11institutetext:  Shanghai Science and Technology Museum, Shanghai 200127,
China 11email<EMAIL_ADDRESS>22institutetext: Department of
Astronomy, Beijing Normal University, Beijing 100875, China 22email:
<EMAIL_ADDRESS>33institutetext: INAF - Osservatorio Astrofisico di Arcetri,
Largo E. Fermi, I-50122 Firenze, Italy 44institutetext: Department of
Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ
08544-1001, USA 55institutetext: INAF - Osservatorio di Astrofisica e Scienza
dello Spazio, via P. Gobetti 93/3, I-40129 Bologna, Italy 66institutetext:
INFN, Sezione di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy

# Physical cool-core condensation radius in massive galaxy clusters††thanks:
Tables 1 and 2, and the tabulated values for the plots shown in Figure B.1,
are available in electronic form at the CDS via anonymous ftp to
cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/cgi-
bin/qcat?J/A+A/

Lei Wang 1122 Paolo Tozzi 33 Heng Yu 22 Massimo Gaspari 44 Stefano Ettori 5566

###### Abstract

Aims. We investigate the properties of cool cores in an optimally selected
sample of 37 massive and X-ray-bright galaxy clusters, with regular
morphologies, observed with Chandra. We started by measuring the density,
temperature, and abundance radial profiles of their intracluster medium (ICM).
From these independent quantities, we computed the cooling ($t_{\rm cool}$),
free-fall ($t_{\rm ff}$), and turbulence ($t_{\rm eddy}$) timescales as a
function of radius.

Methods. By requiring the profile-crossing condition, $t_{\rm cool}/t_{\rm
eddy}=1$, we measured the cool-core condensation radius, $R_{\rm ccc}$, within
which the balancing feeding and feedback processes generate the turbulent
condensation rain and related chaotic cold accretion (CCA). We also
constrained the complementary (quenched) cooling flow radius, $R_{\rm qcf}$,
obtained via the condition $t_{\rm cool}=25\times t_{\rm ff}$, that
encompasses the region of thermally unstable cooling.

Results. We find that in our our massive cluster sample and in the limited
redshift range considered ($1.3\times 10^{14}<M_{500}<16.6\times
10^{14}\,M_{\odot}$, $0.03<z<0.29$), the distribution of $R_{\rm ccc}$ peaks
at $\sim$ 0.01 $r_{500}$ and the entire range remains below $\sim$ 0.07
$r_{500}$, with a very weak increase with redshift and no dependence on the
cluster mass. We find that $R_{\rm qcf}$ is typically three times larger than
$R_{\rm ccc}$, with a wider distribution, and growing more slowly along
$R_{\rm ccc}$, according to an average relation $R_{\rm qcf}\propto R_{\rm
ccc}^{0.46}$, with a large intrinsic scatter.

Conclusions. We suggest that this sublinear relation can be understood as an
effect of the micro rain of pockets of cooled gas flickering in the turbulent
ICM, whose dynamical and thermodynamical properties are referred to as ”macro
weather.” Substituting the classical ad-hoc cool-core radius $R_{\rm
7.7\,Gyr}$, we propose that $R_{\rm qcf}$ is an indicator of the size of
global cool cores tied to the long-term macro weather, with the inner $R_{\rm
ccc}$ closely tracing the effective condensation rain and chaotic cold
accretion (CCA) zone that feeds the central supermassive black hole (SMBH).

###### Key Words.:

galaxies: clusters: intracluster medium (ICM) – X-rays: galaxies: clusters –
hydrodynamics

##  1 Introduction

The hot intracluster medium (ICM) is the largest baryonic component in groups
and clusters of galaxies (Gonzalez et al. 2013). It is observable in the X-ray
band thanks to its strong Bremsstrahlung continuum emission plus emission
lines from highly ionized elements, and it shows temperatures from 1 keV
(groups) to more than 10 keV (massive clusters). The density profile of the
ICM is usually fitted with the $\beta$ profile (Cavaliere & Fusco-Femiano
1978), consisting of a flat core and rapidly decreasing outskirts. However, in
many clusters, the central electron density is not accurately described by a
single $\beta$ profile, but is observed to be sharply peaked, reaching values
significantly larger than the typical $n_{\rm e}$ $\thicksim$ 10-3 cm-3. In
these cases, a double $\beta$ profile is needed to fit the X-ray surface
brightness (as known from ROSAT observations Xue & Wu 2000), effectively
defining a central core where the ICM cooling time, $t_{\rm cool}$ $\propto
kT/n_{e}$, is significantly lower than the typical age of the cluster. This
implies that a large amount of gas should cool completely on a short timescale
(typically less than 1 Gyr) due to radiative losses, leading to a massive
cooling flow with a mass deposition rate of the order of several hundreds up
to thousands M⊙ yr-1. Such large values are obtained directly from the
brightness profile under the assumption of subsonic flow and constant
pressure, as per the so-called ”isobaric cooling flow” model (Fabian & Nulsen
1977; Fabian 1994).

The X-ray luminosity in cluster cores is dominated by the hottest ICM
component (above a few keV), while the coldest one contributes only a few
percent of the total, while being rich in emission lines. At variance with the
”isobaric cooling flow” scenario, high-resolution spectroscopy of bright
clusters with XMM–Newton have not shown any evidence for the multiphase, line-
rich gas predicted by the isobaric cooling model. Instead, it has been
observed that the majority of the ICM typically reaches a temperature plateau
at about one-third of the virial value (Kaastra et al. 2001; Peterson et al.
2001; Tamura et al. 2001; Donahue & Voit 2004), while the cold gas below this
floor is virtually absent. In this framework, the old paradigm of ”cooling
flow” has been abandoned in favour of the ”cool core” scenario (Molendi &
Pizzolato 2001). Despite some amount of cooling gas, possibly associated with
star-forming episodes in the central galaxy, is still allowed by the
observations, the upper limits to the spectroscopic mass-deposition rate
values are at least one order of magnitude lower than the rates expected by
isobaric cooling flow models (McNamara & O’Connell 1989; Makishima et al.
2001; Edge 2001; Edge & Frayer 2003; McNamara & Nulsen 2007). Recently,
spatially resolved spectral analyses of cool cores have confirmed that the
central, isobaric mass-deposition rates are significantly lower than the star-
formation rates observed in the hosted brightest cluster galaxy (BCG; Molendi
et al. 2016). On the other hand, the more convincing cooling flow candidates
are limited to only one well-documented case (the Phoenix cluster, see
McDonald et al. 2012; Tozzi et al. 2015; Pinto et al. 2018; McDonald et al.
2019).

This observational evidence strongly supports the presence of some heating
mechanism that prevents the bulk of the ICM from cooling below about one-third
of the virial temperature in the cool core. This is now considered a standard
condition of the ICM in cool-core clusters, which represent $\sim 70\%$ of the
low-redshift population of X-ray flux-limited sample, according to Hudson et
al. (2010). On the other hand, some amount of cold and multiphase gas is now
commonly observed in the submm and optical band in star forming regions of the
BCG thanks to ALMA (e.g., McNamara et al. 2014; Russell et al. 2017; Temi et
al. 2018; Tremblay et al. 2018; Rose et al. 2019; North et al. 2021) and MUSE
(e.g., Olivares et al. 2019, 2022; Maccagni et al. 2021), respectively. The
presence of multiphase gas suggests that short-lived cooling flows raining all
the way down onto the central supermassive black hole (SMBH) in the BCG have
time to replenish the cold gas reservoir before being quenched by the ensuing
SMBH feedback process. In summary, we are well aware that cool cores are
physical systems where the cooling process is counterbalanced by some global
heating mechanism that strongly suppresses the mass deposition rate, while
still allowing some amount of gas to leak out of the hot phase with a
timescale regulated by a complex feeding (cooling) and feedback (heating)
cycle (Gaspari & Sa̧dowski 2017). A comprehensive and systematic comparison of
the spectroscopic mass deposition rate to the star formation rate in the BCG
and the presence of molecular gas in the cluster core, significantly extending
the small sample explored in Molendi et al. (2016), would provide very
effective constraints on the baryonic cycle in clusters. In addition, the
detection of very diffuse, low-temperature ICM in the center of non-cool-core
clusters by the next generation of X-ray bolometers, may provide support for a
scenario in which cool-core clusters rapidly switch into the non cool-core
phase and vice versa (see Molendi et al. 2023).

Many heating mechanisms have been proposed in the past two decades, among
them: thermal conduction (Zakamska & Narayan 2003), viscous dissipation of
sound waves (Ruszkowski et al. 2004), supernova feedback (Domainko et al.
2004), turbulence combined with conduction (Dennis & Chandran 2005), cosmic
ray–ICM interaction (Guo & Oh 2008; Yang et al. 2019), and feedback from jets
and outflows from the central active galactic nucleus (AGN; e.g., McNamara &
Nulsen 2007; Gaspari et al. 2012b; Barai et al. 2016; Wittor & Gaspari 2020;
McKinley et al. 2022). In particular, the last process is considered the most
likely contributor on the basis of the well-documented interactions between
radio jets and the surrounding ICM. The large amount of mechanical energy
associated with the cavities carved into the ICM by the radio jets may be
eventually transformed into thermal energy of the ICM (Blanton et al. 2010,
for a review) and may stimulate, at the same time, the cooling of some
fraction of the gas (Gaspari et al. 2020, for a review). Therefore, no matter
how many mechanisms are contributing, the central AGN is expected to play an
important role in regulating cooling, ultimately inducing tight scaling
relations between the SMBH and AGN as well as the hot halo properties (Gaspari
et al. 2019; Pasini et al. 2021).

In this work we investigate key physical radii that delimit the spherical
regions where different phases of the complex baryon cycle are actually taking
place, such as the cool-core condensation radius ($R_{\rm ccc}$) and the
quenched cooling flow radius ($R_{\rm qcf}$). In Section 2, we introduce and
discuss our definition of $R_{\rm ccc}$ and $R_{\rm qcf}$. In Section 3, we
derive the typical timescales of relevant processes occurring in galaxy
clusters. In Section 4, we describe the selection of the sample of galaxy
clusters observed with Chandra and used in this work. In Section 5, we discuss
the key parameter represented by the turbulent velocity dispersion of the warm
and cold phase of the diffuse baryons. In Section 6, we describe data
reduction and our analysis strategy. Our results are described in Section 7,
where we show deprojected timescale profiles in each cluster as a function of
radius and we measure the $R_{\rm ccc}$ and the $R_{\rm qcf}$ values, followed
by an investigation of their distribution across the cluster sample. The
physical implication of our findings are discussed in Section 8 and our
conclusions are summarized in Section 9. Throughout the paper, the
cosmological model of reference is a $\Lambda$CDM with parameters $H_{0}$ =
67.8 km s-1 Mpc-1, $\Omega_{\Lambda}$ = 0.692 and $\Omega_{m}$ = 0.308 (Planck
Collaboration et al. 2016). Quoted errors and upper limits correspond to a
1-$\sigma$ confidence level.

##  2 Physical definition of cool-core condensation and quenched cooling-flow
radius

From the observational point of view, it has been well established that the
presence of radio nuclear activity is closely associated with the presence of
a cool core (Dunn & Fabian 2006; Sun 2009). The interactions between the
relativistic electrons and the thermal electrons of the ICM have been
thoroughly studied in spectacular images of few nearby clusters such as
Perseus (Fabian et al. 2003b), Hydra A (McNamara et al. 2000), and few other
clusters at intermediate redshift (Blanton et al. 2011; Ehlert et al. 2011);
in addition, the presence of cavities in the ICM has been explored up to
$z\thicksim 1.2$ (Hlavacek-Larrondo et al. 2015). However, while the energy
budget associated with cavities is sufficient to switch off the cooling in all
the observed cases, the physical mechanism by which the energy of the jet is
transferred isotropically to the ICM is still an issue of debate. Possible
mechanisms include turbulence (e.g., Gaspari 2015) or weak shocks (e.g.,
Fabian et al. 2003a; Mathews et al. 2006) driven by the radio-mode activity of
the central galaxy. The key issue here is to identify a process that regularly
transforms an impulsive and directional energy input of the jet into a gentle
heating, smoothly distributed in time and space, to finally shape the regular
ICM thermodynamical properties observed in cool-core clusters.

All these details cannot be resolved in most cool-core clusters and,
therefore, the actual physical processes can hardly be constrained from the
macroscopic X-ray quantities such as luminosity and temperature. In recent
years, several independent efforts have been devoted to achieve an efficient
observational diagnostics to classify clusters according to the presence of a
cool core and, at the same time, to understand the dominant physical
processes. The many quantities used to define a cool core are all related to
the thermodynamical properties of the ICM but they are associated with
different physical processes: surface brightness excess or cuspiness (Santos
et al. 2008), the temperature gradient (Sanderson et al. 2006; Burns et al.
2008), a steep iron abundance profile (De Grandi et al. 2004; Rasera et al.
2008), steep slopes of central entropy (Pratt et al. 2010), low central
cooling time (O’Hara et al. 2006), classical mass deposition rate (Chen et al.
2007), and (clearly) the slope of the electron density profile (Hudson et al.
2010). All these diagnostics are obtained by combining the same three
independent X-ray observables: surface brightness, temperature, and line
emission.

In a comprehensive overview, Hudson et al. (2010) concluded that the central
cooling time is one of the best diagnostics for identifying and characterizing
cool cores. Furthermore, Gaspari et al. (2018) showed that the ratio of the
cooling time over the turbulence eddy turnover timescale $C\equiv t_{\rm
cool}/t_{\rm eddy}\sim 1$ is a key diagnostic for the condensation extent of
the multiphase rain occurring via chaotic cold accretion (hereafter, CCA). In
other words, whenever the $C$-ratio approaches unity there is enough
turbulence and quick cooling to directly drive non-linear instabilities. This
is confirmed by studies obtained with high-resolution radio/optical telescopes
(Gaspari et al. 2018; Olivares et al. 2019, 2022), which show that multiphase
filamentary structures can be observed within the region enclosed by $t_{\rm
cool}/t_{\rm eddy}\approx 1$. In this region, the turbulent mixing rate is
expected in part to balance the pure cooling flow, in part to drive direct
nonlinear thermal instability that then ends up generating a rainfall. As
shown by theoretical studies, the ICM can be seen as a hierarchical
thermodynamic system that follows a chaotic, top-down multiphase condensation
cascade (Gaspari et al. 2017; Voit et al. 2017). Therefore, we assume that an
appropriate timescale for feedback is provided by the turbulence timescale,
$t_{\rm eddy}$. In this work, we leverage these findings to identify the
approximately spherical region within which the nonlinear multiphase CCA rain
and the triggered feedback response are very effective, via the following
condition:

$C\equiv t_{\rm cool}(R_{\rm ccc})/t_{\rm eddy}(R_{\rm ccc})=1.$ (1)

This relation effectively defines the cool-core condensation radius, $R_{\rm
ccc}$. Within $R_{\rm ccc}$, we expect direct turbulence instability to be
driving localized flickering precipitation and, therefore, the condensation of
the light rain that may be responsible for accretion events onto the central
SMBH in the BCG, and thus leading to star formation episodes in the BCG.

