Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 314 / 3 / b

Table of contents
- Solution b

Solution

Let

u > 0

denote inward radial speed. Steady spherical mass conservation and the radial momentum equation give

\dot{M} = 4 π r^{2} ρ u, (u - \frac{c _{s}^{2}}{u}) \frac{d u}{d r} = \frac{2 c _{s}^{2}}{r} - Φ_{dm}^{'} (r) .

A regular sonic point requires both factors to vanish:

u_{s}^{2} = c_{s, s}^{2}, \frac{2 c _{s, s}^{2}}{r _{s}} = \frac{8 π G Ψ _{0}}{r _{s}^{3/2}},

so

c_{s, s}^{2} = 4 π G Ψ_{0} / r_{s}

and

Φ_{dm} (r_{s}) = - 4 c_{s, s}^{2}

.

The gravitational Bernoulli function equals its value in the uniform gas at infinity:

\frac{u ^{2}}{2} + \frac{c _{s}^{2}}{γ - 1} + Φ_{dm} = \frac{c _{0}^{2}}{γ - 1} .

At the sonic point this gives

c_{s, s}^{2} = \frac{2 c _{0}^{2}}{9 - 7 γ} .

A physical sonic point therefore exists exactly when

1 < γ < \frac{9}{7} .

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