11institutetext: Dipartimento di Fisica e Astronomia “Augusto Righi” – Alma
Mater Studiorum – Università di Bologna, via Gobetti 93/2, I-40129 Bologna
22institutetext: INAF, Osservatorio di Astrofisica e Scienza dello Spazio, via
Piero Gobetti 93/3, 40129 Bologna, Italy 33institutetext: INFN, Sezione di
Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy

# Gas rotation and dark matter halo shape in cool-core clusters of galaxies

T. Bartalesi, 11<EMAIL_ADDRESS>S. Ettori, 2233
<EMAIL_ADDRESS>C. Nipoti 11

(Accepted, October 24, 2023)

###### Abstract

Aims. We study the possibility that the gas in cool-core clusters of galaxies
has non-negligible rotation support, the impact of gas rotation on mass
estimates from current X-ray observations, and the ability of forthcoming
X-ray observatories to detect such rotation.

Methods. We present three representative models of massive cool-core clusters
with rotating intracluster medium (ICM) in equilibrium in cosmologically
motivated spherical, oblate or prolate dark matter halos, represented by
physical density-potential pairs. In the models, the gas follows a composite-
polytropic distribution, and has rotation velocity profiles consistent with
current observational constraints and similar to those found in clusters
formed in cosmological simulations. We show that the models are consistent
with the available measurements of the ICM properties of the massive cluster
population: thermodynamic profiles, shape of surface-brightness distribution,
hydrostatic mass bias and broadening of X-ray emitting lines. Using the
configuration for the microcalorimeter onboard the _XRISM_ satellite, we
generate a set of mock X-ray spectra of our cluster models, which we then
analyze to make predictions on the estimates of the rotation speed that will
be obtained with such an instrument. We then assess what fraction of the
hydrostatic mass bias of our models could be accounted for by detecting
rotation speed with _XRISM_ spectroscopy over the range $(0.1-1)r_{500}$,
sampled with 3 nonoverlapping pointings.

Results. Current data leave room for rotating ICM in cool-core clusters with
peaks of rotation speed as high as $600\,\mathrm{km/s}$. We have shown that
such rotation, if present, will be detected with upcoming X-ray facilities
such as _XRISM_ and that $60-70\%$ of the hydrostatic mass bias due to
rotation can be accounted for using the line-of-sight velocity measured from
X-ray spectroscopy with _XRISM_ , with a residual bias smaller than $3\%$ at
an overdensity of 500. In this way, _XRISM_ will allow us to pin down any mass
bias of origin different from rotation.

###### Key Words.:

Galaxies: clusters: general – Galaxies: clusters: intracluster medium –
X-rays: galaxies: clusters – dark matter

##  1 Introduction

The mass of clusters of galaxies is crucial to understand the formation and
evolution of cosmic structures, and to constrain the parameters that define
the cosmological background (see Pratt et al. 2019 for a review). Clusters of
galaxies are permeated by a hot ($\sim 10^{7}-10^{8}$ K), rarefied ($\sim
10^{-2}-10^{-4}$ particles per cm3), optically thin, gaseous component, known
as intracluster medium (ICM), which emits in the X-rays via thermal
Bremsstrahlung and emission lines from collisional excitation of inner shell
electrons of heavy metals. Assuming that the ICM is in hydrostatic
equilibrium, X-ray observations can thus be used to infer the mass of galaxy
clusters (see Ettori et al. 2013 for a review). Mass estimates obtained in
this way can be very precise, but not accurate (e.g. Ettori et al., 2019),
given that the hydrostatic equilibrium does not account for the residual non-
thermalized (kinetic) energy in the ICM (see e.g. Rasia et al., 2006;
Piffaretti & Valdarnini, 2008; Lau et al., 2009; Suto et al., 2013; Lau et
al., 2013; Biffi et al., 2016; Angelinelli et al., 2020; Gianfagna et al.,
2021). This effect that brings hydrostatic masses to underestimate the ”true”
mass is often referred to as hydrostatic mass bias. Measurements of this bias
can be obtained by comparison with more direct mass estimators (e.g. Zhang et
al., 2010; Mahdavi et al., 2013; Lovisari et al., 2020). In particular, being
the most massive gravitationally bound structures in the Universe, galaxy
clusters are effective gravitational lenses, which provide a complementary,
and typically more accurate method to infer the total (i.e. baryon plus dark
matter) mass (see e.g. Meneghetti et al., 2010; Rasia et al., 2012).
Alternatively, the dynamical mass of a cluster can be estimated by exploiting
measurements of the orbital velocities of its member galaxies (see e.g.
Ferragamo et al., 2021).

Even though in the X-ray observations the gas clumpiness and the use of the
spectroscopic temperature in reconstructing the thermal properties of the ICM
contributed nonnegligibly to the hydrostatic mass bias (see e.g. Rasia et al.,
2006; Roncarelli et al., 2013; Pearce et al., 2020; Towler et al., 2023), most
of this bias is expected to be due to the motions in the ICM: in particular,
turbulence bulk motion, and rotation (see e.g. Nagai et al., 2007b; Nelson et
al., 2014; Biffi et al., 2016; Angelinelli et al., 2020). Most of these
previous works have focused on the relative importance of bulk and random
motions to the total budget of the hydrostatic mass bias, with only a few
studies dedicated to the contribution from the ICM rotational support (e.g.
Fang et al., 2009). There are essentially only two direct ways to measure gas
rotation in galaxy clusters: the rotational kinetic Sunyaev-Zeldovich effect
(Cooray & Chen 2002, Chluba & Mannheim 2002 and also Sunyaev & Zeldovich 1980;
see Baldi et al. 2018 and Altamura et al. 2023b for future perspectives) and
the Doppler shift of the centroids of the X-ray emitting lines or their
Doppler broadening. The latter measurements require X-ray spectrometers at
high energy resolution ($\Delta E\lesssim 10\,\mathrm{eV}$ at $E\approx
6-7\,\mathrm{keV}$ is required to detect a line-of-sight speed of $\approx
500\,\mathrm{km/s}$; e.g. Sunyaev et al. 2003, Bianconi et al. 2013), which
are thus far reached only by calorimeter onboard International X-ray Astronomy
Mission _ASTRO-H/Hitomi_ 111See
https://www.isas.jaxa.jp/en/missions/spacecraft/past/hitomi.html. satellite
(see Hitomi Collaboration et al. 2016 for its results). The loss of _Hitomi_
have prevented us from depicting a comprehensive overview of the kinematics of
the ICM; however, the forthcoming microcalorimeter _Resolve_ onboard the X-Ray
Imaging and Spectroscopy Mission222See https://xrism.isas.jaxa.jp/en/.
(_XRISM_) satellite (with $\Delta E\simeq 7\,\mathrm{eV}$ FWHM at
$E=6-7\,\mathrm{keV}$), launched in September 2023, is expected to provide
some key elements to improve our understanding of the ICM kinematics.
Nowadays, only upper limits on the velocity broadening of X-ray emitting lines
are available: using X-ray Multi-Mirror Mission333See
https://www.cosmos.esa.int/web/xmm-newton. (_XMM-Newton_) Reflection Grating
Spectrometers (RGS) data, Pinto et al. (2015), in most cool cores of clusters,
groups, and massive elliptical galaxies of their observed sample, found
broadening velocities of $\approx 500\,\mathrm{km/s}$ (see also Sanders et al.
2011 and Bambic et al. 2018). Even though some objects have higher upper
limits (of $\approx 1000\,\mathrm{km/s}$), we interpret $500\,\mathrm{km/s}$
as the current upper limit on the rotation speed of the ICM in typical
clusters, which leaves open the possibility that the ICM has nonnegligible
rotation support in relaxed clusters444Indications of rotation support of the
galactic component have been found in some clusters from spectroscopic
observations of member galaxies (see e.g. Oegerle & Hill, 1992; Hwang & Lee,
2007; Ferrami et al., 2023). The differences in the rotation speed profiles of
the ICM and member galaxies are an interesting issue to be explored with
future facilities..

In the cosmological context, the rotation of both dark matter (DM) and gas is
expected to be induced primarily by the large-scale processes involving the
entire cluster (such as tidal torques from neighbouring overdensities; Peebles
1969). In massive clusters (virial masses $\gtrsim 5\times
10^{14}\,\mathrm{M_{\odot}}$) formed in cosmological $N$-body hydrodynamical
nonradiative simulations, Baldi et al. (2017) have found that the rotation
support of the ICM tends to be higher than that of the DM, with values of the
gas spin parameter on average higher by $13\%$ than those of the halo spin
parameter. In principle, the rotation support of the ICM can be further
enhanced by unimpeded radiative cooling, because of conservation of angular
momentum (see e.g. Kley & Mathews, 1995), but in real clusters also heating
mechanisms are at work. In fact, including radiative cooling, Active Galactic
Nucleus (AGN) and stellar feedback models in cosmological simulations, Baldi
et al. (2017) have found that the rotation support of the ICM is similar to
that found in nonradiative simulations.

Based on the properties of the ICM in the central regions, clusters of
galaxies are classified as cool-core and non cool-core clusters (e.g. section
6.4.3 of Cimatti et al., 2019). Given that we are interested in rotation
support of the ICM, in this work we focus on cool-core clusters, which tend to
be relaxed (e.g. Pratt et al., 2010; Mahdavi et al., 2013) and thus good
targets for symmetric equilibrium models of the ICM. By definition, cool-core
clusters are characterized by lower central ICM entropy, which is broadly
interpreted as a signature of cooling. In fact, the measured values of the
central entropy are much higher than predicted in a standard cooling-flow
model (e.g. McDonald et al. 2013). This suggests that, in time-averaged sense,
over $\sim 10\,\mathrm{Gyr}$, radiative cooling is balanced by some form of
heating, a picture also supported by the fact that radiative cosmological
simulations without heating suffer from the ”overcooling” problem, which
produces photometric features inconsistent with observations (e.g. Fang et
al., 2009; Lau et al., 2011, 2012; Nagai et al., 2013). There is growing
consensus that AGN feedback provides the dominant heating contribution in the
cluster inner regions (see McNamara & Nulsen 2012; Hlavacek-Larrondo et al.
2022 for reviews and Nobels et al. 2022; Huško et al. 2022 for recent
results). However, it must be stressed that modeling the complex interplay of
heating and cooling is challenging also for state-of-the art simulations. For
instance, clusters formed in currently available cosmological simulations
including an AGN feedback model can suffer from the ”entropy-core” problem, in
the sense that their inner entropy profiles do not match those observed in
real clusters (Altamura et al., 2023a).

Rotation of the ICM could be relevant also to the energy balance of cool
cores, given that the ICM is known to be weakly magnetized. If the magnetized,
rotating ICM is unstable to the magnetorotational instability (Balbus &
Hawley, 1991), the nonlinear evolution of the instability will lead to
turbulent heating, which could contribute to offsetting the radiative cooling
of the ICM and to halting the cooling flows, lending a hand to the AGN
feedback (see Nipoti & Posti 2014; Nipoti et al. 2015).

In this work, we propose three models representative of typical, nearby,
massive cool-core clusters, with cosmologically motivated dark halos with
different shapes (Sect. 2) and rotating ICM with rotation speed consistent
with observational upper limits (Sect. 3). In Sect. 4, we compare the
intrinsic and observable properties of the ICM in our cluster models to the
observational data of real galaxy clusters. In Sect. 5, we assess the
detectability of the rotation support of our models, building mock X-ray
spectra of the rotating ICM in our cluster models, using the configurations
for _Resolve_. Sect. 6 concludes.

Throughout this article, when using the Hubble parameter
$H(z)=H_{0}E^{1/2}(z)$, where
$E(z)=\sqrt{\Omega_{\Lambda,0}+\Omega_{\mathrm{m},0}(1+z)^{3}}$, we assume
$\Omega_{\mathrm{m},0}=0.3$, $\Omega_{\Lambda,0}=0.7$ and Hubble constant
$H_{0}=70\,\mathrm{km/s/Mpc}$.

##  2 Dark matter halo models

We introduce here the gravitational potentials that we will use to build our
cluster models. Given that the mass content of clusters is dominated by the
dark matter (DM), these gravitational potentials must be essentially
representative of those produced by the cluster DM halos.

