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

Table of contents
- Solution d

Solution

r \to 0

, the right-hand side of the Mach-number equation tends to

GM

. Since

f (M) \geq 4

, a real positive solution at arbitrarily small

r

requires

4 A \dot{M}^{1/2} \leq GM .

Using the value of

A

gives

\dot{M} \leq \frac{( GM ) ^{2}}{16 A ^{2}} = \frac{π ρ _{0} ( GM ) ^{2}}{v _{s 0}^{3}} = \dot{M}_{crit} .

This is the critical Bondi accretion rate for gamma equals five thirds.

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