---
title: Neutron Stars and Pulsars
draft: false
module: Stellar Death and Compact Remnants
moduleNumber: 8
lessonNumber: 4
order: 804
summary: >
  A neutron star is held up by neutron degeneracy and the repulsive nuclear force,
  with a maximum mass, the Tolman-Oppenheimer-Volkoff limit, set by an uncertain
  dense-matter equation of state. Its rotating magnetic dipole sweeps a beam past
  Earth as a pulsar, and magnetic braking traces a track across the period-period-
  derivative diagram. Millisecond pulsars, magnetars, glitches, and the orbital
  decay of the Hulse-Taylor binary follow from the same structure.
topics: [Stellar Death and Compact Remnants]
sources:
  - book: Carroll & Ostlie
    ref: "Ch. 16 — The Degenerate Remnants of Stars; §16.5 Neutron Stars; §16.6 Pulsars"
  - book: Maoz
    ref: "Ch. 4 — Stellar Death and Remnants"
---

The compact object left by most core collapses is a neutron star: a body of roughly
one and a half solar masses compressed into a radius near ten kilometers, with a mean
density exceeding that of an atomic nucleus. It is supported not by electrons but by
degenerate neutrons and the repulsive core of the nuclear force. Discovered as
pulsars — sources of clock-like radio pulses — neutron stars are the most precisely
timed objects in astronomy and the best laboratories for matter at extreme density and
for strong-field gravity. This lesson derives the neutron-star structure and its
maximum mass, develops the rotating-dipole model of pulsars and their spin-down, and
reads the period-period-derivative diagram, the glitches, the recycled millisecond
pulsars, and the orbital decay of the Hulse-Taylor binary that first proved
gravitational radiation exists.

## Neutron degeneracy and structure

When a stellar core collapses past the [white-dwarf
limit](/astrophysics-cosmology/stellar-death-and-compact-remnants/white-dwarfs-and-the-chandrasekhar-limit),
electron capture $e^- + p \to n + \nu_e$ converts protons and electrons to neutrons
until the matter is overwhelmingly neutrons in equilibrium with a small proton-electron
fraction. The neutrons are themselves fermions, and their degeneracy pressure supports
the star. The estimate parallels the white dwarf: replacing the electron mass $m_e$
with the neutron mass $m_n$ and setting $\mu_e \to 1$ in the degenerate mass-radius
relation gives a radius smaller by the factor $m_e/m_n \approx 1/1840$,

$$
R_{\mathrm{NS}} \sim \frac{\hbar^2}{G m_n^{3} M^{1/3}}\cdot(\ldots)
\;\approx\; 10\ \mathrm{km}
\quad\text{for } M \approx 1.4\,M_\odot.
$$

At this radius the mean density is
$\rho \approx 7\times10^{17}\ \mathrm{kg\,m^{-3}}$, several times nuclear density, and
the surface gravity and escape velocity are relativistic:
$GM/Rc^2 \approx 0.2$. Newtonian degeneracy is only a first approximation; the true
structure requires general relativity and a nuclear equation of state.

A neutron star is layered.

- **Outer crust** — a lattice of neutron-rich nuclei in a degenerate electron gas,
  as in a white dwarf, growing more neutron-rich with depth.
- **Inner crust** — beyond the neutron drip density
  $\sim 4\times10^{14}\ \mathrm{kg\,m^{-3}}$, neutrons leak out of nuclei and form a
  free neutron superfluid coexisting with the nuclear lattice.
- **Outer core** — uniform nuclear matter of neutrons with a few percent protons and
  electrons, likely a neutron superfluid and proton superconductor.
- **Inner core** — matter above nuclear density whose composition is unknown: it may
  stay nucleonic or contain hyperons, a pion or kaon condensate, or deconfined quark
  matter. This uncertainty is the central open problem of neutron-star physics.

$$
% caption: The layered structure of a neutron star: a solid neutron-rich crust over a
% superfluid interior and a uniform nuclear-matter core whose innermost composition is
% unknown, drawn as a cutaway quarter.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[very thick] (0,0) -- (3.6,0) arc (0:90:3.6) -- (0,0);
\draw[black] (0,0) ++(0:2.9) arc (0:90:2.9);
\draw[black] (0,0) ++(0:2.3) arc (0:90:2.3);
\draw[acc, very thick] (0,0) ++(0:1.3) arc (0:90:1.3);
\fill[acc!14] (0,0) -- (1.3,0) arc (0:90:1.3) -- (0,0);
\node[acc, font=\scriptsize, anchor=west] at (0.30,0.42) {core};
\node[black, font=\scriptsize, anchor=west] at (0.55,1.25) {inner crust};
\node[black, font=\scriptsize, anchor=west] at (1.15,2.05) {outer crust};
\node[black, font=\scriptsize, anchor=west] at (1.80,2.85) {surface};
\end{tikzpicture}
$$