Cosmological $N$-body DM-only simulations predict for most halos an aspherical
shape, set at the time of the last major merger (Allgood et al. 2006). In
general, the angle-averaged density profile of these simulated halos is well
fitted by the Navarro-Frenk-White (NFW; Navarro et al. 1996) profile

$\rho(r)=\frac{\rho_{\rm n}}{\left(\frac{r}{r_{\rm
s}}\right)\left(1+\frac{r}{r_{\rm s}}\right)^{2}},$ (1)

where $r$ is the distance from the halo center, $\rho_{\rm n}$ is a
characteristic density and $r_{\rm s}$ is the scale radius. The density
distribution of DM in real clusters is also well represented by this profile:
for instance, from X-ray and Sunyaev-Zeldovich effect observations, Ettori et
al. (2019) infer that the NFW profile successfully models the angle averaged
density profiles of the halos of the observed clusters. It is thus natural to
take the NFW density profile (Eq. 1) as reference to build realistic flattened
halo models. In the following sections we will describe how we build
axisymmetric halo models by suitably modifying the spherical NFW model.

Figure 1: Isodensity (dashed) and isopotential (solid) contours in the
meridional plane of the prolate (left panel) and oblate (right panel) NFW
models with $\eta=1/3$. The size of the box is $\approx r_{\rm{200}}/r_{\rm
s}$ (see Sect. 2.2).

###  2.1 Flattened NFW density-potential pairs

Ciotti & Bertin (2005) presented a technique to construct analytic
axisymmetric and triaxial density-potential pairs by modifying a parent
spherical density distribution with given density profile
$\tilde{\rho}(\tilde{r})$, where $\tilde{\rho}=\rho/\rho_{\rm n}$ and
$\tilde{r}=r/r_{\rm s}$, with $\rho_{\rm n}$ a characteristic density and
$r_{\rm s}$ a scale radius. The generic density-potential pair of this family
can be written in Cartesian coordinates $(x,y,z)$ as

$\tilde{\rho}(x,y,z)=\tilde{\rho}(\tilde{r})+\frac{\epsilon\tilde{y}^{2}+\eta\tilde{z}^{2}}{\tilde{r}}\rho^{\prime}(\tilde{r}),$
(2)

where $\tilde{x}=x/r_{\rm s}$, $\tilde{y}=y/r_{\rm s}$, $\tilde{z}=z/r_{\rm
s}$, $\tilde{r}=\sqrt{\tilde{x}^{2}+\tilde{y}^{2}+\tilde{z}^{2}}$ and
$\rho^{\prime}(\tilde{r})={d\tilde{\rho}}/{d\tilde{r}}$, and

$\tilde{\Phi}(x,y,z)=\tilde{\Phi}_{0}(\tilde{r})+(\epsilon+\eta)[\tilde{\Phi}_{1}(\tilde{r})-\tilde{\Phi}_{0}(\tilde{r})]+(\epsilon\tilde{y}^{2}+\eta\tilde{z}^{2})\tilde{\Phi}_{2}(\tilde{r}),$
(3)

where $\tilde{\Phi}=\Phi/(4\pi G\rho_{\rm n}r_{\rm s}^{2})$, $\Phi$ is the
gravitational potential, and $\tilde{\Phi}_{0}$, $\tilde{\Phi}_{1}$ and
$\tilde{\Phi}_{2}$ are functions depending on $\tilde{\rho}(r)$, whose
definitions can be found in Ciotti & Bertin (2005). Here $\epsilon>0$ and
$\eta>0$ are dimensionless parameters which must be such that
$\tilde{\rho}(x,y,z)>0$ everywhere. We note that, though constructed
exploiting the technique of the homeoidal expansion, the density-potential
pairs given by the above formulae do not require $\epsilon$ and $\eta$ to be
much smaller than unity (see section 2 of Ciotti & Bertin, 2005).

Here we assume as parent spherical density profile the NFW model (Eq. 1),
which in dimensionless form reads

$\tilde{\rho}(\tilde{r})=\frac{1}{\tilde{r}\left(1+\tilde{r}\right)^{2}}.$ (4)

Using Equation (4) as $\tilde{\rho}$, Eq. (2) becomes

$\tilde{\rho}(x,y,z)=\frac{1}{\tilde{r}(1+\tilde{r})^{2}}-\frac{\epsilon\tilde{y}^{2}+\eta\tilde{z}^{2}}{\tilde{r}}\frac{1+3\tilde{r}}{\tilde{r}^{2}(1+\tilde{r})^{3}}.$
(5)

The dimensionless gravitational potential generated by the density profile (5)
is given by Eq. (3), where

$\tilde{\Phi}_{0}(r)=-\frac{\ln(1+\tilde{r})}{\tilde{r}},$ (6)
$\tilde{\Phi}_{1}(\tilde{r})=-\frac{1}{6\tilde{r}}+\frac{2}{3\tilde{r}^{2}}-\frac{\ln(1+\tilde{r})}{\tilde{r}^{3}}-\frac{1}{3\tilde{r}^{3}(\tilde{r}+1)}+\frac{1}{3\tilde{r}^{3}}-\frac{1}{3(1+\tilde{r})}$
(7)

and

$\tilde{\Phi}_{2}(\tilde{r})=\frac{1}{2\tilde{r}^{3}}-\frac{2}{\tilde{r}^{4}}+\frac{3\ln(1+\tilde{r})}{\tilde{r}^{5}}+\frac{1}{\tilde{r}^{5}(\tilde{r}+1)}-\frac{1}{\tilde{r}^{5}}.$
(8)

The second term in the r.h.s. of Eq. (5) breaks the spherical symmetry of the
distribution, subtracting density along the directions $\tilde{y}$ and
$\tilde{z}$. It is evident that the dimensionless density distribution (5)
would assume negative values if the directional subtraction of parent density
is sufficiently large. When we consider the NFW as the parent density profile,
the condition that at any point of space $\tilde{\rho}>0$, with $\tilde{\rho}$
given by Eq. (5), imposes $\epsilon,\eta\leq 1/3$ (see Ciotti & Bertin 2005
for the method to limit $\epsilon$ and $\eta$).

In particular, in this work we will consider prolate ($\eta=\epsilon$) and
oblate ($\epsilon=0$) axisymmetric density-potential pairs, having as parent
density distribution Eq. (4), which we will refer to as prolate NFW and oblate
NFW models, respectively. The prolate NFW model ($\eta=\epsilon$), renaming
$x$ as $z$, and viceversa, has density distribution

$\tilde{\rho}(R,z)=\frac{1}{\tilde{r}(1+\tilde{r})^{2}}-\frac{\eta\tilde{R}^{2}}{\tilde{r}}\frac{1+3\tilde{r}}{\tilde{r}^{2}(1+\tilde{r})^{3}}\quad\mbox{(prolate)},$
(9)

(shown for $\eta=1/3$ in the left panel of Fig. 1) and gravitational potential

$\begin{split}\tilde{\Phi}(R,z)=&-\frac{\ln(1+\tilde{r})}{\tilde{r}}+2\eta\left[-\frac{1}{6\tilde{r}}+\frac{2}{3\tilde{r}^{2}}-\frac{\ln(1+\tilde{r})}{\tilde{r}^{3}}\right.\\\
&\left.{}-\frac{1}{3\tilde{r}^{3}(\tilde{r}+1)}+\frac{1}{3\tilde{r}^{3}}-\frac{1}{3(1+\tilde{r})}+\frac{\ln(1+\tilde{r})}{\tilde{r}}\right]\\\
&+\eta\tilde{R}^{2}\left[\frac{1}{2\tilde{r}^{3}}-\frac{2}{\tilde{r}^{4}}+\frac{3\ln(1+\tilde{r})}{\tilde{r}^{5}}+\right.\\\
&\left.{}+\frac{1}{\tilde{r}^{5}(\tilde{r}+1)}-\frac{1}{\tilde{r}^{5}}\right]\quad\mbox{(prolate)}\end{split}$
(10)

(shown for $\eta=1/3$ in the left panel of Fig. 1), where
$R=\sqrt{x^{2}+y^{2}}$ is the radius in the equatorial plane and
$\tilde{R}=R/r_{s}$. The oblate NFW model ($\epsilon=0$), maintaining now the
names of the variables $x$, $y$ and $z$ as in Eq.s (2) and (3), has density
distribution

$\tilde{\rho}(R,z)=\frac{1}{\tilde{r}(1+\tilde{r})^{2}}-\frac{\eta\tilde{z}^{2}}{\tilde{r}}\frac{1+3\tilde{r}}{\tilde{r}^{2}(1+\tilde{r})^{3}}\quad\mbox{(oblate)},$
(11)

(shown for $\eta=1/3$ in the right panel of Fig. 1) and gravitational
potential

$\begin{split}\tilde{\Phi}(R,z)=&-\frac{\ln(1+\tilde{r})}{\tilde{r}}+\eta\left[-\frac{1}{6\tilde{r}}+\frac{2}{3\tilde{r}}-\frac{\ln(1+\tilde{r})}{\tilde{r}^{3}}+\right.\\\
&-\frac{1}{3\tilde{r}^{3}(1+\tilde{r})}+\left.\frac{1}{3\tilde{r}^{3}}-\frac{1}{3(1+\tilde{r})}+\frac{\ln(1+\tilde{r})}{\tilde{r}}\right]\\\
&+\eta\tilde{z}^{2}\left[\frac{1}{2\tilde{r}^{3}}-\frac{2}{\tilde{r}^{4}}+\frac{3\ln(1+\tilde{r})}{\tilde{r}^{5}}+\frac{1}{\tilde{r}^{5}(1+\tilde{r})}+\right.\\\
&\left.{}-\frac{1}{\tilde{r}^{5}}\right]\quad\mbox{(oblate)}\end{split}$ (12)

(shown for $\eta=1/3$ in the right panel of Fig. 1).

In both cases, $z$ is the symmetry axis. Given that the first order terms of
Eq.s (9) and (11) are $\propto\tilde{R}^{2}/\tilde{r}$ or
$\propto\tilde{z}^{2}/\tilde{r}$, respectively, the subtraction of parent
density is more significant in the outer regions. For $\eta\to 1/3$, it
induces a peanutshaped distribution sufficiently far from the center (see Fig.
1).

Figure 2: Profiles of density (upper panel) and gravitational potential (lower
panel) of DMS (red lines), DMO (blue lines) and DMP (green lines) halo models
(see Table 1). In particular, for our axisymmetric models we plot the density
and gravitational potential profiles along the symmetry axis (dashed lines)
and in the equatorial plane (dotted lines). The top axis in both panels
indicates the distance from the center normalized to $r_{\rm{500}}$ of DMS
halo model ($r_{\rm{500}}=1345\,\mathrm{kpc}$; we note that the values of
$r_{\rm{500}}$ of our three halo models differ by less than 2%).

###  2.2 Realistic halo models for massive clusters

A variety of halo shapes are expected from cosmological simulations (e.g. Bett
2012; Henson et al. 2017; see also section 7.5.3 of Cimatti et al. 2019),
depending mainly on the halo merging history. When approximating the halos as
ellipsoids, even if the majority of them is triaxial, the fact that the ratio
of two of the three principal semiaxes is close to unity justifies the use of
the spheroidal approximation for the description of these halos. However, for
one of our models we adopt the spherical approximation, which is appropriate
when the smallest-to-largest axial ratio is close to unity.

Using the density-potential pairs presented in Sect. 2.1, we build our halo
models as follows. The prolate and oblate NFW models (represented by Eq.s 9-10
and 11-12, respectively, which both give for $\eta=0$ the spherical NFW model)
are parameterized by $\rho_{\rm n}$, $r_{\rm s}$ and $\eta$. To be as far as
possible consistent with the predictions of cosmological simulations on the
smallest-to-largest axial ratio (see Allgood et al., 2006), in our spheroidal
halo models, we assume the largest possible flattening ($\eta=1/3$) compatible
with everywhere positive DM density distribution (see Sect. 2.1).

