For $r\ll GM/B$, the Mach number tends to a constant $\mathcal M_1$. Since $A\dot M_{\rm crit}^{1/2}=GM/4$, its algebraic equation is
$$
\boxed{\dot m^{1/2}
\left(\mathcal M_1^{3/2}+3\mathcal M_1^{-1/2}\right)=4},
$$
or equivalently
$$
\boxed{\dot m\,\frac{(\mathcal M_1^2+3)^2}{\mathcal M_1}=16}.
$$
The central Bernoulli balance gives
$$
v_s^2\sim\frac{2GM}{(\mathcal M_1^2+3)r},
$$
so
$$
\boxed{v_s\propto r^{-1/2}},
\qquad
\boxed{\rho\propto v_s^3\propto r^{-3/2}}.
$$
Finally $u=\mathcal M_1v_s$. Using the algebraic relation to rewrite its coefficient gives
$$
\boxed{u\sim
\left(\frac{\dot m\,\mathcal M_1^3}{4}\right)^{1/4}
\left(\frac{GM}{r}\right)^{1/2}}.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 314 / 1 / e / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-314/1/e/solution