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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-314/2/c/solution