When a spherical NFW model is considered in the cosmological context, the
parameters $\rho_{\rm n}$ and $r_{\rm s}$ can be expressed as functions of
other two parameters, the virial mass $M_{\rm{\Delta}}$ and the concentration
$c_{\rm{\Delta}}$, which are routinely measured in cosmological simulations
(e.g. Dutton & Macciò 2014) and estimated for the halos of observed clusters
of galaxies (e.g. Ettori et al. 2010). $M_{\rm{\Delta}}$ is the mass measured
within a sphere of radius $r_{\rm{\Delta}}$, within which the average halo
density is $\Delta\rho_{\rm{crit}}(z)$, where the dimensionless quantity
$\Delta$ is the overdensity and $\rho_{\rm{crit}}(z)=3H^{2}(z)/(8\pi G)$ is
the critical density of the Universe at redshift $z$. The halo concentration
is $c_{\rm{\Delta}}=r_{\rm{\Delta}}/r_{\rm{-2}}$, where $r_{\rm{-2}}$ is the
radius where the logarithmic slope of the angle-averaged density profile is
$-2$. For the spherical NFW model $r_{\rm
s}=r_{\rm{-2}}=r_{\rm{\Delta}}/c_{\rm{\Delta}}$, where

$r_{\rm{\Delta}}=\left[\frac{M_{\rm{\Delta}}}{(4/3)\pi\Delta\rho_{\rm{crit}}(z)}\right]^{1/3},$
(13)

and we infer $\rho_{\rm n}$ from $c_{\rm{\Delta}}$ as

$\rho_{\rm
n}=\frac{\Delta}{3}\frac{\rho_{\rm{crit}}(z)c_{\rm{\Delta}}^{3}}{\ln(1+c_{\rm{\Delta}})-{c_{\rm{\Delta}}}/(1+c_{\rm{\Delta}})}.$
(14)

We now focus on the case of the standard overdensity value $\Delta=200$, and
thus consider $r_{\rm{200}}$, $M_{\rm{200}}$ and
$c_{\rm{200}}=r_{\rm{200}}/r_{\rm{-2}}$. To construct our specific spherical
NFW, hereafter referred to as ”dark matter spherical” (DMS) model, we set
$M_{\rm{200}}=10^{15}\,\mathrm{M_{\odot}}$ and $c_{\rm{200}}=3.98$, in
agreement with the mass-concentration relation of Dutton & Macciò (2014) at
redshift $z\approx 0$.

For the spheroidal halo models, we first compute the mass within the sphere of
radius $r$

$M(r)=4\pi\int_{0}^{r}\left(\int_{0}^{\sqrt{r^{2}-z^{2}}}\rho(R,z)RdR\right)dz,$
(15)

where $\rho(R,z)$ is given by Eqs. (9) or (11) for the prolate and oblate NFW
models, respectively. We then estimate $r_{\rm{200}}$ and $r_{\rm{-2}}$ in the
following way. The average density within the sphere of radius $r$ is
$\langle\rho\rangle(r)=3M(r)/(4\pi r^{3})$; while the angle-averaged density
profile $\rho_{\rm{shell}}(r)$ is estimated by measuring the average density
within concentric spherical shells

$\rho_{\rm{shell}}(r)=\frac{3[M(r+\delta r/2)-M(r-\delta r/2)]}{4\pi[(r+\delta
r/2)^{3}-(r-\delta r/2)^{3}]},$ (16)

where $\delta r=0.8\,\mathrm{kpc}$ is the thickness of the shell centered at
the radius $r$. $r_{\rm{\Delta}}$ is thus defined to be such that
$\langle\rho\rangle(r_{\rm{\Delta}})\simeq\Delta\rho_{\rm{crit}}(z)$, and
$r_{\rm{-2}}$ to be such that

$\left[\frac{d\ln\rho_{\rm{shell}}}{d\ln r}\right]_{r=r_{-2}}\simeq-2.$ (17)

The above equations can thus be used to estimate
$M_{\rm{200}}=M(r_{\rm{200}})$ and $c_{\rm{200}}$ for our flattened halo
models. In practice, to build the oblate and prolate NFW halo models,
hereafter referred to as ”dark matter oblate” (DMO) and ”dark matter prolate”
(DMP) models, respectively, we select pairs of values of $\rho_{\rm n}$ and
$r_{\rm s}$ such that $M_{200}\approx 10^{15}\,\mathrm{M_{\odot}}$ and
$c_{\rm{200}}$ is consistent with the $z\approx 0$ mass-concentration relation
of Dutton & Macciò (2014). The parameters of the halo models DMS, DMP and DMO
are reported in Tab. 1. The corresponding density and gravitational potential
profiles along the symmetry axis and in the equatorial plane are shown in Fig.
2. The upper panel of Fig. 2 shows that, comparing models with approximately
the same mass, because of the outward-increasing directional subtraction of
parent density discussed in Sect. 2.1 (see Fig. 1), the prolate model has
$\rho(R,0)$ steeper and $\rho(0,z)$ shallower than the density profile of the
spherical model, and vice versa for the oblate model. Analogous (but weaker)
trends are found in the gravitational potential profiles (lower panel of Fig.
2).

Table 1: Parameters of the adopted NFW halo models.

$\begin{array}[]{p{0.1\linewidth}lllll}\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr Model&\rho_{\rm n}[\mathrm{g/cm^{3}}]&r_{\rm
s}[\mathrm{kpc}]&\eta&M_{\rm{200}}[\mathrm{M_{\odot}}]&c_{\rm{200}}\\\ \vskip
3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
DMS&4.8\times 10^{-26}&519&0&1.00\times 10^{15}&3.98\\\ DMO&4.6\times
10^{-26}&600&1/3&1.00\times 10^{15}&3.96\\\ DMP&4.8\times
10^{-26}&700&1/3&1.01\times 10^{15}&4.27\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\end{array}$

555We refer to these Navarro-Frenk-White models as ”dark matter spherical”
(DMS), ”dark matter oblate” (DMO), and ”dark matter prolate” (DMP) one,
respectively.

##  3 Building cool-core clusters models with rotating ICM

In this Section we present axisymmetric rotating models of the ICM that, in
the absence of net cooling or heating, is in equilibrium in a given
axisymmetric gravitational potential, representative of an isolated cluster.
The ICM is sufficiently dense to cool in timescales much shorter than the
Hubble time in the cluster core and thus to flow into the center of the
gravitational potential well. However, as already mentioned in the
Introduction, the effect of cooling is expected to be efficiently counteracted
by heating mechanisms, such as AGN and stellar feedback. Thus, the adoption of
stationary models of the ICM is justified as long as there is balance between
cooling and heating in a time-averaged sense (e.g. McCourt et al., 2012),
provided the cluster does not undergo major interactions.

###  3.1 The equilibrium of rotating ICM in a cool-core cluster

Assuming that the total gravitational potential of the cluster $\Phi$ is time-
independent and axisymmetric, we can build simple models of stationary
rotating ICM by considering that the angular velocity of the gas is stratified
over cylinders (and thus that the gas distribution is barotropic, i.e. with
pressure stratified over density666More general (baroclinic) models, not
explored in this work, have vertical gradients of angular velocity, and
pressure not stratified over density.). Under these hypotheses, neglecting
magnetic fields (which are dynamically unimportant for the ICM; see, e.g.,
Bruggen 2013), the gas mass density $\rho_{\rm gas}(R,z)$ and pressure
$p(R,z)$ are related by $\nabla p=-\rho_{\rm gas}\nabla\Phi_{\rm{eff}}$,

$\Phi_{\rm{eff}}(R,z)=\Phi(R,z)-\Phi(R_{\star},z_{\star})-\int_{R_{\star}}^{R}\frac{u_{\phi}^{2}(R^{\prime})}{R^{\prime}}dR^{\prime},$
(18)

is the effective potential and $u_{\phi}(R)$ is the gas rotation velocity and
$(R_{\star},z_{\star})$ a reference point (e.g. Tassoul 1978).

From observations and hydrodynamical simulations, there is evidence that the
ICM is well described by polytropic distributions, essentially independent of
the halo mass (e.g. Ghirardini et al., 2019b), in which the pressure is
stratified over the density as a power law $p=p_{\star}(\rho_{\rm
gas}/\rho_{\rm gas,\star})^{\gamma^{\prime}}$, where $\gamma^{\prime}$ is the
polytropic index, $p_{\star}=p(R_{\star},z_{\star})$ and $\rho_{\rm
gas,\star}=\rho_{\rm gas}(R_{\star},z_{\star})$.

Hereafter, we model the ICM in a cool-core cluster through a two-component
composite polytropic distribution (e.g. Bianconi et al. 2013), by assuming a
polytropic index $\gamma^{\prime}_{\rm{OUT}}>1$ in the outer region and
$\gamma^{\prime}_{\rm{IN}}<1$ in the cool core. It is convenient to adopt
$(R_{\star},z_{\star})=(R_{\rm{break}},0)$, where $R_{\rm{break}}$ is a model
parameter that defines the size of the cool core. For any outward-increasing
axisymmetric potential, defining
$\Delta\Phi_{\rm{eff}}(R,z)=\Phi_{\rm{eff}}(R,z)-\Phi_{\rm{eff}}(R_{\rm{break}},0)$,
we have $\Delta\Phi_{\rm{eff}}(R,z)>0$ in the outer region, and
$\Delta\Phi_{\rm{eff}}(R,z)\leq 0$ in the cool core. Assuming the ideal gas
equation of state, the polytropic distributions of temperature and density of
the ICM, in our models of cool-core clusters, are given by

$n(R,z)=n_{\star}\left[1-\frac{\gamma^{\prime}_{\rm{OUT}}-1}{\gamma^{\prime}_{\rm{OUT}}}\frac{\mu
m_{\rm p}}{k_{\rm
B}T_{\star}}\Delta\Phi_{\rm{eff}}(R,z)\right]^{\frac{1}{\gamma^{\prime}_{\rm{OUT}}-1}}$
(19)

and

$T(R,z)=T_{\star}\left(\frac{n(R,z)}{n_{\star}}\right)^{\gamma^{\prime}_{\rm{OUT}}-1},$
(20)

where $\Delta\Phi_{\rm{eff}}(R,z)>0$, and by

$n(R,z)=n_{\star}\left[1-\frac{\gamma^{\prime}_{\rm{IN}}-1}{\gamma^{\prime}_{\rm{IN}}}\frac{\mu
m_{\rm p}}{k_{\rm
B}T_{\star}}\Delta\Phi_{\rm{eff}}(R,z)\right]^{\frac{1}{\gamma^{\prime}_{\rm{IN}}-1}}$
(21)

and

$T(R,z)=T_{\star}\left(\frac{n(R,z)}{n_{\star}}\right)^{\gamma^{\prime}_{\rm{IN}}-1},$
(22)

where $\Delta\Phi_{\rm{eff}}(R,z)\leq 0$. Here $n=\rho_{\rm gas}/(\mu m_{\rm
p})$ is the gas number density, and $n_{\star}=\rho_{\rm gas,\star}/(\mu
m_{\rm p})$; $\mu$, $m_{\rm p}$, and $k_{\rm B}$ are the mean molecular weight
(taken equal to 0.6), the proton mass and the Boltzmann constant,
respectively.

###  3.2 Rotation law and effective potential

Though the ICM rotation velocity curve is poorly constrained observationally
(see Liu & Tozzi, 2019, for an attempt), it is reasonable to expect that it
could have a relatively steep rise of azimuthal velocity in the cluster
center, a peak at intermediate radii, and a gradual fall in the outskirts (see
Baldi et al., 2017; Altamura et al., 2023b). In particular, following Bianconi
et al. (2013), we adopt the rotation law

$u_{\phi}(R)=u_{\rm{0}}\frac{S}{(1+S)^{2}},$ (23)

where $S\equiv R/R_{\rm{0}}$, $R_{\rm{0}}$ is a reference radius and
$u_{\rm{0}}$ a reference speed.

