The Equation of State and Polytropes

The four structure equations close only when the pressure is expressed in terms of the density, temperature, and composition. This lesson builds that equation of state from its three contributions — ideal gas, radiation, and electron degeneracy — and then studies the special case where pressure is a power law of density alone. That case, the polytrope, turns hydrostatic equilibrium into a single dimensionless differential equation, the Lane-Emden equation, whose solutions capture the gross structure of stars with analytic clarity. The relativistic degenerate polytrope has index $n=3$ and produces a mass fixed independently of the radius, the Chandrasekhar mass. Eddington's standard model uses the same $n=3$ index to describe a star where radiation pressure is a fixed fraction of the total, yielding the quartic equation that relates that fraction to the stellar mass.

The three pressure contributions

Ideal gas. A fully ionized plasma of number density $n$ at temperature $T$ exerts the gas pressure $P_{\text{gas}}=nkT$. Writing the number density through the mass density and the mean molecular weight $\mu$, the mean mass per particle in units of $m_H$, gives

For fully ionized hydrogen $\mu =1/2$ (one proton and one electron share the proton mass); for a general mixture with hydrogen, helium, and metal mass fractions $X$, $Y$, $Z$, the fully ionized value is $\mu \approx (2X+\frac{3}{4}Y+\frac{1}{2}Z{)}^{-1}$, close to $0.6$ for solar composition.

Radiation. The photon gas in thermal equilibrium contributes

Radiation pressure scales as $T^4$ and gas pressure as $\rho T$, so radiation grows in importance at high temperature and low density, dominating the interiors of massive stars.

Electron degeneracy. When the density is high enough that the electron de Broglie wavelengths overlap, the Pauli exclusion principle forces electrons into ever-higher momentum states, producing a pressure that persists even at zero temperature. In the nonrelativistic limit the completely degenerate electron gas gives

where ${\mu }_e$ is the mean molecular weight per free electron (${\mu }_e=2$ for fully ionized helium, carbon, or oxygen). In the ultra-relativistic limit, where the Fermi momentum exceeds $m_{e}c$, the scaling softens to

The total pressure is the sum $P=P_{\text{gas}}+P_{\text{rad}}+P_{\text{deg}}$, with each term dominating in a different region of the density-temperature plane.¹

The polytropic relation and the Lane-Emden equation

A polytrope is a star whose pressure depends on density as a power law,

where $K$ is a constant and $n$ is the polytropic index, related to the exponent by $\gamma =1+1/n$. The relation arises in three important cases: a nonrelativistic degenerate gas has $\gamma =5/3$ ($n=3/2$); an ultra-relativistic degenerate gas or a radiation-dominated gas has $\gamma =4/3$ ($n=3$); and a fully convective star, whose entropy is uniform, follows the adiabat $P\propto {\rho }^{5/3}$ ($n=3/2$).

Insert the polytropic relation into hydrostatic equilibrium combined with mass conservation. Writing Poisson's equation for the gravitational potential $\Phi$ with $\mathbf{d}P/\mathbf{d}r=-\rho \mathbf{d}\Phi /\mathbf{d}r$,

Introduce the dimensionless variables that separate the scale from the shape. Let $\rho ={\rho }_c{\theta }^n$, so that $\theta$ runs from $1$ at the center to $0$ at the surface, and $P=P_c{\theta }^{n+1}$ with $P_c=K{\rho }_c^{1+1/n}$. Scale the radius by

Substituting reduces the structure to a single second-order ordinary differential equation for $\theta (\xi )$, the Lane-Emden equation:

with boundary conditions $\theta (0)=1$ (central density) and ${\theta }^{\prime }(0)=0$ (zero gradient at the center by symmetry). The surface is the first zero ${\xi }_1$ where $\theta ({\xi }_1)=0$; the physical radius is $R=\alpha {\xi }_1$.²

Solutions for n equal to 0, 1, and 5

Three indices admit closed-form solutions, bracketing the physically relevant range.

Uniform density, $n=0$. The equation becomes $\frac{1}{{\xi }^2}({\xi }^2{\theta }^{\prime }{)}^{\prime }=-1$, integrating to

The density is constant ($\rho ={\rho }_c$ everywhere, since ${\theta }^0=1$), recovering the incompressible sphere.

$n=1$. The equation linearizes and its regular solution is a spherical Bessel function,

The $n=1$ polytrope is special because $\alpha$, and therefore the radius $R=\alpha \pi$, is independent of the central density: a polytrope of $\gamma =2$ has a radius fixed by $K$ alone, regardless of mass.

$n=5$. The Schuster solution is

which approaches zero only as $\xi \to \infty$, so the $n=5$ polytrope extends to infinite radius while enclosing a finite mass. It marks the boundary of physically bound polytropes: for $n<5$ the radius is finite, for $n\ge 5$ the configuration is unbounded. All other indices, including the physically central $n=3/2$ and $n=3$, must be integrated numerically, but their solutions lie smoothly between these three.

