# Temperatures of Galaxy Clusters

Recently I've read about clusters of galaxies and have tried to understand how one can measure their mass and temperatures.

Clearly, we can physically measure a couple of things such as the flux of X-rays as a function of photon energy and the distance to the galaxy cluster. Hypothetically speaking, if we had a data set of X-ray flux with the corresponding photon energy how is this used to determine the temperature and mass of the system?

• For thermal X-rays, you can use bremsstrahlung radiation and have power $\sim \sqrt{T}$, so temperature should be on the easy side, no? – Kyle Kanos Feb 8 at 22:15
• @KyleKanos - The only problem is that the temperatures are high enough in some of these places (at least it appears to be) that relativistic effects start to matter for the electrons and inverting the bremsstrahlung spectrum in the relativistic limit is apparently not trivial (never tried but a colleague of mine has been after this for a few years). – honeste_vivere Feb 9 at 18:45
• @honeste_vivere hmm, I suppose that could cause more issues, something I obviously didn't think of in my 10 second response – Kyle Kanos Feb 9 at 21:53
• @KyleKanos - No worries, I made a similar comment to the colleague that has been working on the relativistic corrections to bremsstrahlung and he corrected me. I hadn't thought of the issue either. – honeste_vivere Feb 10 at 17:47

As answered by @Kyle you can estimate the temperature of a galaxy cluster from its X-ray spectra. In addiction to bremsstrahlung radiation, in the spectra there are some emission lines that allows you to estimate $$T$$ (like high ion of $$Fe$$ or $$Ni$$) with persist until $$\sim 10^8 K$$. Remember that i sa $$T(r)$$, i.e. as a function of the distance from the cluster center.

In order to estimate the mass you can assume the hydrostatic equilibrium for the emitting gas, and also that this gas can be approximately follows the perfect gas equation:

$$P= \frac{\rho(r)}{\mu m_H} k T(r)$$; where $$\mu=0.5$$ in case of a gas made only by $$H$$ and

$$\frac{dP}{dr}=-\frac{GM(r)}{r^2}\rho(r)$$ for the hydrostatic equilibrium.

Combining the two equations you find

$$M(r)\propto \frac{d\rho}{dr}T + \frac{dT}{dr}\rho(r)$$

where $$\frac{d\rho}{dr}$$ and $$\frac{dT}{dr}$$ are two observables from the X-rays (you clearly need to assume a density profile).

Note that this is only one method to estimate the mass of a galaxy cluster.