosed within a sphere of radius $r_{\Delta}$ whose
mean density is $\Delta$ times the critical density of the Universe at the
redshift of the galaxy cluster. for six cool-core clusters and found an
approximately universal form in the scaled temperature profiles. Vikhlinin et
al. (2005) measured the ICM temperature profiles from the central region to
the outskirts in 13 nearby, relaxed galaxy clusters and groups, and revealed a
similar trend in the temperature profiles scaled by $r_{180}$. Sanderson et
al. (2006) also measured the ICM temperature profiles in 20 galaxy clusters
and found a possible universal form in the temperature profiles scaled by
$r_{500}$.

Such previous studies aim to investigate possible relations between the
temperature profile and the total mass of galaxy clusters. However, both
cooling and heating processes have a significant impact on cool cores,
indicating that baryon physics in the centers of galaxy clusters likely plays
an important role in shaping the observed characteristics of cool cores.
Therefore, it is essential to focus on cool-core systems and characterize cool
cores by the ICM temperature.

In this paper, we aim to characterize cool cores in our sample using the cool-
core radius that is defined as the turnover radius in the ICM temperature
profile. In addition, we aim to study possible relations between the cool-core
radius and the properties of the host galaxy clusters, including the ICM
thermodynamic properties and cluster mass. To this end, we first analyze the
X-ray spectra of the ICM extracted from annular regions determined by the
morphology of X-ray surface brightness and X-ray photon counts. Then, we
measure the radial profiles of the ICM thermodynamic properties and determine
the cool-core radius by analyzing the ICM temperature profile. Finally, we
measure thermodynamic perturbations in the ICM to constrain the turbulent
velocity of the ICM within the cool cores in our sample.

This paper is organized as follows. Section 2 provides a brief summary of our
sample and their properties. Section 3 presents detailed information regarding
Chandra observations and data reduction procedures. Section 4 describes the
X-ray spectral analysis and measurements of the radial profiles of the ICM
thermodynamic properties. Section 5 presents the study of possible relations
between the cool-core radius and the properties of the host galaxy clusters
and the analysis of thermodynamic perturbations in the ICM. Finally, a summary
is given in Section 6

Throughout the paper, we assume $\Omega_{\rm m}=0.3$,
$\Omega_{\rm\Lambda}=0.7$, and the Hubble constant of $H_{0}=70\,$km s-1
Mpc-1. Unless stated otherwise, quoted errors correspond to 1 $\sigma$
uncertainties.

##  2 Sample

In this paper, we focus on four galaxy clusters: RXCJ1504.1-0248, Abell 3112,
Abell 4059, and Abell 478. These galaxy clusters are included in the HIFLUGCS
catalog (Reiprich & Böhringer, 2002). According to Hudson et al. (2010), all
galaxy clusters in our sample are classified as strong cool-core clusters
based on the radiative cooling time of the ICM at the center. Here, we provide
a brief summary of previous studies on each galaxy cluster. The values of
$M_{500}$ and $r_{500}$ mentioned below are extracted from the Meta-Catalogue
of X-ray detected Clusters of galaxies (MCXC; Piffaretti et al., 2011).

###  2.1 RXCJ1504.1-0248

This cluster is located at a redshift of $z=0.2153$ and is known as one of the
most massive galaxy clusters with $M_{500}=12.47\times 10^{14}$ $M_{\odot}$
($r_{500}=1.52$ Mpc). This cluster hosts an extreme cool core characterized by
a short radiative cooling time (e.g., Bohringer et al., 2005; Hlavacek-
Larrondo & Fabian, 2011). Bohringer et al. (2005) analyzed the X-ray surface
brightness profile using a $\beta$-model (Cavaliere & Fusco-Femiano, 1976,
1978; Ettori, 2000) and found that the core radius of the $\beta$-model is
$\sim 30$ $h_{70}^{-1}$ kpc333$h_{70}$ denotes $h_{70}=H_{0}/(70$ km s-1
Mpc-1) where $H_{0}$ is a Hubble constant used in the literature., which is
significantly smaller than the cooling radius of $\sim 140$ kpc where is
defined as the position at a radiative cooling time of $10$ Gyr. Furthermore,
Bohringer et al. (2005) found a significant drop in the ICM temperature
profile toward the cluster center. The ICM temperature at the center is
measured to be below 5 keV, while the temperature of the ambient ICM is $\sim
10.5$ keV. Hlavacek-Larrondo & Fabian (2011) reported that there is no obvious
X-ray point source associated with the BCG. Giacintucci et al. (2011) found
the presence of a radio mini-halo in the central region of this cluster.

###  2.2 Abell 3112

This cluster is located at a redshift of $z=0.075$ and has a powerful radio
source, PKS 0316-444, in the center. Takizawa et al. (2003) measured the
radial profiles of the ICM thermodynamic properties, including temperature,
abundance, electron density, pressure, and radiative cooling time with
Chandra, and found a temperature drop toward the cluster center. Bulbul et al.
(2012) also studied the temperature profile of the ICM from the center to the
outskirts. The ICM temperature at the center is measured at $\sim 3.4$ keV,
while the peak value in the temperature profile is measured at $\sim 5.1$ keV,
which is consistent with that reported by Ezer et al. (2017). The mass of this
cluster is estimated at $M_{\rm 500}=4.39\times 10^{14}$ $M_{\odot}$
($r_{500}=1.13$ Mpc), which is consistent with that estimated by Nulsen et al.
(2010) and Bulbul et al. (2012).

###  2.3 Abell 4059

This cluster is located at a redshift of $z=0.049$. The ICM temperature
profile shows a temperature drop from $\sim 4$ keV at $\sim 100$ kpc away from
the center to $\sim 2$ keV at the center (e.g., Huang & Sarazin, 1998; Choi et
al., 2004; Reynolds et al., 2008; Mernier et al., 2015). Laganá et al. (2019)
conducted a detailed study of spatial distributions of the ICM temperature,
entropy, pressure, and abundance, respectively. The mass of this cluster is
estimated as $M_{\rm 500}=2.67\times 10^{14}$ $M_{\odot}$ ($r_{500}=0.96$
Mpc), which is consistent with that estimated by Vikhlinin et al. (2009).
Reynolds et al. (2008) reported a slight dip in the pressure profile at $r\sim
15$ kpc, which is likely associated with an X-ray cavity caused by AGN
feedback. Thus, it is considered that a lack of thermal pressure may be
supported by its non-thermal pressure.

###  2.4 Abell 478

This cluster is located at a redshift of $z=0.088$. Sun et al. (2003) analyzed
the central 500 kpc region with Chandra and reported the radial profiles of
the ICM thermodynamic properties, finding that the peak value in the
temperature profile is $\sim 8.5$ keV, while the temperature at the center is
$\sim 3$ keV. These results are in agreement with those measured by
Pointecouteau et al. (2004) and Vikhlinin et al. (2005). X-ray cavities are
observed in the central 15 kpc region, with two weak and small ($\sim$ 4 kpc)
radio lobes spatially associated with the X-ray cavities (Sun et al., 2003).
The mass of this cluster is estimated as $M_{\rm 500}=6.42\times 10^{14}$
$M_{\odot}$ ($r_{500}=1.28$ Mpc).