At the same time, we can identify a region where we assume thermally unstable
cooling may ensue from linear perturbations by establishing a threshold in the
ratio of the cooling, $t_{\rm cool}$, and the free-fall time $t_{\rm ff}$
(e.g., Field 1965). Voit et al. (2015b) found that the minimum value of
$t_{\rm cool}$/$t_{\rm ff}$ fluctuate around values of $10-20$, concluding
that cold clouds start to precipitate out of hot-gas atmospheres when $t_{\rm
cool}$ drops to ten times $t_{\rm ff}$. Later, Hogan et al. (2017) showed that
the minimum of the $t_{\rm cool}$/$t_{\rm ff}$ ratio in a large sample of
observed clusters with constrained nebular emission (tracing the condensed
cool gas) is bound between 10 and 40, with few values below 10, which is also
supported by hydrodynamical simulations with self-regulated AGN jet feedback
(Gaspari et al. 2012b).

Therefore, we argue that an average ratio $\equiv t_{\rm cool}$/$t_{\rm
ff}\sim 25$, despite a large scatter, is a reasonable proxy for tracing the
initial growth of linear thermal instability (TI) in heated cooling flows,
while a value of 10 traces the lower bound of such a criterion111This is often
denoted by TI-ratio, given its relation to linear TI, rather than nonlinear
turbulent condensation.. Overall, we define a quenched cooling flow radius
(and use it here) when the following condition is met:

$t_{\rm cool}(R_{\rm qcf})=25\times\,t_{\rm ff}(R_{\rm qcf})\,.$ (2)

In this framework, the quantity $R_{\rm qcf}$ is expected to be an alternative
definition to the ”classical” cool-core radius defined on the basis of the
cooling time. Indeed, the radius below is often used, whereby the cooling time
is shorter than the reference value of 7.7 Gyr222We note that 7.7 Gyr
corresponds to $z=1$ in the cosmology adopted in Hudson et al. (2010), and,
for consistency with their argument, here we should assume a look-back time of
7.93 Gyr. However, this would negligibly affect our discussion, therefore we
prefer to maintain the nominal reference value of 7.7 Gyr., $R_{\rm
classic}\equiv R_{\rm 7.7\,Gyr}$; Hudson et al. 2010).

Overall, we expect the two newly defined core radii to trace physical
transitions from a macro-scale X-ray emitting ICM atmosphere delimited by a
”classical” cool core radius to a region where a quenched cooling inflow could
potentially develop ($<R_{\rm qcf}$), and, eventually, to a region where
precipitation and feedback are actively vigorous ($<R_{\rm ccc}$). To explore
the behaviour of these two spatial scales, we sought to measure $R_{\rm qcf}$
and $R_{\rm ccc}$ in an optimally selected sample of massive clusters observed
with the Chandra satellite.

##  3 Timescales in the ICM of massive Galaxy Clusters

In this section, we define the timescale for physical processes relevant to
the ICM, which is treated as an optically thin plasma in collisional
ionization equilibrium, despite occasional out-of-equilibrium phases that may
potentially be reached. However, treating the ICM in steady equilibrium is a
fitting approximation and would not bias our results. As previously discussed,
we will also assume spherical symmetry, as a requirement that will affect the
sample selection in certain ways, as discussed in Section 4.

###  3.1 Cooling time

The ICM X-ray emission is composed of thermal bremsstrahlung (free-free
emission) plus line emission from ions of heavy elements. The X-ray luminosity
density (energy emitted per unit time at a unit volume) can therefore be
written as $L_{X}\approx n_{e}^{2}\Lambda(T,Z)$, where $n_{\rm e}$ is the
electron density and $\Lambda(T,Z)$ is the cooling function, which depends on
the temperature of both radiative processes and is also related with the
abundance, $Z,$ of heavy elements in the ICM (see Figure 3 of Peterson &
Fabian 2006). For the hot ICM ($kT>2$ keV), the bremsstrahlung emission
dominates and the approximation $\Lambda(T)\propto T^{1/2}$ is usually
adopted. When temperatures are low, the number of ions (and therefore the
number of possible transitions) strongly increases. As a consequence, the
enhanced contribution from line emission significantly affects the cooling
function. This regime is particularly relevant in cool cores, where low
temperatures are always associated with high metallicity, often reaching
supersolar values (De Grandi et al. 2004; Liu et al. 2020). Therefore, to
describe the cooling efficiency of the ICM accurately at different radii, the
full cooling function must be taken into account.

The main energy loss of the ICM is the thermal radiative emission, which is
mostly observed in the classic 0.5-10 keV X-ray band for the temperature range
we are considering here. Therefore, the cooling time is typically defined as
the timescale after which the ICM entirely loses its internal energy via
bremsstrahlung radiation, down to the point when the gas eventually recombines
and starts loosing energy through other radiative processes. An effective way
to estimate the ICM cooling time is obtained by dividing the gas internal
energy by the luminosity density of the plasma. We can therefore express the
internal energy as $(3/2)\,nkT$, obtaining the following for the cooling time:

$t_{\rm cool}\simeq\frac{3}{2}\frac{nkT}{n_{\rm e}n_{\rm i}\Lambda(Z,T)},$ (3)

where $\Lambda(Z,T)$ is the cooling function for gas with a specific
abundance, $Z,$ and temperature, $kT$. In this work, we compute the cooling
function interpolating the values reported in Sutherland & Dopita (1993). We
note that here we adopt a definition of cooling time based on the internal
energy rather than the enthalpy $(5/2)nkT$ (Peterson & Fabian 2006). The
factor of $5/2$ is assumed to account for the inclusion of the extra work-term
arising from perfect spherically symmetric isobaric compression. However, the
$5/2$ value should be considered as an upper limit, since (under realistic
conditions) there is no perfect isobaric compression and any contribution from
turbulence or AGN heating brings it near the pure 3/2 factor, as shown in
Figure 5 of Gaspari (2015).

For temperatures $kT\gtrsim 2$ keV (and metallicity $0.3\,Z_{\sun}$), only
Bremsstrahlung emission is relevant for such clusters, thus $t_{\rm cool}$ is
well approximated by the expression (see Cavagnolo et al. 2009):

$t_{\rm cool}\simeq 10^{8}\,{\rm yr}\left(\frac{K}{10\,\rm
keV\,cm^{2}}\right)^{3/2}\left(\frac{kT}{\rm 5\,keV}\right)^{-1}\,,$ (4)

where $K\equiv kT/n_{\rm e}^{2/3}$ is the astrophysical (electron) entropy
(Ponman et al. 1999; Tozzi & Norman 2001). We note that using a higher solar
metallicity reduces the normalization in Equation 4 by only 30%. Moreover,
hydrodynamical simulations predict that this cooling time ought to have a
radial dependence approximated by a power law with a slope of $\sim$ 1.3
(Ettori & Brighenti 2008).

###  3.2 Free-fall time

The free-fall time is defined as the timescale of an object falling towards
the center of the cluster without any pressure support or any deceleration due
to viscosity (Binney & Tremaine 1987). The free-fall time can be written as:

$t_{\rm
ff}=\sqrt{\frac{2r}{g(r)}}=\sqrt{\frac{2r\rho_{g}}{dP/dr}}=\sqrt{\frac{2r^{3}}{GM}}\,,$
(5)

where $M$ is the total mass within a spherical radius, $r$. The free-fall
time, therefore, depends on the total mass and not on the properties of the
ICM, such as density. The total mass profile $M(<r)$ can, in turn, be computed
directly from the ICM applying the condition of hydrostatic equilibrium, which
requires the knowledge of the density and temperature profiles of the ICM:

$M(<r)=\,-\,\frac{r^{2}}{G\rho_{g}}\frac{{\rm d}P}{{\rm
d}r}=\,-\,r\frac{kT(r)}{\mu m_{p}G}\left(\frac{{\rm d}\log(n_{e})}{{\rm
d}\log(r)}+\frac{{\rm d}\log(kT)}{{\rm d}\log(r)}\right)\,,$ (6)

where $kT(r)$ is the deprojected temperature profile of the ICM, $n_{e}(r)$
the deprojected electron density profiles, $\mu$ is the mean molecular weight
of ICM (which is usually assumed to be $\mu$ = 0.6). and $m_{p}$ is the proton
mass.

###  3.3 Turbulence timescale

A number of processes, such as cluster mergers, galaxy motions, and AGN
feedback, are capable of producing turbulence in the ICM. The mechanical
energy associated with these processes is very high and X-ray observations
clearly show that the ICM is significantly affected by it (Iapichino et al.
2010; Gaspari et al. 2014; Liu et al. 2015; Lau et al. 2017; Wittor & Gaspari
2020). Combining X-ray and radio observations, it has been shown that radio
jets injects bubbles of relativistic electrons that create cavities within the
ICM on scales of 10-100 kpc (McNamara et al. 2005; Diehl et al. 2008; Blanton
et al. 2011; Hlavacek-Larrondo et al. 2015; Yang et al. 2019). The expectation
is that a relevant fraction of the mechanical energy associated with the
buoyant rise of inflated bubbles (or with bulk motions of infalling halos)
will transfer into the ICM and eventually produce turbulence. Measurements of
turbulence in cluster cores have been reported, in particular, via the power
spectra of density fluctuations (Schuecker et al. 2004; Sanders et al. 2010;
Gaspari & Churazov 2013; Zhuravleva et al. 2014; Hofmann et al. 2016;
Simionescu et al. 2019). The only measurement of turbulence in the ICM at high
spectral resolution has been provided by the Hitomi mission (Hitomi
Collaboration et al. 2016), which showed a few 100 km s-1 in turbulent
velocities in Perseus cluster. The ultimate observational evidence of the
amplitude and distribution of turbulence in the ICM will be provided in the
next future by X-ray high-resolution spectra, obtained with the X-ray
spectrometers Resolve onboard XRISM333The X-ray Imaging and Spectroscopy
Mission (XRISM), formerly named the X-ray Astronomy Recovery Mission (XARM),
is a JAXA/NASA collaborative mission, with ESA participation (see Guainazzi &
Tashiro 2018, and references therein), expected to be launched in 2023. and
X-IFU onboard Athena444Athena stands for Advanced Telescope for High ENergy
Astrophysics, (www.the-athena-x-ray-observatory.eu/). It is the X-ray
observatory mission originally selected by ESA as second L(large)-class
mission within the Cosmic Vision programme and is currently undergoing a
revision process before final adoption. It will address the ”hot and energetic
universe scientific theme” and is due for launch in the second half of the
2030s..

Following Gaspari et al. (2018), we can estimate the turbulence characteristic
timescale via the eddy turnover/mixing555We note that for subsonic turbulence,
as in the ICM, the turbulent dissipation timescale is $\sim 20\times$ longer
than the eddy turnover time. time, such as

$t_{\rm eddy}=2\pi\frac{r^{2/3}L^{1/3}}{\sigma_{v,L}}\,,$ (7)

where $L$ is the energy-injection scale and $\sigma_{v,L}$ is the typical
turbulent velocity measured at the injection scale. The injection scale, $L,$
is related to the AGN feedback influence region and can be approximated via
the typical observed size covered by the pair of cavities inflated in the ICM
(alternatively, by the extent of the H$\alpha$ nebula). A phenomenological
scaling for $L$ is obtained from the large sample by Shin et al. (2016), with
$L\sim 10\,(kT/1\,\rm keV)^{2}$ kpc (see also Gaspari et al. 2019). Given the
strong dependence on the average temperature, it can reach values as high as
$\sim$ 200 kpc, as in the case of MS 0735.6+7421 (Vantyghem et al. 2014).

The other relevant parameter, namely, the turbulent velocity dispersion,
$\sigma_{v,L}$, is preferentially taken from the literature (Gaspari et al.
2018; Olivares et al. 2019) whenever a direct measurement is available. If no
measurements are available, we consider a range of values chosen (as detailed
in Section 5). Another assumption here is that we assume that the three-
dimensional velocity dispersion $\sigma_{v,L}$ is obtained multiplying by
$\sqrt{3}$ the line-of-sight velocity dispersion of the warm and cold gas
measured in (Gaspari et al. 2018; Olivares et al. 2019), which implies
isotropic turbulence. This may not be true when violent gas sloshing is
present. In fact, during the initial phase, sloshing-induced turbulence may
have larger velocity dispersion in the plane of sloshing motion. On the one
hand, estimating and removing the effects of gas sloshing is beyond the
capability of the current analysis. On the other hand, we argue that its
impact is limited. This assumption is based on the fact that sloshing develops
on much larger time scales (see ZuHone et al. 2013; Zuhone & Roediger 2016),
while the AGN feedback is self-regulated on duty cycles corresponding to $\sim
1-10$ Myr (see Gaspari et al. 2012a). Therefore, we expect that the main
contribution to the turbulence level is provided by the AGN feedback, while
sloshing may boost the turbulence on time scales ranging 0.1 – a few Gyr (for
minor and major mergers, respectively; see discussion in Lau et al. 2017).
Within this framework, we adopt the assumption of isotropic turbulence bearing
in mind that $t_{\rm eddy}$ can be slightly affected by anisotropic bulk
motions of the ICM.

##  4 Cluster sample selection

The term ”cool core” typically refers to the central region of a cluster,
approximately in hydrostatic equilibrium, which shows a density profile
significantly peaked toward the center, the temperature profile increasing
with the radius on a scale of $\thicksim$ 50 kpc, an iron abundance profile
decreasing with radius on the same scale, and, therefore, a short cooling
time. In the absence of a recent major merger, a cool core invariably hosts a
BCG at its center.

