---
title: Neutron Stars and Dense Matter
module: Degenerate Fermi Gas
moduleNumber: 9
lessonNumber: 4
order: 904
summary: >
  When a collapsing core passes nuclear density, electron capture converts the
  matter to neutrons and their degeneracy pressure takes over. The same balance
  that fixes a white dwarf, rescaled by the neutron mass, gives a neutron star of a
  few solar masses in a ten-kilometre radius. General relativity is no longer a
  correction: the Tolman-Oppenheimer-Volkoff equation replaces the Newtonian
  balance and sets a maximum mass around two solar masses. This lesson rescales the
  Fermi-gas argument, states where it breaks, and places the compact objects in one
  stability sequence.
topics: [Degenerate Fermi Gas]
sources:
  - book: Pathria & Beale
    ref: "Ch. 8 — Ideal Fermi Systems; §8.5"
  - book: Carroll & Ostlie
    ref: "Ch. 16 — The Degenerate Remnants of Stars"
  - book: Shapiro & Teukolsky
    ref: "Black Holes, White Dwarfs, and Neutron Stars; Ch. 2, 9"
draft: false
---

Above the Chandrasekhar mass, electron capture removes the electrons that hold a
white dwarf up, and the core collapses until the density reaches that of an atomic
nucleus. There the strong interaction and the degeneracy pressure of a new
fermion, the neutron, arrest the collapse. The result is a neutron star: a body of
one to two solar masses compressed into a radius of about ten kilometres, with a
mean density comparable to nuclear matter. The Fermi-gas calculation that fixed
the white dwarf carries over with the neutron mass in place of the electron mass,
but general relativity now enters at leading order.

## Neutron degeneracy pressure

At nuclear density the reaction $p+e^-\to n+\nu_e$ runs to near completion: the
matter is almost pure neutrons, with a small residual proton-electron fraction set
by beta equilibrium. Neutrons are spin-$\tfrac12$ fermions, so the degeneracy
analysis of the ideal Fermi gas applies with degeneracy $g_s=2$. The decisive
change from the white dwarf is that the neutrons now supply **both** the pressure
and the mass. There is no separate heavy species, so the composition factor $\mu_e$
disappears and the mass density is

$$
\rho = m_n\, n,
$$

with $n$ the neutron number density. The nonrelativistic degeneracy pressure is
the standard result with the neutron mass,

$$
P = \frac{(3\pi^2)^{2/3}}{5}\,\frac{\hbar^2}{m_n}\,n^{5/3}
= K_n\,\rho^{5/3},\qquad
K_n = \frac{(3\pi^2)^{2/3}}{5}\,\frac{\hbar^2}{m_n^{8/3}} .
$$

At a fixed number density the neutron pressure is smaller than the electron
pressure by the mass ratio $m_e/m_n\approx1/1839$, because degeneracy pressure
scales as $1/m$. On logarithmic axes the two pressures are parallel lines of slope
$\tfrac53$, the neutron line lying below the electron line.

$$
% caption: Degeneracy pressure scales as $1/m$, so at a fixed number density the electron gas (upper line) is far stiffer than the neutron gas (lower, dashed); both grow as $n^{5/3}$ until their particles turn relativistic.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(6,0) node[right,black]{$\log n$};
\draw[->,black] (0,0)--(0,4.8) node[above,black]{$\log P$};
\draw[acc,very thick] (0.5,2.1)--(4.6,4.3);
\draw[black,very thick,dashed] (0.5,0.5)--(4.6,2.7);
\node[acc,above] at (1.7,2.9) {electrons};
\node[black,below] at (3.6,1.95) {neutrons};
\draw[<->,black] (2.6,1.63)--(2.6,3.23);
\node[black,right] at (2.7,2.75) {mass ratio};
\end{tikzpicture}
$$

## Rescaling the white-dwarf calculation

The Newtonian mass-radius relation transfers directly. Balancing the
nonrelativistic neutron pressure against gravity gives

$$
R \;\sim\; \frac{\hbar^2}{G\,m_n^{8/3}}\;M^{-1/3},
$$

the white-dwarf result with $m_e(\mu_e m_u)^{5/3}$ replaced by $m_n^{8/3}$. At a
fixed mass the radius ratio between the two objects is

$$
\frac{R_{\mathrm{NS}}}{R_{\mathrm{WD}}}
= \frac{m_e\,(\mu_e m_u)^{5/3}}{m_n^{8/3}}
\approx \frac{m_e}{m_n}\left(\frac{2 m_u}{m_n}\right)^{5/3}
\approx 1.7\times10^{-3}.
$$