##  3 Observation and data reduction

We analyzed archival X-ray data of the sample taken with the Advanced CCD
Imaging Spectrometer (ACIS; Garmire et al., 2003) on board the Chandra X-ray
Observatory. The observation identification numbers (ObsIDs) and corresponding
information of Chandra observations in this study are summarized in Table 1.
We used the versions of 4.11 and 4.8.3 for Chandra Interactive Analysis of
Observations (CIAO; Fruscione et al., 2006) and the calibration database
(CALDB), respectively. To ensure data quality, we examined the light curve of
each dataset using the lc_clean task in CIAO, filtering flare data. The
blanksky data provided by the CALDB were adopted as background data for the
spectral analysis. Point sources were identified by the wavdetect task in CIAO
and were subsequently masked. We extracted X-ray spectra of the ICM from each
dataset using the specexctract task in CIAO and combined them after making
individual spectrum, response, and ancillary response files for the spectral
fitting. We used XSPEC version 12.11.0f (Arnaud, 1996) and the atomic database
(AtomDB) for plasma emission modeling version 3.0.9 in the X-ray spectral
analysis, assuming that the ICM is in collisional ionization equilibrium
(Smith et al., 2001; Foster et al., 2012). The abundance table of Anders &
Grevesse (1989) was used in XSPEC. Here, the abundance of a given element is
defined as $Z_{i}=(n_{i,{\rm obs}}/n_{\rm H,obs})/(n_{i,\odot}/n_{\rm
H,\odot})$, where $n_{i}$ and $n_{\rm H}$ represent the number densities of
the $i$th element and hydrogen, respectively. The iron abundance of the ICM is
used to represent the ICM metal abundance, such that the abundance of other
elements is tied to the iron abundance as $Z_{i}=Z_{\rm Fe}$ (Ueda et al.,
2021).

Table 1:  Summary of our sample: cluster name, redshift, net exposure time, physical scale, and datasets taken with Chandra.  Cluster | Redshift | Expo. time (ksec) | Scale (kpc arcsec-1) | ObsID
---|---|---|---|---
RXCJ1504.1-0248 | 0.215 | 161.7 | 3.51 | 4935, 5793, 17197, 17669, 17670
A3112 | 0.075 | 133.7 | 1.43 | 2216, 2516, 6972, 7323, 7324, 13135
A4059 | 0.049 | 120.0 | 0.96 | 897, 5785
A478 | 0.088 | 50.1 | 1.65 | 1669, 6102
Table 2:  Sky coordinates (J2000.0) of the center, the position angle (PA), and the axis ratio (AR) of the ellipse model for the X-ray surface brightness distribution of the sample extracted from Ueda et al. (2021).  Cluster | R.A. | Decl. | PAaa Position angle measured south of east. | AR
---|---|---|---|---
|  |  | (deg) |
RXCJ1504.1-0248 | 15:04:07.48 | -02:48:17.25 | 156 | 0.78
A3112 | 03:17:57.67 | -44:14:17.44 | 100 | 0.75
A4059 | 23:57:00.79 | -34:45:33.44 | 70 | 0.88
A478 | 04:13:25.15 | 10:27:54.94 | 132 | 0.73

##  4 Analysis and Results

To determine the cool-core radius for the sample, our initial goal is to
obtain and analyze the ICM temperature profile of each cluster. We define
annular regions for X-ray spectral analysis based on the morphology of the
X-ray surface brightness of the sample. Next, we extract X-ray spectra of the
ICM from each defined annular region and carry out spectral analysis to
measure the ICM thermodynamic properties. Then, we perform fitting of the ICM
temperature profile to determine the cool-core radius. We also study the
radial profiles of the ICM electron number density, pressure, entropy, and
radiative cooling time.

###  4.1 X-ray imaging analysis

Figure 1 shows the X-ray surface brightness images of the sample in the
$0.5-7.0$ keV band taken with Chandra after subtracting the background and
correcting the exposure time. Since the morphology of their X-ray surface
brightness appears axial symmetric (e.g., Ueda et al., 2021). we adopt the
same elliptical model as that used in Ueda et al. (2021) to define annular
regions for spectral analysis. The position of the center, the position angle,
and the axis ratio of the elliptical model for each cluster are summarized in
Table 2. Note that Ueda et al. (2021) calculated a mean surface brightness by
applying the concentric ellipse fitting algorithm constructed by Ueda et al.
(2017), by minimizing the variance of the X-ray surface brightness relative to
the ellipse model. To determine the width of each radial bin, we ensure that
the net photon counts for each bin falls within the range of $5000-10000$ in
the $0.4-7.0$ keV band for better statistics. For several smaller regions near
the center and outer regions, we adjust the net photon counts in the range of
$1500-5000$. The detailed information regarding our region selection is
summarized in Appendix A (see Table 10).

Figure 1: X-ray surface brightness of the sample: RXCJ1504.1-0248 (top left),
A3112 (top right), A4059 (bottom left), and A478 (bottom right). The X-ray
surface brightness in the $0.5-7.0$ keV band is shown on a logarithm scale in
units of photon sec-1 arcsec-2 cm-2. This image is smoothed with a Gaussian
kernel with $2.3^{\prime\prime}$ FWHM. A dashed, white ellipse shows the cool-
core radius, $r_{\rm cool}$, derived from the best-fit results of the analysis
of the ICM temperature profile.

###  4.2 X-ray spectral analysis

We extract X-ray spectra of the ICM from each elliptical annulus defined in
Section 4.1. The X-ray spectra in the $0.4-7.0$ keV band are analyzed using
the model of phabs * apec in XSPEC. The redshift of each cluster is fixed to
the value in Table 1, which is taken from NASA/IPAC Extragalactic Database
(NED)444http://ned.ipac.caltech.edu/. The column densities of the Galactic
absorption (i.e., $N_{\rm H}$) toward RXCJ1504.1-0248 and A3112 are fixed to
the values measured by HI4PI Collaboration et al. (2016), respectively.
However, for A4059 and A478, Choi et al. (2004) and Sun et al. (2003) reported
that $N_{\rm H}$ varies with radius and is systematically larger than that
derived from HI4PI Collaboration et al. (2016). In fact, the measured values
of $N_{\rm H}$ for A4059 and A478 in our spectral analysis are a factor of 2
larger than those of HI4PI Collaboration et al. (2016), respectively, and
consistent with the previous measurements (Choi et al., 2004; Sun et al.,
2003), respectively. Therefore, we allow $N_{\rm H}$ for A4059 and A478 to
vary in the spectral analysis.

