A pulsar shines like a lighthouse: its strong magnetic field channels beams of radiation out from the magnetic poles, and if the magnetic axis is tilted from the spin axis, the beams sweep around. Each time one crosses Earth we see a pulse. The neutron star is not burning anything; it shines by spending its rotational energy, and we can watch it slow down. The tool is rotational work and power.
💡 Why it works
A spinning body stores $E = \tfrac12I\omega^2$, with $\omega = 2\pi/P$, so $E = 2\pi^2I/P^2$. Differentiate with respect to time: $$\frac{dE}{dt} = I\omega\dot\omega = -\frac{4\pi^2I\dot P}{P^3}.$$ Pulsar timing measures $P$ and its tiny rate of increase $\dot P$ with great precision, and $I \approx 10^{38}$ kg·m² for a neutron star of 1.4 solar masses and about 10 km. If the pulsar has slowed down steadily from a much faster spin, its age is about $\tau = P/(2\dot P)$.
🔑 Key idea
Spin-down power $|\dot E| = 4\pi^2I\dot P/P^3$; characteristic age $\tau = P/(2\dot P)$. A young pulsar loses rotational energy fast enough to power the nebula of glowing gas and particles around it.
A neutron star whose magnetic axis is tilted from its spin axis. Each beam sweeps around once per turn; we see a pulse when one crosses our line of sight.
Worked examples
✏️ Worked example: The Crab pulsar's energy store
The Crab pulsar has $P = 0.0337$ s and $I = 1.0 \times 10^{38}$ kg·m². How much rotational energy does it have?
$\omega = 2\pi/0.0337 = 186$ rad/s.
$E = \tfrac12(1.0 \times 10^{38})(186)^2$.
Answer: About $1.7 \times 10^{42}$ J.
✏️ Worked example: Its spin-down power
The Crab's period grows at $\dot P = 4.2 \times 10^{-13}$ s per second. Find $|\dot E|$.
Answer: About $4.3 \times 10^{31}$ W, about 100,000 times the Sun's luminosity. The Crab Nebula radiates roughly $10^{31}$ W across the spectrum, about a quarter of that.
✏️ Worked example: Its age
Estimate the Crab pulsar's characteristic age.
$\tau = P/(2\dot P) = 0.0337/(8.4 \times 10^{-13})$ s $= 4.0 \times 10^{10}$ s.
Divide by $3.16 \times 10^{7}$ s per year.
Answer: About 1,300 years. Chinese astronomers recorded the supernova in 1054, about 970 years ago: close. The formula assumes the pulsar started out spinning much faster than it does now; the difference tells us the Crab was born with a period not far below today's.
⚠️ Common mistake
Thinking a pulsar shines because it is still hot from the explosion. Its surface is hot, but its thermal glow is far too weak to supply the power we measure. Young pulsars and their nebulae are powered by rotation: the star is a flywheel being slowed down by its magnetic field, and the lost rotational energy, $4\pi^2I\dot P/P^3$, matches what the nebula radiates. Check: the Crab's spin-down power is a few times the nebula's output, as it must be.
Practice
Pass with 3 right answers, the last 2 in a row. Each right answer earns 3 XP.