Substituting the rotation law (23) in Eq. (18), and integrating the rotational
component of the effective potential, we get the analytic effective potential
associated with this rotation law

$\Phi_{\rm{eff}}(R,z)=\Phi(R,z)-\Phi(R_{\star},z_{\star})-\left[I(R)-I(R_{\star})\right],$
(24)

where

$I(R^{\prime})=u_{\rm{0}}^{2}\left[\frac{1}{3}\left(1+\frac{R^{\prime}}{R_{\rm{0}}}\right)^{-3}-\frac{1}{2}\left(1+\frac{R^{\prime}}{R_{\rm{0}}}\right)^{-2}\right].$
(25)

Figure 3: Profiles of ICM rotation speed of our cluster models with spherical
(RMS; blue dashed curve), oblate (RMO; red dotted-dashed curve), and prolate
(RMP; green dotted curve) halos. For comparison, we also show as solid curves
the average rotation speed profiles of the ICM found in clusters formed in
cosmological simulations: in particular, the cyan and orange curves are
obtained from the functional forms given by, respectively, Baldi et al. (2017)
and Altamura et al. (2023b), assuming $r_{\rm{\Delta}}$ and $M_{\rm{\Delta}}$
as in model DMS. In the top axis, the radial coordinate in the meridional
plane is normalized to $r_{\rm{500}}$ of RMS model. The vertical line
indicates $R=R_{\rm{break}}$ for the spherical model, approximately enclosing
the cool-core region (red arrow), which has roughly the same extent in all
models (see Tab. 2). Table 2: Parameters of the cluster models with rotating
ICM.

$\begin{array}[]{p{0.1\linewidth}llllllll}\hline\cr\vskip 3.0pt plus 1.0pt
minus 1.0pt\cr Model&{\rm
Halo}&R_{\rm{break}}[\mathrm{kpc}]&n_{\rm{e,}\star}[\mathrm{cm^{-3}}]&T_{\star}[\mathrm{keV}]&u_{\rm{0}}[\mathrm{km/s}]&R_{\rm{0}}[\mathrm{kpc}]&\gamma^{\prime}_{\rm{IN}}&\gamma^{\prime}_{\rm{OUT}}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr RMS&{\rm DMS}&380&2.5\times 10^{-3}&7.3&1900&280&0.83&1.19\\\
RMO&{\rm DMO}&420&2.2\times 10^{-3}&7.4&1800&180&0.82&1.20\\\ RMP&{\rm
DMP}&360&2.5\times 10^{-3}&7.4&2300&280&0.85&1.19\\\ \vskip 3.0pt plus 1.0pt
minus 1.0pt\cr\hline\cr\end{array}$

777We refer to these cluster models with rotating ICM as ”rotating model
spherical” (RMS), ”rotating model oblate” (RMO) and ”rotating model prolate”
(RMP) depending on the spherical, oblate and prolate DM halo models assumed,
respectively. The corresponding halo models DMS, DMP, and DMO (cited in the
second column) are defined in Sect. 2.2.

###  3.3 Three representative models of massive cool-core clusters with
rotating ICM

Without focusing on a particular cluster, we propose three models with
rotating ICM representative of the observed population of massive
($M_{\rm{200}}\approx 10^{15}{\rm M}_{\odot}$) cool-core clusters, dubbed
”rotating model spherical” (RMS), ”rotating model oblate” (RMO) and ”rotating
model prolate” (RMP). In all these models we assume that the gas follows a
two-component composite polytropic distribution described by Eqs. (19-22), and
that the rotation law has the functional form (23). The effective potential is
thus in the form of Eqs. (24-25). In all cases, to compute the intrinsic and
emission properties of the ICM, we assume metallicity
$Z=0.3\,\mathrm{Z_{\odot}}$ (where $\mathrm{Z_{\odot}}$ is the solar
metallicity reported in Anders & Grevesse 1989), implying $n/n_{\rm{e}}=1.94$,
where $n=n_{\rm{i}}+n_{\rm{e}}$ is the gas number density, $n_{\rm{e}}$ is the
electron number density and $n_{\rm{i}}$ is the ion number density (assuming
full ionization).

In model RMS the total gravitational potential $\Phi$ is given by the
spherical gravitational potential of the halo model DMS described in Sect.
2.2. In models RMO and RMP, the total gravitational potential is axisymmetric,
being, respectively, the potential of the oblate halo model DMO and of the
prolate halo model DMP, described in 2.2. The values of the plasma parameters
$R_{\rm{break}}$, $n_{\rm{e,}\star}$, $T_{\star}$, $\gamma^{\prime}_{\rm{IN}}$
and $\gamma^{\prime}_{\rm{OUT}}$, and of the parameters of the rotation
pattern $R_{\rm{0}}$ and $u_{\rm{0}}$ are reported for all the models in Tab.
2. The ICM rotation speed profiles of the three models, with peak rotation
speeds in the range 400-$600\,\mathrm{km/s}$, are shown in Fig. 3. In the same
figure we plot, for comparison, the average rotation speed profiles of
clusters formed in MUSIC888The synthetic clusters of Baldi et al. (2017) are
selected from the MUSIC-2 sample (Sembolini et al., 2013) having
$M_{200}>5\times 10^{14}h^{-1}\,\mathrm{M_{\odot}}$, where
$h=H_{0}/(100\,\mathrm{km/s/Mpc})$. The corresponding curve in Fig. 3 is built
using data taken from table 4 of Baldi et al. (2017), for gas-VP2b rotation
curve in the so-called AGN simulation. (Baldi et al., 2017) and MACSIS999The
MACSIS cluster sample (Barnes et al., 2017) have friends-of-friends masses at
redshift $z=0$ $\gtrsim 10^{15}\,\mathrm{M_{\odot}}$. The corresponding curve
in Fig. 3 is built using data taken from table B2 of Altamura et al. (2023b)
for the $M_{\rm{500}}<9.7\times 10^{14}\,\mathrm{M_{\odot}}$ subsample in the
so-called gas-aligned case. (Altamura et al., 2023b) cosmological simulations.
Our rotation speed profiles are in between the average profiles found by Baldi
et al. (2017) and Altamura et al. (2023b), and can thus be considered, in this
sense, cosmologically motivated. Moreover, in Sect. 4 we show that our three
rotating models are realistic, in the sense that they have properties
consistent with the currently available observational data of real massive
clusters.

Figure 4: Thermodynamic profiles of the ICM in ”rotating model spherical”
(RMS) model. Upper panels. Radial (dotted) and vertical (dashed) profiles of
electron number density (left panel) and spectroscopic-like temperature (right
panel) for model RMS (blue lines) compared with the corresponding average
observed profiles (red solid lines) and their scatter (red shaded band), taken
from Ghirardini et al. (2019a, G19 in the legends). Here, $n_{\rm{e,shell}}$
(green solid line; left panel) is the angle-averaged (see Sect. 2.2) density
profile of model RMS, and $T_{\rm{sl,mean}}(R_{\rm
P})=\left[T_{\rm{sl}}(R_{\rm P},0)+T_{\rm{sl}}(0,R_{\rm P})\right]/2$ (green
solid line; right panel) is its average spectroscopic-like temperature
profile. Lower panels. Departure of average profile (green solid lines; see
above) of density (left panel) and spectroscopic-like temperature (right
panel) of model RMS from the average observed profiles (red solid lines) with
their scatter (red shaded band). The spherical radius $r$, and the radius in
the plane of the sky $R_{\rm P}=\sqrt{x^{2}+z^{2}}$ are given in kpc in the
bottom axis and normalized to $r_{\rm{500}}\simeq 1345\,\mathrm{kpc}$ in the
top axis. The violet and black dot-dashed vertical lines indicate
$R_{\rm{break}}$ and $r_{\rm{500}}$, respectively. The virial temperature of
this model, defined as in equation 59 of Voit (2005), is $T_{200}\simeq 6.46$
keV.

Figure 5: Density (left panel) and spectroscopic-like temperature (right
panel) profiles of ”rotating model oblate” (RMO) model. The figure display is
the same as Fig. 4, but for RMO model $r_{500}\simeq 1346\,\mathrm{kpc}$ and
$T_{200}\simeq 6.45$ keV.

Figure 6: Density (left panel) and spectroscopic-like temperature (right
panel) profiles of ”rotating model prolate” (RMP) model. The figure display is
the same as Fig. 4, but for RMP model $r_{500}\simeq 1366\,\mathrm{kpc}$ and
$T_{200}\simeq 6.53$ keV.

##  4 Comparison with observations

Here we compare with observational data some properties of the cool-core
cluster models with rotating ICM presented in Sect. 3.3.

###  4.1 Thermodynamic profiles of the ICM

Two directly observable quantities of the ICM are the emission measure, which
is a proxy for gas density, and the spectroscopic temperature ($T_{\rm{sp}}$),
which is the temperature associated with the emission in the X-ray spectrum.
Despite the difficulty to find an analytic expression of the spectroscopic
temperature, Mazzotta et al. (2004) have found a good approximation of it,
called the spectroscopic-like temperature, which, for an axisymmetric cluster
with symmetry axis $z$ orthogonal to the line of sight, is given by

$T_{\rm{sl}}(x,z)=\left.\int_{|x|}^{\infty}\frac{n_{\rm{i}}n_{\rm{e}}T^{1/4}\hat{r}d\hat{r}}{\sqrt{\hat{r}^{2}-x^{2}}}\middle/\int_{|x|}^{\infty}\frac{n_{\rm{i}}n_{\rm{e}}T^{-3/4}\hat{r}d\hat{r}}{\sqrt{\hat{r}^{2}-x^{2}}}\right.,$
(26)

where $T$ is the gas temperature (in this work, given by Eq.s 20, 22) and
$\hat{r}$ is the radius in the plane at height $z$, parallel to the equatorial
plane. Here, $x$ and $z$ are the coordinates in the plane of the sky, with the
origin in the cluster center.

According to the cosmological framework of formation and evolution of cosmic
structures, the population of galaxy clusters is expected to be homogeneous,
with “universal” profiles of the thermodynamic quantities (density,
temperature, pressure, and entropy) of the ICM that depend only on the mass
and redshift of the halo (see e.g. Vikhlinin et al., 2006; Pratt et al., 2010;
Arnaud et al., 2010; Eckert et al., 2012; Ghirardini et al., 2019a; Ettori et
al., 2023). This is particularly true in the regions dominated by the action
of gravity.

Recently, the combination of high-quality data of thermal Sunyaev-Zeldovich
effect (Sunyaev & Zeldovich 1972) and of X-ray observations have allowed
Ghirardini et al. (2019a) to reconstruct the universal thermodynamic profiles
of the XMM Cluster Outskirts Project (X-COP) sample (Eckert et al. 2017) out
to $r_{\rm{200}}$ with an unprecedented accuracy101010The X-COP sample
consists of 13 nearby, massive galaxy clusters selected on the basis of
signal-to-noise ratio of the Sunyaev-Zeldovich effect as resolved in the
_Planck_ maps (Planck Collaboration et al., 2014). Five of these objects are
classified as relaxed, cool-core systems accordingly to their central entropy.
(see also Vikhlinin et al. 2006 and Nagai et al. 2007a for the discussion on
the reliability of the reconstruction method).

We thus compare our models of the rotating ICM in equilibrium in cool-core
clusters of $M_{\rm{200}}\approx 10^{15}\,\mathrm{M_{\odot}}$ with these
thermodynamic profiles in Fig.s 4-6, where the observed temperature is
$T_{\rm{sp}}$. We note that in the inner regions of the cool core (i.e.
$r<60\,\mathrm{kpc}$) the spectroscopic-like temperature of the models departs
significantly from the observed profile of the spectroscopic temperature, this
discrepancy is not very meaningful, given the observational limitations on the
recovery of the thermodynamic properties in such central regions. The
thermodynamic properties of models RMS, RMO and RMP, with different halo
shapes and rotation patterns, are thus reasonably representative of the
average properties of the ICM in massive cool-core clusters.

Once shown that the ICM pressure is stratified over the ICM density following
a power-law function (i.e. that the distribution is polytropic), in the X-COP
sample Ghirardini et al. (2019b) have found polytropic indices that, depending
on the cluster radius, span from 0.75 (in the inner region) to 1.25 (in the
outer region), independent of the cluster mass. The polytropic indices of our
rotating ICM models (RMS, RMO and RMP), $\gamma^{\prime}_{\rm{IN}}\simeq 0.8$
and $\gamma^{\prime}_{\rm{OUT}}\simeq 1.2$ (see Tab. 2), are fully consistent
with those of the observed clusters.

We note that reproducing the observed thermodynamic profiles under the
assumption of rotating ICM is not guaranteed: this is discussed in Appendix A,
where we present an illustrative example of a model with strongly rotating
ICM, which fails to reproduce some characteristic features of the observed
population of massive clusters.