The observed radial profiles of the temperature and electron number density
for the sample are shown in Figures 2 and 3, respectively. To estimate the ICM
electron number density, we assume a line-of-sight length of $L$/1 Mpc.
Additionally, the observed radial profiles of the ICM metal abundance are
presented in Appendix (Figure 10). We also calculate the ICM pressure $p_{\rm
e}$ and entropy $K_{\rm e}$ as

$p_{\rm e}=kT\times n_{\rm e}$ (1)

and

$K_{\rm e}=kT\times n_{\rm e}^{-\frac{2}{3}},$ (2)

respectively, where $kT$ is the ICM temperature and $n_{\rm e}$ is the ICM
electron number density. The radial profiles of the pressure and entropy are
shown in Figures 4 and 5, respectively. In addition, following the approach of
McDonald et al. (2019), we calculate the radiative cooling time of the ICM,
$t_{\rm cool}$, as

$t_{\mathrm{cool}}=\frac{3}{2}\frac{(n_{\mathrm{e}}+n_{\mathrm{p}})kT}{n_{\mathrm{e}}n_{\mathrm{p}}\Lambda(T,Z)},$
(3)

where $n_{\mathrm{p}}$ is the proton number density, and $\Lambda(T,Z)$ is the
cooling function. To convert from $n_{\mathrm{e}}n_{\mathrm{p}}$ to
$n_{\mathrm{e}}$, McDonald et al. (2019) assumed
$n_{\mathrm{e}}/n_{\mathrm{p}}=1.196$, based on the discussion by Markevitch
(2007), namely, $n_{\mathrm{e}}/n_{\mathrm{p}}=1+2x+x_{\mathrm{e}h}$, where
$x\equiv n_{\rm He}/n_{\mathrm{p}}$ represents the helium abundance, and
$x_{\mathrm{e}h}$ represents electrons from elements heavier than helium. The
contribution of $x_{\mathrm{e}h}$ can be negligible, as it is
$x_{\mathrm{e}h}\approx 0.005$ assuming an ICM abundance of $0.3-0.5$ solar
compared to $x=0.098$ from Anders & Grevesse (1989) with 1 solar abundance.
Therefore, we adopt the same approach as McDonald et al. (2019) in calculation
of the radiative cooling time. We compute the cooling function using the
pyatomdb task in AtomDB (Smith et al., 2001; Foster et al., 2012) with the
best-fit parameters of the ICM temperature and abundance in each radial bin.
The radial profiles of the radiative cooling time for the sample are presented
in Figure 6. To show all the radial profiles, we use the distance from the
center along the direction of the major axis of the elliptical model as the
values on the horizontal axis.555In this paper, we adopt
$r=\frac{r_{\mathrm{MAJ,\,inner}}+r_{\mathrm{MAJ,\,outer}}}{2}$ as the
distance from the center, where $r_{\mathrm{MAJ,\,inner}}$ and
$r_{\mathrm{MAJ,\,outer}}$ denote the major axes of the inner and outer
annuli, respectively. Note that we here show the radial profiles of the ICM
properties derived from projected spectral analysis, not spectral deprojection
analysis.

Figure 2: ICM temperature profiles of the sample: RXCJ1504.1-0248 (top left),
A3112 (top right), A4059 (bottom left), and A478 (bottom right). In each
panel, the black error bars show the observed values. The red dashed line and
shaded region display the best-fit profile and $1\sigma$ uncertainty,
respectively. The blue solid line and shaded region correspond to the best-fit
cool-core radius $r_{\rm cool}$ and $1\sigma$ uncertainty, respectively. In
the middle row of each panel, the residuals between the observed and the best-
fit profiles are shown. The mean absolute relative residuals are also
displayed in the bottom row of each panel.

###  4.3 ICM temperature profiles

Figure 2 shows the observed temperature profile of each galaxy cluster. In
good agreement with previous studies, such as Bohringer et al. (2005) for
RXCJ1504.1-0248, Takizawa et al. (2003) for A3112, Reynolds et al. (2008) for
A4059, and Sanderson et al. (2005) for A478, all galaxy clusters in the sample
exhibit a significant drop in the observed ICM temperature profiles toward the
cluster center.

Several empirical models have been proposed to represent the ICM temperature
profile from the center to the outskirts (e.g., Allen et al., 2001; Voigt et
al., 2002; Kaastra et al., 2004; Vikhlinin et al., 2006; O’Sullivan et al.,
2017). For instance, Kaastra et al. (2004) modeled the ICM temperature profile
taking into account the temperature at the center, the peak temperature, and
the cooling radius that represents a characteristic radius in the temperature
profile. However, there is still room for improvement in their model to
account for temperature decrease toward the outskirts.

Motivated by Kaastra et al. (2004) and O’Sullivan et al. (2017), we extend
their models to account for the temperature profile from the center to the
outer region. Therefore, our model can be described as

$T(r)=T_{\rm center}+2\times(T_{\rm peak}-T_{\rm
center})(\frac{x(r)}{1+x(r)})$ (4) $x(r)=\begin{cases}(r/r_{\rm
cool})^{\alpha_{1}}&:r<r_{\rm cool}\\\ (r/r_{\rm cool})^{\alpha_{2}}&:r\geq
r_{\rm cool},\end{cases}$ (5)

where $T_{\rm center}$ is the ICM temperature at the center, $T_{\rm peak}$ is
the peak temperature, $r_{\rm cool}$ is the turnover radius in the temperature
profile (i.e., the peak position; $T(r_{\rm cool})=T_{\rm peak}$), and
$\alpha_{1}$ and $\alpha_{2}$ represent the slopes of the temperature profile
within $r_{\rm cool}$, and beyond $r_{\rm cool}$, respectively. In this paper,
we define the cool-core radius as $r_{\rm cool}$.

We fit the observed temperature profile shown in Figure 2 with our model using
affine-invariant Markov Chain Monte Carlo (MCMC) sampling (Goodman & Weare,
2010) implemented by the emcee python package (Foreman-Mackey et al., 2013).
The log-likelihood function for the fitting is written as

$-2\ln\mathcal{L}=\sum_{i}\frac{[y_{i}-T(r_{i})]^{2}}{\sigma_{y_{i}}^{2}},$
(6)

where $i$ runs over all radial bins in the radial profile, $y_{i}$ and
$\sigma_{y_{i}}$ are the best-fit value and its uncertainty of the temperature
profile in each radial bin, respectively, and $T(r_{i})$ is the model
prediction in each radial bin. We use uninformative uniform priors on $T_{\rm
center}$, $T_{\rm peak}$, $r_{\rm cool}$, $\alpha_{1}$, and $\alpha_{2}$ with
the following ranges: $T_{\rm center}\in(0,30)$ keV, $T_{\rm peak}\in(0,30)$
keV, $r_{\rm cool}\in(0,500)$ kpc, $\alpha_{1}\in(0,2)$, and
$\alpha_{2}\in(-2,0)$. We sample the posterior probability distributions of
the parameters ($T_{\rm center}$, $T_{\rm peak}$, $r_{\rm cool}$,
$\alpha_{1}$, and $\alpha_{2}$) over the full parameter space allowed by the
priors. The mean value and standard deviation of the marginalized
distributions are represented as the best-fit value and its uncertainty,
respectively. For each cluster, the best-fit parameters of the temperature
profile are summarized in Table 3. The best-fit model of the temperature
profile of each cluster is also shown in Figure 2.

