A core-collapse supernova leaves something behind. The collapsing iron core is crushed until its electrons are forced into its protons, turning them into neutrons. What remains is a neutron star: more mass than the Sun in a ball about 20 kilometers across, the size of a city.
Why it works
Two things are conserved as the core shrinks from thousands of kilometers to about 10 km in radius, and both get concentrated. Its angular momentum ($\propto r^2\omega$) stays the same, so as $r$ drops by a factor of hundreds, the spin rate rises by a factor of hundreds of thousands: a core turning once a day ends up spinning several times a second. Its magnetic field is squeezed too, reaching trillions of times the strength of Earth's. Charged particles race along the field lines and beam radiation out from the two magnetic poles. The magnetic axis is usually tilted from the spin axis, so the beams sweep around the sky like a lighthouse's. If one sweeps across Earth, we see a pulse once per rotation: a pulsar.
Key idea
A neutron star is the collapsed core of a massive star: about 1.2 to 2 solar masses, about 20 km across, as dense as an atomic nucleus. Pulsars are spinning neutron stars whose beams sweep past Earth like a lighthouse.
Worked examples
Worked example: How dense?
A neutron star has $1.4$ solar masses ($2.79 \times 10^{30}$ kg) in a radius of $10$ km. Find its density.
- Volume: $\frac{4}{3}\pi (1.0 \times 10^{4})^3 = 4.19 \times 10^{12}\ \text{m}^3$.
- Density: $\frac{2.79 \times 10^{30}}{4.19 \times 10^{12}} = 6.7 \times 10^{17}$ kg/m³.
- One cubic centimeter would have a mass of about 670 million tonnes.
Answer: About $7 \times 10^{17}$ kg/m³.
Worked example: The Crab pulsar
The Crab pulsar, the neutron star at the heart of the Crab Nebula, pulses $30$ times a second. What is its spin period, and how fast is its equator moving if its radius is $10$ km?
- Period: $\frac{1}{30} = 0.033$ s.
- Equator speed: $\frac{2\pi \times 10{,}000\ \text{m}}{0.033\ \text{s}} = 1.9 \times 10^{6}$ m/s, about $0.6\%$ of the speed of light.
Answer: $0.033$ s; about $1.9 \times 10^{6}$ m/s.
Common mistake
Pulsars do not pulse by swelling and shrinking. Nothing that big could vibrate that fast, but a city-sized ball can spin that fast. The pulses are a beam sweeping past, like a lighthouse seen from a ship. Check: pulsar pulses are as regular as atomic clocks and slow down only very gradually as the star loses spin energy.
Jocelyn Bell Burnell, then a graduate student, discovered the first pulsar in 1967 as a strange, regular radio signal.