= Solution
A <stellar polytrope> approximates a spherical self-gravitating star by the structural relation
$$
p=K\rho^{1+1/n},
$$
where $K$ is constant and $n$ is the <polytropic index>. Combining <hydrostatic equilibrium> with <mass conservation> gives
$$
\frac{dp}{dr}=-\frac{Gm(r)\rho}{r^2},\qquad
\frac{dm}{dr}=4\pi r^2\rho.
$$
The <Lane-Emden variables for a stellar polytrope> separate the dimensional scales from the profile:
$$
\rho=\rho_c\theta^n,\quad r=a\xi,\qquad
a^2=\frac{(n+1)K}{4\pi G}\rho_c^{1/n-1}.
$$
For $n>0$ these substitutions give the <Lane-Emden equation>
$$
\boxed{\frac1{\xi^2}\frac d{d\xi}\left(\xi^2\frac{d\theta}{d\xi}\right)=-\theta^n,\qquad
\theta(0)=1,\quad\theta'(0)=0.}
$$
The central conditions express finite central <mass density> and spherical symmetry. If $\xi_1$ is the first zero, a zero-pressure surface has
$$
R=a\xi_1,\qquad
M=4\pi a^3\rho_c[-\xi_1^2\theta'(\xi_1)],
$$
the <Lane-Emden mass formula>. For $0<n<5$ this surface is at finite radius. For example, the <polytrope of index one> has $\theta=\sin\xi/\xi$ and $\xi_1=\pi$, giving the density profile shown in the overview figure.
An isentropic ideal monatomic gas has $p\propto\rho^{5/3}$ and hence $n=3/2$, a useful approximation for a fully convective ideal-gas star. Nonrelativistic <electron degeneracy pressure> has the same density exponent. The exponent $4/3$, giving $n=3$, occurs for ultrarelativistic electron degeneracy and in appropriate radiation-dominated stellar models. The structural index should not be confused with the <adiabatic index> controlling a perturbation unless the model is actually isentropic.
Polytropes give useful mass-radius scalings and central concentration with very few parameters. They do not by themselves specify <nuclear fusion>, energy transport, or the star's thermal evolution. Nor does a structural profile alone establish dynamical stability; that requires the response of <pressure> to the perturbation.
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