## The Tolman-Oppenheimer-Volkoff limit

Just as electron degeneracy caps a white dwarf at the Chandrasekhar mass, neutron
degeneracy and the nuclear force cap a neutron star. The maximum mass follows from the
**Tolman-Oppenheimer-Volkoff (TOV) equation**, the general-relativistic version of
hydrostatic equilibrium,

$$
\frac{\d P}{\d r} = -\frac{G(\rho + P/c^2)(m + 4\pi r^3 P/c^2)}{r^2\left(1 - 2Gm/rc^2\right)},
$$

which reduces to the Newtonian $\d P/\d r = -Gm\rho/r^2$ when pressure and
compactness are small. Each relativistic correction — the pressure's own contribution
to the gravitating energy density $\rho + P/c^2$, the pressure term
$4\pi r^3 P/c^2$, and the metric factor — makes gravity **stronger** than Newtonian.
Above a critical mass no pressure gradient can balance, because increasing the
pressure only increases the source of gravity. The result is a maximum mass, the **TOV
limit**.

> **Definition (TOV mass limit).** The maximum mass of a static neutron star,
> $M_{\mathrm{TOV}}$, above which no equilibrium exists and the object must collapse
> to a black hole. Its value depends on the equation of state of matter above nuclear
> density and lies between about $2.0$ and $2.3\,M_\odot$ for the equations of state
> consistent with observations.

The uncertainty is genuinely about unknown physics: the densest matter is beyond
laboratory reach, so $M_{\mathrm{TOV}}$ is set by an extrapolation of the equation of
state. Two measurements bound it. Pulsars near $2.0\,M_\odot$ have been weighed by
Shapiro delay, so any acceptable equation of state must support at least that,
excluding the softest ones. And the gravitational-wave event
[GW170817](/astrophysics-cosmology/binaries-and-gravitational-waves/multimessenger-astronomy-and-gamma-ray-bursts),
a neutron-star merger, constrained the tidal deformability and hence the radius,
excluding the stiffest ones. The window is narrowing toward $R \approx 11$–$12\
\mathrm{km}$ at $1.4\,M_\odot$.[^co-ns]

$$
% caption: The neutron-star mass-radius relation for a range of equations of state:
% each curve terminates at its own maximum mass, and observations of two-solar-mass
% pulsars and merger tidal limits bracket the allowed band.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.6,0) node[right, black!70] {radius};
\draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {mass};
% stiff EOS (larger radius)
\draw[acc, very thick] (7.6,0.6) .. controls (7.0,2.6) and (6.2,3.7) .. (5.2,3.9)
  .. controls (4.6,3.95) and (4.2,3.6) .. (4.0,3.0);
\node[acc, anchor=west, font=\scriptsize] at (6.4,1.4) {hard};
% soft EOS (smaller radius, lower max)
\draw[black, very thick, densely dashed] (4.6,0.6) .. controls (4.2,2.0) and (3.6,2.9) .. (2.9,3.0)
  .. controls (2.5,3.05) and (2.3,2.7) .. (2.2,2.2);
\node[black, anchor=east, font=\scriptsize] at (3.4,1.2) {soft};
% two-solar-mass line
\draw[black, densely dotted] (0,3.4) -- (8.4,3.4);
\node[black, anchor=south east, font=\scriptsize] at (8.3,3.4) {two solar masses};
\end{tikzpicture}
$$

## The rotating-dipole model of pulsars

A pulsar emits a beam of radio waves that sweeps past Earth once per rotation,
producing a train of pulses as regular as an atomic clock. The **lighthouse model**
explains the pulsing: the magnetic axis is tilted from the rotation axis, so the
radio beam emitted along the magnetic poles rotates with the star, and an observer in
the beam's path sees a pulse each period.

Collapse conserves both magnetic flux and angular momentum, so the neutron star
inherits an extreme spin and field. Flux conservation $B R^2 = \text{const}$ amplifies
a $\sim 10^{-2}\ \mathrm{T}$ stellar field to $\sim 10^{8}\ \mathrm{T}$ as the radius
shrinks by $\sim 10^5$; angular-momentum conservation $I\Omega = \text{const}$ spins a
month-long rotation up to milliseconds-to-seconds.

