Gas Mass Calcuation in Galaxy Cluster

[Earnest Guest]

I found this: http://adsabs.harvard.edu/abs/1992A&A...259L..31B document that describes how the X-Ray emissions from galaxy clusters are used to calculate the mass of the cluster. I'm unable to follow the steps to calculating the gas mass.

Specifically, in section (2) there is a formula for the surface brightness profile that appears to be pretty standard:

S(r) = S_0(1+({\frac{r}{a}})^2)^{(0.5-3\beta)}​

The authors then go on to calculate the central electron density:

n_0 = 2.89 \times 10^{-3} h_{50}^{1/2} cm^{-3}
​

I can't make the connection from the central electron density to a function that provides the mass at a given radius, r. The authors conclude the hot gas mass is

5.1 \times 10^{14} h_{50}^{\frac{-5}{2}} M⊙
​

but I don't see how they get from A to B.

 

[Chalnoth]

Following the references back, this paper gives a little bit more detail.

Unfortunately, the specific math they're using gets pretty complicated as they're using certain assumptions about the distribution of the cluster gas in order to make the problem tractable given their observations. Wish I knew of a source that walked through all of the steps.

 

[Earnest Guest]

Following the references back, this paper gives a little bit more detail.

Unfortunately, the specific math they're using gets pretty complicated as they're using certain assumptions about the distribution of the cluster gas in order to make the problem tractable given their observations. Wish I knew of a source that walked through all of the steps.

Yeah, it's like the gnomes on South Park:
1. Steal Underwear
2. ?
3. Profit!

I've got as far as a density profile but I can't get the next step. Any chance someone can give me an analytical solution to:

\int_0^R k\Big(1+(\frac{r}{a})^2\Big)^p r^2 dr​

 

[Chalnoth]

Argh, somehow I left out the link I meant to post:
http://adsabs.harvard.edu/full/1976A&A...49..137C

Not sure if that helps or not.