= Solution
Hydrostatic equilibrium and mass conservation are
$$
\frac{dP}{dr}=-\frac{Gm\rho}{r^2},
\qquad
\frac{dm}{dr}=4\pi r^2\rho.
$$
For $P=K\rho^{1+1/n}$, set
$$
\rho=\rho_c\theta^n,\qquad r=a\xi,
\qquad
a^2=\frac{(n+1)K}{4\pi G}\rho_c^{1/n-1}.
$$
Eliminating $m$ gives the <Lane-Emden equation>
$$
\boxed{\frac1{\xi^2}\frac d{d\xi}
\left(\xi^2\frac{d\theta}{d\xi}\right)=-\theta^n}.
$$
If $\xi_1$ is its first zero, then
$$
R=a\xi_1,\qquad
M=4\pi a^3\rho_c[-\xi_1^2\theta'(\xi_1)].
$$
Eliminating $\rho_c$ at fixed $K$ gives
$$
\boxed{M^{\,n-1}R^{\,3-n}=C},
$$
so 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.
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