A neutron star is about a thousand times smaller than a white dwarf of equal mass:
the same solar mass that fills an Earth-sized white dwarf sits in a radius near
$10$–$12\,\mathrm{km}$. The mean density then reaches
$\rho\sim10^{17}$–$10^{18}\,\mathrm{kg\,m^{-3}}$, comparable to the nuclear
saturation density $\rho_0\approx2.3\times10^{17}\,\mathrm{kg\,m^{-3}}$. A neutron
star is, to order of magnitude, a single nucleus with the mass of a star.[^nsscale]

> **Worked example.** Estimate the radius of a $1.4\,M_\odot$ neutron star from the
> nonrelativistic degeneracy balance. The dimensional relation gives
>
> $$
> R \sim \frac{\hbar^2}{G\,m_n^{8/3}\,M^{1/3}}
> = \frac{(1.055\times10^{-34})^2}
> {(6.67\times10^{-11})(1.675\times10^{-27})^{8/3}(2.8\times10^{30})^{1/3}}
> \approx 3\,\mathrm{km}.
> $$
>
> Including the same $O(1)$ Lane-Emden coefficient as the white dwarf brings this to
> $\sim10\,\mathrm{km}$, the observed scale. The Fermi momentum at this density is a
> sizeable fraction of $m_n c$, so the neutrons are mildly relativistic and the
> nonrelativistic estimate is already at the edge of its validity.

[^nsscale]: The rescaling and the nuclear-density comparison follow Pathria & Beale,
§8.5, and Carroll & Ostlie, Ch. 16. The nuclear saturation density is
$\rho_0\approx2.3\times10^{17}\,\mathrm{kg\,m^{-3}}$ ($n_0\approx0.16\,\mathrm{fm^{-3}}$).

$$
% caption: A neutron star has roughly the density of an atomic nucleus, $\sim10^{17}\,\mathrm{kg\,m^{-3}}$, some fourteen orders of magnitude above ordinary matter; it is a nucleus-density object with the mass of a star and a radius of order ten kilometres.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[acc,thick] (0,0) circle (0.55);
\fill[acc!16] (0,0) circle (0.55);
\node[acc,align=center] at (0,-1.15) {nucleus};
\draw[black,thick] (4.2,0) circle (1.2);
\fill[black] (4.2,0) circle (1.2);
\node[black,align=center] at (4.2,-1.7) {neutron star};
\draw[->,black] (0.75,0.55)--(2.85,0.55);
\node[black,above,align=center] at (1.8,0.6) {same density};
\node[black,align=center] at (0,1.15) {nuclear\\density};
\node[black,align=center] at (4.2,1.65) {radius near 10 km};
\end{tikzpicture}
$$

## Where the ideal-gas argument breaks

Two effects make the Newtonian ideal-neutron-gas estimate quantitatively wrong,
even as it gets the scale right.

- **General relativity.** The compactness of a neutron star,
  $GM/Rc^2\sim0.2$, is not small; for a white dwarf it is $\sim10^{-4}$ and for the
  Sun $\sim10^{-6}$. Newtonian hydrostatic equilibrium is replaced by the
  **Tolman-Oppenheimer-Volkoff equation**, in which pressure itself gravitates and
  the effective gravity is stronger than the Newtonian value. General relativity is
  destabilizing: it lowers the maximum mass relative to the Newtonian prediction.
- **Nuclear interactions.** At and above $\rho_0$ the neutrons are not free. The
  strong interaction is repulsive at short range and stiffens the equation of
  state well beyond the ideal $P\propto\rho^{5/3}$, while its attractive part and
  the appearance of new particles (hyperons, possibly deconfined quarks) soften it.
  The pressure-density relation at these densities is not known from first
  principles.

The pure ideal-neutron-gas calculation with the TOV equation, carried out by
Oppenheimer and Volkoff, gives a maximum mass near $0.7\,M_\odot$ — below observed
neutron-star masses. The discrepancy is the signature that interactions dominate:
the measured masses require a stiffer equation of state than the free gas
provides.[^tov]

The size of the general-relativistic correction is measured by the compactness. At
$GM/Rc^2\sim0.2$ the surface gravitational redshift is of order $30\%$ and the
escape velocity is a third of the speed of light. In the TOV equation this enters
three ways: the source of gravity is $\rho c^2+P$ rather than $\rho c^2$, so
pressure itself gravitates; the enclosed mass includes the gravitational binding
energy; and the metric factor $(1-2Gm/rc^2)^{-1}$ steepens the pressure gradient.
Each effect strengthens the effective gravity, so a Newtonian star that would be
stable becomes unstable in general relativity. Unlike the Chandrasekhar limit,
which follows from the relativistic softening of the electron pressure, the
neutron-star maximum mass is set jointly by that softening and by relativistic
gravity.

[^tov]: The Oppenheimer-Volkoff limit for a free neutron gas and the role of the
equation of state are developed in Shapiro & Teukolsky, Ch. 9; Pathria & Beale,
§8.5, gives the ideal-gas version.