To achieve a robust measurement of radial timescale profiles down to a few
kpc, we decided to perform our study with Chandra data, thanks to its angular
resolution unparalleled among current X-ray facilities, which allows us to
resolve a scale below 10 kpc virtually at redshifts up to $z\sim 1$. In this
way, the effective resolution of our analysis will be determined solely by the
surface brightness of the ICM emission and the exposure depth of the data. To
optimally select a sample of clusters for our study, we considered the Chandra
archive to collect all the targets where we can measure the timescale profiles
well inside the potential cool core region with an accuracy sufficient to
identify the transition radii, $R_{\rm qcf}$ and $R_{\rm ccc}$. As a rule of
thumb, we know that the typical size of cool cores is around 40 kpc (Santos et
al. 2008), therefore, as the first criterion, we require enough counts within
this radius to extract at least two rings with a minimum of 3000 net counts
each in the 0.5-7 keV band. This will allow us to measure the temperature with
an error lower than $\sim 20\%$ in the spatial bins within 40 kpc.

On the other hand, we also need to measure temperature and density out to
300–400 kpc, in order to track how timescale profiles behave before
approaching the cooling region. Due to the limited Chandra field of view (16′
$\times$ 16′ for ACIS-I, 8′ $\times$ 8′ for ACIS-S), the requirement to reach
$\sim$ 400 kpc in the FOV translates in a conservative lower limit $z>0.03$
and $z>0.1$ for ACIS-I and ACIS-S, respectively. We also require $z<0.3$ to
have at least a width of 3 arcsec for the minimum ring width of $\sim 15$ kpc.

We start from a total sample of 1144 galaxy clusters or groups, consisting of
all the publicly available archival data under this category. Our requirement
on the redshift range brings this number down to 456 clusters. After applying
our criterion on the net number of counts and removing those clusters that
strongly depart from a relaxed, a spherical morphology after a visual
inspection, we obtained a total of 37 clusters at 0.03 $<z<$ 0.3 satisfying
our criteria. With the knowledge that a simple visual inspection does not
guarantee a relaxed dynamics, with this step, we excluded all the clusters
with disturbed morphologies that would make a spherical deprojection
unreliable. We also note that the requirement on the net counts within 40 kpc
strongly favors cool-core clusters, since to reach the same angular resolution
in our spectral analysis, non-cool-core clusters need to have a significantly
deeper exposure to match the same number of photons in the inner 40 kpc as
compared to cool-core clusters. Therefore, non cool-core clusters are expected
to be underrepresented in our final sample. However, since we are aiming at
investigating the correlation between core properties and other observables
(and not the distribution across the cluster population), the predominance of
cool-core clusters does not strongly affect our conclusions. The list of
clusters, with redshift, position, ObsID, and effective total exposure time
after data reduction, is shown in Table 1. We note that not all the available
exposures for each target are used. In particular, we removed all the off-
centered pointings (aiming at the cluster outskirts) and we discarded some
pointings with very low exposures ($\sim 5$ ks) or with an observation date
very far from the bulk of the other observations (which would imply a
significantly different effective area). Usually, we keep observations in both
ACIS detectors, but in some cases, we discarded the observations in one of the
detectors when the corresponding exposure is a minority of the total exposure
time; this was done to achieve a simpler spectral analysis at the cost of a
negligible loss in signal.

Table 1: Sample of clusters observed with Chandra and satisfying the criteria defined in Section 3. Column 1: Cluster name; column 2: Redshift; column 3: R.A. of cluster center; column 4: Declination of the cluster center; column 5: Obsid used in the analysis (ACIS-S and ACIS-I observations are marked with ”S” and ”I,” respectively); column 6: Exposure time after data reduction; column 7: Net photon counts within 40 kpc in the 0.5-7.0 keV band. Cluster | z | RA | DEC | ObsID | $t_{\rm exp}(\rm ks)$ | Net photon counts
---|---|---|---|---|---|---
A2199 | 0.0302 | 16h28m38.4s | +39d33m03.6s | 497,498(S) | 158.3 | 78044.9
|  |  |  | 10748,10803 |  |
|  |  |  | 10804,10805(I) |  |
A496 | 0.0329 | 04h33m37.92s | -13d15m43.2s | 931,3361,4976(S) | 104.0 | 60790.1
2A0335+096 | 0.0363 | 03h38m40.56s | +09d58m12s | 919,7939,9792(S) | 103.0 | 93744.0
A2589 | 0.0414 | 23h23m57.12s | +16d46m44.4s | 3210,6948,7190 | 92.3 | 5413.7
|  |  |  | 7340(S) |  |
MKW3S | 0.0450 | 15h21m51.84s | +07d42m32.4s | 900 (I) | 57.3 | 51816.4
Hydra A | 0.0548 | 09h18m05.7s | -12d05m44s | 4969,4970(S) | 98.82 | 214993.0
A85 | 0.0551 | 00h41m45.84s | -09d22m44.4s | 904,15173,15174 | 195.2 | 45290.9
|  |  |  | 16263,16264 (I) |  |
A2626 | 0.0553 | 23h36m30.48s | +21d08m45.6s | 3192,16136(S) | 135.6 | 11042.6
A133 | 0.0566 | 01h02m56.4s | -22d08m27.6s | 2203(S) | 154.3 | 34627.3
|  |  |  | 9897,13518(I) |  |
SERSIC159-03 | 0.0580 | 23h13m58.32s | -42d43m33.6s | 1668(S),11758(I) | 107.7 | 16595.1
A1991 | 0.0587 | 14h54m31.44s | +18d38m31.2s | 3193(S) | 38.3 | 28575.4
A3112 | 0.0753 | 03h17m57.6s | -44d14m16.8s | 2216,2516(S),13135(I) | 66.4 | 10847.0
A2029 | 0.0773 | 15h10m56.16s | +05d44m38.4s | 891,4977(S) | 126.9 | 167014.4
|  |  |  | 6101,10434,10435 |  |
|  |  |  | 10436,10437(I) |  |
A2597 | 0.0852 | 23h25m19.68s | -12d07m26.4s | 922,6934,7329(S) | 151.64 | 79071.1
A3921 | 0.0928 | 22h50m02.88s | -64d21m54s | 4973(I) | 29.4 | 3211.7
A2244 | 0.0968 | 17h02m42.72s | +34d03m36s | 4179(S) | 57.0 | 21639.1
RXCJ1558.3-1410 | 0.0970 | 15h58m21.84s | -14d09m57.6s | 9402(S) | 40.1 | 18550.6
PKS0745-19 | 0.1028 | 07h47m31.2s | -19d17m38.4s | 12881(S) | 118.1 | 175872.9
RXCJ1524.2-3154 | 0.1028 | 15h24m12.72s | -31d54m25.2s | 9401(S) | 40.9 | 33188.5
RXCJ0352.9+1941 | 0.1090 | 03h52m58.8s | +19d40m58.8s | 10466(S) | 27.2 | 11261.1
A1664 | 0.1283 | 13h03m42.48s | -24d14m42s | 1648,7901,17172 | 110.0 | 11873.3
|  |  |  | 17173,17557,17568(S) |  |
A2204 | 0.1522 | 16h32m47.04s | +05d34m33.6s | 499(S) | 96.82 | 74069.5
|  |  |  | 6104,7940(I) |  |
A907 | 0.1527 | 09h58m21.36s | -11d03m39.6s | 535,3185,3205(I) | 106.1 | 8276.3
HerculesA | 0.1550 | 16h51m08.16s | +04d59m34.8s | 5796,6257(S) | 97.1 | 5347.0
RXJ2014.8 | 0.1612 | 20h14m51.6s | -24d30m21.6s | 11757(S) | 19.91 | 16628.1
A1204 | 0.1706 | 11h13m18s | +17d36m10.8s | 2205(I) | 23.6 | 7863.1
Zw2701 | 0.2140 | 09h52m48.96s | +51d53m06s | 12903(S) | 95.8 | 11649.9
RXCJ1504-0248 | 0.2153 | 15h04m08.4s | -02d48m25.2s | 4935,5793,17197 | 150.0 | 13281.1
|  |  |  | 17669,17670(I) |  |
RXCJ1459.4-1811 | 0.2357 | 14h59m28.8s | -18d10m44.4s | 9428(S) | 39.6 | 9297.7
4C+55.16 | 0.2411 | 08h34m54.96s | +55d34m22.8s | 4940(S) | 96.0 | 16780.3
CL2089 | 0.2492 | 09h00m36.96s | +20d53m42s | 10463(S) | 40.6 | 9861.6
RXJ2129.6+0005 | 0.2499 | 21h29m40.08s | +00d05m24s | 552,9370(I) | 40.0 | 3797.8
A1835 | 0.2532 | 14h01m01.92s | +02d52m40.8s | 6880,6881,7370(I) | 194.0 | 44128.0
RXCJ1023.8-2715 | 0.2533 | 10h23m50.16s | -27d15m21.6s | 9400(S) | 36.7 | 10127.0
CL0348 | 0.2537 | 01h06m49.2s | +01d03m21.6s | 10465(S) | 48.9 | 10354.3
MS1455.0+2232 | 0.2578 | 14h57m14.4s | +22d20m38.4s | 543,4192(I) | 101.7 | 17724.3
ZW3146 | 0.2906 | 10h23m39.6s | +04d11m09.6s | 909,9371(I) | 86.2 | 10715.8

##  5 Turbulent velocity dispersion estimate

Figure 1: Top panel: Turbulent velocity values measured in the warm or cold
phase by Gaspari et al. (2018, blue points) and Olivares et al. (2019, orange
points). The reference dotted line $\sigma\propto T^{1/3}$ corresponds to
Equation 9, with the cyan-shaded area showing the 1 $\sigma$ uncertainty. The
solid lined corresponds to the constant average value given by Equation 8,
with the orange-shaded area showing its 1 $\sigma$ uncertainty. Bottom panel:
the histogram distribution of the clusters with $\sigma_{v}$ measurement
(blue) and of the complementary subsample (green).

We collected all the measurements available (in the literature) of the
turbulent velocity at a particular injection (5-10 kpc) of the warm or cold
phase for our selected clusters, converting the line-of-sight measurement into
a three-dimensional (3D) value, assuming isotropic turbulence, as discussed in
Section 3.3. Here, we assume that the turbulent velocity dispersion of the ICM
is very close to the value measured for the cold and warm medium. This
correlation is shown to have a slope and a normalization close to unity, with
a scatter of $\sim$30% (see Fig 1 in Gaspari et al. 2018). We retrieved
average ensemble measurements of $\sigma_{v}$ for 14 clusters from Gaspari et
al. (2018) that overlap with our sample. From Olivares et al. (2019), we
collected two other new measurements. We note that Olivares et al. (2019)
provided other measurements in clusters overlapping with Gaspari et al.
(2018); however, they are tied to pencil-beam values, while ensemble values,
such as those listed in Table 1 of Gaspari et al. (2018), should be used here,
since they better trace the dynamical and thermodynamical properties of the
turbulent medium, which we indicate as ”macro weather.”

Therefore, 40% of our sample has values of 3D $\sigma_{v}$ and associated
uncertainties (directly derived from the observed line-of-sight $\sigma_{v}$)
that can be directly inserted in Equation 7 for the corresponding cluster. In
Figure 1 (left panel), we show the 16 clusters in the $\sigma_{v}-kT$ plane.
We note a large intrinsic scatter with no clear trend with the temperature.
Such a large scatter likely dilutes any weak dependence on the (core-excised)
temperature, given the current low statistics. The measured values are
scattered in the range $200<\sigma_{v}<400$ km/s, with the average value and
rms given by:

$\langle\sigma_{v}\rangle=306\pm 78\,\,{\tt km/s}\,.$ (8)

The constant value $\langle\sigma_{v}\rangle$ and its uncertainty are shown as
an orange shaded area in Figure 1 (left panel). The question is then whether
the turbulent velocity in the clusters without direct measurements can be
assumed to be within this range. As a first step, we compared the temperature
distribution of the 18 clusters with measured $\sigma_{v}$ to that of the 25
clusters without $\sigma_{v}$. The two distributions are shown in Figure 1
(right panel), where it is possible to see that the clusters without
$\sigma_{v}$ measurement are hotter (hence, more massive) than those with
$\sigma_{v}$. Therefore, assuming a constant values for $\sigma_{v}$, despite
the large uncertainty, may not fully track the dependence on the temperature-
mass scale.

The dependence of $\sigma_{v,L}$ on the mass scale has been investigated only
in a few works. Gaspari et al. (2018) find $\sigma_{v}\propto M^{0.2}\propto
T_{\rm x}^{0.3}$ in the larger sample of 72 groups and clusters, by leveraging
ensemble-beam optical spectra, which are shown to linearly trace hot-gas
turbulent velocities within a 0.13 dex scatter. By using a completely
different method and band based on X-ray brightness and density fluctuations
in 33 groups and clusters, Hofmann et al. (2016) retrieved a slightly negative
slope in the hot-gas turbulent Mach numbers (defined as $\sigma_{v}/c_{\rm
s}$, with the sound speed being $c_{\rm s}\propto T_{\rm x}^{1/2}$),
translating again to $\sigma_{v}\propto T_{\rm x}^{0.3}$. Overall, if we fit
the normalization of a relation with a slope of $0.3$ to the available data,
we find the following relation

$\sigma_{v}=(315\pm 86)\left(\frac{kT}{4.85\,\rm keV}\right)^{0.3}\,{\tt
km/s}\,,$ (9)

shown as the light-blue shaded area in Figure 1. Given the current data, we
cannot estimate the 3D $\sigma_{v}$ in massive clusters with better
approximation than that discussed here. To compute $t_{\rm eddy}(r)$ in the
clusters without $\sigma_{v}$ measurement, we preferred to use the
$\sigma_{v}-kT$ relation given by Equation 9; from a theoretical and
simulation perspective, we do expect self-regulated AGN feedback (hence,
turbulence) scaling at some level with halo mass from poor to rich clusters.
Nevertheless, we also discuss the results obtained with both Equations 8 and 9
a posteriori to keep the systematics under control, finding that differences
due to the different scaling relations are negligible when compared with the
statistical uncertainties.

