Hydrostatic equilibrium with non-thermal pressure

In galaxy clusters, the total pressure of the system will comprise the thermal pressure, $p_\mathrm{th}$ , exerted by the intracluster gas, as well as the NTP, $p_\mathrm{nt}$ , arising due to gravitational shocks and mergers, or due to feedback processes (e.g. powerful outflows driven by active galactic nuclei) in the central regions. For galaxy clusters in hydrostatic equilibrium, the acceleration exerted by this pressure on the intracluster gas will be balanced by the gravitational force generated by the cluster’s mass, at any halo radius.

In the idealised case of a spherically symmetric cluster, the hydrostatic equilibrium condition at any halocentric radius, r, is:

(1) \begin{equation} \frac{\mathrm{d} }{\mathrm{d}r}\left[p_\mathrm{th}(r) + p_\mathrm{nt}(r)\right] = - \unicode{x03C1}_\mathrm{gas}(r)\frac{G M(r)}{r^2},\end{equation}

in terms of the radial derivatives of the thermal pressure profile, $p_\mathrm{th}(r)$ , and the NTP profile, $p_\mathrm{nt}(r)$ ; the halo’s enclosed mass, M(r); the density profile of the intracluster gas, $\unicode{x03C1}_\mathrm{gas}(r)$ ; and the gravitational constant, G. We note that this assumption of spherical symmetry is not necessary always true for a large population of real clusters (see, e.g. Campitiello et al. Reference Campitiello2022); however, we will assume this holds hereafter.

In the simplest case, considering only gas and dark matter within the galaxy cluster, the enclosed halo mass, M(r), is given by integrating the sum of the density profiles for the dark matter halo, $\unicode{x03C1}_\mathrm{dm}(r)$ , and the intracluster gas, $\unicode{x03C1}_\mathrm{gas}(r)$ , within the spherical volume of radius r, as:

(2) \begin{equation} M(r) = 4\pi \int_0 ^r \left[ \unicode{x03C1}_\mathrm{dm}(r^\prime) + \unicode{x03C1}_\mathrm{gas}(r^\prime) \right] r^\prime{} ^2 \mathrm{d}r^\prime.\end{equation}

To solve for the temperature profile, T(r), of the intracluster gas, the thermal pressure profile must be related to the gas’ state variables, which for an ideal gas, obeys the relation:

(3) \begin{equation} p_\mathrm{th} = \frac{k_\mathrm{B}T}{\unicode{x03BC} m_\mathrm{p}} \unicode{x03C1}_\mathrm{gas},\end{equation}

where $k_\mathrm{B}$ is the Boltzmann constant, $\unicode{x03BC}$ is the mean molecular weight, and $m_\mathrm{p}$ is the proton mass. Subsequently, Equation (1) can be expressed as:

(4) \begin{equation} \frac{\mathrm{d}}{\mathrm{d}r}\left[ \unicode{x03C1}_\mathrm{gas}(r) T(r) + \frac{\unicode{x03BC} m_\mathrm{p}}{k_\mathrm{B}} p_\mathrm{nt} (r)\right] = - \frac{G\unicode{x03BC} m_\mathrm{p}}{k_\mathrm{B}} \frac{\unicode{x03C1}_\mathrm{gas}(r) M(r)}{r^2},\end{equation}

which can be solved for T(r), given some radial parameterisation for the NTP profile, $p_\mathrm{nt}(r)$ .

When observationally estimating the cluster mass, the NTP term in Equation (4) is generally assumed to be zero (e.g. Vikhlinin et al. Reference Vikhlinin, Kravtsov, Forman, Jones, Markevitch, Murray and Van Speybroeck2006, Reference Vikhlinin2009), circumventing the need to assume the form of $p_\mathrm{nt}(r)$ , which is not well constrained. We took this approach (i.e. neglecting the contribution of $p_\mathrm{nt}$ ) in S24b. Without a NTP term, the hydrostatic equilibrium state of the cluster is assumed to be entirely balanced by the gas’ thermal pressure; this requires the gas to be hotter than it would otherwise be if NTP was present. In this study, hereafter, we will refer to the hydrostatic state without NTP as ‘pristine equilibrium’, to differentiate from the ‘real equilibrium’ state that will include NTP.

