 Solution

ID: past-exam-of-the-short-options-paper-shared-by-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2019/s2/3/c/solution

Assume an isotropic, stationary spherical galaxy cluster. Its three-dimensional velocity dispersion obeys

⟨ v^{2} ⟩ = 3 σ_{los}^{2}

. For a uniform sphere, the virial theorem gives

3 M σ_{los}^{2} = 3 G M^{2} / (5 R)

, hence

M_{dyn} ≃ \frac{5 R σ _{los}^{2}}{G} = 9.4 \times 1 0^{14} M_{⊙} .

A different density profile changes the coefficient, so

1 0^{15} M_{⊙}

is the meaningful scale. For isothermal X-ray gas in hydrostatic equilibrium,

M (r) = - \frac{k _{B} T r}{G μ m _{p}} \frac{d l n ρ _{g}}{d l n r} .

Using

μ = 0.6

and a representative slope

- d ln ρ_{g} / d ln r = 2

–

3

1 Mpc

gives

M_{X} ≃ (1.3

–

1.9) \times 1 0^{15} M_{⊙}

. The temperature and extent alone cannot fix this coefficient without a density profile. The total luminosity is

5 \times 1 0^{12} L_{⊙}

, so the previously inferred mass-to-light ratio gives

M_{⋆} ≃ 4.50 (5 \times 1 0^{12}) M_{⊙} = 2.25 \times 1 0^{13} M_{⊙} .

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