##  6 Data reduction and analysis

The data reduction was performed using the CIAO software (version 4.12) with
CALDB 4.9. We appllied a charge transfer inefficiency correction, time-
dependent gain adjustment, grade correction, and pixel randomization. First of
all, each observation was reprocessed using the chandra$\\_$repro function.
The script reads data from the standard data distribution and creates a new
bad pixel file, a new level=2 event file, and a new level=2 type PHA file for
each selected region, with the appropriate response files. We removed high-
energy background flares from the event files with the deflare command. In
addition, the background is reduced with the VFAINT cleaning whenever
possible. The final exposure times are typically lower than the nominal
exposure time only by a few percent. The level 2 files obtained in this way
are reprojected to match the coordinates of the observation with the longest
exposure for each clusters. The merged files are used only for imaging
analysis, while spectra are extracted from each single Obsid.

The annular regions that were used to extract the spectra are centered on the
emission peak. First, we identified the peak of the surface brightness profile
on the total band image smoothed on a 3 arcsec scale and then the center of
circle whose photometry maximize the S/N. Usually the two positions differ by
few arcsec, given the regular shape of the selected clusters. If the
difference is less than 2 arcsec, we used the position of the peak to fit the
surface brightness; otherwise, we considered the difference of the two
centroids to be significant and then we adopted the maximization of the S/N as
a more robust estimate of the cluster center.

We adaptively chose the width of each annulus to guarantee at least 3000 net
counts in the 0.5-7 keV band. Thanks to the exquisite angular resolution of
Chandra, we do not need to correct any effects caused by the point spread
function (PSF) when analyzing the spectra, since the PSF size is much smaller
than the bin width. Before creating the spectra, we manually remove all the
unresolved sources (mostly foreground and background AGN) visible in the soft,
hard and total-band images. For each region, we produce response matrix (RMF)
and ancillary response (ARF) files from each Obsid using CIAO. In this way, we
keep track of all the differences in the ACIS effective area among different
Obsid and, of course, among the different type of ACIS detectors.

Since each galaxy cluster has a different size, we need to define a normalized
radius to express our results and compare different clusters. In this work,
the radius $r_{500}$, defined as the radius enclosing an average total mass
density 500 times larger than the critical density at the cluster redshift, is
estimated for each cluster with the relation provided by Vikhlinin et al.
(2006):

$r_{500}=\frac{0.792}{hE(z)}\Bigg{(}\frac{T_{\rm x}}{\rm
5\,keV}\Bigg{)}^{0.527}\,\rm Mpc\,,$ (10)

where the cosmological evolution factor is
$E(z)=\sqrt{\Omega(1+z)^{3}+\Omega_{\Lambda}}$ and $h\equiv H_{0}/(100{\rm
kms}^{-1}{\rm Mpc}^{-1})$. Here $T_{\rm x}$ is the global X-ray temperature,
estimated using the emission-weighted average of the temperature measured at
$r>50$ kpc with a single temperature apec model, which includes thermal
bremsstrahlung and line emission. This value, obtained from the core-excised
total emission, is considered to be a robust proxy of the more formally-
defined virial temperature, since it is not affected by the prominent emission
in the core where the temperature may decrease significantly.

We divided the inner $\sim$ 400 kpc into annuli with about 3000 net counts
each in the total (0.5-7 keV) band. The spectra were analyzed with XSPEC
v12.8.2 (Arnaud 1996). To model the X-ray emission in each ring, we used a
single-temperature apec model where the ratio between the elements refers to
the solar elemental abundances as in Asplund et al. (2009). Galactic
absorption is modeled with tbabs (Wilms et al. 2000), where the Galactic
column density, $n_{\rm H}$, at the cluster position is initially set as
$n_{\rm H,tot}$ from Willingale et al. (2013). The temperature, metal
abundance, and normalization are left as free parameters. The projected
temperature and metallicity profiles are given by the spectral fit. The
normalization values of the spectra, $K,$ are linked to the 3D density profile
through the relation:

$K=\frac{10^{-14}}{4\pi D^{2}_{A}(1+z)^{2}}\int n_{e}n_{H}dV\,,$ (11)

where $D_{A}$ is the angular diameter distance, $n_{e}$ and $n_{H}$ are the 3D
density profiles of electrons and hydrogen atoms, respectively, and the volume
integral is performed on the projected annulus and along the line of sight.
Assuming spherical symmetry, it is possible to obtain the deprojected electron
density averaged in the spherical shell corresponding to the projected radius
as:

$n_{e}^{2}(r)=-\frac{4\times
10^{14}}{0.82\pi}{D^{2}_{A}(1+z)^{2}}\int_{r}^{\infty}\frac{\rm d}{{\rm
d}s}\Big{(}\frac{K(s)}{s\Delta s}\Big{)}\frac{{\rm
d}s}{\sqrt{s^{2}-r^{2}}}\,.$ (12)

where $r$ and $s$ are the physical and projected radii, respectively. The full
derivation of Equation 12 is shown in Appendix A. Here, we assumed that the
ratio of hydrogen nuclei to electron density is 0.82, which is appropriate for
a fully ionized plasma (Ettori et al. 2002). In order to avoid spurious noise
amplification, we applied Equation 12 to the analytical fit to the function
$\sqrt{K(s)/s\Delta s}$,where $\Delta s$ is the bin width. The fitting
function is assumed (for simplicity) to be the single or double $\beta$ model
that is used for the fit of the deprojected density profile, where a single
$\beta$ model component is expressed as:

$n_{e}(r)=n_{0}\left[1+\left(\frac{r}{r_{c}}\right)^{2}\right]^{-\frac{3}{2}\beta}\,,$
(13)

where $n_{0}$ is the value of the central density. Cool-core clusters show a
pronounced peak in the center, with a plateau typically at $r<2-3\times
10^{-2}r_{500}$. On the other hand, non-cool-core clusters have a profile that
flattens around $r\sim 0.1\times r_{500}$, and therefore the values of the
central density are significantly lower. The uncertainty in the density
profiles, corresponding to a formal 1-$\sigma$ confidence level, is obtained
by computing a large $\sim 10^{4}$ set of profiles corresponding to a Monte
Carlo sampling of the best-fit parameters given their statistical uncertainty.

The 3D temperature profiles for relaxed galaxy clusters can be modeled with an
analytical function obtained as the product of two different regimes,
corresponding to the core and the outer region, with opposite slopes (see
Vikhlinin et al. 2005, 2006):

$T_{\rm 3D}(r)=T_{0}\,t_{\rm inner}(r)\,t_{\rm outer}(r)\,,$ (14)

where the function $t_{\rm outer}(r)$ describe a gentle decrease at large
radii of the form:

$t_{\rm outer}(r)=\frac{({r}/{r_{t}})^{-a}}{[1+({r}/{r_{t}})^{b}]^{c/b}}\,,$
(15)

with $a,b,$ and $c$ defined positive. The central part of the temperature
profile instead, requires a function $t_{\rm inner}(r),$ which is
parameterized as follows:

$t_{\rm inner}(r)=(x+T_{\rm min}/T_{0})/(x+1),\ \ x=(r/r_{c})^{a_{\rm
inner}}\,,$ (16)

with $a_{\rm inner}$ defined positive. We note that the temperature profile is
described by eight free parameters, which we deem are too many to be
meaningfully constrained by our profiles. However, here we are interested
mostly in the temperature profiles itself, while we are not directly using the
parameters $r_{t}$ and $r_{c}$ since they are strongly degenerate with the
other parameters.

To derive the deprojected temperature profile $T_{\rm 3D}$, we fit the
projected temperature profile with Equation 14 and then deprojected the best-
fit profile numerically through a standard onion-peeling technique (see Ettori
et al. 2002, for details), where the temperature in shells is recovered by
correcting the spectral estimate with the emission observed in each ring along
the line of sight, assuming a spherically symmetric ICM distribution. We
assume that this step does not introduce additional errors in the deprojected
temperature profiles, so the the uncertainty is equal to that of the projected
profile. Clearly, this is correct if we assume that the best-fit analytical
profile provides an accurate description, and avoids the typical noise
amplification that we would obtain with a straightforward deprojection of the
measured projected temperature. We argue that our uncertainty on the
temperature is only slightly underestimated since we have selected
preferentially spherically symmetric clusters, for which we do expect a smooth
projected profile.

The mass profile is then obtained from Equation 6. The temperature, density,
and total mass 3D profiles have been already obtained for several clusters in
the literature, and therefore are not presented nor discussed here. We compare
our profiles with those in the literature when possible, and find reasonable
agreement with some residual uncertainty of the order of 10-20% in a few
cases, possibly due to the different calibration, different set of exposures
used, and different binning. With respect to a simple review of the
literature, our profiles have the advantage to be consistently computed with
the same assumptions and calibration. With our deprojected 3D electron density
profile, temperature profile, and total mass profile, we are now able to
derive the timescale profiles defined in Section 3.

Figure 2: Profiles of $t_{\rm eddy}$, $t_{\rm cool}$ and $25\times t_{\rm ff}$
as a function of the physical radius for RXCJ1504 (top panel). Distribution of
intersection radii defined by the condition $t_{\rm cool}(R_{\rm cc})=t_{\rm
eddy}(R_{\rm cc})$, obtained by sampling random profiles extracted according
to the uncertainty on the best-fit profile (tottom panel). The red dashed line
shows the Gaussian fit to the distribution used to derive the central value
for $R_{\rm ccc}$ and its 1$\sigma$ uncertainty.

##  7 Results

The profiles of the timescales, $t_{\rm cool}$, $t_{\rm eddy}$, and $t_{\rm
ff}$ were obtained by plugging the temperature, density, and mass 3D profiles
into Equation 4, 5, and 7. The timescale profiles are shown in Appendix B for
all the clusters in our sample. We note that $t_{\rm ff}$ profiles can be
fairly approximated with a power law, while $t_{\rm cool}$ profiles show more
significant changes in slope. We remark that $t_{\rm cool}$ is very steep in
the presence of a strong cool core (with temperature dropping and density
increasing toward the center), while it flattens when the core is less
prominent. In some cases, $t_{\rm cool}$ has a slight increase toward the
center. We notice that in some cases this is associated with the presence of a
central AGN and we conclude that this effect may be uniquely due to the
combination of AGN wings that have not been properly removed and of the size
of the spatial bin that tend to smooth the central density peak. Since this
does not affect our measurements of $R_{\rm ccc}$ and $R_{\rm qcf}$, we do not
further comment on the shape of $t_{\rm cool}$ and $t_{\rm ff}$ with regard to
the initial few spatial bins.

We also note that in a few systems (A1991, PKS0745, RXCJ1504, CL2089, and
A1835) we see TI ratios that drop significantly below 10, usually near values
of 7-8. This appear to be in contradiction with previous works, where profiles
have never observed below the threshold $t_{\rm cool}/t_{\rm ff}=10$ (see
Figure 7 of Hogan et al. 2017). While the majority of clusters show a minimum
value of the TI-ratio in the 10-40 range, we verify that in all the cases
where the profile drops below 10 our results are robust and can be ascribed to
our updated analysis and calibration methods. Indeed, we recall that
hydrodynamical simulations of self-regulated feeding and feedback in clusters
do show the TI ratio falling to 5-10 during major CCA triggering periods
(e.g., see Figure 3 in Gaspari et al. 2012b). We argue that as samples expand
and become deeper, we expect to identify several more systems to fill such
regions where the TI-ratio falls shortly below 10.

Once we fitted the timescale profiles with Polynomial fitting, we compared the
best-fit function for $t_{\rm cool}$ with that for $t_{\rm eddy}$ and $t_{\rm
ff}$. We estimate $R_{\rm ccc}$ and $R_{\rm qcf}$ at the radii where $t_{\rm
cool}$ crosses $t_{\rm eddy}$ and $25\,t_{\rm ff}$, respectively (see Section
2). The best-fit values of $R_{\rm ccc}$ and $R_{\rm qcf}$ and the
corresponding errors are computed as follows. We extracted randomly $1000$
profiles according to the statistical uncertainties on the best-fit parameters
of each profile. For each cluster, we obtained a distribution of intersection
radii $R_{\rm ccc}$ and $R_{\rm qcf}$. Then, we fit the distribution of
$R_{\rm ccc}$ and $R_{\rm qcf}$ with a Gaussian, deriving the best-fit value
(the center of the Gaussian) and the 1-$\sigma$ uncertainty. An example is
given in Figure 2 for the measurement of $R_{\rm ccc}$ in RXCJ1504.
Incidentally, we note that some $t_{\rm cool}$ profiles significantly change
slopes (see, e.g., 2A0335, Hydra, RXCJ1558, and PKS0745), so that, in
principle, we have two intersections. However, the one close to the center is
typically associated with a region of the profile with larger uncertainties.
When looking at the distribution of intersection radii obtained by considering
all the intersections between the randomized time-scale profiles, we fit only
the most prominent peak, so that we naturally discarded these few potentially
ambiguous cases where a secondary peak is present. We visually checked that
there were no cases where two peaks of comparable significance are present in
the distribution of intersection radii.

In some other cases, the best fit profiles do not overlap666For four systems
with $t_{\rm eddy}$ very close to $t_{\rm cool}$, we use as effective crossing
the C-ratio including the intrinsic scatter band, $C$ $\sim$ 0.5 - 2.0 (e.g.
Maccagni et al. 2021).. Nevertheless, the same procedure allows us to obtain
an upper limit at a given confidence level. We set the 1-$\sigma$ upper limit
to $R_{\rm ccc}$ and $R_{\rm qcf}$, measuring the value below which we find
the 84% of the of the measured values. The best fit values for $R_{\rm ccc}$
and $R_{\rm qcf}$, together with the core-excised, emission weighted
temperature (for $r>40$ kpc) and total mass with $r_{500}$, are listed in
Table 2.