Table 3:  Parameter constraints of the temperature profile model derived from the MCMC analysis.  Cluster |  $T_{\rm center}$ (keV) |  $T_{\rm peak}$ (keV) |  $r_{\rm cool}$ (kpc) | $\alpha_{1}$ | $\alpha_{2}$
---|---|---|---|---|---
RXCJ1504.1-0248 | $4.13\pm 0.24$ | $12.00\pm 0.56$ | $325\pm 55$ | $0.97\pm 0.11$ | $-0.66\pm 0.48$
A3112 | $2.21\pm 0.16$ | $5.41\pm 0.11$ | $89.4\pm 8.0$ | $0.88\pm 0.10$ | $-0.35\pm 0.14$
A4059 | $1.21\pm 0.13$ | $4.70\pm 0.08$ | $77.1\pm 8.7$ | $0.90\pm 0.10$ | $-0.05\pm 0.05$
A478 | $1.54\pm 0.65$ | $7.62\pm 0.32$ | $245\pm 66$ | $0.55\pm 0.13$ | $-0.45\pm 0.42$

We calculate residuals between the observed and the best-fit profiles in each
radial bin. The residuals are computed as $(y_{i}-T_{\rm b}(r_{i}))$, where
$T_{\rm b}(r_{i})$ is the best-fit profile in each radial bin. Based on these
residuals, we also compute the mean absolute relative residual within each set
of five radial bins within $r_{\rm cool}$ as well as for all the data points
for the outer region (i.e., $r>r_{\rm cool}$) using $|(y-T_{\rm b})/T_{\rm
b}|$. The profiles of the residuals and the mean absolute relative residuals
are shown in the middle and bottom row of each panel in Figure 2,
respectively. The calculation of residuals will also be applied to the radial
profiles of the other components.

We also calculate the ratios of the parameters of the temperature profile for
RXCJ1504.1-0248, A3112, and A478 to those for A4059, since A4059 has the
smallest values for each parameter. These ratios are summarized in Table 4.

Table 4:  Ratios of the parameters of the temperature profile for RXCJ1504.1-0248, A3112, and A478 to those of A4059.  Cluster | $T_{\rm center}/T_{\rm center,A4059}$ | $T_{\rm peak}/T_{\rm peak,A4059}$ | $r_{\rm cool}/r_{\rm cool,A4059}$
---|---|---|---
RXCJ1504.1-0248 | $3.42\pm 0.42$ | $2.55\pm 0.13$ | $4.21\pm 0.82$
A3112 | $1.83\pm 0.24$ | $1.15\pm 0.03$ | $1.16\pm 0.16$
A478 | $1.27\pm 0.55$ | $1.62\pm 0.07$ | $3.18\pm 0.92$

###  4.4 ICM electron number density profiles

In the same manner as the analysis of the ICM temperature profiles, we also
analyze the radial profiles of the electron number density for the sample. A
$\beta$-model is frequently used to model the density profile of the ICM
(e.g., Cavaliere & Fusco-Femiano, 1976, 1978; Ettori, 2000). For cool-core
clusters, a double $\beta$-model is preferred to account for the observed
profile, because of the strong excess at the cluster center (e.g.,
Pointecouteau et al., 2004; Santos et al., 2008; Henning et al., 2009; Santos
et al., 2010; Ota et al., 2013). Therefore, to fit the observed number density
profile for each cluster, we adopt a double $\beta$-model expressed as

$\displaystyle n(r)$ $\displaystyle=n_{1}(r)+n_{2}(r)$ (7)
$\displaystyle=n_{0,\,1}\left[1+\left(\frac{r}{r_{\rm
c,\,1}}\right)^{2}\right]^{-\alpha_{1}}+n_{0,\,2}\left[1+\left(\frac{r}{r_{\rm
c,\,2}}\right)^{2}\right]^{-\alpha_{2}},$

where $n_{0,\,1}$ and $n_{0,\,2}$ represent the values at the center,
respectively, $r_{\rm c,\,1}$ and $r_{\rm c,\,2}$ are the core radii of each
$\beta$-model, respectively, and $\alpha_{1}$ and $\alpha_{2}$ are the slopes
of each $\beta$ model, respectively.

Following the procedures of the MCMC analysis in Section 4.3, we use
uninformative uniform priors $n_{0,\,1},\,n_{0,\,2}\in(0,1)$ cm -3, $r_{\rm
c,\,1},\,r_{\rm c,\,2}\in(0,1000)\,{\rm kpc}$,
$\alpha_{1},\,\alpha_{2}\in(0.5,3.5]$, and ensure $n_{0,\,1}>n_{0,\,2}$. Thus,
we fit the observed profile and sample the posterior probability distributions
of these six parameters over the full parameter space allowed by the priors.
The best-fit parameters obtained from the MCMC analysis are summarized in
Table 5. The best-fit profile of each cluster along with its uncertainty is
shown in Figure 3.

Table 5:  Parameter constraints of the double $\beta$-model derived from the MCMC analysis.  Cluster | $n_{\rm 0,1}$ | $r_{\rm c,1}$ | $\alpha_{1}$ | $n_{\rm 0,2}$ | $r_{\rm c,2}$ | $\alpha_{2}$
---|---|---|---|---|---|---
| ($10^{-2}$ cm -3) | (kpc) |  | ($10^{-2}$ cm -3) | (kpc) |
RXCJ1504.1-0248 | $4.10\pm 0.15$ | $30.0\pm 2.0$ | $0.72\pm 0.08$ | $0.24\pm 0.12$ | $329\pm 111$ | $1.71\pm 0.83$
A3112 | $0.80\pm 0.02$ | $53.9\pm 2.6$ | $0.62\pm 0.01$ | $0.78\pm 0.03$ | $19.1\pm 2.9$ | $2.16\pm 0.39$
A4059 | $0.40\pm 0.01$ | $72.5\pm 4.6$ | $0.60\pm 0.03$ | $0.38\pm 0.04$ | $14.7\pm 8.8$ | $2.86\pm 0.51$
A478 | $1.54\pm 0.09$ | $19.8\pm 2.1$ | $0.64\pm 0.10$ | $0.59\pm 0.08$ | $161\pm 17$ | $0.85\pm 0.08$

Figure 3: Same as Figure 2 but for the ICM electron number density profiles of
the sample: RXCJ1504.1-0248 (top left), A3112 (top right), A4059 (bottom
left), and A478 (bottom right).

###  4.5 ICM pressure and entropy profiles

Based on the best-fit profiles for the ICM temperature and electron number
density, we compute the predicted profiles for the ICM pressure and entropy
profiles using Equations 1 and 2, respectively. Here, we do not perform direct
fitting of the ICM pressure and entropy profiles. The observed ICM pressure
and entropy profiles and their predicted profiles are shown in Figures 4 and
5, respectively.

