 Solution

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 $\beta =-1/3$.