The pristine equilibrium temperature of the gas, which we denote $T_\mathrm{eq}(r)$ , is then given by the general solution:

(5) \begin{equation} T_\mathrm{eq}(r) = \frac{G\unicode{x03BC} m_\mathrm{p}}{k_\mathrm{B}} \frac{1}{\unicode{x03C1}_\mathrm{gas}(r)} \int _r^\infty \frac{M(r^\prime) \unicode{x03C1}_\mathrm{gas}(r^\prime) \mathrm{d}r^\prime}{r^\prime{} ^2}.\end{equation}

When including NTP in the cluster’s hydrostatic equilibrium state, that is, Equation (4), the real equilibrium gas temperature, T(r), will instead be given by:

(6) \begin{equation} T(r) = T_\mathrm{eq}(r) \left[1 - \mathcal{F}(r)\right],\end{equation}

where $\mathcal{F}(r)$ parameterises the fraction of NTP to total pressure in the system:

(7) \begin{equation} \mathcal{F}(r) \equiv \frac{p_\mathrm{nt}}{p}(r).\end{equation}

By this definition, the gas’ thermal pressure, $p_\mathrm{th}(r)$ , will be related to the NTP fraction, $\mathcal{F}(r)$ , by:

(8) \begin{equation} p_\mathrm{th}(r) = p(r) \left[1 - \mathcal{F}(r) \right],\end{equation}

where p(r) is the cluster’s total pressure. We assume this total pressure will always be given by the hydrostatic equilibrium condition, Equation (1), defining the equilibrium pressure:

(9) \begin{equation} p_\mathrm{eq}(r) = G \int _r^\infty \frac{M(r^\prime) \unicode{x03C1}_\mathrm{gas}(r^\prime) \mathrm{d}r^\prime}{r^\prime{} ^2},\end{equation}

which, in the pristine equilibrium assumption, will also be the thermal pressure of the gas.

The gas entropy

The definition of the intracluster gas entropy, K, is:

(10) \begin{equation} K \equiv \frac{k_\mathrm{B}T}{n_\mathrm{e}^{2/3}},\end{equation}

in terms of the Boltzmann constant, $k_\mathrm{B}$ , the gas temperature, T, and the electron number density, $n_\mathrm{e}$ , which is given by:

(11) \begin{equation} n_\mathrm{e} = \frac{\unicode{x03C1}_\mathrm{gas}}{\unicode{x03BC}_\mathrm{e} m_\mathrm{p}};\end{equation}

here $\unicode{x03BC}_\mathrm{e}$ is the mean molecular weight of electrons and $m_\mathrm{p}$ is the proton mass. For a spherically symmetric cluster, the radial gas entropy profile, K(r), is then:

(12) \begin{equation} K(r) = \left[\frac{\unicode{x03BC}_\mathrm{e} m_\mathrm{p}}{\unicode{x03C1}_\mathrm{gas}(r)}\right]^{2/3}k_\mathrm{B}T(r).\end{equation}

We can assign a pristine equilibrium gas entropy, $K_\mathrm{eq}(r)$ , to a cluster that is in pristine equilibrium, which will be defined as:

(13) \begin{equation} K_\mathrm{eq}(r) = \left[\frac{\unicode{x03BC}_\mathrm{e} m_\mathrm{p}}{\unicode{x03C1}_\mathrm{gas}(r)}\right]^{2/3}k_\mathrm{B}T_\mathrm{eq}(r),\end{equation}

in terms of the pristine equilibrium temperature of the gas, $T_\mathrm{eq}(r)$ .

