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

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.