As mentioned in Section 2.3, Reynolds et al. (2008) observed a slight dip in
the pressure profile of A4059 at $r\sim 15$ kpc. We also observe this dip in
our pressure profile, which is in agreement with that reported by Reynolds et
al. (2008). The X-ray cavity found by Reynolds et al. (2008) may be associated
with this pressure dip and give non-thermal pressure support for this region.

Figure 4: Same as Figure 3 but for the ICM pressure profiles of the sample:
RXCJ1504.1-0248 (top left), A3112 (top right), A4059 (bottom left), and A478
(bottom right).

Figure 5: Same as Figure 3 but for the ICM Entropy profiles of the sample:
RXCJ1504.1-0248 (top left), A3112 (top right), A4059 (bottom left), and A478
(bottom right).

Figure 6: Radial profiles of the ICM radiative cooling time: RXCJ1504.1-0248
(top left), A3112 (top right), A4059 (bottom left), and A478 (bottom right).
The blue, vertical solid line and shaded region correspond to the cool-core
radius and $1\sigma$ uncertainty for each cluster.

##  5 Discussion

We have conducted an analysis of the radial profiles of the ICM thermodynamic
properties, including temperature, electron number density, pressure, entropy,
and radiative cooling time for our sample: RXCJ1504.1-0248, A3112, A4059, and
A478. We have also determined the cool-core radius for each cluster by
analyzing their observed ICM temperature profiles.

In this section, we discuss the characteristics of the cool-core systems
defined by the cool-core radius in the sample. We also explore possible
relations between the cool-core radius and the properties of the host galaxy
clusters, as well as investigate potential universal forms in the radial
profiles scaled by the cool-core radius. Furthermore, we study perturbations
in the ICM thermodynamic properties within the cool cores.

###  5.1 Cool-core radius

Cool cores are characterized by a significant drop in the ICM temperature
toward the cluster center (Molendi & Pizzolato, 2001). Since the cool-core
radius has been defined as the turnover radius in the ICM temperature profile,
the cool-core radius corresponds to a boundary region where the cooling of the
ICM becomes dominant. Therefore, the cool-core radius is an important aspect
for understanding the underlying physics of cool cores.

We have determined the cool-core radius for the sample as summarized in Table
3. Among our sample, RXCJ1504.1-0248 has the largest cool-core radius ($r_{\rm
cool}=324\pm 55$ kpc), while A4059 has the smallest cool-core radius ($r_{\rm
cool}=77\pm 8$ kpc). A3112 and A478 exhibit the relatively small and large
cool-core radii, respectively.

We find that the radiative cooling time of the ICM at the cool-core radius
exceeds 10 Gyr for all galaxy clusters in our sample (see Figure 6). In
particular, RXCJ1504.1-0248 exhibits a radiative cooling time of
$32^{+5}_{-11}$ Gyr at its cool-core radius. These time scales are
significantly longer than the inferred age of low-$z$ galaxy clusters. Our
results indicate that the ICM temperature starts dropping toward the cluster
center from a region exhibiting such a long radiative cooling time. Although
it is difficult to predict the past thermodynamic properties of the ICM at the
same region as the present cool-core radius, it is expected that the effect of
radiative cooling in such regions is minimal or negligible. Furthermore, our
results show that the radiative cooling time of the ICM gradually increases
toward the outskirts, indicating no apparent feature or discontinuity at the
cool-core radius. Therefore, our findings indicate that mechanisms for
determining the size of cool cores rely not only on radiative cooling but also
on additional mechanisms.

Gas sloshing can induce displacement of cool gas originally in a cool core
toward the outskirts, leading to mixing of cool gas with ambient hot gas
(e.g., Ascasibar & Markevitch, 2006; ZuHone et al., 2010; Keshet, 2012; Naor &
Keshet, 2020; Keshet et al., 2023). Since the cool-core systems in the sample
have developed well as indicated by the observed temperature drop, radiative
cooling is expected to play a dominant role in generating cool gas in the
inner regions of the cool cores. Such cool gas is likely to be displaced
toward the outer regions by gas sloshing. In fact, evidence of gas sloshing
has been observed in the cool cores in our sample (Ueda et al., 2021),
supporting this hypothesis.

###  5.2 Relation between the cool-core radius and the cluster mass

To explore possible relations between the cool-core radius and the cluster
mass, we extract the $M_{500}$ and $r_{500}$ values for our sample from the
MCXC catalog (Piffaretti et al., 2011). These values are summarized in Table
6. Since $r_{500}$ is commonly used as a scaling factor for the radial
profiles of the ICM thermodynamic properties, this study provides insights
into the connection between the cool-core radius and $r_{500}$.

Table 6:  Cool-core radius, cluster mass $M_{500}$, fiducial radius $r_{500}$, and those scaled by the values of A4059.  Cluster | $r_{\rm cool}$ | $r_{\rm cool}/r_{\rm cool,A4059}$ | $r_{\rm 500}$ | $r_{\rm cool}/r_{500}$ | $r_{\rm 500}/r_{\rm 500,A4059}$ | $M_{\rm 500}$ | $M_{\rm 500}/M_{\rm 500,A4059}$
---|---|---|---|---|---|---|---
| (kpc) |  | (Mpc) |  |  | ($10^{14}$ $M_{\odot}$) |
RXCJ1504.1-0248 | $325\pm 55$ | $4.21\pm 0.82$ | 1.52 | $0.21\pm 0.04$ | 1.58 | 12.47 | 4.68
A3112 | $89.4\pm 8.0$ | $1.16\pm 0.16$ | 1.13 | $0.080\pm 0.007$ | 1.17 | 4.39 | 1.65
A4059 | $77.1\pm 7.7$ | 1.00 | 0.96 | $0.080\pm 0.008$ | 1.00 | 2.67 | 1.00
A478 | $245\pm 66$ | $3.18\pm 0.92$ | 1.28 | $0.19\pm 0.05$ | 1.32 | 6.42 | 2.41

To investigate the relations of the cool-core radius to $M_{500}$ and to
$r_{500}$, we conduct a regression analysis using a model expressed as

$\frac{r_{\rm cool}}{\hat{r}_{\rm
cool}}=N_{i}\left(\frac{x_{i}}{\hat{x}_{i}}\right)^{\alpha_{i}},$ (8)

where $N_{i}$ is a normalization of the model defined as $N_{i}=10^{A_{i}}$,
$x_{i}$ represents either $M_{500}$ or $r_{500}$, and $\alpha_{i}$ is the
slope of the model. Here, $i$ denotes the values for $M_{500}$ or $r_{500}$.
We center the relation on the pivot values $\hat{r}_{\rm cool}=167$ kpc,
$\hat{M}_{500}=5.41\times 10^{14}$ $M_{\odot}$, and $\hat{r}_{500}=1.20$ Mpc
set at the median of the distributions of $r_{\rm cool}$, $M_{500}$, and
$r_{500}$, respectively.

