---
title: The Equation of State and Polytropes
draft: false
module: Stellar Structure
moduleNumber: 4
lessonNumber: 3
order: 403
summary: >
  Stellar pressure comes from gas, radiation, and, at high density, degenerate
  electrons. When pressure depends on density as a power law, hydrostatic
  equilibrium reduces to the Lane-Emden equation, whose solutions describe
  polytropes of index n. The relativistic degenerate case, n equal to three,
  gives a mass independent of radius, the Chandrasekhar mass. Eddington's
  standard model treats a radiation-supported star as an n equal to three
  polytrope and yields the quartic relating radiation fraction to mass.
topics: [Stellar Structure]
sources:
  - book: Carroll & Ostlie
    ref: "Ch. 10 — The Interiors of Stars; §10.2 Pressure Equation of State; Ch. 16 §16.3 Polytropes"
  - book: Maoz
    ref: "Ch. 3 — Stellar Physics; Ch. 4 (electron degeneracy)"
---

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 $P_{\text{gas}} = n k T$. 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

$$
P_{\text{gas}} = \frac{\rho\, k T}{\mu m_H}.
$$

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 $\mu \approx (2X + \tfrac{3}{4}Y + \tfrac{1}{2}Z)^{-1}$,
close to $0.6$ for solar composition.

**Radiation.** The photon gas in thermal equilibrium contributes

$$
P_{\text{rad}} = \frac{1}{3} a T^4,
\qquad a = \frac{4\sigma}{c} = 7.566 \times 10^{-16}\ \text{J}\,\text{m}^{-3}\,\text{K}^{-4}.
$$

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

$$
P_{\text{deg}} = K_{\text{NR}} \left(\frac{\rho}{\mu_e}\right)^{5/3},
\qquad
K_{\text{NR}} = \frac{(3\pi^2)^{2/3}}{5}\,\frac{\hbar^2}{m_e m_H^{5/3}},
$$

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

$$
P_{\text{deg}} = K_{\text{UR}} \left(\frac{\rho}{\mu_e}\right)^{4/3},
\qquad
K_{\text{UR}} = \frac{(3\pi^2)^{1/3}}{4}\,\frac{\hbar c}{m_H^{4/3}}.
$$

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.[^co-eos]

$$
% caption: Which term dominates the pressure depends on density and temperature:
% radiation at high T and low density, degeneracy at high density, ideal gas between.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.4,0) node[right, black!70] {log density};
\draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {log temperature};
% radiation region boundary (P_rad = P_gas): T ~ rho^{1/3}, a rising line
\draw[acc, very thick] (0.3,1.4) -- (7.8,3.6);
\node[acc, anchor=south] at (2.0,3.4) {radiation};
% degeneracy boundary (nonrel): T ~ rho^{2/3}, steeper rising line to the right
\draw[black, very thick, densely dashed] (3.4,0.2) .. controls (4.8,1.4) and (6.2,2.6) .. (7.6,3.9);
\node[black, anchor=west] at (6.4,1.4) {degenerate};
% ideal gas region label
\node[black, anchor=center] at (3.0,1.6) {ideal gas};
\end{tikzpicture}
$$

## The polytropic relation and the Lane-Emden equation

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

$$
P = K \rho^{\gamma} = K \rho^{1 + 1/n},
$$

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
$\d P/\d r = -\rho\, \d\Phi/\d r$,

$$
\frac{1}{r^2}\frac{\d}{\d r}\!\left(\frac{r^2}{\rho}\frac{\d P}{\d r}\right) = -4\pi G \rho.
$$

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

$$
r = \alpha\,\xi, \qquad
\alpha^2 = \frac{(n+1) P_c}{4\pi G \rho_c^2}
         = \frac{(n+1) K}{4\pi G}\,\rho_c^{(1-n)/n}.
$$

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

$$
\frac{1}{\xi^2}\frac{\d}{\d \xi}\!\left(\xi^2 \frac{\d\theta}{\d\xi}\right) = -\theta^n,
$$

with boundary conditions $\theta(0) = 1$ (central density) and $\theta'(0) = 0$
(zero gradient at the center by symmetry). The surface is the first zero $\xi_1$
where $\theta(\xi_1) = 0$; the physical radius is $R = \alpha\,\xi_1$.[^co-poly]

## 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 $\tfrac{1}{\xi^2}(\xi^2
\theta')' = -1$, integrating to

$$
\theta(\xi) = 1 - \frac{\xi^2}{6}, \qquad \xi_1 = \sqrt{6}.
$$

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,

$$
\theta(\xi) = \frac{\sin\xi}{\xi}, \qquad \xi_1 = \pi.
$$

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

$$
\theta(\xi) = \left(1 + \frac{\xi^2}{3}\right)^{-1/2},
$$

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.

$$
% caption: Lane-Emden solutions theta(xi) for several indices: larger n concentrates
% mass toward the center and pushes the surface zero outward.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.6,0) node[right, black!70] {xi};
\draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {theta};
\foreach \y/\lab in {0/0,2/0.5,4/1.0}
  \draw[black] (-0.08,\y) -- (0.08,\y) node[left, black, xshift=-3pt] {\lab};
% n=0 parabola down to sqrt6 ~ 2.449 -> x-scale: 8/6*xi
\draw[acc, very thick] (0,4.0) .. controls (1.2,3.6) and (2.4,2.4) .. (3.27,0.0);
\node[acc, anchor=south west] at (2.4,2.4) {n = 0};
% n=1 sinc to pi ~ 3.1416 -> x=8/6*pi ~ 4.19
\draw[black, very thick, densely dashed] (0,4.0) .. controls (1.4,3.5) and (3.0,1.8) .. (4.19,0.0);
\node[black, anchor=south west] at (3.4,1.4) {n = 1};
% n=5 approaching zero slowly, never reaching within frame
\draw[black, very thick, densely dotted] (0,4.0) .. controls (2.0,2.6) and (4.5,1.2) .. (8.2,0.5);
\node[black, anchor=south] at (6.4,0.65) {n = 5};
\end{tikzpicture}
$$

