 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-66/4/a/ii/solution

The hot intracluster gas emits X-rays, which constrain its temperature and density profile. If its pressure is predominantly thermal and it is in hydrostatic equilibrium,

\frac{d P}{d r} = - ρ_{g} \frac{GM ( < r )}{r ^{2}}, P = \frac{ρ _{g} k _{B} T}{μ m _{p}} .

Eliminating the pressure gives

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

Here

μ m_{p}

is the mean mass per gas particle. A gas at the observed cluster temperatures would not remain confined without a deep gravitational potential; the inferred total mass exceeds the gas and stellar mass together. Nonthermal pressure, mergers, gas clumping and temperature-profile errors can bias this estimate, but do not turn X-ray luminosity itself into a direct measurement of total mass.

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