**1. The problem statement, all variables and given/known data**

Hi, I am having difficulties with a first-year astronomy question that was issued on a final a few years back.

A group of astronomers make various observations of a cluster spanning 10arcmin (600arcsec) on the sky. X-ray astronomers found the cluster to be full of hot gas, with λmax=0.0377nm.

Optical astronomers found the bolometric apparent magnitude of the whole cluster to be, m=10.4.

The mass to light ration of the stellar populations in the galaxies in the cluster was equal to 2. Meanwhile, radio astronomers found the neutral hydrogen line associated with the cluster (λHI)=27.3cm. Finally, for this problem use the Hubble constant of 70km/s/Mpc and the density of the intercluster medium to be equal to 10^-27g/cm^3

Find the following:

1. The redshift, z and distance to the cluster d, in pc.

2. The radius of the cluster r, in pc.

3. The mass of the hot gas in the cluster, Mgas.

4. Luminosity and mass of the stellar content, in solar units.

5. Cluster’s total mass Mtot

6. Percentage of cluster’s mass in dark matter.

**2. Relevant equations**

Stated in the question it says use basic formula and do not use modified inverse square laws for distant galaxies for brightness or size.

This question applies many formulas, I am not sure which to use for every part.

**3. The attempt at a solution**

1. z(of hydrogen line, using normal atomic hydrogen radius of 21cm found in textbook):

z=(λs-λr)/λr = (27.3cm-21.0cm)/21.0cm = 0.3

d=1/p = 1/600" = 0.00167pc.

2. Not sure what to do here.

3. Mgas= (r(Vgas)^2)/G

Vgas(X-ray)=140(sqrt(T)), where T is found by 2.9e6/λmax to be 7.69e7K. Thus, Vgas is equal to 1.23e6m/s, and therefore Mgas = r(1.23e6)^2/6.67e-11

4. L/Lsun=100^(4.75-M)/5

M=m+5-5log(d)

M=10.4+5-5log(0.00167) = 29.3, thus L=1.23e-5Lsun?

I don’t know if this was done correctly?

5. Not sure what to do here.

6. I believe that Mtot-Mstars=Mdark matter then I just find its percentage but still don’t have either Mtotal or Mstars.

Thanks so much!

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