## 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:

$$
M = \int_0^R 4\pi r^2 \rho\,\d r
  = 4\pi \alpha^3 \rho_c \int_0^{\xi_1} \xi^2 \theta^n\,\d\xi
  = -4\pi \alpha^3 \rho_c\, \xi_1^2 \theta'(\xi_1),
$$

using $\xi^2\theta^n = -(\xi^2\theta')'$. 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 $n$:

$$
M \propto R^{(3 - n)/(1 - n)}, \qquad \text{equivalently} \qquad
R \propto M^{(1 - n)/(3 - n)}.
$$

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.

$$
% caption: The polytropic mass-radius relation: nonrelativistic degeneracy gives
% R proportional to M to the minus one-third, so heavier white dwarfs are smaller.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.4,0) node[right, black!70] {mass};
\draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {radius};
% R ~ M^{-1/3}: decreasing curve
\draw[acc, very thick] (0.5,3.9) .. controls (1.6,2.4) and (3.2,1.4) .. (5.6,0.9)
  .. controls (6.6,0.72) and (7.2,0.6) .. (7.8,0.5);
\node[acc, anchor=north east] at (4.2,1.6) {n = 3/2 degenerate};
% vertical asymptote near Chandrasekhar mass
\draw[black, densely dashed] (7.4,0) -- (7.4,4.0);
\node[black, anchor=south, rotate=90] at (7.65,2.0) {Chandrasekhar mass};
\end{tikzpicture}
$$

## 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:

$$
M_{\text{Ch}} = 4\pi \left(\frac{K_{\text{UR}}}{\pi G}\right)^{3/2}
                \big(-\xi_1^2 \theta'\big)_{n=3},
$$

with the $n = 3$ Lane-Emden solution supplying $\xi_1 = 6.897$ and $-\xi_1^2
\theta'(\xi_1) = 2.018$. Inserting $K_{\text{UR}} = (3\pi^2)^{1/3}\hbar c/(4
m_H^{4/3})$ collapses the constants to

$$
M_{\text{Ch}} = \frac{5.83}{\mu_e^2}\,M_\odot \approx 1.44\ M_\odot
\quad (\mu_e = 2).
$$

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.[^co-chandra] The scaling $M_{\text{Ch}} \propto
(\hbar c/G)^{3/2}/(m_H^2 \mu_e^2)$ 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 $P_{\text{gas}} = \beta P = \rho k T/(\mu m_H)$ and
$P_{\text{rad}} = (1-\beta)P = \tfrac{1}{3}aT^4$ gives

$$
P = \left[\frac{3 k^4 (1-\beta)}{a\,\mu^4 m_H^4\,\beta^4}\right]^{1/3} \rho^{4/3}
  \equiv K\,\rho^{4/3},
$$

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,

$$
1 - \beta = 0.003\,\left(\frac{M}{M_\odot}\right)^2 \mu^4 \beta^4,
$$

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.

$$
% caption: The degeneracy boundary in the log T, log rho plane: degenerate below the
% line, ideal gas above; the main-sequence and white-dwarf regimes sit on either side.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.4,0) node[right, black!70] {log density};
\draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {log temperature};
% degeneracy boundary T ~ rho^{2/3}
\draw[acc, very thick] (0.4,0.6) .. controls (3.0,1.8) and (5.4,3.0) .. (7.9,4.1);
\node[acc, anchor=south east, rotate=27] at (5.6,2.9) {degeneracy boundary};
% ideal-gas region (upper left)
\node[black, anchor=center] at (1.9,3.4) {ideal gas};
\node[black, anchor=center] at (2.0,3.0) {main sequence};
% degenerate region (lower right)
\node[black, anchor=center] at (6.2,1.1) {degenerate};
\node[black, anchor=center] at (6.2,0.7) {white dwarf};
\end{tikzpicture}
$$

## Summary

The stellar equation of state sums ideal-gas pressure $\rho k T/(\mu m_H)$, radiation
pressure $\tfrac{1}{3}aT^4$, 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 $M_{\text{Ch}} \approx 1.44\,M_\odot$, and the same
index underlies Eddington's standard model and its quartic for the radiation fraction.
The [standard solar
model](/astrophysics-cosmology/stellar-structure/the-standard-solar-model) applies the
full numerical machinery to the one star whose interior we can test in detail.

[^co-eos]: Carroll & Ostlie, §10.2 — The Pressure Equation of State: ideal-gas, radiation, and degeneracy pressure and the mean molecular weight.
[^co-poly]: Carroll & Ostlie, §16.3 — Polytropes and the Lane-Emden Equation: the polytropic relation, the dimensionless reduction, and the solutions for $n = 0, 1, 5$.
[^co-chandra]: Carroll & Ostlie, §16.3 — the Chandrasekhar limiting mass from the $n = 3$ relativistic polytrope.