Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 315 / 3

For a spherical stellar polytrope with $P=K{\rho }^{1+1/n}$, the Lane-Emden equation givesEliminating the central density at fixed composition and entropy gives the polytropic mass-radius relationAn incompressible rocky body has $n=0$ and $\beta =1/3$. A moderately massive gas giant is approximately an $n=1$ polytrope and has $\beta =0$, explaining its weak radius dependence on mass. In a more strongly degenerate nonrelativistic regime, $n=3/2$ gives $\beta =-1/3$.

Assemble a uniform-density sphere from shells. Since $m(r)=M(r/R{)}^3$ and $dm=3M{r}^{2}dr/R^3$,Hydrostatic equilibrium gives its central pressureThus, relative to the same uniform-density estimate for Earth,Using $(M,R{)}_{\mathrm{N}}\simeq (17.1,3.88)$ gives $P_{c,\mathrm{N}}/P_{c,\oplus }\simeq 1.3$, while $(M,R{)}_{\mathrm{J}}\simeq (318,11.2)$ gives $P_{c,\mathrm{J}}/P_{c,\oplus }\simeq 6.4$. Real central pressures differ because all three planets are compressible and compositionally stratified.

For uniform density, Kelvin-Helmholtz contraction releases binding energy $|U|=3G{M}^2/(5R)$. If mass and luminosity are constant and stellar heating is negligible at 90 au, the contraction age isEnergy conservation, $L=-dU/dt$, givesThe virial theorem places roughly half of the released gravitational energy into internal heat. Including that effect gives $t_{\mathrm{KH}}\simeq 3G{M}^2/(10RL)$ and $\dot{R}\simeq -10L{R}^2/(3G{M}^2)$.