A clock that runs down and says what it is

A pulsar’s timing is the most precise measurement in astronomy. Pulse arrival times for the best objects are modelled to within a hundred nanoseconds over decades, which is a fractional precision no other astronomical quantity approaches.

What comes out of that precision is two numbers: the rotation period $P$, and its rate of change $\dot P$. Everything else a catalogue lists for a pulsar — its age, its magnetic field, the power it radiates — is computed from those two through a model, and it is worth being precise about which entries are measurements and which are inferences dressed as measurements.

What is being timed #

A pulsar is a rotating neutron star with a magnetic axis that is not its rotation axis, and a beam of radio emission that sweeps past the Earth once per rotation. That is the model and it has been the model since 1968; what makes it credible is the stability.

The rotation of a neutron star is the steadiest clock in nature for a straightforward reason. Its moment of inertia is around $10^{45}$ g cm² — twenty kilometres of radius carrying more than a solar mass — and its angular momentum is enormous relative to any torque acting on it. Nothing on the star’s surface, and very little around it, can change the spin appreciably. What can, and does, is the electromagnetic torque of the rotating magnetic field.

So $\dot P$ is positive. Every isolated pulsar is slowing down, and the slowing is what the whole of this essay is about.

Three numbers from two #

Assume the star is a rotating magnetic dipole radiating into a vacuum. The radiated power goes as the fourth power of the rotation rate and the square of the perpendicular component of the magnetic moment, and equating it to the loss of rotational kinetic energy gives

$$\dot\nu = -k\nu^3, \qquad\text{equivalently}\qquad P\dot P = \text{constant}.$$

From that, three quantities follow.

The characteristic age, obtained by integrating the spin-down back to zero period:

$$\tau_c = \frac{P}{2\dot P}.$$

The surface dipole field, obtained by inverting the radiated-power expression with an assumed radius and moment of inertia:

$$B = 3.2\times10^{19}\sqrt{P\dot P}\ \text{gauss}.$$

The spin-down luminosity, the rate at which rotational energy is being lost:

$$\dot E = 4\pi^2 I \frac{\dot P}{P^3}.$$

Each of those is quoted in every pulsar catalogue and each is a model. The age assumes the braking law and a birth period of zero. The field assumes a radius, a moment of inertia, an orthogonal dipole and a vacuum. The luminosity assumes only the moment of inertia, which makes it the most trustworthy of the three and still not a measurement.

The test #

The braking law can be tested, and the test is unambiguous. Write the spin-down as $\dot\nu = -k\nu^n$ for an unspecified exponent. Then

$$n = \frac{\nu\ddot\nu}{\dot\nu^2},$$

and if $\ddot\nu$ can be measured, $n$ follows with no assumption about $k$, the field, the radius or anything else. Magnetic dipole braking into a vacuum demands exactly 3.

Measuring $\ddot\nu$ requires a long, clean timing baseline, and only a handful of pulsars are steady enough to supply one. Every measurement gives less than 3: the Crab at 2.51, B1509−58 at 2.839, B0540−69 at 2.140, Vela at 1.4.

The model is wrong, and the way it is wrong is informative. An index below 3 means the star is losing angular momentum more efficiently at low rotation rates than a vacuum dipole would — consistent with a magnetosphere filled with plasma, a magnetic field that is growing, or an inclination angle that is changing. All three have been proposed and none is settled.

That birth period is a real result and worth pausing on. It is not measured and it is not assumed: it is the third number implied when two measured quantities meet a date in a chronicle. A pulsar was born spinning at nineteen milliseconds, and it is known because Chinese astronomers wrote down when.

How wrong is a characteristic age #

The Crab’s error is 29 per cent, which sounds tolerable. It is not typical.

The two assumptions push in opposite directions, which is why the errors are not one-sided. A birth period comparable to the present period makes the true age shorter than $\tau_c$ — the star has spun down less than the integral assumes. A braking index below 3 makes it longer. Which dominates depends on the object, and neither is known in advance.