###  4.2 Flattening of the X-ray surface-brightness distributions

The gas rotation and halo flattening leave a trace in the shape of the X-ray
surface-brightness distribution. Here, we compare the shape of the X-ray
surface-brightness distribution in our models and in real massive clusters.
One way to account for the departure of the iso-surface brightness contours
from the circular shape is through an average axial ratio, based on the
inertia’s tensor of surface brightness distribution (see Buote & Canizares,
1992, 1994).

Assuming to observe our models edge-on (i.e. with symmetry axis orthogonal to
the line of sight), the surface brightness is

$\Sigma(x,z)=2\int_{|x|}^{\infty}\frac{n_{\rm{i}}n_{\rm{e}}\Lambda(T)\hat{r}d\hat{r}}{\sqrt{\hat{r}^{2}-x^{2}}},$
(27)

where $\Lambda(T)$ is the cooling function (in particular we take $\Lambda$
from Tozzi & Norman 2001, for $Z=0.3\,\mathrm{Z_{\odot}}$).

Figure 7: Surface-brightness maps of models RMO (left panel) and RMP (right
panel). The boxes (with origin in the cluster center) extend out to $\approx
r_{\rm{500}}$ (see white arrows).

Using Eq. (27), we compute the surface-brightness distribution of our models,
which is shown in Fig. 7 for models RMO and RMP.

Given that the inertia’s tensor of the surface brightness distribution is in
diagonal form for a cluster observed edge-on, its diagonal terms are
$I_{20}=\sum_{i=1}^{P}\Sigma_{i}x_{i}^{2}$ and
$I_{02}=\sum_{i=1}^{P}\Sigma_{i}z_{i}^{2}$, where $\Sigma_{i}$ is the surface
brightness (given by Eq. 27) at the grid point of plane-of-the-sky coordinates
$(x_{i},z_{i})$, called hereafter pixel, and $P$ is the total number of
pixels. From the definition of diagonal terms, it follows that the average
axial ratio is $\zeta={I_{\min}}/{I_{\max}}$, where
$I_{\max}=\max\\{I_{20},I_{02}\\}$ and $I_{\min}=\min\\{I_{20},I_{02}\\}$.

Figure 8: Comparison of the average axial ratio of surface-brightness
distribution of models (vertical lines) RMS (black), RMO (yellow) and RMP
(red) with the cumulative distribution of the average axial ratios measured
for the CHEX-MATE clusters (blue) by Campitiello et al. (2022).

In this work, we compare our models to the results obtained for the XMM
Cluster Heritage Project (CHEX-MATE) sample111111The CHEX-MATE sample is a
signal-to-noise limited sample of 118 galaxy clusters detected by Planck via
their Sunyaev-Zel’dovich effect; it is composed by two subsamples: the Tier-1,
including the population of clusters at the most recent time ($z<0.2$) and the
Tier-2, with the most massive objects to have formed thus far in the history
of the Universe; see http://xmm-heritage.oas.inaf.it/ for further details.
(CHEX-MATE Collaboration et al. 2021), which contains both cool-core and non-
cool-core clusters observed within their $r_{\rm{500}}$. To match the clusters
of this sample, we compute the average axial ratio of our cluster models only
in the plane-of-the-sky region defined by $R_{\rm{break}}\leq|x|\leq
r_{\rm{500}}$ and $R_{\rm{break}}\leq|z|\leq r_{\rm{500}}$. In Fig. 8 we
present the cumulative distribution of average axial ratios of CHEX-MATE
clusters, where the 25th, 50th, and 75th percentiles are $\zeta=0.77$,
$\zeta=0.85$, and $\zeta=0.89$, respectively (see also figure B.1 of
Campitiello et al. 2022).

The models RMS, RMO and RMP have, respectively, $\zeta=0.96$, $\zeta=0.87$,
and $\zeta=0.93$, corresponding to the 93th, 62th and 85th percentiles of the
distribution of the CHEX-MATE sample and thus are consistent with the less
flattened population of massive clusters. The halos formed in cosmological
simulations (having average ellipticity $\approx 0.5$; e.g. Allgood et al.
2006) tend to be more flattened than our aspherical halo models (having
ellipticity $\sim 0.3$). The relatively high values of $\zeta$ of our cluster
models are a consequence of the method adopted to build the density-potential
pairs of our oblate and prolate halo models: given the requirement of
everywhere positive halo density, the Ciotti & Bertin (2005) method prevents
from building highly flattened halos (see Sect. 2). However, the flattening of
our ICM models is due only to rotation and halo shape, while mergers,
substructures, and anisotropic turbulence, all neglected in our models, are
likely present in real clusters, where they can contribute to lower $\zeta$.

Figure 9: Profiles of the hydrostatic mass bias of models RMS (blue dashed
line), RMO (red dashed-dotted line) and RMP (green dotted line), compared to
the observational measurements (points): in particular, to average hydrostatic
mass biases, taken from figure 5 of Ettori et al. (2019, black points), from
tables 2 and 4 of Lovisari et al. (2020, orange points), and from table 4 of
Mahdavi et al. (2013, brown points). The vertical error bars of observational
data indicate 1$\sigma$ uncertainty on $b$. In the bottom axis, the radius is
normalized to $r_{\rm{500}}$ of the corresponding model, with the RMS model
radii $r_{2500}$, $r_{1000}$, $r_{500}$, and $r_{200}$ reported on the plot
top. The top axis reports the corresponding overdensity $\Delta$ of the RMS
model (for models RMO and RMP the overdensity values are very similar to those
of model RMS). The radius corresponding to a given overdensity is computed
from the true mass profile, which, in the case of observational data, is
assumed to be obtained from the weak gravitational lensing analysis. The red
arrow indicates the extent of the cool core.

###  4.3 Hydrostatic mass bias

The mass recovered under the assumption of hydrostatic equilibrium and
spherically symmetric gravitational potential is (e.g. Lau et al., 2013)

$M_{\rm{HE}}(<r)=-\frac{r^{2}}{G\rho_{\rm{gas,shell}}}\frac{\partial
p_{\rm{shell}}}{\partial r},$ (28)

where $p_{\rm{shell}}$ and $\rho_{\rm{gas,shell}}$ are, respectively, the
angle-averaged (see Sect. 2.2) pressure and density profiles. The hydrostatic
mass bias profile is

$b(r)=1-\frac{M_{\rm{HE}}(<r)}{M_{\rm{true}}(<r)},$ (29)

where $M_{\rm{true}}$ is the angle-averaged mass (Eq. 15) of the halo model
that generates the gravitational potential, into which the ICM is in
equilibrium. Using Eqs. (28) and (29), we compute $b(r)$ for our cluster
models, which we plot in Fig. 9, finding in all cases that the hydrostatic
mass bias, except for the central region, tends to decrease with radius.

The mass estimates from weak gravitational lensing are believed to be
significantly less biased than those from X-ray observations (e.g. Meneghetti
et al., 2010; Lee et al., 2018), at least for nonmerging clusters (Lee et al.,
2023). Thus, when we consider the hydrostatic mass bias of real clusters, we
take the cluster mass from weak lensing as an estimate of $M_{\rm{true}}$. In
particular, in Fig. 9 we compare the hydrostatic mass bias of our cluster
models to the following measurements:

  * •

The error-weighted average of the hydrostatic mass biases of the massive
clusters in the X-COP sample, which are classified as relaxed, at true (i.e.
obtained from weak lensing measurements) $r_{\rm{500}}$ and $r_{\rm{200}}$.
The hydrostatic and weak lensing masses are determined by Ettori et al. (2019)
and Herbonnet et al. (2020), respectively.

  * •

The average hydrostatic mass bias of a large subsample of the _Planck_
Sunyaev-Zeldovich effect galaxy clusters (62 clusters of true masses in the
range $3\times 10^{14}-2\times 10^{15}\,\mathrm{M_{\odot}}$, at $z<0.5$) at
$1\,\mathrm{Mpc}$ and true $r_{\rm{500}}$. The hydrostatic masses are
determined by Lovisari et al. (2020), while the weak lensing masses are taken
from Sereno (2015).

  * •

The average hydrostatic mass bias of the relaxed cluster subsample (most of
which are found to have prominent cool cores) of the Canadian Cluster
Comparison Project (50 clusters at $0.15<z<0.55$, selected with the X-ray
spectroscopic temperature $>3\,\mathrm{keV}$), at true $r_{2500}$, $r_{1000}$
and $r_{\rm{500}}$. The hydrostatic and weak lensing masses are determined by
Mahdavi et al. (2013) and Hoekstra et al. (2012), respectively.

As shown by Fig. 9, the rotation support assumed in our cluster models is
realistic, in the sense that it induces hydrostatic mass bias comparable to or
lower than those detected in real clusters (with the exception of the estimate
of Mahdavi et al. 2013 at $r_{2500}$; see Sect. 5.4 for a discussion). On the
basis of the comparison of the thermodynamic profiles of the ICM, shape of
surface-brightness distribution, and hydrostatic mass bias of our cluster
models with observations, we conclude that our models are consistent with the
main cluster observables that are currently able to constrain the rotation
speed of the ICM in cool-core clusters.

##  5 Measuring rotation with X-ray spectroscopy

In the near future, the advent of the microcalorimeters, soft X-ray
spectrometers such as _Resolve_ onboard _XRISM_ , a JAXA/NASA collaborative
mission with ESA participation, will provide us with X-ray spectra at high
spectral resolution (Tashiro et al., 2018), allowing us to measure the line-
of-sight component of the ICM velocity (e.g. Ota et al. 2018) and thus
estimate its rotation support. In this Section, using the configurations for
_Resolve_ , we present a set of mock X-ray spectra of the rotating ICM in our
cluster models and we assess the detectability of rotation with X-ray
spectroscopy.

Table 3: Characteristics of the mock pointings.

$\begin{array}[]{p{0.25\linewidth}lll}\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr
Region&\left|x\right|\,[\mathrm{kpc}]&z\,[\mathrm{kpc}]&Radius\,[\mathrm{kpc}]\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr R1&200&0&100\\\ R2&650&0&150\\\ R3&1150&0&250\\\ \vskip 3.0pt plus
1.0pt minus 1.0pt\cr\hline\cr\end{array}$

121212Coordinates (second and third columns) and radius (fourth column) of
circular regions of the mock observations in the plane of the sky (with the
origin in the cluster center).

###  5.1 Building mock spectra of the rotating ICM

Here, we present our mock spectra, focusing primarily on the kinematic
signatures. Given that, for a temperature of the ICM higher than
$3\,\mathrm{keV}$, a mock multitemperature source spectrum (i.e. constructed
from a multitemperature model) and the best fit to this spectrum with a
singletemperature model are indistinguishable in the X-rays (Mazzotta et al.
2004), we directly simulate the X-ray thermal emission of the ICM of our
models through a singletemperature model. In particular, we use the velocity
Broadened Astrophysical Plasma Emission
Code131313https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/node136.html.
(BAPEC), where a parameter accounts for a general broadening of the X-ray
emission lines, including the thermal broadening of the ionized metals, and
any other contribution in the form of “Doppler broadening” due to the
cumulative effect of the different Doppler shifts caused by a distribution of
the velocities of the ions. With this model, the Doppler shift of the lines is
parametrized by an effective redshift ($z_{\rm{eff}}$), which can be different
from the cluster’s redshift $z_{0}$ due to the action of a coherent, bulk
motion, and their equivalent width is regulated by the metallicity which we
fix to $0.3\,\mathrm{Z_{\odot}}$. We observe our models of cool-core clusters
edge-on, to maximize the contribution of rotation to the l.o.s. velocity,
which is thus

