Neutron Stars and Pulsars

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, 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 $m_e/m_n\approx 1/1840$,

At this radius the mean density is $\rho \approx 7\times 10^{17}\mathrm{kg}{\mathrm{m}}^{-3}$, several times nuclear density, and the surface gravity and escape velocity are relativistic: $GM/R{c}^2\approx 0.2$. 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 $\sim 4\times 10^{14}\mathrm{kg}{\mathrm{m}}^{-3}$, 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.

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,

which reduces to the Newtonian $\mathbf{d}P/\mathbf{d}r=-Gm\rho /r^2$ 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.

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 $2.0M_{\odot }$ 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, 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 $1.4M_{\odot }$.¹

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

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

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

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.

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.²

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

with $f(e)$ an enhancement factor for the orbital eccentricity $e$. The measured decay, ${\dot{P}}_b\approx -2.4\times 10^{-12}\mathrm{s}{\mathrm{s}}^{-1}$, 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.

Summary

A neutron star supports $\sim 1.4M_{\odot }$ 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 $2.3M_{\odot }$ 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.