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For the nonrelativistic white dwarf, combine the hydrostatic pressure support equation and enclosed mass equation:Eliminating $m$ yieldsIntroduce the Lane-Emden variables for a stellar polytrope,The resulting Lane-Emden equation isThe white dwarf surface is at the first zero ${\xi }_1$ of $\theta$, with $R=a{\xi }_1$.

Radiative diffusion in a star can be writtenDividing by hydrostatic equilibrium giveswhich is constant by assumption. Since both $P_{\mathrm{rad}}$ and $P$ vanish at the surface, integration gives $P_{\mathrm{rad}}=(1-\beta )P$ with constant $\beta$.

Eliminating $T$ between the gas and radiation equations of state now givesThus the star is an $n=3$ stellar polytrope. PutThe structure equations become the Lane-Emden equationThe surface is the first zero ${\xi }_1\simeq 6.89685$. Writingthe Lane-Emden mass formula givesThereforewhereThe additional radiative-equilibrium relation is $1-\beta =\nu L/(4\pi cGM)$.

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 $\beta =-1/3$.

The radiative-pressure equation isDividing by hydrostatic equilibrium and substituting
$\kappa ={\kappa }_0(M_r/L_r)(L/M)$ givesa constant. Since both pressures vanish at the surface,
$P_{\mathrm{rad}}=(1-\beta )P$ and $\beta$ is constant throughout the star. Part i therefore haswith spatially constant $K$.

Writing $\rho ={\rho }_c{\theta }^3$ andgives the $n=3$ Lane-Emden equationThe surface is the first zero $\theta ({\xi }_1)=0$. The $n=3$ equation has no elementary closed-form solution; among nonnegative indices, the standard analytic cases are $n=0,1,5$. This Eddington standard model approximates chemically homogeneous massive main-sequence stars with substantial radiation pressure.

For $P=K{\rho }^{1+1/n}$, introduce the Lane-Emden equation variablesCombining mass conservation with hydrostatic equilibrium givesRegularity and normalization imposeThe stellar surface is the first positive zero ${\xi }_1$ of $\theta$. Integrating the equation once givesand hence

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.

A stellar polytrope of index $n$ obeysCombining hydrostatic equilibrium, $dP/dr=-Gm\rho /r^2$, with mass conservation, $dm/dr=4\pi r^2\rho$, gives the Lane-Emden equationRegularity and normalization at the stellar centre requireIntegrating the Lane-Emden equation from the centre then gives the enclosed mass

For $n=5$, direct substitution into the equation findsso $\beta =1/3$ and $\gamma =-1/2$. This profile has no finite first zero and hence has infinite radius, but its total mass converges:Its mean density over the full, infinite configuration is consequently zero.

For a perfect gas, $T\propto P/\rho \propto {\rho }^{1/5}$, so $T/T_c=\theta$. The luminosity from CNO cycle burning with $ϵ={ϵ}_0\rho T^{11}$ is thereforewhereThus $A$ is of order unity. The model is physically poor because the $n=5$ polytrope has infinite radius and zero mean density, while strongly temperature-sensitive CNO burning changes the thermal gradient and commonly creates convection. Real cores also have evolving composition, non-polytropic opacity and energy transport, and boundaries supplied by the surrounding star.

The regular $n=5$ Lane-Emden equation solution has no finite first zero. Its Lane-Emden mass formula gives $m(\xi )=4\pi {\rho }_c{\alpha }^3{\xi }^3/[3(1+{\xi }^2/3{)}^{3/2}]$, so its infinite-radius total mass is $4\pi \sqrt{3}{\rho }_c{\alpha }^3$. Its enclosed mean mass density is ${\rho }_c(1+{\xi }^2/3{)}^{-3/2}$ and tends to zero. A finite externally confined core instead requires a specified truncation pressure.

The regular $n=0$ solution of the Lane-Emden equation is $\theta =1-{\xi }^2/6$, with first zero $\sqrt{6}$. Its interior density is constant and its pressure varies as $\theta$. This is the structural incompressible limit; $P=K{\rho }^{1+1/n}$ itself is singular at $n=0$, and does not assign the perturbative stellar adiabatic exponent of a thermally stratified ideal-gas model.

A stellar polytrope obeys $P=K{\rho }^{1+1/n}$ for constant $K$ and polytropic index $n$. Its dimensionless density profile satisfies the Lane-Emden equation.