$v_{\rm{los}}(x,z)=\frac{1}{\Sigma(x,z)}\int_{|x|}^{\infty}\frac{n_{\rm{i}}n_{\rm{e}}\Lambda(T)|x|u_{\phi}(\hat{r})d\hat{r}}{\sqrt{\hat{r}^{2}-x^{2}}},$
(30)

where $\Sigma(x,z)$ is given by Eq. (27), $u_{\phi}(|x|)$ by (23) with
parameters $u_{\rm{0}}$ and $R_{\rm{0}}$ reported in Tab. 2. To decouple the
rotation from the contributions to the broadening of X-ray emitting lines, we
observe sufficiently large regions, to be spatially resolved by the
spectrometer _Resolve_ , where the ICM is either approaching or receding: in
particular, we simulate the observation of regions R1, R2, and R3, reported in
Tab. 3. The l.o.s. speed of the ICM in our cluster models is consistent with
the observational upper limit on the rotation speed of $500\,\mathrm{km/s}$ in
the cool cores of real galaxy clusters (e.g. Sanders et al., 2011; Pinto et
al., 2015; Bambic et al., 2018): for all models, $|v_{\rm{los}}|\lesssim
450\,\mathrm{km/s}$ in region R1, which belongs to the inner region. We find
that the energy shift of a $6\,\mathrm{keV}$ line due to the rotation speed of
$400\,\mathrm{km/s}$ is $8\,\mathrm{eV}$. _Resolve_ , thanks to its energy
resolution of $\sim 7\,\mathrm{eV}$ at
$E=6\,\mathrm{keV}$141414https://xrism.isas.jaxa.jp/research/analysis/manuals/xrqr_v2.1.pdf,
has the potential to detect such an energy shift, unlike the currently
available X-ray CCD detectors with energy resolution in the order of $\approx
100\,\mathrm{eV}$. Assuming $v_{\rm{los}}$ positive for approaching ICM, we
compute $z_{\rm{eff}}$ as

$z_{\rm{eff}}=(1+z_{0})\sqrt{\frac{1+\frac{\langle
v_{\rm{los}}\rangle}{c}}{1-\frac{\langle v_{\rm{los}}\rangle}{c}}}-1,$ (31)

where we always take $z_{0}=0.05$. In this Section, $\langle...\rangle$ refers
to the average along all the lines of sights, which cross one of the regions
of Tab. 3: following Roncarelli et al. (2018), we use as a weight for the
average along the l.o.s. $n_{\rm{i}}n_{\rm{e}}\Lambda(T)$, except for the
spectroscopic-like temperature, which is defined by Eq. (26).

Table 4: Input parameters of our mock spectra.

$\begin{array}[]{p{0.3\linewidth}llll}\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr Model -
Region&T[\mathrm{keV}]&norm&z_{\rm{eff}}(R)&z_{\rm{eff}}(B)\\\ \vskip 3.0pt
plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr RMS -
R1&6.36&0.0072&0.0512&0.0488\\\ RMS - R2&6.01&0.0014&0.0511&0.0489\\\ RMS -
R3&4.96&0.0007&0.0509&0.0491\\\ RMO - R1&6.23&0.0078&0.0512&0.0488\\\ RMO -
R2&6.18&0.0014&0.0509&0.0491\\\ RMO - R3&5.07&0.0007&0.0506&0.0494\\\ RMP -
R1&6.55&0.0061&0.0515&0.0485\\\ RMP - R2&6.00&0.0012&0.0514&0.0486\\\ RMP -
R3&4.92&0.0005&0.0511&0.0489\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\end{array}$

151515The parameter $norm$ accounts for the normalization of the spectrum. We
quote the effective redshift ($z_{\rm{eff}}$) both for receding (identified by
R) and approaching (by B) ICM.

At $E>2\,\mathrm{keV}$, the strongest and better-modeled lines of the X-ray
spectra are due to the transitions of inner shell electrons of the iron in the
ICM (see e.g. Zhuravleva et al. 2012; Ota et al. 2018, and Fig.10, where we
show a typical spectrum of the ICM, discussed in detail below). The iron thus
represents the reference element for the calculations on the line broadening.
Previous works have shown that, though being formally independent of the line
broadening, the best-fitting Doppler shift of X-ray emitting lines is
decisively affected by their broadening. In particular, on the basis of the
results of the fits to mock observations of the rotating ICM, Bianconi et al.
(2013) point out that, at a fixed signal-to-noise ratio, the best-fitting
Doppler shift of the centroids of the X-ray emission lines suffers from a
higher error when increasing their overall broadening above $\approx
300\,\mathrm{km/s}$. Such a consideration brings us to take into account the
following contributions to the broadening of the strong iron emitting lines:

  * •

The random motion of iron ions produces the thermal broadening
($\sigma_{\rm{TH}}$), which is accounted for by the spectroscopic-like
temperature (Eq. 26) in BAPEC model. In our mock spectra,
$90\,\mathrm{km/s}<\sigma_{\rm{TH}}<110\,\mathrm{km/s}$. We notice that the
adopted value of the spectroscopic-like temperature represents a weighted-
average of the observed distribution in the integrated spectra, with typical
dispersions around this central value in the range $(0.37-0.53)$ keV for all
the models.

  * •

The turbulence, which is believed to be ubiquitous in galaxy clusters on the
basis of hydrodynamical simulations (e.g. Vazza et al., 2017) and observations
(e.g. Schuecker et al., 2004), is expected to induce a nonnegligible
contribution (in the order of a few hundred km/s) to the broadening of the
iron emitting lines, known as turbulent broadening $\sigma_{\rm{turb}}$ (e.g.
Zhuravleva et al. 2012). In the following analysis, we consider a
$\sigma_{\rm{turb}}$ of both 0 and 500 km/s, the latter one considered as an
upper limit on the turbulent velocity dispersion in typical galaxy clusters
(see e.g. Pinto et al., 2015).

Figure 10: Typical spectrum of the ICM in equilibrium in our cluster models.
Left panel. Fit (upper panel; red line) and residuals (lower panel) to the
mock spectrum of the receding ICM (black crosses), in equilibrium in RMS
model, for the plane-of-the-sky region R1, taking
$\sigma_{\rm{turb}}=500\,\mathrm{km/s}$. FeXXV and FeXXVI (with rest-frame
energies close to 6.7 and $6.9\,\mathrm{keV}$, respectively), referred to as
Fe-K, are the prominent emission lines in the upper panel. In the lower panel,
the green line traces the null residuals. We rebinned the data for display
purposes (we recall that using the C-Statistics the data without background
are not binned). Right panel. Zoom in the range $6-7\,\mathrm{keV}$ of the
yellow band in the left panel, where the emission lines FeXXV and FeXXVI are
emphasized. Here, we rebin the data in a way different from the left panel for
display purposes.

In order to mimic an observation as far as possible realistic, we introduce a
typical absorption due to the Milky Way ($N_{\mathrm{H}}=5\times
10^{20}\,\mathrm{cm^{-2}}$; e.g. HI4PI Collaboration et al. 2016), using the
PHotoelectric ABSorption
model161616https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/XSmodelPhabs.html.
(PHABS). Assuming also the parameters of Table 4, an exposure time of
100$\,\mathrm{ks}$, and convolving in the range $0.5$-$8\,\mathrm{keV}$ with
instrumental response functions of _Resolve_ 171717See
https://heasarc.gsfc.nasa.gov/docs/xrism/proposals/. in Xspec181818See
https://heasarc.gsfc.nasa.gov/xanadu/xspec/. (Arnaud 1996), we build mock
spectra of the rotating ICM of our cluster models (see an example in Fig. 10).
We do not consider any background in our mock spectra, working in the ideal
condition of the analysis of very bright regions. To account for the different
behavior of response matrices at different energies, for any region under
consideration, we present two mock spectra: one for “approaching”/blue-shifted
ICM and another for “receding”/red-shifted ICM, with typical differences in
energy of the line centroids of a few tens of eV (see Fig. 10). Moreover, to
assess the impact of the turbulence on the fit to the shape of the emitting
lines, for any region under consideration we present a couple of mock spectra:
one with $\sigma_{\rm{turb}}=500\,\mathrm{km/s}$ and another without
turbulence. The emission at 6-$7\,\mathrm{keV}$ (yellow vertical band in the
left-hand panel of Fig. 10) provides the most valuable information to measure
the l.o.s. speed (see also Ota et al. 2018), because of the relatively high
emissivity of iron emitting lines FeXXV and FeXXVI (see also the right-hand
panel of Fig. 10).

Using the C-statistics (Cash, 1979), as suggested by Ota et al. 2018 (see also
Humphrey et al., 2009; Kaastra, 2017), and thawing all the parameters except
$N_{\mathrm{H}}$, we then fit the absorbed model BAPEC to the mock spectrum in
Fig. 10. With the purpose of studying the _Resolve_ ability to detect the ICM
rotation (see Sect. 5.2), in Tab. 5 we report the expectation values and the
statistical errors of the parameters of the fit to the X-ray emission lines:
the effective redshift $z_{\rm{eff}}$ (that regulates the energy shift of
their centroids), the turbulent velocity $\sigma_{\rm{turb}}$ (that
contributes to their broadening), the metallicity $Z$ (that regulates their
intensity), and the spectroscopic temperature $T$ (that is related to a
contribution to their broadening).

Table 5: Results of the spectral analysis of the mock spectra of the ICM.

$\begin{array}[]{p{0.13\linewidth}l|llll}\hline\cr\vskip 3.0pt plus 1.0pt
minus 1.0pt\cr Model -
Region&\sigma_{\rm{turb}}[\textrm{km/s}]&z_{\rm{eff}}(S_{\rm{eff}})&\sigma_{\rm{turb}}[\mathrm{km/s}]&Z[Z_{\odot}](S_{\rm{Z}})&T[\mathrm{keV}](S_{\rm{T}})\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr RMS - R1-R&0&0.0512\pm 0.0001(0.0)&1\pm 46&0.28\pm 0.02(1.0)&6.37\pm
0.16(0.0)\\\ RMS - R1-R&500&0.0513\pm 0.0002(0.5)&478\pm 57&0.29\pm
0.02(0.5)&6.16\pm 0.15(1.3)\\\ RMS - R2-R&0&0.0511\pm 0.0001(0.0)&4\pm
66&0.27\pm 0.05(0.6)&5.96\pm 0.37(0.1)\\\ RMS - R2-R&500&0.0515\pm
0.0005(0.8)&522\pm 151&0.36\pm 0.06(1.0)&6.07\pm 0.31(0.2)\\\ RMS -
R3-R&0&0.0510\pm 0.0002(0.5)&3\pm 79&0.31\pm 0.06(0.2)&5.13\pm 0.26(0.7)\\\
RMS - R3-R&500&0.0517\pm 0.0005(1.6)&826\pm 252&0.31\pm 0.06(0.2)&5.17\pm
0.34(0.6)\\\ \\\\[0.28453pt] RMO - R1-R&0&0.0512\pm 0.0001(0.0)&47\pm
43&0.28\pm 0.02(1.0)&6.33\pm 0.15(0.7)\\\ RMO - R1-R&500&0.0512\pm
0.0002(0.0)&486\pm 49&0.30\pm 0.02(0.0)&6.28\pm 0.15(0.3)\\\ RMO -
R2-R&0&0.0509\pm 0.0001(0.0)&0\pm 79&0.33\pm 0.05(0.6)&5.82\pm 0.32(1.1)\\\
RMO - R2-R&500&0.0503\pm 0.0005(1.2)&477\pm 123&0.26\pm 0.04(1.0)&5.80\pm
0.33(1.2)\\\ RMO - R3-R&0&0.0507\pm 0.0002(0.5)&0\pm 105&0.31\pm
0.06(0.2)&5.59\pm 0.41(1.3)\\\ RMO - R3-R&500&0.0504\pm 0.0004(0.5)&299\pm
161&0.38\pm 0.07(1.1)&4.87\pm 0.32(0.6)\\\ \\\\[0.28453pt] RMP -
R1-R&0&0.0515\pm 0.0001(0.0)&83\pm 67&0.27\pm 0.02(1.5)&6.53\pm 0.18(0.1)\\\
RMP - R1-B&0&0.0485\pm 0.0001(0.0)&3\pm 69&0.29\pm 0.02(0.5)&6.83\pm
0.20(1.4)\\\ RMP - R1-B&500&0.0484\pm 0.0002(0.5)&445\pm 62&0.29\pm
0.02(0.5)&6.49\pm 0.17(0.4)\\\ RMP - R2-R&0&0.0514\pm 0.0001(0.0)&2\pm
83&0.32\pm 0.05(0.4)&5.36\pm 0.29(2.2)\\\ RMP - R2-B&0&0.0486\pm
0.0001(0.0)&7\pm 68&0.30\pm 0.05(0.0)&6.21\pm 0.38(0.6)\\\ RMP -
R2-B&500&0.0483\pm 0.0005(0.6)&565\pm 169&0.37\pm 0.06(1.2)&5.72\pm
0.34(0.8)\\\ RMP - R3-R&0&0.0512\pm 0.0002(0.5)&107\pm 85&0.37\pm
0.08(0.9)&5.86\pm 0.49(1.9)\\\ RMP - R3-B&0&0.0489\pm 0.0003(0.0)&8\pm
158&0.23\pm 0.07(1.0)&5.20\pm 0.44(0.6)\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\end{array}$

191919Input conditions of the ICM (two columns on the left of the black
vertical line): the name of the model of the rotating ICM (RMS, RMO, RMP), the
name of the regions where the spectra are integrated (R1, R2 or R3 described
in Tab. 3; first column), with the positive (identified by -R) or negative (by
-B) input $\langle v_{\rm{los}}\rangle$ (first column), and the assumed
turbulent velocity dispersion (second column). Output parameters (four columns
on the right of the vertical black line) and their statistical errors (at
$1\sigma$ of confidence level) of the best fits in our mock spectra of the
rotating ICM in equilibrium in model RMS, reporting in brackets the
significance of the corresponding ”best-fit” results (computed by using Eq.
32).

