Question

A cluster of galaxies has a radius of about 4.8 million light-years (4.5×10²² m ) and an intracluster medium with a temperature of 6.5×10⁷ K . [Hint: Consult Mathematical Insight 23.3.]

A) Estimate the mass of the cluster. Give your answer in both kilograms and solar masses. Suppose the combined luminosity of all the stars in the cluster is 7.6×10¹² LSun .

B) What is the cluster's mass-to-light ratio?

Answer 1

given

cluster of galaxies

radius, r = 4.8 *10^6 LY = 4.5*10^22 m

temperature, T = 6.5*10^7 K

A. luminosity, S = sigma*A*T^4

sigma = 5.67*10^-8

A = 4*pi*r^2

hence

S = 2.633*10^70 W

now, from mass luminosity relation

S/S. = 32000*M/M.

where S. is luminosity of sun, S. = 3.828*10^26 W

M. = mass of sun = 2*10^30 kg

hence

given

S/S. = 7.6*10^12

hence

M = 237500000M. =4.75*10^38 kg

B. cluster's mass to light ratio = 4.75*10^38/7.6*10^12*3.828*10^26 = 0.1632706374085 kg / W