The tilted magnetic dipole radiates electromagnetically, extracting rotational energy
and slowing the spin. The **magnetic-dipole spin-down** power is

$$
\dot E = -\frac{\mu_0}{6\pi c^3}\,\Omega^4 m^2 \sin^2\alpha,
$$

with $m$ the magnetic dipole moment and $\alpha$ the tilt angle. Equating this to the
loss of rotational kinetic energy $\tfrac{1}{2}I\Omega^2$ gives the spin-down of the
period $P = 2\pi/\Omega$,

$$
P\dot P = \frac{\mu_0\, m^2 \sin^2\alpha}{6\pi c^3 I}\;(2\pi)^2 = \text{const},
$$

so a pulsar's period and its derivative together measure its magnetic field. Solving
for the field at the pole,

$$
B \approx 3.2\times10^{15}\,\sqrt{P\dot P}\ \ \mathrm{T}
\quad (P \text{ in seconds}).
$$

$$
% caption: The lighthouse model: the magnetic dipole axis is tilted from the spin
% axis, so the radio beams along the magnetic poles sweep a cone across the sky and
% an observer in the path records one pulse per rotation.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[very thick] (2.4,2.0) circle (0.8);
% spin axis vertical
\draw[black, densely dashed] (2.4,0.7) -- (2.4,3.9);
\node[black, anchor=south, font=\scriptsize] at (2.4,3.9) {spin axis};
% magnetic axis tilted
\draw[acc, very thick] (1.35,1.1) -- (3.45,2.9);
% beams (cones) along magnetic axis
\draw[acc, thick] (3.45,2.9) -- (5.4,4.0);
\draw[acc, thick] (3.45,2.9) -- (5.5,3.2);
\fill[acc!10] (3.45,2.9) -- (5.4,4.0) -- (5.5,3.2) -- cycle;
\draw[acc, thick] (1.35,1.1) -- (-0.4,0.1);
\draw[acc, thick] (1.35,1.1) -- (-0.5,0.9);
\node[acc, anchor=west, font=\scriptsize] at (5.0,3.7) {beam};
% observer
\node[black, font=\scriptsize] at (7.4,3.6) {to Earth};
\draw[->, black, densely dotted] (7.2,3.5) -- (5.7,3.6);
\end{tikzpicture}
$$

## The period-period-derivative diagram

Plotting each pulsar's period $P$ against its period derivative $\dot P$ organizes the
whole population, the neutron-star analog of the H-R diagram. Lines of constant
$B \propto \sqrt{P\dot P}$ and constant **characteristic age**
$\tau = P/2\dot P$ run diagonally across it.

- **Ordinary pulsars** cluster at $P \sim 0.1$–$1\ \mathrm{s}$ and
  $\dot P \sim 10^{-15}$, with fields $\sim 10^{8}\ \mathrm{T}$ and ages $\sim$ Myr.
  They drift down and to the right as they spin down.
- The **death line** marks where the voltage across the polar cap drops too low to
  sustain pair production and the radio emission shuts off. Pulsars that cross it
  enter the pulsar graveyard.
- **Magnetars** occupy the upper right: long periods and huge $\dot P$, implying
  fields $\sim 10^{11}\ \mathrm{T}$, the strongest known. Their emission is powered by
  magnetic-field decay rather than rotation, and they produce soft-gamma repeater
  bursts.
- **Millisecond pulsars** sit at the lower left: $P \sim$ a few milliseconds and tiny
  $\dot P \sim 10^{-20}$, implying weak fields $\sim 10^{4}\ \mathrm{T}$ and ages of
  gigayears.

> **Definition (Characteristic age).** The spin-down age
> $\tau = P/2\dot P$, obtained by integrating the dipole spin-down law assuming a
> constant field and an initial period much shorter than the present one. It estimates
> a pulsar's age from its timing alone, accurate to a factor of a few.

$$
% caption: The period-period-derivative diagram: ordinary pulsars in the bulk drift
% toward the death line as they spin down, magnetars sit at high field in the upper
% right, and recycled millisecond pulsars occupy the low-field lower left.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.6,0) node[right, black!70] {log period};
\draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {log period derivative};
% bulk population
\draw[black] (3.6,2.6) ellipse (1.3 and 0.9);
\node[font=\scriptsize] at (3.6,2.6) {ordinary};
% magnetars upper right
\draw[black] (6.6,4.0) ellipse (0.9 and 0.5);
\node[black, font=\scriptsize] at (6.6,4.0) {magnetars};
% millisecond lower left
\draw[black] (1.4,0.7) ellipse (0.8 and 0.45);
\node[black, font=\scriptsize] at (1.4,0.7) {ms pulsars};
% death line diagonal
\draw[black, densely dashed] (4.4,0.2) -- (8.2,2.4);
\node[black, anchor=north west, font=\scriptsize, rotate=25] at (5.6,0.9) {death line};
\end{tikzpicture}
$$

## Glitches and recycled pulsars

Two timing phenomena reveal the interior. **Glitches** are sudden spin-ups: the period
abruptly shortens by a part in $10^6$ against the steady spin-down, then relaxes.
They are attributed to the superfluid interior. The crust spins down under magnetic
braking, but the interior neutron superfluid, whose rotation is quantized into
vortices pinned to the crustal lattice, lags behind. When the lag grows large enough
the vortices unpin catastrophically and transfer angular momentum to the crust,
spinning it up. Glitches are direct evidence for a pinned superfluid inside neutron
stars.