Mass, radius, and the mass-radius relation

Once $\theta (\xi )$ is known, the total mass follows from integrating the density, which the Lane-Emden equation lets us write in terms of the surface derivative alone:

using ${\xi }^2{\theta }^n=-({\xi }^2{\theta }^{\prime }{)}^{\prime }$. With $R=\alpha {\xi }_1$, eliminating the central density between the mass and radius expressions gives a homology mass-radius relation for a polytrope of index $n$:

The exponent captures the qualitative behavior across the physical range. For a nonrelativistic degenerate star, $n=3/2$, the relation gives $R\propto M^{-1/3}$: more massive white dwarfs are smaller, the inverted mass-radius relation that characterizes degenerate matter. For $n=3$ the exponent diverges, meaning the mass is independent of the radius — the situation of the Chandrasekhar limit below. For $n=1$ the radius is independent of the mass, as noted from the Bessel solution.

The Chandrasekhar mass from the n equal to 3 polytrope

At the highest densities the electrons become ultra-relativistic and the degenerate pressure follows $P=K_{\text{UR}}(\rho /{\mu }_e{)}^{4/3}$, a polytrope of $\gamma =4/3$, $n=3$. For $n=3$ the mass expression loses its dependence on the central density entirely. Writing ${\alpha }^3{\rho }_c$ with ${\alpha }^2\propto {\rho }_c^{(1-n)/n}={\rho }_c^{-2/3}$ for $n=3$, one finds ${\alpha }^3{\rho }_c\propto {\rho }_c^{-1}{\rho }_c={\rho }_c^0$, a constant. The mass is fixed by the constant $K_{\text{UR}}$ alone:

with the $n=3$ Lane-Emden solution supplying ${\xi }_1=6.897$ and $-{\xi }_1^2{\theta }^{\prime }({\xi }_1)=2.018$. Inserting $K_{\text{UR}}=(3{\pi }^2{)}^{1/3}\hslash c/(4m_H^{4/3})$ collapses the constants to

The Chandrasekhar mass is the maximum mass a white dwarf can have: at $M_{\text{Ch}}$ the electrons are fully relativistic and can no longer stiffen to resist gravity, so no equilibrium exists above it.³ The scaling $M_{\text{Ch}}\propto (\hslash c/G{)}^{3/2}/(m_H^2{\mu }_e^2)$ shows the limit is set by fundamental constants: it is the mass at which the electron degeneracy pressure, itself a quantum-relativistic effect, is overwhelmed by self-gravity.

Electron degeneracy and Eddington's standard model

Onset of degeneracy. Degeneracy sets in when the thermal energy per electron falls below the Fermi energy, or equivalently when the ideal-gas pressure drops below the degenerate pressure at fixed density. Equating $P_{\text{gas}}=P_{\text{deg}}$ in the nonrelativistic case gives a boundary in the $\log T$-$\log \rho$ plane of slope $T\propto {\rho }^{2/3}$: below this line (high density, low temperature) the gas is degenerate, above it (low density, high temperature) it is an ideal gas. The boundary locates where white-dwarf cores, degenerate helium cores of red giants, and the inner regions of the lowest-mass stars sit relative to normal main-sequence interiors.

Eddington's standard model. For a star supported by gas plus radiation, define the ratio $\beta =P_{\text{gas}}/P$, so that $P_{\text{rad}}=(1-\beta )P$. Eddington observed that if $\beta$ is constant throughout the star, the total pressure becomes a pure power law of density with $\gamma =4/3$, making the star an $n=3$ polytrope. Eliminating $T$ between $P_{\text{gas}}=\beta P=\rho kT/(\mu m_H)$ and $P_{\text{rad}}=(1-\beta )P=\frac{1}{3}a{T}^4$ gives

confirming the $n=3$ form with a $\beta$-dependent constant $K$. Matching this $K$ to the $n=3$ polytropic mass yields the Eddington quartic, a relation between the radiation fraction and the mass,

which shows that radiation pressure is a negligible fraction of the total in low-mass stars and grows toward dominance in the most massive stars, where $1-\beta$ approaches unity and the star nears the Eddington luminosity limit. The standard model captures the essential $M$-dependence of the radiation fraction with a single polytrope.

Summary

The stellar equation of state sums ideal-gas pressure $\rho kT/(\mu m_H)$, radiation pressure $\frac{1}{3}a{T}^4$, and degeneracy pressure $K(\rho /{\mu }_e{)}^{5/3}$ or $K(\rho /{\mu }_e{)}^{4/3}$, each dominating a distinct region of the density-temperature plane. When pressure is a power law $P=K{\rho }^{1+1/n}$, hydrostatic equilibrium reduces to the Lane-Emden equation, solved in closed form for $n=0,1,5$ and numerically otherwise, giving the polytropic mass-radius relation $R\propto M^{(1-n)/(3-n)}$. The relativistic degenerate case $n=3$ makes the mass independent of radius, the Chandrasekhar mass $M_{\text{Ch}}\approx 1.44M_{\odot }$, and the same index underlies Eddington's standard model and its quartic for the radiation fraction. The standard solar model applies the full numerical machinery to the one star whose interior we can test in detail.