The gas entropy slope

By taking the logarithmic derivative of Equation (12) with respect to the halocentric radius, r, we define the ‘entropy slope’, k(r), in terms of the logarithmic derivatives of the gas’ temperature and density, as:

(14) \begin{equation} k(r) \equiv \frac{\mathrm{d} \ln K(r)}{\mathrm{d} \ln r} = \frac{\mathrm{d} \ln T(r)}{\mathrm{d} \ln r} - \frac{2}{3} \frac{\mathrm{d} \ln \unicode{x03C1}_\mathrm{gas}(r)}{\mathrm{d} \ln r}.\end{equation}

For a cluster in pristine equilibrium, the associated pristine equilibrium entropy slope, $k_\mathrm{eq}(r)$ , is obtained from Equation (13), as:

(15) \begin{equation} k_\mathrm{eq}(r) \equiv \frac{\mathrm{d} \ln K_\mathrm{eq}(r)}{\mathrm{d} \ln r} = \frac{\mathrm{d} \ln T_\mathrm{eq}(r)}{\mathrm{d} \ln r} - \frac{2}{3} \frac{\mathrm{d} \ln \unicode{x03C1}_\mathrm{gas}(r)}{\mathrm{d} \ln r}.\end{equation}

By relating the gas’ real equilibrium temperature, T(r), to its pristine equilibrium temperature, $T_\mathrm{eq}(r)$ , by Equation (6), these entropy slopes can be related to the NTP fraction, $\mathcal{F}(r)$ , via the differential equation:

(16) \begin{equation} k(r) = k_\mathrm{eq}(r) + \frac{\mathrm{d} \ln \left[1 - \mathcal{F}(r)\right]}{\mathrm{d} \ln r}.\end{equation}

The NTP fraction is thus constrained if both k(r) and $k_\mathrm{eq}(r)$ are known.

Table 1. Summary of the five parameters in the ideal baryonic cluster halo model: their symbol, definition, and physical values when $\Delta=500$ .

Constraining the non-thermal pressure fraction

One approach for solving Equation (16) is to relate the real entropy slope to its pristine equilibrium value via a weighting function, w(r), such that:

(17) \begin{equation} k(r) = w(r) \cdot k_\mathrm{eq}(r).\end{equation}

We choose the weighting function such that k(r) matches literature values for the entropy slope over an appropriate range of halocentric radii. Given some form for this weighting function, w(r), we solve Equation (16) in the form:

(18) \begin{equation} k_\mathrm{eq}(r)\left[w(r) - 1\right] = \frac{\mathrm{d} \mathcal{F}(r)}{\mathrm{d}r}\frac{r}{\left[\mathcal{F}(r) - 1\right]}, \end{equation}

which requires a boundary condition on $\mathcal{F}(r)$ . We introduce the parameter $\mathcal{F}_0 \equiv \mathcal{F}(r=0)$ , as the cluster’s central NTP fraction, such that Equation (18) can be integrated to give:

(19) \begin{equation} \mathcal{F}(r) = 1 + (\mathcal{F}_{0} - 1) \cdot \mathrm{e} ^{\int_{0}^r k_\mathrm{eq}(r^\prime) \left[w(r^\prime) - 1\right]\frac{\mathrm{d}r^\prime}{r^\prime}}.\end{equation}

A scale-free approach

We define the cluster’s virial mass, $M_\mathrm{vir}$ , in terms of its virial radius, $r_\mathrm{vir}$ , such that:

(20) \begin{equation} M_\mathrm{vir} \equiv \frac{4}{3} \pi r_\mathrm{vir}^3 \Delta \unicode{x03C1}_\mathrm{crit,0};\end{equation}

$M_\mathrm{vir}$ is the mass enclosing an average density of $\Delta$ times the present-day critical density of the universe, $\unicode{x03C1}_\mathrm{crit,0}$ , with the convention $\Delta=500$ usually assumed in studies of galaxy clusters. We therefore use $M_{500}$ as the virial mass and $r_{500}$ as the virial radius. We then define a scale-free dimensionless halocentric radius, s, as:

(21) \begin{equation} s \equiv \frac{r}{r_\mathrm{vir}},\end{equation}

where $r_\mathrm{vir}$ depends on this choice of $\Delta$ .

In terms of s, the NTP fraction solution in Equation (19) can be expressed as:

(22) \begin{equation} \mathcal{F}(s) = 1 + (\mathcal{F}_{0} - 1) \cdot \mathrm{e} ^{\int_{0}^s k_\mathrm{eq}(s^\prime) \left[w(s^\prime) - 1\right]\frac{\mathrm{d}s^\prime}{s^\prime}}.\end{equation}

This can be solved, given a scale-free profile for the pristine equilibrium gas entropy slope, $k_\mathrm{eq}(s)$ ; a scale-free weighting function, w(s); and a prescription for the cluster’s central NTP fraction, $\mathcal{F}_0$ .

