Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 314 / 2 / c

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- Solution c

Solution

At the Bondi sonic point,

u_{s} = c_{s}

and

GM / r_{s} = 2 c_{s}^{2}

. Evaluating the Bernoulli function there gives

\frac{c _{s}^{2}}{2} + c_{s}^{2} ln ρ_{s} - 2 c_{s}^{2} = c_{s}^{2} ln ρ_{0},

and hence

ρ_{s} = e^{3/2} ρ_{0} .

The conserved mass accretion rate is therefore

\dot{M} = 4 π r_{s}^{2} ρ_{s} c_{s} = π e^{3/2} ρ_{0} \frac{( GM ) ^{2}}{c _{s}^{3}} .

Thus

A = π e^{3/2} .

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