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

The approximation requires a nonrelativistic monatomic gas whose particles collide often enough for local thermodynamic equilibrium, with a mean free path much shorter than every macroscopic scale. Translational motion must dominate its heat capacity, giving the adiabatic exponent $\gamma =5/3$. Heat conduction, viscosity, shocks, radiative heating and cooling, ionization, chemical reactions, and excitation of internal degrees of freedom must be negligible over the flow time. Under those conditions entropy is advected and the gas behaves as an adiabatic perfect gas.

Let $u>0$ be the inward radial speed. Steady spherical Bondi accretion conserves mass:For $\gamma =5/3$, $p=K{\rho }^{5/3}$ and the adiabatic sound speed satisfies $v_s^2=\gamma p/\rho \propto {\rho }^{2/3}$. Matching the reservoir therefore givesThe Bernoulli integral isWriting the Mach number as $\mathcal{M}=u/v_s$ and eliminating $v_s$ with mass conservation givesMultiplication of the Bernoulli equation by $r/2$ now yieldswhere

The functionhas its unique minimum at $\mathcal{M}=1$, where $f(1)=4$. At large $r$, mass conservation and the reservoir boundary conditions give $u\propto r^{-2}$ and hence $\mathcal{M}\to 0$. The physical solution therefore begins on the subsonic branch $0<\mathcal{M}<1$. It cannot pass smoothly to $\mathcal{M}>1$ without reaching the minimum, where the two algebraic branches meet. For $\gamma =5/3$ that meeting can occur only in the limiting central behavior of the critical solution, so the flow remains subsonic for every $r>0$.

As $r\to 0$, the right-hand side of the Mach-number equation tends to $GM$. Since $f(\mathcal{M})\ge 4$, a real positive solution at arbitrarily small $r$ requiresUsing the value of $A$ givesThis is the critical Bondi accretion rate for gamma equals five thirds.

For $r\ll GM/B$, the Mach number tends to a constant ${\mathcal{M}}_1$. Since $A{\dot{M}}_{\mathrm{crit}}^{1/2}=GM/4$, its algebraic equation isor equivalentlyThe central Bernoulli balance givessoFinally $u={\mathcal{M}}_1{v}_s$. Using the algebraic relation to rewrite its coefficient gives