 Solution

Hydrostatic equilibrium and mass conservation areFor $P=K{\rho }^{1+1/n}$, setEliminating $m$ gives the Lane-Emden equationIf ${\xi }_1$ is its first zero, thenEliminating ${\rho }_c$ at fixed $K$ givesso one convenient choice is $\boxed{\alpha =n-1,\beta =3-n}$. Equivalently, one may take $\beta =1$ and $\alpha =(n-1)/(3-n)$.

The index $n=0$ describes an incompressible constant-density body, approximating a weakly compressed small rocky planet. The index $n=3/2$ describes a fully convective monatomic ideal gas or nonrelativistic electron degeneracy, as in a low-mass star, brown dwarf, or nonrelativistic white dwarf.