## The maximum mass

The TOV equation with a given equation of state yields a mass-radius curve with a
maximum: a turning point beyond which no stable star exists, the relativistic
analog of the Chandrasekhar limit. Its location depends on the uncertain dense-matter
physics, so it is quoted as a band rather than a single number. Two observational
handles pin it down:

- Pulsars with masses near $2\,M_\odot$ measured from binary timing require the
  equation of state to be stiff enough to support them, ruling out the softest
  models.
- The tidal deformability inferred from the neutron-star merger GW170817 bounds the
  stiffness from above, disfavoring the hardest models.

Together these place the maximum neutron-star mass, the **TOV limit**, at roughly
$2.2$–$2.3\,M_\odot$. A remnant heavier than this cannot be supported by any known
pressure and collapses to a black hole.

$$
% caption: The neutron-star mass-radius relation is nearly vertical over a spread of masses and turns over at the TOV maximum mass; the exact curve is a band because the dense-matter equation of state is uncertain.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(5.6,0) node[right,black]{$R$};
\draw[->,black] (0,0)--(0,4.2) node[above,black]{$M$};
\fill[acc!14] plot[domain=0.6:3.4,samples=60] (\x,{3.2*exp(-((\x-2.2)/1.05)^2)+0.18})
  -- plot[domain=3.4:0.6,samples=60] (\x,{3.2*exp(-((\x-2.2)/1.28)^2)-0.18}) -- cycle;
\draw[black,thick,dashed] (0,3.4)--(4.4,3.4);
\node[black,right] at (2.9,3.68) {TOV limit};
\node[acc] at (2.2,1.4) {stable};
\end{tikzpicture}
$$

## The compact-object sequence

The degenerate remnants form a sequence ordered by mass, each supported by a
different pressure and each with its own upper bound.

| Remnant | Support | Mass scale | Radius |
|---------|---------|------------|--------|
| White dwarf | electron degeneracy | up to $M_{\mathrm{Ch}}\approx1.4\,M_\odot$ | $\sim10^4\,\mathrm{km}$ |
| Neutron star | neutron degeneracy $+$ nuclear forces | $\sim1.1$–$2.2\,M_\odot$ | $\sim10\,\mathrm{km}$ |
| Black hole | none | above the TOV limit | horizon $\sim GM/c^2$ |

Each degeneracy pressure caps the mass it can support: electrons at the
Chandrasekhar mass, neutrons at the TOV limit. A collapsing core that exceeds the
neutron-star bound has no remaining source of pressure and forms a black hole. The
progression follows the same physics throughout: a Fermi gas whose zero-temperature
pressure holds off gravity until relativistic softening, and finally relativistic
gravity itself, exhausts its ability to do so.

$$
% caption: The compact remnants lie in order of increasing mass along one axis, each supported by a distinct degeneracy pressure and each ending at the mass where that pressure fails.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(11,0) node[right,black]{$M$};
\draw[acc,line width=3pt] (0.3,0)--(3.0,0);
\draw[black,line width=3pt] (3.4,0)--(6.6,0);
\draw[black,line width=3pt] (7.0,0)--(10.4,0);
\draw[black,dashed] (3.2,-0.6)--(3.2,0.9);
\node[black,above] at (3.2,0.9) {$M_{\mathrm{Ch}}$};
\draw[black,dashed] (6.8,-0.6)--(6.8,0.9);
\node[black,above] at (6.8,0.9) {TOV};
\node[acc,below,align=center] at (1.6,-0.25) {white\\dwarf};
\node[black,below,align=center] at (5.0,-0.25) {neutron\\star};
\node[black,below,align=center] at (8.7,-0.25) {black\\hole};
\end{tikzpicture}
$$

## Summary

- Above the Chandrasekhar mass, electron capture converts the core to neutrons,
  whose degeneracy pressure ($\rho=m_n n$, $P\propto n^{5/3}$) supports a neutron
  star; the pressure is smaller than the electron gas by $m_e/m_n$ at fixed density.
- Rescaling the white-dwarf balance with the neutron mass gives a radius near
  $10\,\mathrm{km}$ at a solar mass, a density comparable to nuclear matter
  $\rho_0\approx2.3\times10^{17}\,\mathrm{kg\,m^{-3}}$.
- The ideal-gas estimate is only qualitative: the compactness $GM/Rc^2\sim0.2$
  forces the general-relativistic TOV equation, and nuclear interactions set the
  equation of state, so the free-gas Oppenheimer-Volkoff mass ($\sim0.7\,M_\odot$)
  falls short of observed masses.
- Observations bracket the TOV maximum mass near $2.2$–$2.3\,M_\odot$; the
  remnants form the sequence white dwarf $\to$ neutron star $\to$ black hole,
  ordered by the mass each degeneracy pressure can hold.