At the <Bondi sonic point>, $u_s=c_s$ and $GM/r_s=2c_s^2$. Evaluating the <Bernoulli function> there gives
$$
\frac{c_s^2}{2}+c_s^2\ln\rho_s-2c_s^2
=c_s^2\ln\rho_0,
$$
and hence
$$
\rho_s=e^{3/2}\rho_0.
$$
The conserved <mass accretion rate> is therefore
$$
\dot M=4\pi r_s^2\rho_sc_s
=\pi e^{3/2}\rho_0\frac{(GM)^2}{c_s^3}.
$$
Thus
$$
\boxed{A=\pi e^{3/2}}.
$$

Solution

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

{title2=$\dot M$}

The mass accretion rate is the mass entering an accretor per unit time. In a steady spherical flow, $\dot M=4\pi r^2\rho u$ is independent of radius.


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