###  5.2 Significativity of the recovered observable quantities

In this Section, we discuss how the BAPEC parameters $z_{\rm{eff}}$,
$\sigma_{\rm{turb}}$, $Z$, and $T$ are recovered from the fit of our mock
spectra, once convolved with the _Resolve_ response matrices in the X-rays.

We thus introduce the significativity of the ”best-fit” quantity
$Q^{\rm{out}}$ (reported in Tab. 5):

$S_{\rm{Q}}=\frac{\left\lvert
Q^{\rm{out}}-Q^{\rm{in}}\right\rvert}{err_{\rm{Q}}},$ (32)

where $Q^{\rm{in}}$ and $err_{\rm{Q}}$ are the input parameter (reported in
Tab. 4) and the error of $Q^{\rm{out}}$ to $\simeq 68\%$ of confidence
(reported in Tab. 5), respectively. $S_{\rm{Q}}$ measures at which level of
confidence the ”best-fit” parameters match the input values: $S_{\rm{Q}}\leq
1$ means that the spectral analysis recovers the input parameter $Q^{\rm{in}}$
within $\simeq 68\%$ of confidence. A lower $S_{\rm{Q}}$ thus corresponds to a
better recovery of the observable property $Q^{\rm{in}}$ via the spectral
best-fitting. Using $Q=\\{T,z_{\rm{eff}},Z\\}$ in Eq. (32), we estimate their
significance, reported in Tab. 5, where we refer to the significance of
$z_{\rm{eff}}$ as $S_{\mathrm{eff}}$. The input parameters of the
spectroscopic temperature, metallicity, and effective redshift in most
spectral analyses are recovered within $1\sigma$ of confidence level. To
illustrate the results of these mock observations, we focus on the best and
worst recoveries of the rotation speed of the ICM. First, we compare the
effective redshift measured in R1 region of RMS cluster model with receding,
nonturbulent ICM (see the third column and the first row of Tab. 5) to the
corresponding input $z_{\rm{eff}}$ (see the fifth column and the first row of
Tab. 4): the output $z_{\rm{eff}}$ perfectly matches the input $z_{\rm{eff}}$
(i.e. $S_{\mathrm{eff}}=0$, using Eq. 32). Second, we compare the effective
redshift measured in the region R3 of the RMS cluster model with receding,
turbulent ICM (see the third column and the sixth row of Tab. 5) to the
corresponding input $z_{\rm{eff}}$ (see the fifth column and the third row of
Tab. 4): the output $z_{\rm{eff}}$ matches the input $z_{\rm{eff}}$ at
$1.6\sigma$. Though each measurement depends on the signal-to-noise ratio,
this exercise shows the ability of _Resolve_ to measure the rotation speed of
the ICM at high significance, assuming that the cluster cosmological redshift
and Milky Way absorption are known. We note that the statistical errors
associated to the ”best-fit” spectroscopic temperature, effective redshift,
and metallicity (Tab. 5) depend on the signal-to-noise ratio: these errors
decrease by raising the signal-to-noise ratio, i.e. by increasing the exposure
time (here assumed to be 100 ks) and enlarging the plane-of-the-sky exposure
region (see Tab. 3). For instance, comparing the RMS-R3-R spectral analyses
with $\sigma_{\rm{turb}}=0$ and $\sigma_{\rm{turb}}=500\,\mathrm{km/s}$ (see
Tab. 5), we note that, keeping fixed the signal-to-noise ratio, the increase
in $\sigma_{\rm{turb}}$ induces a higher error of ”best-fit” effective
redshift and turbulent velocity. Most importantly, in this case the input
$z_{\rm{eff}}$ is recovered within $1\sigma$ if input $\sigma_{\rm{turb}}=0$
and out of $1\sigma$ if input $\sigma_{\rm{turb}}=500\,\mathrm{km/s}$. From
the entire set of our results, the significativity of effective redshift
appears to be sensitive to the input turbulent velocity dispersion: the
spectral best-fitting recovers, on average, the input $z_{\rm{eff}}$ with
higher $S_{\rm{eff}}$ (i.e. within a higher confidence level), when we
increase the input $\sigma_{\rm{turb}}$. This outcome is in line with the
picture that emerged from the X-ray mock observations of galaxy clusters from
hydrodynamical simulations, where the increase in the complexity of the
velocity field (here, obtained with increasing turbulent velocity dispersion,
at fixed rotation speed) reduces our ability to recover the kinematic
properties of the ICM (e.g. Roncarelli et al., 2018).

We have also studied the covariance among the BAPEC best-fit parameters. We
obtain that the off-diagonal correlation coefficients are significantly lower
than 0.2, implying no relevant cross-correlation, between $z_{\rm{eff}}$,
$\sigma_{\rm{turb}}$, $Z$, and $T$. A partial exception is the $\approx 0.2$
correlation coefficient between $Z$ and $\sigma_{\rm{turb}}$ for all the
models: this weak correlation is due to the way $Z$ is measured ($Z$ is
estimated by measuring the equivalent width of emitting lines). In conclusion,
we find that the cross-correlations have a negligible impact on our
measurements of the ICM rotation speed.

Using the configurations for _Resolve_ , we conclude that, even in the
presence of the turbulence of $500\,\mathrm{km/s}$, the l.o.s. component of
the rotation velocity is recovered through the fitting procedure within
$1\sigma$ of confidence level in most analyses of the mock spectroscopic data.
The analysis of our cool-core cluster models shows that current observational
constraints, such as the rotation speed of the ICM based on the upper limits
on the broadening of the X-ray emitting lines, the measurements of the
thermodynamic profiles, the flattening of the surface brightness distribution
and of the hydrostatic mass bias, leave room for rotation of the ICM up to
$600\,\mathrm{km/s}$ in typical clusters. Further tests of our cluster models
with rotating ICM will be provided by future measurements of the l.o.s.
velocity with _XRISM/Resolve_ that will put stringent and direct constraints
on the intrinsic kinematics of the ICM in galaxy clusters.

###  5.3 Assessing the hydrostatic mass bias with X-ray spectroscopy

In our cluster models, the ICM is in equilibrium and departs from the
hydrostatic condition owing only to rotation. Here, we point out the
perspectives and limitations on the use of X-ray spectroscopy for the mapping
of nonnegligible rotation support of the ICM.

As discussed above, the l.o.s. velocity $v_{\rm{los}}(x,z)$ (see Eq. 30) can
be recovered from the measurements of the properties of the X-ray emitting
lines (see e.g. Biffi et al., 2013; Roncarelli et al., 2018). Thus, a proxy
for the rotational contribution to the hydrostatic mass bias, defined in Eq.
(29), is $M_{\rm{rot}}/M_{\rm{true}}$, where

$M_{\rm{rot}}(<r)=\frac{v_{\rm{los}}^{2}(r,0)r}{G}$ (33)

is the mass associated with the gas rotation support, and $M_{\rm{true}}$ the
same halo mass as in Eq. (29).

Using in Eq. (33) the true l.o.s. rotation speed $v_{\rm{los}}$, given by Eq.
(30), we compute the $M_{\rm{rot}}$ profiles of our cluster models (see curves
in Fig. 11). Then, to find the l.o.s. velocity $\langle v_{\rm{los}}\rangle$
as measured from the best-fits to our mock spectra, we use Eq. (31), where
$z_{\rm{eff}}$ is now the best-fit value to the mock spectrum of the receding
ICM without turbulence (reported in Tab. 5). Substituting $\langle
v_{\rm{los}}\rangle$ instead of $v_{\rm{los}}(r,0)$ in Eq. (33), where we
consider the radius $r$ equal to the value of the plane-of-the-sky
$x$-coordinate (reported in Tab. 3 for the region under consideration), we
estimate the mass associated with gas rotation support at the centers of the
regions chosen for our mock observations. Following this method, from the
normal distribution with mean and standard deviation equal to the best-fit
effective redshift and its error (both reported in Tab. 5), respectively, we
infer the errors (as 16th and 84th percentiles) on $M_{\rm{rot}}$ as estimated
from X-ray spectroscopy for our cluster models.

Fig. 11 shows that $M_{\rm{rot}}$ estimated from the best-fit $z_{\rm{eff}}$
recovers within $1\sigma$ statistical errors (vertical error bars) the mass
associated with the rotation support estimated from the true l.o.s. velocity
(Eq. 30). This is consistent with the fact that the best-fit effective
redshift from the spectral analysis recovers within $1\sigma$ of confidence
level the input effective redshift (Sect. 5.2 ). However, as shown by the
curves in Fig. 11, $M_{\rm{rot}}/M_{\rm{true}}$ based on Eq. (33), where we
take the true l.o.s. speed, is lower than the hydrostatic mass bias $b$,
measured from the theoretical angle-averaged pressure profile of the ICM (see
Sect. 4.3). The reason for this discrepancy (pointed out also by Ota et al.,
2018) is that the mean l.o.s. velocity at a projected distance $d$ from the
symmetry axis is lower than the rotation speed of the ICM at an intrinsic
distance $d$ from the symmetry axis.

Figure 11:  Fraction of the hydrostatic mass bias due to rotation ($b$) of RMS
(blue), RMO (red) and RMP (green) models that we recover from the true (dashed
lines) and best-fit (points) l.o.s. velocities using Eq. (33). The horizontal
error bars indicate the extent of the region of the mock observations, while
the vertical error bars the $1\sigma$ errors on $M_{\rm{rot}}$ estimated from
X-ray spectroscopy. Radius is normalized to $r_{\rm{500}}$ as in Fig. 9, with
the RMS radii $r_{2500}$, $r_{1000}$ and $r_{500}$ indicated in the top axis.

Focusing on the hydrostatic mass biases of our cluster models as measured from
X-ray spectroscopy (points with error bars in Fig. 11), we conclude that the
estimates of the rotation support over the range $(0.1-1)r_{\rm{500}}$
obtained through the ”best-fit” l.o.s. rotation speed resolved by _Resolve_
are able to account for $55-70\%$ of the hydrostatic mass bias of our models.
It follows that a _Resolve_ -like correction for the rotation support of the
ICM is expected to leave a residual hydrostatic mass bias due to rotation
smaller than $3\%$ at $r_{\rm{500}}$ for systems similar to our model
clusters. Error bars in Fig. 11 are larger in the outermost bin for all the
models: this is a consequence of the increase of the statistical uncertainties
of the spectral parameters due to the lower signal present in those regions of
our cluster models.

Moreover, the poor angular resolution of _Resolve_ (with a Point-Spread-
Function with a half power diameter of $\approx 1.7^{\prime}$) prevents us
from sampling the hydrostatic mass bias profile in a larger independent number
of radial bins. This will be possible in the future with the Advanced
Telescope for High Energy Astrophysics202020The ESA satellite _ATHENA_ , is
scheduled to be launched not before 2036 (see https://www.the-athena-x-ray-
observatory.eu/en). (_ATHENA_ ; Nandra et al. 2013), thanks to its expected
arcsec resolution combined with the performance of the onboard X-ray
microcalorimeter _X-IFU_ (see e.g. Roncarelli et al., 2018).

