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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?

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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.

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  • $\begingroup$ What is the name of this law? Do you have a link that can give me the background on the derivation? $\endgroup$
    – user32023
    Commented Oct 5, 2015 at 12:30
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    $\begingroup$ @DonaldRoyAirey It's not a "law". However you could look at the section on thermal bremsstrahlung in Radiative Processes in Astrophysics by Rybicki & Lightman (p.160-162), or in fact there is a wikipedia page on coronal radiative losses in the solar corona. The hot cluster gas is similar, but usually a bit hotter and thus more bremsstrahlung dominated. en.wikipedia.org/wiki/Coronal_radiative_losses $\endgroup$
    – ProfRob
    Commented Oct 5, 2015 at 14:10
  • $\begingroup$ I don't see how this works. If the radiation was bremsstrahlung, then the X-Ray radiation would eventually cool down to CMBR. What renews the energy such that the output is constant? $\endgroup$
    – user32023
    Commented Oct 5, 2015 at 14:54
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    $\begingroup$ Here's a neat lecture from Chris Flynn on bremsstrahlung, which includes a computation of the cooling time for a rather large gas cloud emitting only brems. rad.; it would indeed cool rapidly... if there were no new source for energy. $\endgroup$
    – Kyle Kanos
    Commented Oct 5, 2015 at 16:02
  • $\begingroup$ @KyleKanos You are right - for some reason I thought cluster cooling times were very long. $\endgroup$
    – ProfRob
    Commented Oct 5, 2015 at 17:50

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