The ideal baryonic cluster halo profiles

In general, a model for a cluster’s pristine equilibrium entropy slope, $k_\mathrm{eq}(s)$ , in scale-free form requires a scale-free structural parameterisation for a cluster’s intracluster gas and dark matter halo, to solve for its hydrostatic state. We use the analytic model derived in S24b, which we briefly summarise.

We obtained an ‘ideal baryonic cluster halo’ in S24b in terms of scale-free density profiles for the dark matter halo, $\unicode{x03C1}_\mathrm{dm}(s)$ , and the intracluster gas, $\unicode{x03C1}_\mathrm{gas}(s)$ . For the dark matter, this profile was taken as a generalisation to the NFW (Navarro, Frenk, & White Reference Navarro, Frenk and White1995, Reference Navarro, Frenk and White1996, Reference Navarro, Frenk and White1997) profile:

(23) \begin{equation} \frac{\unicode{x03C1}_\mathrm{dm} (s, c, \unicode{x03B1}, \unicode{x03B7})}{\Delta \unicode{x03C1}_\mathrm{crit, 0}} = \frac{(1 - \unicode{x03B7} f_\mathrm{b, cos}) u(c, \unicode{x03B1})}{3s^{\unicode{x03B1}} (1 + cs)^{3-\unicode{x03B1}}},\end{equation}

and for the intracluster gas, by the similarly generalised profile:

(24) \begin{equation} \frac{\unicode{x03C1}_\mathrm{gas}(s, c, \unicode{x03B1}, \unicode{x03B7}, d, \varepsilon)}{\Delta \unicode{x03C1}_\mathrm{crit,0}} = \frac{\unicode{x03B7} f_\mathrm{b, cos}\mathcal{U}(c, \unicode{x03B1}, d, \varepsilon)}{3s^{\varepsilon} [1 + \mathcal{C}(c, \unicode{x03B1}, d, \varepsilon) s ]^{3-\varepsilon}}.\end{equation}

These density profiles are each a function of the dimensionless halocentric radius, s, and taken in a dimensionless ratio to some overdensity, $\Delta$ , times the present-day critical density of the universe, $\unicode{x03C1}_\mathrm{crit,0}$ . The five parameters that specify these density profiles are summarised in Table 1, along with their recommended value or range in values (cf. S24b). The parameter functions in Equations (23) and (24) are then specified in terms of these parameters:

(25) \begin{equation} u(c, \unicode{x03B1}) \equiv \left[\int _0 ^1 \frac{s^{2-\unicode{x03B1}} \mathrm{d}s}{(1 + cs)^{3-\unicode{x03B1}}}\right]^{-1},\end{equation}

(26) \begin{equation} \mathcal{C}(c, \unicode{x03B1}, d, \varepsilon) \equiv \frac{d(\unicode{x03B1} - \varepsilon) + c(3 - \varepsilon)}{3-\unicode{x03B1}},\end{equation}

and:

(27) \begin{equation} \mathcal{U}(c, \unicode{x03B1}, d, \varepsilon) \equiv \left[\int _0 ^{1} \frac{s^{2-\varepsilon}\mathrm{d}s}{[1 + \mathcal{C}(c, \unicode{x03B1}, d, \varepsilon) s]^{3-\varepsilon}} \right] ^{-1}.\end{equation}

We adopt a cosmological baryon fraction of $f_\mathrm{b, cos}=0.158$ (Planck Collaboration et al. 2016).

Figure 1. The pristine equilibrium gas entropy profiles, in scale-free form $K_\mathrm{eq}/K_\mathrm{500}$ , shown in the top row, and the pristine equilibrium gas entropy slopes, $k_\mathrm{eq} \equiv \mathrm{d} \ln K_\mathrm{eq} / \mathrm{d} \ln r$ , shown in the bottom row, each traced over the scaled halocentric radius $r/r_\mathrm{500}$ , as predicted for the ideal baryonic cluster halo model. The halo concentration, c, and dilution, d, are both fixed parameters, whilst each column varies the gas inner slope, $\varepsilon$ . Within each box, each colour varies the halo inner slope, $\unicode{x03B1}$ , with the solid coloured lines tracing a fraction of cosmological baryon content of $\unicode{x03B7}=0.8$ , and the shaded colour region around each solid line (not visible for all curves) tracing this value continuously between $\unicode{x03B7}=0.6$ and $\unicode{x03B7}=1$ .