###  5.4 Discussion on the hydrostatic bias

In Section 4.3, we discussed how the measurements of the hydrostatic mass bias
can be used to limit the rotation speed of the ICM. In Table 6, we quote the
hydrostatic mass bias due to rotation in our cluster models at some
characteristic overdensities available to observations. A general trend is
that the observed hydrostatic mass bias decreases with increasing overdensity
(see e.g. Zhang et al., 2010; Mahdavi et al., 2013; Sereno & Ettori, 2015;
Lovisari et al., 2020). A similar trend is also recovered in hydrodynamical
simulations (see e.g. Nagai et al., 2007b; Lau et al., 2009; Meneghetti et
al., 2010; Rasia et al., 2012; Gianfagna et al., 2021). This behavior results
in tension with the hydrostatic mass bias profiles recovered from our models,
which increase with increasing $\Delta$ (see Fig. 9). Cosmological
hydrodynamical simulations show that the support from turbulence in galaxy
clusters increases with radius (see e.g. Fang et al., 2009; Lau et al., 2009;
Towler et al., 2023), overcoming the rotational contribution well within
$r_{500}$. Thus, the observed trend of the hydrostatic mass bias is expected
to follow the increase in the turbulent support of the ICM moving outward,
with a non-negligible contribution from the rotation only in the inner
regions. Indeed, the few data available at $r_{2500}$ (see Fig. 9), where
hydrodynamical simulations suggest comparable support from rotation and
turbulence, suggest a hydrostatic bias marginally consistent (within
$2\sigma$) with the predictions of our models.

In the near future, new instruments and space telescopes will permit more
accurate determinations of the hydrostatic mass bias at different
overdensities in a larger sample of galaxy clusters. In particular, the
aforementioned _XRISM_ and _eRosita_ 212121See https://www.mpe.mpg.de/eROSITA.
(onboard the Spectrum-Roentgen-Gamma mission and, only in the future, the
observatory _Athena_), together with currently available X-ray observatories
(_XMM-Newton_ and _Chandra_ 222222See https://chandra.harvard.edu/.), will
continue to provide the measurements of the hydrostatic mass through X-ray
observations. The ESA optical/infrared space telescope _Euclid_ 232323See
https://sci.esa.int/web/euclid. and other ground-based campaigns will
complement with weak lensing mass estimates the information on the mass budget
in larger samples of galaxy clusters, allowing us to refine our comprehension
of the statistical properties of the hydrostatic mass bias.

Table 6: Characteristic values of the hydrostatic mass bias of our cluster
models.

$\Delta$ | $b$
---|---
RMS | RMO | RMP
2500 | 0.09 | 0.07 | 0.13
1000 | 0.06 | 0.04 | 0.09
500 | 0.05 | 0.03 | 0.07
242424We quote the hydrostatic mass biases ($b$) of our clusters models (RMS,
RMO and RMP) at $r_{\rm{\Delta}}$ with $\Delta=\\{2500,1000,500\\}$.

##  6 Conclusions

In this work, we have presented three representative, realistic models of
massive ($M_{\rm{200}}\approx 10^{15}\,\mathrm{M_{\odot}}$) cool-core galaxy
clusters with rotating ICM in equilibrium in dark matter halos consistent with
observational findings and theoretical predictions on the halo shape and mass-
concentration relation (Sect. 2). While one of the models has a spherical NFW
halo, the other two have, respectively, physically consistent oblate and
prolate NFW halos, built analytically using the method of Ciotti & Bertin
(2005). Our cool-core cluster models, which have barotropic ICM rotation with
velocity peaks as high as $600\,\mathrm{km/s}$ (see Fig. 3), have ICM
temperature and density profiles consistent with the corresponding universal
profiles of real clusters. Cosmological hydrodynamical simulations can also be
used to calibrate these analytic models (for instance, on the location of
$R_{\rm break}$, the parameter that defines the size of the cool core) once
any overcooling problem (see e.g. Kravtsov & Borgani, 2012) is properly
solved, and realistic cooling cores are produced in systems that did not
experience a major merger in the central region (e.g. Rasia et al., 2015). The
shape of surface-brightness contours, the discrepancy between hydrostatic and
true masses, and the broadening of X-ray emission lines of the models are also
consistent with currently available observations.

We obtained a set of mock X-ray spectra of the rotating ICM from the
aforementioned three cluster models, using the configuration for the
microcalorimeter _Resolve_ onboard _XRISM_ , for different turbulence
conditions. In this way, we estimated how well the rotation speed and the
hydrostatic mass bias due to rotation are recovered based on the results of
_Resolve_ -like spectral analysis (Sect. 5).

The main conclusions of this work are the followings:

  * •

The existence of realistic cluster models with the peaks of the rotation speed
of the ICM in the range 400-$600\,\mathrm{km/s}$ leaves open the possibility
that the rotation support of the ICM is nonnegligible in real cool-core galaxy
clusters.

  * •

Even with turbulent velocity dispersion as high as $500\,\mathrm{km/s}$, a
_Resolve_ -like X-ray spectral analysis recovers the input l.o.s. rotation
speed at high significance.

  * •

Measuring the __line-of-sight velocity from X-ray spectroscopy with _XRISM_
accounts for $55-70\%$ of the hydrostatic mass bias due to rotation. In this
way, _XRISM_ will allow us to pin down any mass bias of origin different from
rotation (for instance, due to turbulence; see e.g. Ettori & Eckert 2022).

On one side, improving spatial and spectral resolution in X-rays will open a
new window in which the combination of the intrinsic thermodynamic profiles
with the rotation and turbulent velocity dispersion profiles can be used to
validate models of the ICM, providing robust estimates of the cluster mass. On
the other side, Sect. 5.3 shows the need for a functional form that properly
maps the intrinsic rotation speed through the line-of-sight rotation speed as
resolved in massive clusters. Most of the limitations of this mapping come
from the possible degeneracy present in the interpretation of the
observational data. Possible contaminants that can limit our interpretation of
the physical state of the ICM are, for example, unresolved gas clumps,
multiphase gas, metallicity inhomogeneities, and complex velocity fields not
properly mapped both in the plane of the sky and along the line of sight (see
also Sect. 5.2). We postpone further study on this topic to future work.

X-ray observations will enable us to guess both the rotation axis and the
maximal rotation speed (see e.g. Ota et al., 2018; Liu & Tozzi, 2019) in some
favorable conditions (broadly speaking, bright enough source and X-ray
detector with sufficient spatial and spectral resolution). Once these X-ray
observations are available, the kinetic Sunyaev-Zeldovich (see e.g. some
observational constraints in Sayers et al., 2013, 2019; Mroczkowski et al.,
2019, for a review) can be resolved (thanks also to the forthcoming ground-
based Simons Observatory; Ade et al. 2019) and compared to the X-ray
constraints to provide a consistent picture of the ICM peculiar velocity along
the line of sight.

The presented results strongly encourage future spectroscopic observations of
relaxed galaxy clusters with _XRISM/Resolve_ (in the forthcoming decade)
and/or _ATHENA/X-IFU_ (in the far future; see also Roncarelli et al. 2018) to
quantify the level of the ICM rotation speed, and to reduce as far as possible
the hydrostatic mass bias in real clusters, with important implications for
the use of galaxy clusters as accurate cosmological proxies (see e.g. Pratt et
al., 2019).

As pointed out by Nipoti & Posti (2014) and Nipoti et al. (2015), if the ICM
is weakly magnetized (as found by the observational works reviewed by Bruggen
2013) and significantly rotating, the magnetorotational instability could also
have relevant effects. Thus, the possibility that the ICM has nonnegligible
rotation support with a speed as high as $600\,\mathrm{km/s}$ in real clusters
acquires a great interest for the implications not only on the mass estimates,
which are needed to use galaxy clusters as cosmological probes, but also for
our understanding of the energy balance and evolution of the cool cores,
because the magnetorotational instability could play a role in regulating
their energetic budget.

###### Acknowledgements.

We thank the referee Edoardo Altamura for useful suggestions. S.E.
acknowledges the financial contribution from the contracts ASI-INAF Athena
2019-27-HH.0, “Attività di Studio per la comunità scientifica di Astrofisica
delle Alte Energie e Fisica Astroparticellare” (Accordo Attuativo ASI-INAF n.
2017-14-H.0), and from the European Union’s Horizon 2020 Programme under the
AHEAD2020 project (grant agreement n. 871158).

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##  Appendix A An extreme cluster model with rotating ICM

Table 7: Parameters of ”rotating model extreme” (RME) model.

$\begin{array}[]{p{0.2\linewidth}llllllll}\hline\cr\vskip 3.0pt plus 1.0pt
minus 1.0pt\cr Model&{\rm
Halo}&R_{\rm{break}}[\mathrm{kpc}]&n_{\rm{e,}\star}[\mathrm{cm^{-3}}]&T_{\star}[\mathrm{keV}]&u_{\rm{0}}[\mathrm{km/s}]&R_{\rm{0}}[\mathrm{kpc}]&\gamma^{\prime}_{\rm{IN}}&\gamma^{\prime}_{\rm{OUT}}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr RME&{\rm DMO}&420&2.5\times 10^{-3}&7.8&3000&100&0.60&1.55\\\ \vskip
3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\end{array}$

252525Same as Tab. 2, but for model RME.

In this appendix, with the purpose of illustrating the effect of strong ICM
rotation on observable properties of galaxy clusters, we present a cluster
model (of $M_{\rm{200}}\approx 10^{15}\,\mathrm{M_{\odot}}$) with rotating
ICM, which, different from the three models presented in Sect. 3.3, is not
realistic, because, though having realistic gas density distribution, has a
temperature distribution substantially different from that of real clusters.
This extreme cluster model, which we refer to as ”rotating model extreme”
(RME), has gravitational potential generated by the oblate halo model DMO (see
Sect. 2.2) and gas rotation law given by Eq. (23) with values of the
parameters $u_{\rm{0}}$ and $R_{\rm{0}}$ (see Tab. 7) such that rotation speed
peak is $\simeq 750\,\mathrm{km/s}$ at a radius $\simeq 160\,\mathrm{kpc}$
(see bottom panel of Fig. 12). The values of the other gas parameters
($R_{\rm{break}}$, $\gamma^{\prime}_{\rm{IN}}$, $\gamma^{\prime}_{\rm{OUT}}$,
$n_{\rm{e,}\star}$ and $T_{\star}$; see Tab. 7) are chosen so that the angle-
averaged gas density profile of model RME is consistent with the universal gas
density profile of observed cool-core clusters (top panel of Fig. 12).
However, this choice of the values of the parameters implies that, due to the
strong rotation support, the temperature profile of model RME is grossly
inconsistent with the universal temperature profile derived for observed cool-
core clusters (middle panel of Fig. 12).

Given the rotation speed curve and the gravitational potential assumed for
model RME, we were not able to find a combination of values of the plasma
parameters such that both the density and the spectroscopic-like temperature
profiles are consistent with those observed for massive cool-core clusters.
Though this does not allow us to place an upper limit on the peak of the
rotation speed of the ICM, it is a strong indication that rotation speeds
higher than $\approx 600$ km s-1 are problematic not only for the
spectroscopic constraints on the broadening of the X-ray emission line, but
also for constraints imposed by the shape of the universal thermodynamic
profiles. Comparing further the polytropic indices $\gamma^{\prime}_{\rm{IN}}$
and $\gamma^{\prime}_{\rm{OUT}}$ of our ICM distributions, to those observed,
we also note that model RME, which requires lower $\gamma^{\prime}_{\rm{IN}}$
and higher $\gamma^{\prime}_{\rm{OUT}}$ than our realistic models (see Tab.s 2
and 7), is in tension with the results of Ghirardini et al. (2019b) on the
polytropic indices of observed clusters (see Sect. 4.1).

Figure 12: Thermodynamic (upper and central panels) and rotation speed
profiles (lower panel) of the ICM in ”rotating model extreme” (RME) model.
Upper and central panels. Same as the left and right panels of Fig. 4,
respectively, but for model RME. Lower panel. Rotation speed profile of the
ICM in model RME. The red arrow indicates approximately the extent of the cool
core, defined as in Fig. 3.