I am currently studying X-ray-emitting plasma pervading clusters of galaxies, and I see many studies employ an ideal equation of state to infer the pressure of the plasma. Is this the correct thing to do? Apart from this, it seems like an ideal equation of state is thrown around in astrophysics many times without proper justification. What do you think?
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$\begingroup$ State equation defines conditions upon which fluid solution special cases of Einstein field equations are based. So it's not "thrown around [...] without proper justification" like you say. Albeit, I can't give exact derivations of fluid solutions. In addition, modifying $w$ parameter in $p =w \rho c^2$ is easy to account for baryonic matter, radiation, dark energy, etc., unless you want to through away these concepts in astronomy and use MOND or something like that. $\endgroup$– Agnius VasiliauskasCommented Sep 28, 2023 at 14:03
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1$\begingroup$ State equations are more general and go beyond "defining the conditions upon which fluid solution special cases of Einstein field equations are based." My question is very specific about the usability of a special "easy" equation of state, the ideal equation P=nkT. People, especially in the observation community, use it to infer the plasma pressure in clusters without properly justifying the applicability. For all we know, the plasma could be a two-phase medium, or it may not be in equilibrium (winds/ cooling inflows). $\endgroup$– bhoutikCommented Sep 29, 2023 at 18:41
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An ideal equation of state assumes point-like, non-interacting particles.
Given the size of fundamental particles, we can compare this with $n^{-1/3}$ (where $n$ is the particle number density), which represents a typical particle separation in the gas/plasma.
The intra-cluster medium has a typical electron (and proton) number density of around $10^3$ m$^{-3}$. Electrons are of course point-like particles, but even protons, with a size $\sim 10^{-15}$ m are tiny compared with the typical particle separation of 0.1 m.
The second assumption can be tested by comparing the thermal or kinetic energy of the particles with any interaction energy. Providing the typical particle kinetic energy is much larger than the potential energy of their interactions, then they will behave like free particles as demanded by an ideal gas assumption.
In the intracluster medium, the typical particle kinetic energies will be $\sim 3k_BT/2$, which is $10^{-15}$ J for gas with $k_B T \sim 5$ keV. The potential energy between a proton and an electron separated by 0.1 m is $\sim 2\times 10^{-27}$ J. So again, the ideal gas approximation of no interactions is satisfied by many orders of magnitude.
In terms of cooling times etc. - the thin gas of the intracluster medium has a cooling time longer than the age of the universe, but may be shorter in the dense cores of clusters. There can also be a cold molecular gas in the cores of clusters. Turbulence can provide some pressure support in the central regions. All of these are considered in detailed simulations of clusters - but still use an ideal equation of state (e.g. Mohapatra & Sharma 2019).
In astrophysical scenarios I would say intracluster gas is possibly the most ideal of ideal gases - there are other places (e.g. the centre of the Sun) where the approximation is good, but not as good.
