= Solution
A <stellar polytrope> replaces a detailed <equation of state> and thermal structure by $P=K\rho^{1+1/n}$ for $n>0$, with spatially constant $K$ and <polytropic index> $n$. Together with spherical <hydrostatic equilibrium> and mass conservation, it gives a tractable family of stellar density profiles. Introduce the <Lane-Emden variables for a stellar polytrope>
$$
\rho=\rho_c\theta^n,\qquad r=b\xi,\qquad b^2=\frac{(n+1)K}{4\pi G}\rho_c^{1/n-1}.
$$
Combining $dP/dr=-Gm\rho/r^2$ with $dm/dr=4\pi r^2\rho$ eliminates $m$ to give $r^{-2}d[r^2\rho^{-1}(dP/dr)]/dr=-4\pi G\rho$. Substitution of the pressure law and the dimensionless variables gives the <Lane-Emden equation>
$$
\frac1{\xi^2}\frac{d}{d\xi}\left(\xi^2\frac{d\theta}{d\xi}\right)=-\theta^n,\qquad\theta(0)=1,\quad\theta'(0)=0.
$$
A regular finite-radius model for $0\leq n<5$ ends at the first zero $\xi_1$. Its radius and mass are
$$
R=b\xi_1,\qquad M=4\pi b^3\rho_c[-\xi_1^2\theta'(\xi_1)].
$$
These formulas supply central-to-mean density ratios and <stellar homology> scalings without solving a full evolutionary model. Eliminating $\rho_c$, for $0<n<5$ and $n\ne3$, gives the <polytropic mass-radius relation>
$$
R\propto K^{n/(3-n)}G^{-n/(3-n)}M^{(1-n)/(3-n)}.
$$
An efficient-convection <monatomic gas> has $n=3/2$. The same index describes a cold, nonrelativistic <electron degeneracy pressure> law; at fixed composition it gives $R\propto M^{-1/3}$ for a <white dwarf>. Ultrarelativistic <electron degeneracy pressure> has $n=3$, for which the mass is independent of $\rho_c$ and scales as $(K/G)^{3/2}$. This is the origin of the <Chandrasekhar limit>. A radiation-dominated, approximately constant-entropy interior also approaches $P\propto\rho^{4/3}$ and $n=3$. The <polytrope of index zero> is an incompressible comparison model, useful for illustrating the role of central concentration.
The <gravitational energy of a stellar polytrope>, $\Omega=-3GM^2/[(5-n)R]$, and the <stellar virial theorem> make polytropes useful for contraction-energy estimates. They also provide structural models for perturbation and stability calculations. \b[A polytrope is a structural approximation, not a complete stellar evolution calculation.] One must still specify <stellar nuclear fusion>, <opacity>, energy transport, composition and boundary conditions to determine the <luminosity> and history. Moreover, the structural exponent $1+1/n$ need not equal the <adiabatic exponent> relevant to a rapid perturbation. Real stars can have different effective indices in their cores and envelopes, as well as <ionization> zones or varying <electron degeneracy pressure>.
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