Understanding X-Ray emitting gas calculations in Galaxy Clusters This is a follow up to a question I asked about the Bullet (and Coma) cluster.
I've read several papers now about the fact that galaxy clusters collect overdense areas of hydrogen which emits X-ray radiation.  These overdense areas can make up close to 90% of the baryonic mass according to some calculations.  However, the physics of this calculation elude me.  I understand the ideal gas law:
$$PV = nRT$$
And I also understand the part where they take a picture of the sky, measure the bright area, project it into 3D for the volume, then measure the temperature, but this is where I get stuck.  Without knowing the pressure, how do you make the connection that allows you to calculate the number of moles (total mass) in the cloud?  For example, I've found a calculation of 3.0 e13 M⊙ for the mass of the X-Ray gasses in the Coma cluster from this Gursky et al.
Won't half the mass, 1.5 e13 M⊙, emit the same temperature if the pressure is doubled?  How can we estimate the mass of a gas cloud if we don't know the pressure?
 A: But you do know the pressure. The X-ray flux of a hot, optically thin gas is proportional to the emission measure (electron density squared times volume) and to the emissivity of the hot gas (and also divided by a geometric factor to account for the distance to the object).
The frequency integrated emissivity of very hot gas is approximately that of thermal bremsstrahlung (in practice the calculations are more precise, including ionised species, Gaunt factors etc) and depends on $T^{1/2}$.
$$ F_x = \Lambda(T) n_{e} n_i V/4\pi d^2$$
So you measure the flux $F_x$ and temperature $T$ from the X-ray spectrum; calculate the emissivity $\Lambda$; estimate the gas volume $V$, estimate distance $d$ from the redshift and Hubble parameter and rearrange to get $n_e n_i \simeq n_e^2$. The pressure is roughly $2n_e k_B T$ (the gas is almost fully ionised and mostly hydrogen).
If we let the mass units per electron be $\mu_e$ (roughly 1.2 for a standard H, He mixture), then density $\rho = \mu_e m_u n_e$; multiply by the volume and you have the gas mass.
