Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 316 / 4 / v / Solution

For a spherical planet of mass density ${\rho }_p$, $R_p=(3M_p/4\pi {\rho }_p{)}^{1/3}$. The point-comet surface escape velocity and circular Kepler orbit speed therefore giveRaise this equation to the power $3/2$ and solve for $M_p$:Thus the constant of proportionality is $\boxed{\sqrt{3/(32\pi )}}$, and the gravitational constant cancels. Dimensional checking gives mass on the right: $M_{\star }^{3/2}{a}_p^{-3/2}{\rho }_p^{-1/2}$ has that dimension.

This uses the planet's own escape velocity. If instead $\chi$ is defined at finite-comet contact, an additional factor $(1+R/R_p{)}^{3/2}$ appears on the right, and $R_p$ itself depends on $M_p$; the quoted pure-planet scaling is then only the $R/R_p\to 0$ limit.