Table 2: Results of our analysis for all the clusters in our sample. Column 1: Cluster name; column 2: Value of $r_{500}$; column 3: Core-excised temperature (for $r>40$ kpc and $r<r_{500}$); column 4: Total mass within R500; column 5: Value of $\sigma_{v}$ calculated from Equation 9; column 6: Cool-core condensation radius defined by the condition $t_{\rm cool}(R_{\rm ccc})/t_{\rm eddy}(R_{\rm ccc})=1$ ; column 7: Quenched cooling flow radius defined by the condition $t_{\rm cool}(R_{\rm qcf})/[25\,t_{\rm ff}(R_{\rm qcf})]=1$.; columns 8 and 9: $R_{\rm ccc}$ and $R_{\rm qcf}$ in units of $10^{-2}\times r_{500}$; column 10: Reference of $\sigma_{v}$ measurements to each cluster. where (a) is Gaspari et al. (2018), (b) is Olivares et al. (2019), and (c) is this work using Equation 9. Cluster | $r_{500}$ | T | $M_{500}$ | $\sigma_{v}$ | $R_{\rm ccc}$ | $R_{\rm qcf}$ | $R_{\rm ccc}$ | $R_{\rm qcf}$ | ref
---|---|---|---|---|---|---|---|---|---
| (kpc) | (keV) | $(10^{14}M_{\odot})$ | (km/s) | (kpc) | (kpc) | $(10^{-2}\,r_{500})$ | $(10^{-2}\,r_{500})$ |
A2199 | 1087 | $4.75_{-0.27}^{+0.28}$ | $3.8_{-0.3}^{+0.3}$ | $216.5_{-6.9}^{+6.9}$ | $3.0_{-0.0}^{+0.4}$ | $23.0_{-1.3}^{+1.4}$ | $0.3_{-0.0}^{+0.0}$ | $2.1_{-0.1}^{+0.1}$ | (a)
A496 | 1218 | $5.90_{-0.53}^{+0.62}$ | $5.4_{-0.7}^{+0.8}$ | $326.6_{-89.2}^{+89.2}$ | $\leq 10.0$ | $23.6_{-0.6}^{+0.8}$ | $\leq 0.8$ | $1.9_{-0.1}^{+0.1}$ | (c)
2A0335 | 1042 | $4.41_{-0.27}^{+0.29}$ | $3.4_{-0.3}^{+0.3}$ | $316.0_{-6.9}^{+6.9}$ | $26.8_{-0.8}^{+0.7}$ | $48.4_{-0.8}^{+0.8}$ | $2.6_{-0.1}^{+0.1}$ | $4.6_{-0.1}^{+0.1}$ | (b)
A2589 | 997 | $4.07_{-0.30}^{+0.42}$ | $3.0_{-0.3}^{+0.5}$ | $297.1_{-81.1}^{+81.1}$ | $\leq 15$ | $\leq 15$ | $\leq 1.5$ | $\leq 1.5$ | (c)
MKW3S | 896 | $3.33_{-0.20}^{+0.22}$ | $2.2_{-0.2}^{+0.2}$ | $284.8_{-77.7}^{+77.7}$ | $\leq 10$ | $\leq 10$ | $\leq 1.1$ | $\leq 1.1$ | (c)
Hydra | 955 | $3.79_{-0.18}^{+0.18}$ | $2.6_{-0.2}^{+0.2}$ | $382.8_{-5.2}^{+5.2}$ | $11.0$ | $43.8_{-0.9}^{+0.8}$ | $\leq 1.2$ | $4.6_{-0.1}^{+0.1}$ | (a)
A85 | 1124 | $5.16_{-0.22}^{+0.24}$ | $4.3_{-0.3}^{+0.3}$ | $258.1_{-6.9}^{+6.9}$ | $7.1_{-0.9}^{+0.9}$ | $28.1_{-2.3}^{+1.9}$ | $0.6_{-0.1}^{+0.1}$ | $2.5_{-0.2}^{+0.2}$ | (a)
A2626 | 875 | $3.21_{-0.17}^{+0.17}$ | $2.0_{-0.2}^{+0.2}$ | $277.7_{-75.8}^{+75.8}$ | $\leq 10$ | $\leq 20$ | $\leq 1.1$ | $\leq 2.3$ | (c)
A133 | 1014 | $4.25_{-0.23}^{+0.26}$ | $3.2_{-0.3}^{+0.3}$ | $211.3_{-3.5}^{+3.5}$ | $10.0_{-2.4}^{+1.2}$ | $32.5_{-1.5}^{+1.3}$ | $1.0_{-0.2}^{+0.1}$ | $3.2_{-0.1}^{+0.1}$ | (a)
SERSIC159 | 788 | $2.64_{-0.12}^{+0.13}$ | $1.5_{-0.1}^{+0.1}$ | $277.1_{-3.5}^{+3.5}$ | $14.5_{-1.1}^{+1.5}$ | $74.4_{-2.3}^{+2.9}$ | $1.8_{-0.1}^{+0.2}$ | $9.4_{-0.3}^{+0.4}$ | (a)
A1991 | 753 | $2.43_{-0.14}^{+0.13}$ | $1.3_{-0.1}^{+0.1}$ | $188.8_{-6.9}^{+6.9}$ | $31.0_{-1.1}^{+1.2}$ | $38.8_{-1.1}^{+1.2}$ | $4.1_{-0.2}^{+0.2}$ | $5.1_{-0.1}^{+0.2}$ | (a)
A3112 | 1060 | $4.70_{-0.33}^{+0.37}$ | $3.7_{-0.4}^{+0.4}$ | $446.9_{-5.2}^{+5.2}$ | $5.0_{-0.0}^{+0.0}$ | $32.7_{-3.2}^{+3.1}$ | $0.5_{-0.0}^{+0.0}$ | $3.1_{-0.3}^{+0.3}$ | (a)
A2029 | 1441 | $8.42_{-0.39}^{+0.41}$ | $9.2_{-0.6}^{+0.7}$ | $365.9_{-99.9}^{+99.9}$ | $5.0_{-0.0}^{+0.0}$ | $23.4_{-1.6}^{+1.5}$ | $0.3_{-0.0}^{+0.0}$ | $1.6_{-0.1}^{+0.1}$ | (c)
A2597 | 948 | $3.83_{-0.19}^{+0.20}$ | $2.6_{-0.2}^{+0.2}$ | $417.4_{-1.7}^{+1.7}$ | $10.6_{-1.3}^{+1.1}$ | $64.2_{-1.6}^{+1.5}$ | $1.1_{-0.1}^{+0.1}$ | $6.8_{-0.2}^{+0.2}$ | (a)
A3921 | 1134 | $5.42_{-0.41}^{+0.48}$ | $4.6_{-0.5}^{+0.6}$ | $331.2_{-90.4}^{+90.4}$ | $\leq 30$ | $\leq 30$ | $\leq 2.6$ | $\leq 2.6$ | (c)
A2244 | 1149 | $5.57_{-0.34}^{+0.42}$ | $4.7_{-0.4}^{+0.5}$ | $329.6_{-90.0}^{+90.0}$ | $\leq 10$ | $\leq 100$ | $\leq 0.9$ | $\leq 8.7$ | (c)
RXCJ1558.3 | 1088 | $5.02_{-0.28}^{+0.33}$ | $4.0_{-0.3}^{+0.4}$ | $239.0_{-5.2}^{+5.2}$ | $\leq 30.0$ | $62.0_{-2.7}^{+2.6}$ | $\leq 2.8$ | $5.7_{-0.2}^{+0.2}$ | (a)
PKS0745 | 1742 | $12.34_{-1.07}^{+1.30}$ | $16.6_{-2.2}^{+2.6}$ | $339.0_{-5.2}^{+5.2}$ | $12.5_{-1.0}^{+1.1}$ | $55.2_{-0.7}^{+0.7}$ | $0.7_{-0.1}^{+0.1}$ | $3.2_{-0.0}^{+0.0}$ | (a)
RXCJ1524.2 | 1058 | $4.79_{-0.28}^{+0.31}$ | $3.7_{-0.3}^{+0.4}$ | $332.5_{-5.2}^{+5.2}$ | $18.4_{-2.6}^{+1.9}$ | $42.9_{-1.2}^{+1.3}$ | $1.7_{-0.2}^{+0.2}$ | $4.1_{-0.1}^{+0.1}$ | (b)
RXCJ0352.9 | 825 | $3.00_{-0.14}^{+0.15}$ | $1.8_{-0.1}^{+0.1}$ | $329.1_{-5.2}^{+5.2}$ | $39.5_{-2.0}^{+1.8}$ | $58.1_{-1.5}^{+1.9}$ | $4.8_{-0.2}^{+0.2}$ | $7.0_{-0.2}^{+0.2}$ | (a)
A1664 | 877 | $3.43_{-0.16}^{+0.16}$ | $2.2_{-0.2}^{+0.2}$ | $279.0_{-76.2}^{+76.2}$ | $31.0_{-4.3}^{+4.0}$ | $75.1_{-2.1}^{+2.6}$ | $3.5_{-0.5}^{+0.5}$ | $8.6_{-0.2}^{+0.3}$ | (c)
A2204 | 1531 | $10.09_{-0.93}^{+1.18}$ | $11.8_{-1.6}^{+2.1}$ | $382.1_{-104.3}^{+104.3}$ | $29.6_{-0.9}^{+0.9}$ | $65.7_{-1.3}^{+1.5}$ | $1.9_{-0.1}^{+0.1}$ | $4.3_{-0.1}^{+0.1}$ | (c)
A907 | 1076 | $5.17_{-0.33}^{+0.38}$ | $4.1_{-0.4}^{+0.4}$ | $323.0_{-88.2}^{+88.2}$ | $10.0_{-0.8}^{+0.0}$ | $36.5_{-6.6}^{+5.8}$ | $0.9_{-0.1}^{+0.0}$ | $3.4_{-0.6}^{+0.5}$ | (c)
HerculesA | 998 | $4.49_{-0.27}^{+0.30}$ | $3.3_{-0.3}^{+0.3}$ | $293.8_{-80.2}^{+80.2}$ | $9.6_{-3.6}^{+2.4}$ | $43.1_{-7.9}^{+7.7}$ | $1.0_{-0.4}^{+0.2}$ | $4.3_{-0.8}^{+0.8}$ | (c)
RXJ2014.8 | 1250 | $6.92_{-0.45}^{+0.67}$ | $6.5_{-0.6}^{+0.9}$ | $356.8_{-5.2}^{+5.2}$ | $18.5_{-2.1}^{+2.0}$ | $58.6_{-2.9}^{+3.5}$ | $1.5_{-0.2}^{+0.2}$ | $4.7_{-0.2}^{+0.3}$ | (a)
A1204 | 846 | $3.33_{-0.18}^{+0.17}$ | $2.0_{-0.2}^{+0.2}$ | $382.8_{-5.2}^{+5.2}$ | $24.4_{-3.3}^{+5.6}$ | $99.0_{-6.6}^{+7.6}$ | $2.9_{-0.4}^{+0.7}$ | $11.7_{-0.8}^{+0.9}$ | (a)
Zw2701 | 1020 | $4.94_{-0.25}^{+0.25}$ | $3.7_{-0.3}^{+0.3}$ | $312.3_{-85.3}^{+85.3}$ | $\leq 10.0$ | $73.3_{-4.1}^{+5.2}$ | $\leq 1.0$ | $7.2_{-0.4}^{+0.5}$ | (c)
RXCJ1504 | 1191 | $6.63_{-0.50}^{+0.56}$ | $5.9_{-0.7}^{+0.7}$ | $346.3_{-94.5}^{+94.5}$ | $46.1_{-2.6}^{+2.6}$ | $103.3_{-4.7}^{+4.8}$ | $3.9_{-0.2}^{+0.2}$ | $8.7_{-0.4}^{+0.4}$ | (c)
RXCJ1459.4 | 1031 | $5.15_{-0.28}^{+0.32}$ | $3.9_{-0.3}^{+0.4}$ | $307.3_{-83.9}^{+83.9}$ | $40.0_{-4.8}^{+1.4}$ | $84.0_{-2.6}^{+3.1}$ | $3.9_{-0.5}^{+0.1}$ | $8.1_{-0.2}^{+0.3}$ | (c)
4C+55.16 | 997 | $4.85_{-0.25}^{+0.26}$ | $3.6_{-0.3}^{+0.3}$ | $306.5_{-83.7}^{+83.7}$ | $14.9_{-2.2}^{+2.8}$ | $73.8_{-3.6}^{+4.4}$ | $1.5_{-0.2}^{+0.3}$ | $7.4_{-0.4}^{+0.4}$ | (c)
CL2089 | 881 | $3.86_{-0.19}^{+0.19}$ | $2.5_{-0.2}^{+0.2}$ | $289.3_{-79.0}^{+79.0}$ | $45.0_{-2.5}^{+2.9}$ | $77.9_{-2.4}^{+3.0}$ | $5.1_{-0.3}^{+0.3}$ | $8.8_{-0.3}^{+0.3}$ | (c)
RXJ2129.6 | 1148 | $6.39_{-0.46}^{+0.52}$ | $5.5_{-0.6}^{+0.7}$ | $197.4_{-5.2}^{+5.2}$ | $21.7_{-5.2}^{+10.7}$ | $95.2_{-8.8}^{+12.3}$ | $1.9_{-0.5}^{+0.9}$ | $8.3_{-0.8}^{+1.1}$ | (a)
A1835 | 1274 | $7.81_{-0.57}^{+0.68}$ | $7.5_{-0.8}^{+1.0}$ | $358.3_{-97.8}^{+97.8}$ | $37.5_{-1.7}^{+1.6}$ | $87.8_{-1.8}^{+2.3}$ | $2.9_{-0.1}^{+0.1}$ | $6.9_{-0.1}^{+0.2}$ | (c)
RXCJ1023.8 | 998 | $4.91_{-0.32}^{+0.37}$ | $3.6_{-0.4}^{+0.4}$ | $316.9_{-86.5}^{+86.5}$ | $25.0_{-0.0}^{+2.2}$ | $109.5_{-5.5}^{+6.3}$ | $2.5_{-0.0}^{+0.2}$ | $11.0_{-0.6}^{+0.6}$ | (c)
CL0348 | 799 | $3.23_{-0.13}^{+0.15}$ | $1.9_{-0.1}^{+0.1}$ | $273.6_{-74.7}^{+74.7}$ | $53.2_{-2.9}^{+2.8}$ | $97.9_{-3.6}^{+4.9}$ | $6.7_{-0.4}^{+0.3}$ | $12.2_{-0.4}^{+0.6}$ | (c)
MS1455.0 | 1014 | $5.09_{-0.25}^{+0.27}$ | $3.8_{-0.3}^{+0.3}$ | $316.5_{-86.4}^{+86.4}$ | $32.5_{-5.1}^{+3.9}$ | $110.9_{-3.6}^{+3.5}$ | $3.2_{-0.5}^{+0.4}$ | $10.9_{-0.4}^{+0.3}$ | (c)
ZW3146 | 1187 | $7.09_{-0.54}^{+0.57}$ | $6.3_{-0.7}^{+0.8}$ | $345.0_{-94.2}^{+94.2}$ | $32.8_{-2.7}^{+2.6}$ | $123.8_{-6.2}^{+7.5}$ | $2.8_{-0.2}^{+0.2}$ | $10.4_{-0.5}^{+0.6}$ | (c)

The distributions of $R_{\rm ccc}$ and $R_{\rm qcf}$ in our sample are shown
in Figure 3. To obtain the distributions of $R_{\rm qcf}$ and $R_{\rm ccc}$,
first we resampled the $R_{\rm ccc}$ and $R_{\rm qcf}$ values of each cluster
1000 times by randomly varying the profiles of $t_{\rm cool}$, $t_{\rm eddy}$,
and $t_{\rm ff}$ according to their uncertainty. Eventually, we averaged the
number of values falling in each bin and obtained the final histogram
distributions of $R_{\rm qcf}$ and $R_{\rm ccc}$ that properly take into
account the uncertainties in the timescale profiles.