Following the procedures of the MCMC analysis in Section 4.3, we use
uninformative uniform priors $A_{r},\,A_{M}\in(-5,5)$ and
$\alpha_{r},\,\alpha_{M}\in(0,10)$. We fit the data in the log-log space and
sample the posterior probability distributions of the parameters over the full
parameter space allowed by the priors. The best-fit parameters for the
relations of $r_{\rm cool}$ to $M_{500}$ and to $r_{500}$ are summarized in
Table 7. The best-fit relations for $M_{500}$ and $r_{500}$ with $1\sigma$
uncertainty are shown in Figure 7.

Table 7:  Best-fit parameters of the relation of $r_{\rm cool}$ to $M_{500}$ and to $r_{500}$.  $r_{\rm cool}$ vs. $M_{500}$ |  $r_{\rm cool}$ vs. $r_{500}$
---|---
$A_{M}$ | $N_{M}=10^{A_{M}}$ | $\alpha_{M}$ | $A_{r}$ | $N_{r}=10^{A_{r}}$ | $\alpha_{r}$
$-0.10\pm 0.04$ | $0.79_{-0.07}^{+0.08}$ | $0.94\pm 0.18$ | $-0.10\pm 0.04$ | $0.80_{-0.07}^{+0.08}$ | $3.12\pm 0.59$

Figure 7: Relations of the cool-core radius ($r_{\rm cool}$) to $M_{500}$ and
to $r_{500}$. Left: the relation between $r_{\rm cool}$ and $M_{500}$. The red
solid line and shaded region shows the best-fit relation obtained from the
fitting in the log-log space and its 1 $\sigma$ uncertainty, respectively.
Right: Same as the left panel but for the relation between $r_{\rm cool}$ and
$r_{500}$.

We find that the best-fit relation of the cool-core radius to $M_{500}$ is
consistent with a linear relation (i.e., $\alpha_{M}=0.94\pm 0.18$). Since
$M_{500}$ is proportional to $r_{500}^{3}$, the best-fit slope for the $r_{\rm
cool}$-$r_{500}$ relation is also consistent with this dependency (i.e.,
$\alpha_{r}=3.12\pm 0.59$).

If we assume $r_{\rm cool}\propto L_{\rm X}$, meaning that the size of cool
cores depends on the X-ray luminosity of their host galaxy clusters, can we
infer a slope for the $r_{\rm cool}$-$M_{500}$ relation? Since the cluster
mass $M_{500}$ was estimated using the luminosity-mass scaling relation from
Arnaud et al. (2010) ($L_{\rm X}\propto M^{1.64}$), with the X-ray luminosity
in the $0.1-2.4$ keV band within $r_{500}$ being used in the calculation (see
Piffaretti et al., 2011), a slope of $\sim 1.6$ may be expected for the
$r_{\rm cool}$-$M_{500}$ relation. Previous studies found that the observed
slope for the luminosity-mass scaling relation falls within the range of
$1.3\sim 2$ (see a review Giodini et al., 2013, and references therein).
Furthermore, the self-similar model predicts a slope of $4/3$ for the
luminosity-mass relation. On the other hand, if $r_{\rm cool}\propto t_{\rm
cool}$, $t_{\rm cool}\propto n^{-1}\sqrt{T}$ may yield a slope of $1/3$ for
the $r_{\rm cool}$-$M_{500}$ relation. However, the best-fit slope is
inconsistent with these slopes.

The observed linear relation between $r_{\rm cool}$ and $M_{500}$ indicates
that not only baryon physics such as cooling, heating, and heat transport is
crucial for the formation of cool cores, but also the size of cool cores is
linked to the evolution of their host galaxy clusters. Once the mass of a
galaxy cluster increases owing to a cluster merger, the ICM density at the
center is expected to increase after the relaxation of merger events. Then,
radiative cooling becomes more efficient, leading to the accumulation of a
large amount of cool gas in the central region. Since galaxy clusters grow
through continuous accretion of material, including cluster mergers, from
their surrounding large-scale environments, gas sloshing is expected to take
place in cool cores continuously. In fact, this hypothesis is supported by
recent comprehensive studies of cool cores (Ueda et al., 2020, 2021).
Therefore, the continuous occurrence of gas sloshing may play a crucial role
not only in determining the size of cool cores but also in suppressing runaway
cooling of the ICM. Cool cores may coevolve with their host galaxy clusters.

For the relation between the cool-core radius and a fiducial radius, Vikhlinin
et al. (2005) reported that the projected temperature as a function of radius
reaches a maximum at $r\sim 0.1-0.2\,r_{180}$. Rasmussen & Ponman (2007)
studied the relation between the peak position in the temperature profile
(i.e., $r_{\rm cool}$) and $r_{500}$ using a sample of 15 nearby galaxy groups
observed with Chandra. Assuming a linear relation, they fitted the data and
obtained $r_{\rm cool}=(0.20\pm 0.02)\,(r_{500}/{\rm kpc})-(46.7\pm
15.1)\,{\rm kpc}$. O’Sullivan et al. (2017) also studied the relation between
the turnover radius and $r_{500}$ using a sample of high-richness local galaxy
groups, finding that the observed relation in their sample is mostly
consistent with that presented by Rasmussen & Ponman (2007). In addition,
Rasmussen & Ponman (2007) and O’Sullivan et al. (2017) pointed out that the
relation for galaxy groups differs from that observed in a sample of local
galaxy clusters (Vikhlinin et al., 2005), suggesting that this discrepancy may
be attributed to differences in the physical properties between galaxy
clusters and groups. In fact, our best-fit relation between $r_{\rm cool}$ and
$r_{500}$ seems to deviate from that observed for galaxy groups. Since galaxy
groups have a lower temperature gas compared to galaxy clusters, radiative
cooling becomes more efficient owing to emission lines from the metals in the
ICM. Additionally, mergers may have a more significant impact on a cool core
of the primary galaxy group. Perhaps, the $r_{\rm cool}$-$r_{500}$ relation
may have a break. Further studies are required to arrive at a firm conclusion
regarding the similarities and differences for the $r_{\rm cool}$-$r_{500}$
relation between galaxy clusters and groups.

###  5.3 Scaled radial profiles of the ICM thermodynamic properties

To look deeply into the ICM thermodynamic properties within the cool cores,
we, here, scale the observed radial profiles. These scaled profiles are
presented by normalizing the values on the horizontal and the vertical axes by
the cool-core radius and the value of each component at the cool-core radius,
respectively. Figure 8 shows the scaled radial profile of the ICM temperature.
Figure 9 presents the scaled profiles of the other components.