The pristine equilibrium gas entropy and gas entropy slope of the ideal baryonic cluster halos

Taking the expression for the pristine equilibrium gas entropy, Equation (13), we can predict the entropy profiles for the ideal baryonic cluster halo model:

(28) \begin{equation}\begin{aligned} \frac{K_\mathrm{eq}(s, c, \unicode{x03B1}, \unicode{x03B7}, d, \varepsilon)}{K_\mathrm{vir}} &= \frac{\left\{3s^{\varepsilon} \left[1 + \mathcal{C}(c, \unicode{x03B1}, d, \varepsilon)s\right]^{3-\varepsilon}\right\}^{5/3} }{\left[\unicode{x03B7} \, \mathcal{U}(c, \unicode{x03B1}, d, \varepsilon)\right]^{2/3}} \\ & \quad \times \mathcal{I}(s, c, \unicode{x03B1}, \unicode{x03B7}, d, \varepsilon),\end{aligned}\end{equation}

as a function of the dimensionless halocentric radius, s; the five structural parameters from Table 1; and the integral function:

(29) \begin{equation}\begin{aligned} \mathcal{I}(s, c, \unicode{x03B1}, \unicode{x03B7}, d, \varepsilon)&\equiv \int _s ^\infty \frac{ \mathrm{d}s^\prime \Biggl\{ (1 - \unicode{x03B7} f_\mathrm{b, cos})u(c, \unicode{x03B1}) \cdot \int _0 ^{s^\prime} \frac{s^\prime{}^\prime{} ^{2-\unicode{x03B1}} \mathrm{d}s^\prime{}^\prime}{(1 + cs^\prime{}^\prime )^{3-\unicode{x03B1}}} }{s^\prime{}^{2 + \varepsilon} \left[1 + \mathcal{C}(c, \unicode{x03B1}, d, \varepsilon)s^\prime \right]^{3 - \varepsilon}} \\ & \hspace{-15mm} + \unicode{x03B7} f_\mathrm{b, cos} \mathcal{U}(c, \unicode{x03B1}, d, \varepsilon) \cdot \int _0 ^{s^\prime} \frac{s^\prime{}^\prime{}^{2-\varepsilon} \mathrm{d}s^\prime{}^\prime}{\left[1 + \mathcal{C}(c, \unicode{x03B1}, d, \varepsilon)s^\prime{}^\prime \right]^{3-\varepsilon}} \Biggr\}.\end{aligned}\end{equation}

In this expression, the gas entropy is scaled by the virial entropy, $K_\mathrm{vir}$ , which we define as:

(30) \begin{equation} K_\mathrm{vir} \equiv \left[\frac{\unicode{x03BC}_\mathrm{e} m_\mathrm{p}}{f_\mathrm{b, cos} \Delta \unicode{x03C1}_\mathrm{crit,0}}\right]^{2/3} k_\mathrm{B}T_\mathrm{vir},\end{equation}

in terms of the virial temperature, $T_\mathrm{vir}$ , defined as:

(31) \begin{equation} T_\mathrm{vir} \equiv \frac{1}{3} \frac{\unicode{x03BC} m_\mathrm{p}}{k_\mathrm{B}} \frac{GM_\mathrm{vir}}{r_\mathrm{vir}}.\end{equation}

When $\Delta=500$ in Equation (30), this allows us to define the entropy and temperature values of $K_{500}$ and $T_{500}$ .

