Let $u>0$ denote inward radial speed. Steady spherical mass conservation and the radial momentum equation give
$$
\dot M=4\pi r^2\rho u,
\qquad
\left(u-\frac{c_s^2}{u}\right)\frac{du}{dr}
=\frac{2c_s^2}{r}-\Phi_{\rm dm}'(r).
$$
A regular <sonic point> requires both factors to vanish:
$$
u_s^2=c_{s,s}^2,\qquad
\frac{2c_{s,s}^2}{r_s}
=\frac{8\pi G\Psi_0}{r_s^{3/2}},
$$
so $c_{s,s}^2=4\pi G\Psi_0/\sqrt{r_s}$ and $\Phi_{\rm dm}(r_s)=-4c_{s,s}^2$.

The gravitational <Bernoulli function> equals its value in the uniform gas at infinity:
$$
\frac{u^2}{2}+\frac{c_s^2}{\gamma-1}+\Phi_{\rm dm}
=\frac{c_0^2}{\gamma-1}.
$$
At the sonic point this gives
$$
c_{s,s}^2=\frac{2c_0^2}{9-7\gamma}.
$$
A physical sonic point therefore exists exactly when
$$
\boxed{1<\gamma<\frac97}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-314/3/b/solution