A possible universal form is found in the scaled temperature profiles. To
clarify this point, we simultaneously fit the scaled temperature profiles
using the same model as that used in Section 4.3 with the MCMC method. In this
fitting, we fix $r_{\rm cool}$ in Equation 4 at 1.0 since the temperature
profile has already been scaled by $r_{\rm cool}$. Instead, we add a new
parameter, intrinsic scatter, into the log-likelihood function. Therefore, the
log-likelihood function can be expressed as

$-2\ln\mathcal{L}=\sum_{i}\ln{(2\pi\sigma_{i}^{2})}+\sum_{i}\frac{[y_{i}-T(r_{i})]^{2}}{\sigma_{i}^{2}},$
(9)

where $i$ runs over all annulus in which sample, $y_{i}$ and $T(r_{i})$ are
the scaled value and the model prediction of the scaled temperature profile in
each radial bin, respectively, and $\sigma_{i}$ includes the observational
uncertainty $\sigma_{y_{i}}$ and lognormal intrinsic scatter $\sigma_{\rm
int}$,

$\sigma_{i}^{2}=\sigma_{y_{i}}^{2}+\sigma_{\rm int}^{2}.$ (10)

The $\sigma_{\rm int}$ parameter accounts for the intrinsic scatter around the
mean relation due to unaccounted errors and/or astrophysics. We continue to
use uninformative uniform priors on $T_{\rm center}$, $T_{\rm peak}$,
$\alpha_{1}$, $\alpha_{2}$ and $\ln{\sigma_{\rm int}}$ as $T_{\rm
center}\in(0,2)$, $T_{\rm peak}\in(0,2)$, $\alpha_{1}\in(0,2)$,
$\alpha_{2}\in(-2,0)$ and $\ln{\sigma_{\rm int}}\in(-20,1)$. We sample the
posterior probability distributions of the parameters over the full parameter
space allowed by the priors. The best-fit parameters are summarized in Table
8.

Figure 8:  Temperature profile of each cluster scaled by $r_{\rm cool}$ and $T_{\rm peak}$ for the horizontal and the vertical axes, respectively. The red dashed line and shaded region display the best-fit profile and 1 $\sigma$ uncertainty, respectively. The blue, vertical solid line indicates $r_{\rm cool}=1$. The residuals and mean absolute relative residuals are shown in the middle and bottom rows, respectively.  Table 8:  Best-fit parameters derived from the simultaneous fitting of the scaled temperature profiles.  $T_{\rm center}$ | $T_{\rm peak}$ | $r_{\rm cool}$ | $\alpha_{1}$ | $\alpha_{2}$ |  $\sigma_{\mathrm{int}}$(%)
---|---|---|---|---|---
$0.34\pm 0.13$ | $0.99\pm 0.03$ | 1 (fixed) | $0.78\pm 0.12$ | $-0.14\pm 0.10$ | $3.25_{-0.79}^{+1.04}$

We find that the ratio of the temperature at the center to the peak value is
$0.34\pm 0.13$, which is in good agreement with the well-known observational
trend in the temperature profiles of cool-core clusters (e.g., Sanderson et
al., 2006; Hudson et al., 2010; Simionescu et al., 2011). The slopes of the
model for the regions inside and outside the cool cores are measured at
$\alpha_{1}=0.78\pm 0.12$ and $\alpha_{2}=-0.14\pm 0.10$, respectively. The
intrinsic scatter $\sigma_{\rm int}$ is obtained at
$3.25^{+1.04}_{-0.79}\,\%$.

A possible universal form in the temperature profiles scaled by a fiducial
radius ($r_{2500}$ or $r_{500}$) within cool cores has been proposed (e.g.,
Allen et al., 2001; Sanderson et al., 2006). On the other hand, Vikhlinin et
al. (2005) and Hudson et al. (2010) reported no such universal form. Our
results indicate a possible universal form in the scaled temperature profile.
Since all cool cores in the sample are classified into strong cool cores
(Hudson et al., 2010), such a universal form may be observed in strong cool
cores only. Additionally, we have scaled the temperature profiles by the cool-
core radius, rather than a fiducial radius like $r_{500}$. It is possible that
the cool-core radius is a more appropriate factor than a fiducial radius for
scaling the radial profiles of the ICM thermodynamic properties, as indicated
by the observed intrinsic scatter. The ratio of $r_{\rm cool}$ to $r_{500}$
varies within our sample (see Table 6), emphasizing the significance of the
chosen radius for scaling in revealing a universal form in the temperature
profile.

To investigate possible universal forms seen in the other components, we also
scale the radial profiles using the same manner as that for the scaled
temperature profiles. Figure 9 shows the scaled radial profiles of the ICM
electron number density, pressure, entropy, and radiative cooling time. In
contrast to the scaled temperature profiles, we find that there is no
universal form in the scaled profiles within the cool cores. The values of
each profile at the center are highly scattered, suggesting that the ICM
temperature is likely the most fundamental factor for characterizing cool
cores. However, further studies are required to reveal possible universal
forms in the radial profiles of the ICM thermodynamic properties. An approach
involving forward model fitting will be useful to conduct a simultaneous
analysis of the ICM temperature and density profiles (e.g., Umetsu et al.,
2022).

In the region outside the cool cores, a universal form is found in the scaled
entropy profile. Such a universal form is known as the universal entropy
profile (Voit et al., 2005). However, in the region inside the cool cores, the
scaled entropy profiles for our sample start varying toward the cluster
center, indicating that the cool-core radius is influenced by the interplay
between cooling and heating processes.

Figure 9:  Scaled radial profiles of the ICM electron number density (top
left), pressure (top right), entropy (bottom left), and radiative cooling time
(bottom right) in the sample. The radial profiles are normalized in the same
manner as in Figure 8.

###  5.4 Analysis of thermodynamic perturbations

It has been studied that thermodynamic perturbations in the ICM are a good
proxy for examining gas motions including turbulence (e.g., Gaspari et al.,
2014; Churazov et al., 2016; Hofmann et al., 2016; Ueda et al., 2018;
Zhuravleva et al., 2018; Kitayama et al., 2020; Ueda et al., 2021; Zhuravleva
et al., 2023). Gaspari et al. (2014) presented that entropy perturbations in
the ICM can be use to infer one-dimensional Mach numbers of turbulence ($\cal
M_{\rm 1D}$), assuming that pressure perturbations in the ICM are negligible.
Following Gaspari et al. (2014), Hofmann et al. (2016) measured pressure and
entropy perturbations in the ICM, and estimated $\cal M_{\rm 1D}$ within the
cool cores in a sample of 33 galaxy clusters.

Motivated by Gaspari et al. (2014) and Hofmann et al. (2016), we investigate
thermodynamic perturbations in the ICM to constrain gas motions in the cool
cores in our sample. We have measured the residuals and mean absolute relative
residuals between the observed and the best-fit profiles (see the middle and
bottom panels of Figures 2, 3, 4, and 5), which can be converted into average
fractional perturbations. We calculate the average fractional perturbations in
the ICM thermodynamic properties within the cool cores, as summarized in Table
9. Our results are consistent with those measured by Hofmann et al. (2016) for
their sample. In particular, the average fractional perturbations in the ICM
pressure in our sample is lower than 0.15, which agrees with their
measurements ($0.09\pm 0.06$). Thus, the observed perturbations are nearly
isobaric. Therefore, similar to Hofmann et al. (2016), the entropy
perturbations in the ICM can be used to infer $\cal M_{\rm 1D}$.