We find that the distribution of $R_{\rm ccc}$ is peaked at $\sim
0.01\,r_{500}$, and is entirely included within $\sim 0.07\,r_{500}$. We also
notice an apparent bimodality in the distribution of $R_{\rm qcf}$, which is
peaked at $\sim 0.04\,r_{500}$, and shows a second peak at $0.07\,r_{500}$. In
principle the bimodality in the distribution of $R_{\rm qcf}$ is expected to
be significant, since it is visible after the randomization of the $R_{\rm
qcf}$ values which would have erased the dip at $\sim 0.05\,r_{500}$ – as this
was only due do discreteness effects. However, as we show in the lower panel
of Figure 3, the bimodality disappear when the distribution is plotted as a
function of the physical radius. Therefore, we do not investigate further this
feature. In addition, we recall that, since our sample selection is biased
towards cool-core clusters, some features in the distribution of $R_{\rm ccc}$
and $R_{\rm qcf}$ in our sample should not be ascribed to general properties
of the cluster population. Finally, we note that the distribution of $R_{\rm
ccc}$ is shifted at lower values by a factor of $\sim 3$ with respect to the
distribution of $R_{\rm qcf}$.

We note that, historically, the classic cooling radius is often defined as the
radius, where $t_{\rm cool}=t_{\rm age}$, and $t_{\rm age}$ is a typical ”age”
of the object, roughly estimated as $t(z_{o})-t(z\sim 2)\sim 1/H_{0}$, $t$ is
the age of the universe at different redshifts $z$ and $z\sim 2$ is assumed to
mark the epoch of galaxy cluster formation. Another arbitrary definition of
cooling radius is obtained by directly comparing the cooling time to some
absolute reference timescale, for instance, 1 Gyr, or a specific value such as
7.7 Gyr (Hudson et al. 2010). These criteria allow one to derive a simple and
immediate order-of-magnitude estimate of the actual cooling radius, at
variance with our definition based on a local, physical criterion, independent
from the cosmic epoch. Therefore, we will refer to the ”classical” cool-core
definition adopting the fixed threshold $t_{\rm cool}<7.7$ Gyr as the radius
$R_{7.7}$, and compare it to the physically motivated values of $R_{\rm ccc}$
and $R_{\rm qcf}$. The ”classical” cool-core $R_{7.7}$ distribution is shifted
to larger values with respect to $R_{\rm ccc}$ and $R_{\rm qcf}$, distributed
in the broad range of $0.1-0.2\,r_{500}$, as shown in Figure 3. Despite the
wide distribution, $R_{7.7}$ is clearly disconnected from the physically
motivated values for $R_{\rm ccc}$ and $R_{\rm qcf}$. We argue that these
values describe more accurately the condensation and the intermittent cooling
flow-regions, as we further discuss in the next section (Section 8).

Figure 3: Histogram of the novel cool-core radii $R_{\rm ccc}$ and $R_{\rm
qcf}$, compared with the classical cool-core radius (within which $t_{\rm
cool}<7.7$ Gyr), as fractions of $r_{500}$ (top) and in kpc unit (bottom).

In Figure 4, we investigate in more details the relation between $R_{\rm qcf}$
and $R_{\rm ccc}$. We can directly verify that $R_{\rm qcf}>R_{\rm ccc}$ for
all the clusters, as expected, with a few cases for which $R_{\rm qcf}\sim
R_{\rm ccc}$ within the errors. With the bootstrap fitting method, and using
the relation:

${\rm log}\,R_{\rm qcf}=A\times{\rm log}\,R_{\rm ccc}+\gamma,$ (17)

we obtain $A=0.46^{+0.02}_{-0.03}$ and $\gamma=-0.41^{+0.04}_{-0.05}$. We
included also the upper limits in this analysis, shown with arrows in Figure
4. The correlation is statistically very significant, but also shows a
considerable intrinsic scatter in addition to the statistical noise. The
interesting result is that the relation is significantly different from a
linear relation, $R_{\rm qcf}\propto R_{\rm ccc}$. From this, we conclude that
larger quenched cool cores have a larger probability to host a larger cool-
core condensation region, implying more vigorous precipitation events. In
other words, if we assume that a large cool core ($R_{\rm qcf}\sim
0.1\,r_{500}$) is the signature of a cooling and feedback activity ongoing for
a long time, we may expect that the precipitation condition (and therefore the
condensation region) is extended to the entire cool-core region ($R_{\rm
ccc}\sim R_{\rm qcf}$), while in smaller cool cores, the condensation region
is progressively smaller (we discuss this further in Section 8).

Figure 4: Quenched cooling flow radius $R_{\rm qcf}$ plotted against the cool-
core condensation radius $R_{\rm ccc}$, including the upper limits in $R_{\rm
qcf}$ and $R_{\rm ccc}$ (green arrows). The solid line is the best-fit power
law relation (with 1-$\sigma$ uncertainty shown by the shaded area), while the
dashed line shows $R_{\rm qcf}\propto R_{\rm ccc}$ with arbitrary
normalization. The color of each point is color coded according to the
temperature of each cluster. Error bars correspond to 1-$\sigma$ confidence
level.

In Figure 5, we plot the normalized (left panel) and physical (right panel)
quenched cool-core radius, $R_{\rm qcf}$, and cool-core condenstation radius,
$R_{\rm ccc}$, versus the total mass. The plots show a substantial scatter
with no clear dependence on the mass over the range spanned by our sample,
except for a small hint of lower values at larger masses (relatively to
$r_{\rm 500}$, but not as absolute values). The large intrinsic scatter may
reflect the different history of each cluster, with no clear dependence on the
total mass (the integrated accretion history) but, rather, on the occurrence
of recent major mergers, which have the effect of erasing the cool core and
restarting the cooling process and the formation of a new cool core. The
smaller $R_{\rm ccc}$ and $R_{\rm qcf}$ values at $M_{500}\sim
10^{15}M_{\odot}$ are affected by small number statistics and probably by a
selection bias: smaller cool cores survive the criterion on the net counts
within 40 kpc because they are substantially brighter. On the other hand,
smaller clusters must have larger cool cores to pass the selection criteria.
Extending our sample to the lower mass, galaxy group will be important to
understand potential mass trends, which are nevertheless expected to be weak
(e.g., Gaspari et al. 2018).

Figure 5: Quenched cool core radius $R_{\rm qcf}$ (red) and cool-core
condensation radius $R_{\rm ccc}$ (blue) plotted versus the total mass
$M_{500}$. Error bars correspond to 1-$\sigma$ confidence level. In the top
(bottom) panel the radii are in units of $r_{500}$ (physical kpc). No dominant
trend is observed. Figure 6: Quenched cooling flow radius $R_{\rm qcf}$ (red)
and the cool-core condensation radius $R_{\rm ccc}$ (blue) plotted versus
redshift. The black dots show the size of the innermost bins, to visualize of
how well we are able to resolve the inner regions compared with the $R_{\rm
qcf}$ values.

The dependence of $R_{\rm ccc}$ and $R_{\rm qcf}$ on redshift is shown in
Figure 6. In both cases, there is an increasing trend with redshift. We do not
attempt to interpret this relation in terms of evolution of the cluster
population due to the limited size of our sample and our selection technique.

##  8 Discussion

Cool cores are one of the most important features of galaxy clusters. As
introduced in Section 1, they represent the heart of the ICM atmosphere that
fill the galaxy cluster potential well. They roughly demark the reservoir of
gas out of which the central SMBH can feed recursively over the cosmic
evolution (see the multiscale diagram in the review by Gaspari et al. 2020).
However, they are often arbitrarily defined, by comparing the cooling time
with ad-hoc timescales such as 1 or 7.7 Gyr (see Hudson et al. 2010 for a
detailed discussion). Previous studies also focused on visual definitions,
such as the peak in the temperature profile (e.g., Vikhlinin et al. 2006), or
classical pure or unheated cooling flow rates (e.g., McDonald et al. 2018).
Inspired by the results of Gaspari et al. (2018), we investigated here a more
physically-driven way to establish the ”cooling-flow” region, namely, where
the ICM plasma actually condenses, subsequently forming multiphase filaments
and clouds that rain onto the SMBHs via CCA (e.g., Tremblay et al. 2018; Rose
et al. 2019; Juráňová et al. 2019; Olivares et al. 2019, 2022; Temi et al.
2022). As a result of our investigation, we propose that the ratio of crucial
ICM feeding and feedback timescales provides a dimensionless and unbiased way
to assess such a cooling region, which can potentially differ from the above
definition, namely, that of the classical ad hoc ”cool core.” Here, we discuss
the implications of our main findings, namely: 1) definition and meaning of
the condensation cool-core radius $R_{ccc}$; 2) definition and meaning of the
quenched cooling flow radius $R_{qcf}$; 3) $R_{ccc}$-$R_{qcf}$ relation and
constraints on the AGN feeding-feedback duty cycle.

###  8.1 Definition and meaning of th condensation cool-core radius

We find that a more realistic definition for the roughly spherical region
where condensation and precipitation are strongly effective, is provided by
the condensation cool-core radius $R_{\rm ccc}$. Here, we find typical values
$r\sim 0.01<R_{\rm ccc}/r_{500}<0.05$, that is, about 5-10$\times$ smaller
than the above classical definitions of the cool core radius (see upper panel
of Figure 3). This has been achieved by leveraging the $C$-ratio crossing
around unity, which is a simple and clear physical threshold that
characterizes the triggering of the turbulent non-linear thermal instability
and related top-down multiphase condensation (e.g., Gaspari et al. 2017; Voit
2018; Olivares et al. 2022). In our sample of massive galaxy clusters, $C\sim
1$ yields a distribution of $R_{\rm ccc}$ peaked at $\sim$ 20 kpc (see lower
panel of Figure 3), which is comparable to the values found in the upper
envelope of the Gaspari et al. (2018) sample (including poor clusters groups
as well). It is important to note that such a novel cool-core condensation
radius is not an hypothetical region over which a massive pure cooling flow
would ensue, but the actual physical region set by the observed balance of
feeding and feedback processes. The scattered but rather flat $R_{\rm
ccc}-M_{500}$ relation (see Figure 5) suggests that more local properties
(such as the BCG mass) may drive the condensation core evolution. We defer
this issue to a subsequent study.

###  8.2 Definition and meaning of the quenched cooling flow radius

We further explored a complementary criterion $t_{\rm cool}/t_{\rm ff}<25$ to
trace the region over which a quenched cooling flow may potentially develop
from linear TI (Gaspari et al. 2012b; Sharma et al. 2012; Voit et al. 2015a).
Despite the phenomenological non-unitary threshold that is still hard to
understand in a comprehensive theoretical framework, the related quenched
cooling flow radius, $R_{\rm qcf}$, is another valuable indication of the
potentially condensing region out of the heated macro cooling flow. Although
it is affected by larger fluctuations in the distribution function, ranging
from $0.02$ to $0.13R_{500}$ (or from 20 to 130 kpc; see Figure 3), $R_{\rm
qcf}$ tends to be on average larger than $R_{\rm ccc}$ by factor up to $\sim$
3\. We thus find that $R_{\rm ccc}$ is the inner part of the condensation
region affected by direct turbulent precipitation and CCA, while $R_{\rm qcf}$
represents the more extended ”weather” over which we expect the secular
development of a quenched cooling flow. We expect each cluster to be in
different weather stages of the self-regulated feeding and feedback cycle,
oscillating between extended and more localized condensation rainfalls
($<0.05\,r_{500}$). On the other hand, $R_{\rm qcf}$ only traces the global
quenched cooling flow region, up to $\geq 0.1,r_{500}$, which does not
necessarily imply actual condensation. Indeed, most nebular emission and warm
gas is typically contained within smaller radii, which are comparable to
$R_{\rm ccc}$ (e.g., Gaspari et al. 2018). Therefore, in comparison to the
quenched cooling flow radius, $R_{\rm qcf}$, $R_{\rm ccc}$ represents a more
reliable and stable indicator for the effective cool core in terms of
condensed matter, which can be leveraged in simulations and semi-analytical
models, as well as in the interpretation of observations.

Incidentally, we also explored an alternative definition of $R_{\rm qcf}$,
corresponding to the condition $t_{\rm cool}/t_{\rm ff}=10$. In this case, we
find that less than ten clusters would have a well-defined crossing value,
while the majority would instead have an undefined value, due to their
profiles slowly approaching the condition $t_{\rm cool}/t_{\rm ff}=10$ in the
flat part of the profile. We note, however, that the condition $t_{\rm
cool}/t_{\rm ff}=10$ must be considered as a lower bound to the $R_{\rm qcf}$
criterion, as we already mentioned. We are aware that the actual threshold is
more likely a range of values, rather than a single-value threshold, and that
as a consequence, the actual $R_{\rm qcf}$ distribution may be more scattered
than that presented in this work. Nevertheless, we argue that an average ratio
$t_{\rm cool}$/$t_{\rm ff}\sim 25$ is a reasonable proxy to trace the initial
growth of linear thermal instability (TI) in heated cooling flows (see Hogan
et al. 2017). Finally, considering a range of values for the threshold, apart
from introducing a large scatter, would not change the qualitative aspects of
our results.