Taking the logarithmic derivative of Equation (28) with respect to s, we find that the pristine equilibrium entropy slope, $k_\mathrm{eq}(s)$ , for this model can be solved as:

(32) \begin{equation}\begin{aligned} & k_\mathrm{eq}(s, c, \unicode{x03B1}, \unicode{x03B7}, d, \varepsilon) = \frac{5}{3} \left[\varepsilon + (3-\varepsilon) \frac{\mathcal{C}(c, \unicode{x03B1}, d, \varepsilon)s}{\left[1 + \mathcal{C}(c, \unicode{x03B1}, d, \varepsilon)s\right]}\right] \\ &\qquad - \frac{\Biggl\{ (1 - \unicode{x03B7} f_\mathrm{b, cos}) u(c, \unicode{x03B1}) \cdot \int_0 ^s \frac{s^\prime{}^{2-\unicode{x03B1}} \mathrm{d}s^\prime}{(1 + cs^\prime)^{3-\unicode{x03B1}}} }{\mathcal{I}(s, c, \unicode{x03B1}, \unicode{x03B7}, d, \varepsilon) s^{\varepsilon+1}\left[1 + \mathcal{C}(c, \unicode{x03B1}, d, \varepsilon)s\right]^{3-\varepsilon}} \\ &\qquad + \unicode{x03B7} f_\mathrm{b, cos} \mathcal{U}(c, \unicode{x03B1}, d, \varepsilon) \cdot \int_0^s \frac{s^\prime{}^{2-\varepsilon} \mathrm{d}s^\prime}{\left[1 + \mathcal{C}(c, \unicode{x03B1}, d, \varepsilon)s^\prime\right]^{3-\varepsilon}} \Biggr\}.\end{aligned}\end{equation}

Over the parameter space detailed in Table 1, the profiles for the pristine equilibrium gas entropy, in the form $K_\mathrm{eq}/K_{500}$ , and the corresponding slopes, $k_\mathrm{eq}$ , are traced within Fig. 1, as a function of the dimensionless halocentric radius $s \equiv r/r_{500}$ .

Fig. 1 shows how varying the gas profile’s inner slope, $\varepsilon$ , drives the behaviour of the gas entropy in the central region. Gas cores, $\varepsilon=0$ , in the left column, produce high central entropy, characteristic of NCC clusters; weak gas cusps, $\varepsilon=0.5$ , in the centre column, attain a central entropy slope of $k_\mathrm{eq} \simeq 0.6-0.9$ , which is roughly consistent with observational constraints in CC clusters (e.g. Babyk et al. Reference Babyk, McNamara, Nulsen, Russell, Vantyghem, Hogan and Pulido2018). This association between cuspy gas inner slopes and CCs is a well known observational correlation (see, e.g. Hudson et al. Reference Hudson, Mittal, Reiprich, Nulsen, Andernach and Sarazin2010) and is well reproduced in these panels. In the right panel, showing NFW-like gas cusps, $\varepsilon=1$ , the gas entropy becomes increasingly steep towards the centre of the cluster, implying a rapid drop in the gas entropy in the core. We note that such steep gradients are not generally observed or predicted.

Throughout the parameter space traced in Fig. 1, the gas entropy slope converges to a constant value of $k_\mathrm{eq} \simeq 1.4$ in the cluster’s outskirts, beyond halocentric radii $r \gtrsim 0.8 r_{500}$ . In comparison to the consensus in the literature, where the gas entropy slope is expected to attain a constant value of $k \simeq 1.1$ beyond $r \gtrsim 0.6 r_{500}$ , this pristine equilibrium model systematically overestimates the gas entropy in the cluster’s outer region. In particular, by Equation (15), when treating the intracluster gas density profile as fixed, this overestimate in $k_\mathrm{eq}(s)$ reflects an overestimate in the logarithmic derivative of the pristine equilibrium gas temperature, $T_\mathrm{eq}(s)$ , which, as the gas temperature will be decreasing in the cluster’s outer region, implies that $T_\mathrm{eq}(s)$ is decreasing too gently with radius in the outskirts. If the gas temperature falls more rapidly, this implies that NTP is required, specifically as an increasing function of halocentric radius, to ensure that the cluster remains in hydrostatic equilibrium.

Choosing a weighting function

To relate the pristine and real entropy slopes and thus predict the required NTP function, we must prescribe the weighting function, w(s). At large radii, when $k_\mathrm{eq} \simeq 1.4$ , we require a weighting of $w \simeq 0.8$ , such that the entropy slope is reduced to $k \simeq 1.1$ , as is observed in both simulations and observations. Leaving the entropy slopes unchanged in the inner region, where slopes consistent with CCs and NCCs are relatively well established, implies that the weighting function must take the form of a continuous step function, transitioning between $w = 1$ and $w = 0.8$ as a function of halocentric radius.