The arithmetic is worth writing out, because it puts both assumptions into one expression and makes their characters visible. Integrating $\dot\nu = -k\nu^n$ from a birth frequency to the present gives

$$t = \frac{P}{(n-1)\dot P}\left[1 - \left(\frac{P_0}{P}\right)^{n-1}\right].$$

Setting $n = 3$ and $P_0 = 0$ recovers $P/2\dot P$, which is the catalogued number. The two corrections that recovery discards behave quite differently.

The bracket is a bounded correction. It cannot exceed one, so a non-zero birth period can only make the true age smaller, and it stops mattering as soon as the pulsar has spun down by a large factor: at ten times the birth period, with $n = 3$, the bracket is 0.99. A characteristic age is nearly exact in its birth-period assumption for an old pulsar and badly wrong for a young one, which is the reverse of how ages usually behave.

The prefactor is unbounded. Replacing the 2 by $n-1$ multiplies the age by $2/(n-1)$, which is 1.32 at the Crab’s measured index and 5.0 at Vela’s. Nothing constrains the index from below except that it exceed one, and a star whose torque is dominated by a particle wind rather than by a dipole can approach that value, where the expression diverges.

So the young pulsars — the ones with remnants, the ones whose ages can be checked at all — are exactly the population in which both corrections are largest, and the old ones, where the formula is nearly right, are the ones with nothing left to check it against.

The lesson generalises past pulsars. An expression derived by integrating a rate backwards to an assumed initial condition is a measurement of the rate and an assumption about the beginning, and where the beginning can be checked independently it is usually wrong.

The kinematic ages in that figure deserve an explanation of their own, because the method exists only for a population with an unusual property.

A pulsar is not at rest relative to its birthplace. Neutron stars have a mean space velocity of about four hundred kilometres a second, an order of magnitude above their progenitors’, and the excess is a kick delivered at the moment of core collapse by an asymmetry in the explosion or in the neutrino emission. Measure a pulsar’s proper motion and its distance, project the motion backwards, and the time since it left an association or the geometric centre of a remnant is an age with no rotational model anywhere in it.

That is the only clock in the subject genuinely independent of the spin-down, which is why those seven points exist at all. It carries its own assumptions — the birthplace has to be identifiable, the motion has to have been ballistic, and the distance is usually the weakest link — but none of them is the braking law, so it can be used to test it.

The kicks matter well beyond the dating. Four hundred kilometres a second exceeds the Galaxy’s escape speed for a substantial fraction of the population, so a pulsar born in the disc reaches a height of hundreds of parsecs above it within a few million years and some leave altogether. That is why the old, slow objects at the lower right of the plane are found far from the plane of the Galaxy, and it is a second and independent argument that they are old: a pulsar’s height above the disc is a clock too, running on the same kick its proper motion measures.

It also explains a selection effect built into the diagram. A high velocity carries a pulsar away from its remnant, and a remnant fades within tens of thousands of years, so the association between a pulsar and the debris of the star that made it is available only briefly. Most of the plane’s occupants have long outlived any means of dating them except their own.

The energy budget #

The spin-down luminosity is the number with the fewest assumptions in it, and it can be checked against something observed.

For the Crab, $\dot E = 4.5\times10^{38}$ erg per second, which is 120,000 times the Sun’s total output. The Crab Nebula radiates about $1.3\times10^{38}$ erg per second across the spectrum, from radio synchrotron to gamma rays. The rotating star is supplying the nebula’s luminosity with a factor of three to spare, and that agreement — made by Pacini and by Gold in 1967 and 1968, before pulsars were understood — is the reason the model is believed at all despite the braking index refuting its details.

The comparison is worth its own line because it is the only place in this essay where an inference is tested against an independent observation rather than against another inference. Everything else is arithmetic on $P$ and $\dot P$.

The population, and what the plane says #

The $P$–$\dot P$ diagram is the pulsar equivalent of the Hertzsprung–Russell diagram: a plot of two observables in which populations separate.