###  8.3 $R_{ccc}$-$R_{qcf}$ relation and constraints on the AGN duty cycle

The $R_{\rm qcf}-R_{\rm ccc}$ relation can give us further insight into the
AGN feeding-feedback duty cycle. Notably, there is no linear relation between
the two condensation radii: they tend to become comparable at large 100 kpc
values but non-linearly diverge as they approach smaller sizes (Fig. 4). The
divergence of such radii toward smaller scales can be interpreted through the
micro and meso precipitation having a more flickering duty cycle than the
macro (ensemble) weather. Such a trend is qualitatively consistent with the
CCA variability found in high-resolution 3D hydrodynamical simulations
(Gaspari et al. 2017), showing that the characteristic power spectral
frequency of the rains has a negative slope (flicker noise). Notably, such
feature has been probed in other recent cluster surveys and observations
(e.g., McDonald et al. 2021; Somboonpanyakul et al. 2021). Extrapolating below
the cluster regime, the sublinear slope implies that smaller halos are
expected to have a larger $R_{\rm qcf}-R_{\rm ccc}$ separation; indeed, recent
multiwavelength constraints suggest that the raining region is substantially
more compact in isolated galaxies (Temi et al. 2022) compared with the global
cooling flow zone. Furthermore, we suggest that such differences in ”cool-
core” size could allow to reconcile better the differences between the cooling
rates $\dot{M}_{\rm cool}$ retrieved via imaging out of the classical large
cool core with those constrained via spectroscopy, with the latter usually
limited within smaller regions similar to $R_{\rm ccc}$ (e.g., Molendi et al.
2016).

To recap, the above combined evidences suggest a global core radii picture
based on $R_{\rm ccc}\la R_{\rm qcf}\ll R_{\rm classic}$. As shown in Fig. 3,
each cluster has a ”classical” cool-core region $R_{\rm classic}$, which
envelopes a sphere with radius up to $\sim 0.2-0.3\,r_{500}$ (or several 100
kpc). However, such $R_{\rm classic}$ is purely based on an arbitrary
cosmological threshold $t_{\rm cool}<7.7$ Gyr (Hudson et al. 2010). Within
such classical cool core where most of the X-ray radiation is emitted (but the
hot gas is not able to actually condense into the lower gas phases) we find
the long-term quenched cool core $R_{\rm qcf}$, where feeding and feedback
processes balance out secularly, akin to an extended macro weather. Inside
such cool core, we find the effective condensation rain and flickering CCA
traced via $R_{\rm ccc}$ and directly tied to the nebular emission and warm or
cold gas detections.

##  9 Conclusions

We computed the profiles of the cooling time $t_{\rm cool}(r)$, free-fall time
$t_{\rm ff}(r)$, and turbulence cascade timescale $t_{\rm eddy}(r)$ in 37
massive ($2\times 10^{14}<M_{500}<2\times 10^{15}M_{\odot}$) galaxy clusters
observed with Chandra with high S/N at $0.03<z<0.29$. The profiles have been
obtained from observables entirely derived from X-ray datasets, such as
temperature, electron density, and metal abundance profiles. We derive for
each cluster a condensation core radius $R_{\rm ccc}$ defined as the radius
where $t_{\rm cool}=t_{\rm eddy}$, and a (quenched) cooling flow radius
$R_{\rm qcf}$ defined as $t_{\rm cool}=25\times t_{\rm ff}$. We accurately
evaluate the statistical uncertainties on these timescales and radii, and
explore the distribution of their values across our cluster sample. Our main
results are as follows.

  * •

The distribution of the condensation core $R_{\rm ccc}$ peaks at $\sim
0.01-0.02\,r_{500}$ and entirely included within $\sim 0.07\,r_{500}$. The
distribution of $R_{\rm qcf}$ is broader and shifted on average to $\sim$
$3\times$ larger values with respect to the $R_{\rm ccc}$ distribution.

  * •

We find no significant correlation between total mass and the two radii
$R_{\rm ccc}$ and $R_{\rm qcf}$, with a hint of larger values (relatively to
$r_{\rm 500}$, but not as absolute values) toward low masses (below $3\times
10^{14}M_{\odot}$, which shall be further explored with larger samples
including galaxy groups. Instead, both and $R_{\rm ccc}$ appear to moderately
increase with redshift.

  * •

Supported by theoretical models and high-resolution hydrodynamical simulations
of the multiphase condensation rain and CCA (e.g., Gaspari et al. 2017, 2018),
we find that $R_{\rm qcf}$ is a tracer of the extended quenched cooling flow
or macro weather. The smaller cool-core condensation radius $R_{\rm ccc}$
traces instead the actual inner rain, which also drives CCA feeding episodes
onto the SMBH. Both proposed core radii are an order of magnitude smaller than
the classical and ad-hoc cool-core definition $R_{7.7\,\rm Gyr}$, where the
halo is emitting strong X-ray radiation, but not condensing or inflowing.

  * •

We find that the correlation between $R_{\rm qcf}$ and $R_{\rm ccc}$ is non-
linear, diverging at lower values up to a factor of 3 over an order of
magnitude in $r_{500}$. This relation between the two scales allows us to
infer some features of the duty cycle and related appearance or disappearance
of the local cooling flow. Specifically, the slope of the $R_{\rm qcf}-R_{\rm
ccc}$ relation is measured to be $0.46^{+0.02}_{-0.03}$, and suggests that the
micro CCA rain is flickering on and off in the macro weather, in agreement
with hydrodynamical simulations (Gaspari et al. 2017).

As shown above, to describe the fate of the multiscale cooling gas and its
interplay with the cluster atmosphere, we need a combination of high spatial
resolution and good spectroscopic capabilities. For instance, among the future
generation of X-ray instruments, those onboard Athena will play an important
role to help in this direction. It is expected to achieve a PSF below 10
arcsec over the entire field of view and an effective area at 1 keV a factor
$>$ 5 larger than the current detectors, along with the Wide Field Imager
(WFI) providing sensitive wide field imaging and spectroscopy and the X-ray
Integral Field Unit (X-IFU) delivering spatially resolved high-resolution
X-ray spectroscopy – altogether, these instruments will allow for an
unprecedented level of accuracy in mapping the state of the gas within 0.1
$r_{500}$ and cool cores in systems at $T>2$ keV up to a redshift of $\la
0.4$. Further, high angular-resolution X-ray facilities such as
Lynx777https://www.lynxobservatory.com/ and AXIS888http://axis.astro.umd.edu
are proposing to investigate with subarcsecond resolution over a FoV of
400-500 arcmin2 the X-ray sky, improving this capability of a factor $\sim
100$ with respect to Chandra ACIS-I. All such core missions will enable
crucial steps forward in the characterization and understanding of the weather
in galaxy cluster cores by linking the micro, meso, and macro scales as well
as above key condensation processes.

## Acknowledgments

We thank the Referee for their constructive criticism and helpful suggestions.
This work was supported by the Bureau of International Cooperation, Chinese
Academy of Sciences under the grant GJHZ1864. We acknowledge financial
contribution from the agreement ASI-INAF n.2017-14-H.0. P.T. acknowledges
financial support through grant PRIN-MIUR 2017WSCC32: “Zooming into dark
matter and proto-galaxies with massive lensing clusters”. M.G. acknowledges
partial support by NASA HST GO-15890.020/023-A; this work is part of the
broader BlackHoleWeather program. S.E. acknowledges financial contribution
from the contracts ASI-INAF Athena 2019-27-HH.0, INAF mainstream project
1.05.01.86.10, and funding from the European Union Horizon 2020 Programme
under the AHEAD2020 project (grant agreement n. 871158). The Chandra raw data
used in this paper are available to download at the Chandra Data Archive
website999https://cxc.cfa.harvard.edu/cda/. The reduced data are also
available upon request from the corresponding author.