There are two parameters that need to be chosen for such a function: the steepness of the transition and the radius at which the transition occurs. We set the mid-point weight of $w = 0.9$ to occur at a halo radius of $r \simeq 0.4 r_{500}$ , with the steepness set by an amplitude of 5 in the exponent. This choice ensures that the entropy slope $k \simeq 1.1$ is reached and remains fixed, above halocentric radii $r \gtrsim 0.6 r_{500}$ , whilst $k \simeq 1$ is a better fit to its value within the region $0.2r_{500} \lesssim r \lesssim 0.4 r_{500}$ .

Figure 2. The weighted gas entropy slopes, $k \equiv \mathrm{d} \ln K / \mathrm{d} \ln r$ , traced over the scaled halocentric radius $r/r_\mathrm{500}$ , derived as a modification to the pristine equilibrium profiles from Fig. 1, when weighted by the weighting function from Equation (33). The halo concentration, c, and dilution, d, are both fixed parameters, whilst each column varies the gas inner slope, $\varepsilon$ . Within each box, each colour varies the halo inner slope, $\unicode{x03B1}$ , with the solid coloured lines tracing a fraction of cosmological baryon content of $\unicode{x03B7}=0.8$ , and the shaded colour region around each solid line (not visible for all curves) tracing this value continuously between $\unicode{x03B7}=0.6$ and $\unicode{x03B7}=1$ . The faded profiles in the background of each panel correspond to the associated pristine equilibrium entropy slopes, $k_\mathrm{eq} \equiv \mathrm{d} \ln K_\mathrm{eq} / \mathrm{d} \ln r$ , from the top row of Fig. 1.

Figure 3. The non-thermal pressure (NTP) fraction, $\mathcal{F} \equiv p_\mathrm{nt} / p$ , traced over the scaled halocentric radius $r/r_{500}$ , that solves the entropy slope constraints via Equation (22). In each box, the cluster’s structural parameters are varied over the entire parameter space from Table 1, producing the turquoise shaded regions, given a choice in the cluster’s central NTP fraction, $\mathcal{F}_0$ , which is set to $\mathcal{F}_0 = 0$ in the left panel, and $\mathcal{F}_0 = 0.1$ in the right panel. The black dotted line in each box is the best-fit to the functional form proposed in Nelson et al. (Reference Nelson, Lau and Nagai2014), given in Equation (34), with its best-fitting parameters specified in Table 2. We compare our predictions to numerical fits: from Nelson et al. (Reference Nelson, Lau and Nagai2014), shown by the orange line; and from Angelinelli et al. (Reference Angelinelli, Vazza, Giocoli, Ettori, Jones, Brunetti, Brüggen and Eckert2020), shown by the light blue and blue dashed lines, corresponding to different contributions of the gas motion. We also compare to observational constraints: from the Hitomi Collaboration et al. (2018), as given by the pink shaded region, with the 4% value from Hitomi Collaboration et al. (2016) shown by the pink solid line; from Eckert et al. (Reference Eckert2019), shown by the red error bars; and from Dupourqué et al. (Reference Dupourqué, Clerc, Pointecouteau, Eckert, Ettori and Vazza2023), shown by the purple error bars.

This scale-free weighting function is then specified by the continuous step function:

(33) \begin{equation} w(s) = 0.8 + \frac{1}{5\left[1 + \mathrm{e}^{5\left[\log_{10}(s) + 0.4\right]}\right]}.\end{equation}

We emphasise that this choice in parameters within the step function is not unique and could be altered in both the steepness of the transition and its radial occurrence, both of which exhibit a degree of degeneracy to one another, and each of which can quantitatively impact the predicted NTP profile. However, as we have ensured that our choice produces entropy slopes that are consistent with the values in the literature, we do not consider other choices hereafter.

These new entropy slopes, calculated by weighting each of the pristine equilibrium entropy slopes, $k_\mathrm{eq}$ , from Fig. 1, are shown in Fig. 2. For all parameter configurations, these weighted entropy slopes now converge to $k = 1.1$ in the cluster’s outskirts, as ensured.