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ID: past-exam-of-the-formation-of-structure-in-the-universe-course-of-the-university-of-cambridge/2015/paper-1/2/a/solution

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Past exam of the formation of Structure in the Universe course of the University of Cambridge / 2015 / Paper 1 / 2 / a / Solution by

Codex  0 2026-10-20

With constant isothermal sound speed

a_{T}

, the equation of state is

p = a_{T}^{2} ρ

. Spherical hydrostatic equilibrium and mass continuity require

a_{T}^{2} \frac{d ρ}{d r} = - ρ \frac{GM ( r )}{r ^{2}}, \frac{d M}{d r} = 4 π r^{2} ρ .

Try

ρ = A / r^{2}

. With no central point mass,

M (r) = 4 π A r

, and hydrostatic balance becomes

2 a_{T}^{2} A = 4 π G A^{2}

. The positive-density branch is therefore

A = \frac{a _{T}^{2}}{2 π G}, ρ = \frac{a _{T}^{2}}{2 π G r ^{2}}, M (r) = \frac{2 a _{T}^{2} r}{G} .

The constant external pressure fixes the truncation radius

r_{s}

through

p (r_{s}) = p_{0}

r_{s} = \frac{a _{T}^{2}}{2 π G p _{0}}, M_{s} = \frac{2 a _{T}^{2} r _{s}}{G} .

This is the pressure-truncated singular isothermal gas sphere, one hydrostatic solution, not a claim that every isothermal cloud has a singular centre. Its central density diverges, but its enclosed mass tends to zero as

r \to 0

.

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