## References

  * Arnaud (1996) Arnaud, K. A. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. Jacoby & J. Barnes, 17
  * Asplund et al. (2009) Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481
  * Barai et al. (2016) Barai, P., Murante, G., Borgani, S., et al. 2016, MNRAS, 461, 1548
  * Binney & Tremaine (1987) Binney, J. & Tremaine, S. 1987, Galactic Dynamics (Princeton Series in Astrophysics)
  * Blanton et al. (2010) Blanton, E. L., Clarke, T. E., Sarazin, C. L., Randall, S. W., & McNamara, B. R. 2010, Proceedings of the National Academy of Science, 107, 7174
  * Blanton et al. (2011) Blanton, E. L., Randall, S. W., Clarke, T. E., et al. 2011, ApJ, 737, 99
  * Burns et al. (2008) Burns, J. O., Hallman, E. J., Gantner, B., Motl, P. M., & Norman, M. L. 2008, ApJ, 675, 1125
  * Cavagnolo et al. (2009) Cavagnolo, K. W., Donahue, M., Voit, G. M., & Sun, M. 2009, ApJS, 182, 12
  * Cavaliere & Fusco-Femiano (1978) Cavaliere, A. & Fusco-Femiano, R. 1978, A&A, 70, 677
  * Chen et al. (2007) Chen, Y., Reiprich, T. H., Böhringer, H., Ikebe, Y., & Zhang, Y. Y. 2007, A&A, 466, 805
  * De Grandi et al. (2004) De Grandi, S., Ettori, S., Longhetti, M., & Molendi, S. 2004, A&A, 419, 7
  * Dennis & Chandran (2005) Dennis, T. J. & Chandran, B. D. G. 2005, ApJ, 622, 205
  * Diehl et al. (2008) Diehl, S., Li, H., Fryer, C. L., & Rafferty, D. 2008, ApJ, 687, 173
  * Domainko et al. (2004) Domainko, W., Gitti, M., Schindler, S., & Kapferer, W. 2004, A&A, 425, L21
  * Donahue & Voit (2004) Donahue, M. & Voit, G. M. 2004, in Clusters of Galaxies: Probes of Cosmological Structure and Galaxy Evolution, ed. J. S. Mulchaey, A. Dressler, & A. Oemler, 143
  * Dunn & Fabian (2006) Dunn, R. J. H. & Fabian, A. C. 2006, MNRAS, 373, 959
  * Edge (2001) Edge, A. C. 2001, MNRAS, 328, 762
  * Edge & Frayer (2003) Edge, A. C. & Frayer, D. T. 2003, ApJ, 594, L13
  * Ehlert et al. (2011) Ehlert, S., Allen, S. W., von der Linden, A., et al. 2011, MNRAS, 411, 1641
  * Ettori & Brighenti (2008) Ettori, S. & Brighenti, F. 2008, MNRAS, 387, 631
  * Ettori et al. (2002) Ettori, S., De Grandi, S., & Molendi, S. 2002, A&A, 391, 841
  * Fabian (1994) Fabian, A. C. 1994, ARA&A, 32, 277
  * Fabian & Nulsen (1977) Fabian, A. C. & Nulsen, P. E. J. 1977, MNRAS, 180, 479
  * Fabian et al. (2003a) Fabian, A. C., Sanders, J. S., Allen, S. W., et al. 2003a, MNRAS, 344, L43
  * Fabian et al. (2003b) Fabian, A. C., Sanders, J. S., Crawford, C. S., et al. 2003b, MNRAS, 344, L48
  * Field (1965) Field, G. B. 1965, ApJ, 142, 531
  * Gaspari (2015) Gaspari, M. 2015, MNRAS, 451, L60
  * Gaspari et al. (2012a) Gaspari, M., Brighenti, F., & Temi, P. 2012a, MNRAS, 424, 190
  * Gaspari & Churazov (2013) Gaspari, M. & Churazov, E. 2013, A&A, 559, A78
  * Gaspari et al. (2014) Gaspari, M., Churazov, E., Nagai, D., Lau, E. T., & Zhuravleva, I. 2014, A&A, 569, A67
  * Gaspari et al. (2019) Gaspari, M., Eckert, D., Ettori, S., et al. 2019, ApJ, 884, 169
  * Gaspari et al. (2018) Gaspari, M., McDonald, M., Hamer, S. L., et al. 2018, ApJ, 854, 167
  * Gaspari et al. (2012b) Gaspari, M., Ruszkowski, M., & Sharma, P. 2012b, ApJ, 746, 94
  * Gaspari & Sa̧dowski (2017) Gaspari, M. & Sa̧dowski, A. 2017, ApJ, 837, 149
  * Gaspari et al. (2017) Gaspari, M., Temi, P., & Brighenti, F. 2017, MNRAS, 466, 677
  * Gaspari et al. (2020) Gaspari, M., Tombesi, F., & Cappi, M. 2020, Nature Astronomy, 4, 10
  * Gonzalez et al. (2013) Gonzalez, A. H., Sivanandam, S., Zabludoff, A. I., & Zaritsky, D. 2013, ApJ, 778, 14
  * Guainazzi & Tashiro (2018) Guainazzi, M. & Tashiro, M. S. 2018, arXiv e-prints, arXiv:1807.06903
  * Guo & Oh (2008) Guo, F. & Oh, S. P. 2008, MNRAS, 384, 251
  * Hitomi Collaboration et al. (2016) Hitomi Collaboration, Aharonian, F., Akamatsu, H., et al. 2016, Nature, 535, 117
  * Hlavacek-Larrondo et al. (2015) Hlavacek-Larrondo, J., McDonald, M., Benson, B. A., et al. 2015, ApJ, 805, 35
  * Hofmann et al. (2016) Hofmann, F., Sanders, J. S., Nandra, K., Clerc, N., & Gaspari, M. 2016, A&A, 585, A130
  * Hogan et al. (2017) Hogan, M. T., McNamara, B. R., Pulido, F. A., et al. 2017, ApJ, 851, 66
  * Hudson et al. (2010) Hudson, D. S., Mittal, R., Reiprich, T. H., et al. 2010, A&A, 513, A37
  * Iapichino et al. (2010) Iapichino, L., Maier, A., Schmidt, W., & Niemeyer, J. C. 2010, in American Institute of Physics Conference Series, Vol. 1241, American Institute of Physics Conference Series, ed. J.-M. Alimi & A. Fuözfa, 928–934
  * Juráňová et al. (2019) Juráňová, A., Werner, N., Gaspari, M., et al. 2019, MNRAS, 484, 2886
  * Kaastra et al. (2001) Kaastra, J. S., Ferrigno, C., Tamura, T., et al. 2001, A&A, 365, L99
  * Lau et al. (2017) Lau, E. T., Gaspari, M., Nagai, D., & Coppi, P. 2017, ApJ, 849, 54
  * Liu et al. (2020) Liu, A., Tozzi, P., Ettori, S., et al. 2020, A&A, 637, A58
  * Liu et al. (2015) Liu, A., Yu, H., Tozzi, P., & Zhu, Z.-H. 2015, ApJ, 809, 27
  * Maccagni et al. (2021) Maccagni, F. M., Serra, P., Gaspari, M., et al. 2021, A&A, 656, A45
  * Makishima et al. (2001) Makishima, K., Ezawa, H., Fukuzawa, Y., et al. 2001, PASJ, 53, 401
  * Mathews et al. (2006) Mathews, W. G., Faltenbacher, A., & Brighenti, F. 2006, ApJ, 638, 659
  * McDonald et al. (2012) McDonald, M., Bayliss, M., Benson, B. A., et al. 2012, Nature, 488, 349
  * McDonald et al. (2018) McDonald, M., Gaspari, M., McNamara, B. R., & Tremblay, G. R. 2018, ApJ, 858, 45
  * McDonald et al. (2021) McDonald, M., McNamara, B. R., Calzadilla, M. S., et al. 2021, ApJ, 908, 85
  * McDonald et al. (2019) McDonald, M., McNamara, B. R., Voit, G. M., et al. 2019, ApJ, 885, 63
  * McKinley et al. (2022) McKinley, B., Tingay, S. J., Gaspari, M., et al. 2022, Nature Astronomy, 6, 109
  * McNamara & Nulsen (2007) McNamara, B. R. & Nulsen, P. E. J. 2007, ARA&A, 45, 117
  * McNamara et al. (2005) McNamara, B. R., Nulsen, P. E. J., Wise, M. W., et al. 2005, Nature, 433, 45
  * McNamara & O’Connell (1989) McNamara, B. R. & O’Connell, R. W. 1989, AJ, 98, 2018
  * McNamara et al. (2014) McNamara, B. R., Russell, H. R., Nulsen, P. E. J., et al. 2014, ApJ, 785, 44
  * McNamara et al. (2000) McNamara, B. R., Wise, M., Nulsen, P. E. J., et al. 2000, ApJ, 534, L135
  * Molendi et al. (2023) Molendi, S., De Grandi, S., Rossetti, M., et al. 2023, A&A, 670, A104
  * Molendi & Pizzolato (2001) Molendi, S. & Pizzolato, F. 2001, ApJ, 560, 194
  * Molendi et al. (2016) Molendi, S., Tozzi, P., Gaspari, M., et al. 2016, A&A, 595, A123
  * North et al. (2021) North, E. V., Davis, T. A., Bureau, M., et al. 2021, MNRAS, 503, 5179
  * O’Hara et al. (2006) O’Hara, T. B., Mohr, J. J., Bialek, J. J., & Evrard, A. E. 2006, ApJ, 639, 64
  * Olivares et al. (2019) Olivares, V., Salome, P., Combes, F., et al. 2019, A&A, 631, A22
  * Olivares et al. (2022) Olivares, V., Salomé, P., Hamer, S. L., et al. 2022, A&A, 666, A94
  * Pasini et al. (2021) Pasini, T., Finoguenov, A., Brüggen, M., et al. 2021, MNRAS, 505, 2628
  * Peterson & Fabian (2006) Peterson, J. R. & Fabian, A. C. 2006, Phys. Rep, 427, 1
  * Peterson et al. (2001) Peterson, J. R., Paerels, F. B. S., Kaastra, J. S., et al. 2001, A&A, 365, L104
  * Pinto et al. (2018) Pinto, C., Bambic, C. J., Sanders, J. S., et al. 2018, MNRAS, 480, 4113
  * Planck Collaboration et al. (2016) Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2016, A&A, 594, A13
  * Ponman et al. (1999) Ponman, T. J., Cannon, D. B., & Navarro, J. F. 1999, Nature, 397, 135
  * Pratt et al. (2010) Pratt, G. W., Arnaud, M., Piffaretti, R., et al. 2010, A&A, 511, A85
  * Rasera et al. (2008) Rasera, Y., Lynch, B., Srivastava, K., & Chandran, B. 2008, ApJ, 689, 825
  * Rose et al. (2019) Rose, T., Edge, A. C., Combes, F., et al. 2019, MNRAS, 489, 349
  * Russell et al. (2017) Russell, H. R., McDonald, M., McNamara, B. R., et al. 2017, ApJ, 836, 130
  * Ruszkowski et al. (2004) Ruszkowski, M., Bruggen, M., & Begelman, M. C. 2004, in American Astronomical Society Meeting Abstracts, Vol. 204, American Astronomical Society Meeting Abstracts #204, 09.01
  * Sanders et al. (2010) Sanders, J. S., Fabian, A. C., Smith, R. K., & Peterson, J. R. 2010, MNRAS, 402, L11
  * Sanderson et al. (2006) Sanderson, A. J. R., Ponman, T. J., & O’Sullivan, E. 2006, MNRAS, 372, 1496
  * Santos et al. (2008) Santos, J. S., Rosati, P., Tozzi, P., et al. 2008, A&A, 483, 35
  * Schuecker et al. (2004) Schuecker, P., Finoguenov, A., Miniati, F., Böhringer, H., & Briel, U. G. 2004, A&A, 426, 387
  * Sharma et al. (2012) Sharma, P., McCourt, M., Quataert, E., & Parrish, I. J. 2012, MNRAS, 420, 3174
  * Shin et al. (2016) Shin, J., Woo, J.-H., & Mulchaey, J. S. 2016, ApJS, 227, 31
  * Simionescu et al. (2019) Simionescu, A., ZuHone, J., Zhuravleva, I., et al. 2019, Space Sci. Rev., 215, 24
  * Somboonpanyakul et al. (2021) Somboonpanyakul, T., McDonald, M., Gaspari, M., Stalder, B., & Stark, A. A. 2021, ApJ, 910, 60
  * Sun (2009) Sun, M. 2009, ApJ, 704, 1586
  * Sutherland & Dopita (1993) Sutherland, R. S. & Dopita, M. A. 1993, ApJS, 88, 253
  * Tamura et al. (2001) Tamura, T., Kaastra, J. S., Peterson, J. R., et al. 2001, A&A, 365, L87
  * Temi et al. (2018) Temi, P., Amblard, A., Gitti, M., et al. 2018, ApJ, 858, 17
  * Temi et al. (2022) Temi, P., Gaspari, M., Brighenti, F., et al. 2022, ApJ, 928, 150
  * Tozzi et al. (2015) Tozzi, P., Gastaldello, F., Molendi, S., et al. 2015, A&A, 580, A6
  * Tozzi & Norman (2001) Tozzi, P. & Norman, C. 2001, ApJ, 546, 63
  * Tremblay et al. (2018) Tremblay, G. R., Combes, F., Oonk, J. B. R., et al. 2018, ApJ, 865, 13
  * Vantyghem et al. (2014) Vantyghem, A. N., McNamara, B. R., Russell, H. R., et al. 2014, MNRAS, 442, 3192
  * Vikhlinin et al. (2006) Vikhlinin, A., Kravtsov, A., Forman, W., et al. 2006, ApJ, 640, 691
  * Vikhlinin et al. (2005) Vikhlinin, A., Markevitch, M., Murray, S. S., et al. 2005, ApJ, 628, 655
  * Voit (2018) Voit, G. M. 2018, ApJ, 868, 102
  * Voit et al. (2015a) Voit, G. M., Bryan, G. L., O’Shea, B. W., & Donahue, M. 2015a, ApJ, 808, L30
  * Voit et al. (2015b) Voit, G. M., Donahue, M., Bryan, G. L., & McDonald, M. 2015b, Nature, 519, 203
  * Voit et al. (2017) Voit, G. M., Meece, G., Li, Y., et al. 2017, ApJ, 845, 80
  * Willingale et al. (2013) Willingale, R., Starling, R. L. C., Beardmore, A. P., Tanvir, N. R., & O’Brien, P. T. 2013, MNRAS, 431, 394
  * Wilms et al. (2000) Wilms, J., Allen, A., & McCray, R. 2000, ApJ, 542, 914
  * Wittor & Gaspari (2020) Wittor, D. & Gaspari, M. 2020, MNRAS, 498, 4983
  * Xue & Wu (2000) Xue, Y.-J. & Wu, X.-P. 2000, MNRAS, 318, 715
  * Yang et al. (2019) Yang, H.-Y. K., Gaspari, M., & Marlow, C. 2019, ApJ, 871, 6
  * Zakamska & Narayan (2003) Zakamska, N. L. & Narayan, R. 2003, ApJ, 582, 162
  * Zhuravleva et al. (2014) Zhuravleva, I., Churazov, E., Schekochihin, A. A., et al. 2014, Nature, 515, 85
  * ZuHone et al. (2013) ZuHone, J. A., Markevitch, M., Ruszkowski, M., & Lee, D. 2013, ApJ, 762, 69
  * Zuhone & Roediger (2016) Zuhone, J. A. & Roediger, E. 2016, Journal of Plasma Physics, 82, 535820301

##  Appendix A Derivation of the deprojected electron density (Equation 12)

When fitting the spectra of the projected annuli, we obtain the projected,
emission-weighted temperature and metallicity, as well as the projected
normalization of the emission measure, corresponding to the normalization of
the spectrum. The normalization parameter $K$ of of the mekal spectrum in
Xspec is defined as:

$K=\frac{10^{-14}}{4\pi D_{A}^{2}(1+z)^{2}}\int n_{e}(r)n_{H}(r){\rm
d}V\sim\frac{10^{-14}0.82}{4\pi D_{A}^{2}(1+z)^{2}}\int n_{e}(r)^{2}{\rm d}V,$
(18)

where 0.82 is the cosmic ratio, $n_{H}/n_{e}$, and $n_{e}(r)$ is the radial
density profile of the electron density, which is the quantity we want to
derive from the observed (projected) values. In the assumption of spherical
symmetry, the differential volume element ${\rm d}V$ corresponding to a
projected annulus of finite width, $\Delta s$, depends on the projected
radius, $s,$ and on the coordinate $z$ along the line of sight, simply as
$dV=2\pi s\Delta sdz$. Since $z=\sqrt{(r^{2}-s^{2})}$, we obtain:

${\rm d}V=2\pi s\Delta s\frac{r{\rm d}r}{\sqrt{r^{2}-s^{2}}}\,.$ (19)

If we define the function $f(r)$ as

$f(r)\equiv\frac{0.82\,10^{-14}}{4D_{A}^{2}(1+z)^{2}}n_{e}^{2}(r)\,,$ (20)

then we have rewritten $K$ as a function of the projected radius, $s,$ in a
compact expression:

$\frac{K(s)}{s\Delta s}=2\int_{s}^{\infty}\frac{rf(r){\rm
d}r}{\sqrt{r^{2}-s^{2}}}\,.$ (21)

If we apply the Abel transform, we can write:

$f(r)=-\frac{1}{\pi}\int_{r}^{\infty}\frac{d}{ds}\Big{(}\frac{K(s)}{s\Delta
s}\Big{)}\frac{{\rm d}s}{\sqrt{s^{2}-r^{2}}}\,.$ (22)

This allows us to write the square of the electron density with an expression
based on observables. Combining Equations A.3 and A.5, we obtain

$n_{e}^{2}(r)=-\frac{4\times
10^{14}}{0.82\pi}{D^{2}_{A}(1+z)^{2}}\int_{r}^{\infty}\frac{d}{ds}\Big{(}\frac{K(s)}{s\Delta
s}\Big{)}\frac{{\rm d}s}{\sqrt{s^{2}-r^{2}}}\,,$ (23)

which is the same as Equation 12 given in Section 6.

##  Appendix B Timescale profiles

In Fig. 1, we show the profiles of the cooling time, $t_{\rm cool}$,
turbulence eddy time, $t_{\rm eddy}$, and free-fall time, $t_{\rm ff}$. Each
row refers to a cluster following the same order of Table 2. In the left
panels, we show the points corresponding to the spatial bins of the spectral
fits, with the points obtained from the best-fit deprojected temperature and
density profiles for $t_{\rm cool}$ and $t_{\rm ff}$. In addition, we plot the
polynomial best fit to the time scale profiles as a continuous line, showing
the 1-$\sigma$ uncertainty with a shaded area. In the right column, we show
the ratio of the best-fit function of $t_{\rm cool}$, $t_{\rm eddy}$, and
$t_{\rm ff}$. In particular, $t_{\rm cool}/t_{\rm ff}$ defines $R_{\rm qcf}$
when $t_{\rm cool}/t_{\rm ff}=25$, while $t_{\rm cool}/t_{\rm eddy}$ defines
$R_{\rm ccc}$ when $t_{\rm cool}/t_{\rm eddy}=1$. In Fig. 2, we can see that
the results of using Equation 8 and Equation 9 to find $R_{\rm ccc}$ are
basically the same, and the error between Equation 9 and the real data is also
small, so we use Equation 9 to calculate $R_{\rm qcf}$ for those galaxy
clusters without measurement data.

Figure 1: Profiles of the cooling time $t_{\rm cool}$, the turbulence eddy
turnover time $t_{\rm eddy}$, and the free fall time $t_{\rm ff}$ (left).
Ratio of $t_{\rm cool}/t_{\rm ff}$ that defines $R_{\rm qcf}$, when $t_{\rm
cool}/t_{\rm ff}=25$, and $t_{\rm cool}/t_{\rm eddy}$, which defines $R_{\rm
ccc}$ when $t_{\rm cool}/t_{\rm eddy}=1$ (right).

Figure 1:  \- continued .

Figure 1:  \- continued .

Figure 1:  \- continued .

Figure 1:  \- continued .

Figure 1:  \- continued .

Figure 1:  \- continued .

Figure 1:  \- continued .

Figure 1:  \- continued .

Figure 1:  \- continued .

Figure 2: Comparison chart of the results(16 galaxy clusters) of calculating
$R_{\rm ccc}$ with Equation 8 and measured data (left). Comparison chart of
the results(16 galaxy clusters)of calculating $R_{\rm ccc}$ with Equation 9
and measured data (right). Comparison chart of the results of calculating
$R_{\rm ccc}$ with Equations 8 and 9 (bottom).