Assuming that the perturbations are isobaric, the observed entropy
perturbations can directly be converted into values of $\cal M_{\rm 1D}$. A478
exhibits a slightly larger $\cal M_{\rm 1D}$ with $0.077\pm 0.069$, while the
inferred values of $\cal M_{\rm 1D}$ are comparable in the sample. Thus, the
corresponding three dimensional Mach numbers ($\cal M_{\rm 3D}$) of the sample
is lower than ${\cal M}_{\rm 3D}<0.25$, which is significantly lower than
unity and is consistent with those measured by the previous studies (Hofmann
et al., 2016; Hitomi Collaboration et al., 2018; Ueda et al., 2021). We expect
that XRISM will be able to achieve direct measurements of the turbulent
velocity for our sample (Tashiro et al., 2018). Additionally, the Athena X-ray
Observatory will provide us with a great opportunity to measure turbulent
velocities in cool cores (Nandra et al., 2013; Barcons et al., 2017).

Table 9:  Average fractional perturbations in the temperature, electron number density, pressure, and entropy profiles within the cool cores.  Cluster | $\langle|\mathrm{d}T|/T\rangle$ | $\langle|\mathrm{d}n|/n\rangle$ | $\langle|\mathrm{d}P|/P\rangle$ | $\langle|\mathrm{d}K|/K\rangle$
---|---|---|---|---
RXCJ1504.1-0248 | $0.047\pm 0.041$ | $0.019\pm 0.015$ | $0.046\pm 0.044$ | $0.051\pm 0.044$
A3112 | $0.021\pm 0.017$ | $0.029\pm 0.051$ | $0.032\pm 0.060$ | $0.030\pm 0.030$
A4059 | $0.046\pm 0.035$ | $0.028\pm 0.019$ | $0.031\pm 0.027$ | $0.060\pm 0.043$
A478 | $0.075\pm 0.066$ | $0.015\pm 0.010$ | $0.073\pm 0.060$ | $0.077\pm 0.069$

##  6 Summary and Conclusions

In this paper, we have conducted a detailed study of the radial profiles of
the ICM thermodynamic properties in cool-core systems in a sample of four
galaxy clusters (RXCJ1504.1-0248, A3112, A4059, and A478) using archival X-ray
data from the Chandra X-ray Observatory. The goal of this study was to observe
the characteristics of the cool cores in the sample and explore mechanisms to
generate the observed characteristics. To this end, we have measured the
turnover radius in the radial profile of the ICM temperature and defined the
cool-core radius as the turnover radius. We have also studied the
thermodynamic properties of the ICM within the cool-core radius and the
relation between the cool-core radius and the cluster mass. The main
conclusions of this paper are summarized as follows:

  1. 1.

Since cool cores are characterized by a significant drop in the ICM
temperature toward the cluster center, we defined the cool-core radius as the
turnover radius in the ICM temperature profile, allowing us to study the ICM
thermodynamic properties in the regions inside and outside the cool cores. We
found no apparent feature at the cool-core radius in the radial profiles of
the ICM electron density, pressure, entropy, and radiative cooling time. These
results indicate that the boundary between inside and outside cool cores is
primarily identified in the temperature profile, suggesting that the ICM
temperature is the most fundamental factor for characterizing cool cores.

  2. 2.

In our sample, the radiative cooling time of the ICM at the cool-core radius
exceeds 10 Gyr, with RXCJ1504.1-0248 exhibiting a radiative cooling time of
$32^{+5}_{-11}$ Gyr at its cool-core radius. Such a long time scale indicates
that not only radiative cooling but also additional mechanisms may be required
to explain the observed properties. Gas sloshing is possible to displace cool
gas generated by radiative cooling away from the center and induce mixing
between such cool gas and the ambient hot gas in the outer region, leading to
a temperature drop.

  3. 3.

We found that the best-fit relation between the cool-core radius and the
cluster mass $M_{500}$ in our sample is consistent with a linear relation. Our
findings suggest that cool cores are linked to the evolution of galaxy
clusters. Cool cores may coevolve with their host galaxy clusters. Since the
cluster mass increases owing to mergers, which in turn induce gas sloshing, it
is plausible that gas sloshing plays a significant role in the evolution of
cool cores.

  4. 4.

A possible universal form in the temperature profiles scaled by the cool core
radius is found in the cool cores in our sample. However, the scaled profiles
of the other components within the cool cores are highly scattered, indicating
that there is no universal form.

  5. 5.

The one-dimensional Mach numbers of turbulence ($\cal M_{\rm 1D}$) in the cool
cores in the sample are constrained by analyzing the entropy perturbations in
the ICM. The inferred $\cal M_{\rm 3D}$ is significantly lower than unity,
suggesting that subsonic gas motions are dominant in the cool cores.

We thank Keiichi Umetsu for fruitful discussions. We also thank H.-Y. Karen
Yang for helpful comments. The scientific results of this paper are based in
part on data obtained from the Chandra Data Archive: ObsID 4935, 5793, 17197,
17669, 17670, 2216, 2516, 6972, 7323, 7324, 13135, 897, 5785, 1669, and 6102.
S.U. acknowledges the support from the National Science and Technology Council
of Taiwan (NSTC 111-2811- M-007-008 and 111-2112-M-001-026-MY3). We thank the
ASIAA Summer Student Program 2019 for hospitality and providing us with an
opportunity to launch the project.

##  Appendix A Region selection for X-ray spectral analysis

Here, we summarize detailed information regarding the region selection and
radial bin sizes of the regions for the X-ray spectral analysis.

Table 10:  Summary of our region selection for the X-ray spectral analysis: region interval, region range, the number of regions, and the maximum size for our region selection.  Cluster | Region interval (arcsec)aa Radial bin size between the inner and outer major axes of each elliptical annulus region.  | Region range (arcsec) | Number of regions | Maximum size (arcsec)bb Size at the outermost region in our region selection.
---|---|---|---|---
RXCJ1504.1-0248 | 1 | 0–3 | 3 | 180
0.5 | 3–20 | 34
1 | 20–36 | 16
2 | 36–60 | 12
5 | 60–90 | 6
10 | 90–180 | 9 |
A3112 | 2 | 0–60 | 30 | 180
5 | 60–100 | 8
10 | 100–180 | 8
A4059 | 5 | 0–5 | 1 | 192
4 | 5–25 | 5
3 | 25–160 | 45
4 | 160–192 | 8
A478 | 4 | 0–4 | 1 | 320
2 | 4–110 | 53
4 | 110–210 | 25
10 | 210–320 | 11

##  Appendix B Radial profiles of the ICM metal abundance

Here, we show the radial profiles of the ICM metal abundance for the sample.
We find that the ICM metal abundance in A3112 starts slightly decreasing at
$\sim 15$ kpc toward the center, A4059 also exhibits a decrease of the ICM
metal abundance from $\sim 20$ kpc to the center, which is consistent with
that measured by Choi et al. (2004).

Figure 10: Radial profiles of the ICM metal abundance: RXCJ1504.1-0248 (top
left), A3112 (top right), A4059 (bottom left), and A478 (bottom right). The
blue, vertical solid line and shaded region correspond to the cool-core radius
and $1\sigma$ uncertainty for each cluster.

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