**Recycled (millisecond) pulsars** are old neutron stars spun back up by accretion. A
neutron star in a binary that has spun down and crossed the death line can be reborn:
mass transferred from a companion carries angular momentum, spinning the star up to
millisecond periods, while accretion buries and weakens the magnetic field. The
recycled pulsar returns to activity at the lower-left corner of the diagram, now a
stable millisecond clock. The presence of millisecond pulsars almost exclusively in
binaries, and the observation of accreting millisecond X-ray pulsars, confirms the
recycling scenario.[^maoz-ns]

## The Hulse-Taylor binary and gravitational radiation

The binary pulsar PSR B1913+16, discovered by Hulse and Taylor in 1974, is a pulsar in
a $7.75\ \mathrm{hour}$ orbit with another neutron star. The pulsar's clock-like pulses
let the orbit be tracked with extraordinary precision. General relativity predicts that
such a system loses energy to gravitational radiation and that the orbit slowly
shrinks; the quadrupole formula gives an orbital-period decay rate

$$
\dot P_b = -\frac{192\pi G^{5/3}}{5 c^5}
\left(\frac{P_b}{2\pi}\right)^{-5/3}
\frac{m_1 m_2}{(m_1 + m_2)^{1/3}}\,f(e),
$$

with $f(e)$ an enhancement factor for the orbital eccentricity $e$. The measured
decay, $\dot P_b \approx -2.4\times10^{-12}\ \mathrm{s\,s^{-1}}$, matches the
general-relativistic prediction to better than a fraction of a percent, and the
cumulative shift in the time of orbital periastron has grown to tens of seconds over
decades, tracking the predicted parabola exactly. This was the first evidence, though
indirect, that gravitational waves exist and carry energy — recognized with the 1993
Nobel Prize and later confirmed directly by the
[detection of merging
binaries](/astrophysics-cosmology/binaries-and-gravitational-waves/gravitational-waves-from-inspiraling-binaries).

$$
% caption: The Hulse-Taylor orbital decay: the accumulated shift in periastron time
% follows the general-relativistic parabola from gravitational-wave energy loss, with
% the data points lying on the curve over decades.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,4.2) -- (0,0) node[below, black!70] {periastron shift};
\draw[->, black] (0,4.2) -- (8.6,4.2) node[right, black!70] {year};
% GR parabola (downward)
\draw[acc, very thick] (0.4,4.0) .. controls (3.0,3.4) and (5.0,2.0) .. (8.0,0.3);
\node[acc, anchor=south west, font=\scriptsize] at (4.4,2.4) {GR prediction};
% data points on the curve
\foreach \x/\y in {1.2/3.85,2.6/3.55,4.0/2.85,5.4/1.95,6.8/1.1,7.8/0.45}
  \fill[acc] (\x,\y) circle (1.7pt);
\end{tikzpicture}
$$

## Summary

A neutron star supports $\sim 1.4\,M_\odot$ in a $\sim 10\ \mathrm{km}$ radius by
neutron degeneracy and the repulsive nuclear force, layered into a crust, a superfluid
interior, and a core of uncertain composition. Its maximum mass, the TOV limit, is set
by the dense-matter equation of state and bracketed between $\sim 2.0$ and
$2.3\,M_\odot$ by heavy pulsars and merger constraints. A tilted rotating magnetic
dipole makes the star a pulsar, and dipole spin-down with $P\dot P = \text{const}$
measures the field $B \propto \sqrt{P\dot P}$ and the age $\tau = P/2\dot P$. The
period-period-derivative diagram orders the population from ordinary pulsars drifting
toward the death line, to high-field magnetars, to recycled millisecond pulsars spun
up by accretion. Glitches expose the pinned interior superfluid, and the Hulse-Taylor
binary's orbital decay matched the gravitational-wave prediction, the first evidence
for gravitational radiation.

[^co-ns]: Carroll & Ostlie, §16.5–16.6 — Neutron Stars and Pulsars: the structure, the TOV limit, the rotating-dipole model, and the period-period-derivative diagram.
[^maoz-ns]: Maoz, Ch. 4 — neutron stars: superfluid glitches, recycled millisecond pulsars, and the Hulse-Taylor binary as gravitational-wave evidence.