Quantitative observations of Pandora cluster

1. The problem statement, all variables and given/known data
Hey, if anyone could help with this question I am very stuck on (not homework) I would really appreciate it.
A group of astronomers make observations of the Pandora cluster of galaxies spanning 600 arcseconds in the sky. X-Ray astronomers found it to have a λmax of 0.0377nm. It has a bolometric apparent magnitude, m of 10.4. It has a mass to light ratio of 2. Radio astronomers found neutral hydrogen lines of 27.3cm. Adopt Hubble’s constant, Ho of 70km/s/Mpc and interstellar density of 10^-27g/cm^3.

1.Find redshift,z and distance,d to the cluster in pc.
2.Find radius of cluster, r in pc
3.Find mass of hot gas in the cluster, Mgas in solar units
4.Find Luminosity of stellar content, Lstars in solar units and its corresponding mass of stellar content in solar units.
5.Find total cluster mass and what percent is dark matter?
2. Relevant equations
>Do not use modified inverse square laws for brightness or sizes since this is a distant galaxy.
>Knowing from lectures and such dark matter should be >85% of total mass.
Many formulas can be used in this problem, but it states use basic formulae only; including:
M=rv^2/g | Vgas=140m/s(sqrt(T) | z=Δλ/λ | v=dHo | L/Lsun=100(4.75-M)/5 , etc…
3. The attempt at a solution

1. z is found with atomic hydrogen lines from radio guys, z=27.3cm-21cm/21cm = 0.3
d is found with hubble’s law, v=dHo, d=v/Ho where v=cz in km/s.
d is found to be 1286Mpc

2.I’m not 100% sure I did this part correct,
θ=600"/60 = 10arcmins .. divide by 60 again and get 0.1667degrees,
R=sin(0.1667)*1286e6 = 3740817pc
I am not sure if this is the correct radius value, I have seen other formulas used that say this may be diameter, looking for clarification.

3. Mgas=R(Vgas)^2/G.
>Have radius, R and G is a constant and Vgas can be found with formula above, knowing temperature from x-ray astronomers wavelength calculations (T=2.9e6nm/λ[nm])
… skipping the math, Mgas=6.52e14Msun

where M(abs magnitude)=m+5-5logd, have distance and m, bolometric apparent magnitude.. M is equal to ~-30. L is therefore roughly 9.1e13Lsun I believe.

Knowing this and the mass-light ratio I believe I am to do this: (M/L) =2= (M/9.1e13Lsun), thus M=2(9.1e13)=1.82e14Msuns(stellar content)

5.I don’t understand why this part is asked because I have already found the mass of hot gas which I thought was equal to it’s total mass, just really confused here.



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