The main population sits between periods of 0.1 and 5 seconds with fields around $10^{12}$ gauss, and it is the ordinary product of core collapse. A pulsar enters at the top left, young and fast, and drifts down and right as it slows — the trajectory being a line of slope $-1$ if $P\dot P$ is constant, which is what the field contours are.

The magnetars sit at the top right: periods of seconds to tens of seconds, derivatives a thousand times larger, and inferred fields of $10^{14}$ to $10^{15}$ gauss. Their X-ray luminosities exceed their spin-down luminosities, so they are not rotation-powered at all — the energy comes from the decay of the field itself, and the model of this essay does not apply to them.

The millisecond pulsars sit at the lower left: periods of a few milliseconds, derivatives of $10^{-20}$, and fields of $10^{8}$ gauss. Their characteristic ages are billions of years and their true ages are, for once, comparable. They are old neutron stars that were spun back up by accreting from a companion — which is why nearly all of them are in binaries and why the field is low, having decayed over the intervening gigayears.

The regions are not disjoint #

Those three populations were described as though the plane sorted them. The objects found since the diagram was first drawn have made the boundaries porous in both directions.

There are ordinary radio pulsars with magnetar-strength inferred fields. A handful sit above $4\times10^{13}$ gauss — the level at which the field exceeds the quantum critical value and the magnetar mechanism is supposed to take over — and behave as rotation-powered radio sources, with none of the X-ray outbursts that define the class.

And there are magnetars that have emitted radio pulses. Several of the objects identified by their X-ray bursting have, in the months after an outburst, switched on as transient radio pulsars and then faded, with a spectrum unlike an ordinary pulsar’s and a profile that changes from rotation to rotation.

So a position in the plane does not determine the behaviour. Something not on either axis does — most probably the geometry of the field near the surface, which the dipole formula averages over and which the plane therefore cannot show.

The rotating radio transients make the same point from a third direction: sources detected only as isolated dispersed pulses at intervals of minutes to hours, whose underlying rotation periods place them among the ordinary population. Whether they are a distinct class or an extreme of the nulling every pulsar shows is unresolved, and two coordinates cannot resolve it.

A diagram that separates populations is evidence that two variables matter, not that they are the only two. The Hertzsprung–Russell diagram survives that test, because a star’s luminosity and temperature really do fix its structure at a given composition. The $P$–$\dot P$ plane does not, and the objects sitting on the wrong side of every line drawn across it are the demonstration.

Two of the three quantities the plane is built from are worth reading at other settings.

The death line #

Below and to the right of the main population is a boundary past which pulsars are not found, and it has a mechanism.

Radio emission requires a supply of electron–positron pairs, produced when curvature radiation from accelerated particles reaches the pair-production threshold in the star’s own magnetic field. The potential available scales as $B/P^2$, so a pulsar that has slowed enough, or whose field has decayed enough, can no longer make pairs and goes dark. Setting the threshold gives a line of slope 3 in the $P$–$\dot P$ plane.

The line is drawn on every version of the diagram and it has an embarrassment sitting under it. PSR J2144−3933 has a period of 8.51 seconds — the longest of any ordinary radio pulsar — and lies well below every version of the death line that has been proposed. It is nevertheless a radio pulsar, observed repeatedly since 1999.

A boundary that a known object is on the wrong side of is a boundary with a missing term in it, and the candidates are an unusually favourable geometry, a field far from a dipole near the surface, or a pair-production mechanism that has been mis-specified. The discovery since of several radio-emitting sources with periods of tens of minutes has made the question sharper rather than settling it.

Where the model stops #

Four assumptions, each of which fails somewhere.

The star is a vacuum dipole, and it is not: the magnetosphere is filled with plasma, the field near the surface has multipole structure, and the torque comes from currents flowing in that plasma rather than from radiation into empty space. The braking index measures the failure.

The moment of inertia is assumed. $10^{45}$ g cm² is a model value with the equation of state of neutron-star matter inside it, and that equation of state is not known to better than a factor of order unity in the relevant regime. Every field and every luminosity in this essay carries that uncertainty.

The spin-down is smooth, and for young pulsars it is not. Glitches — sudden spin-ups of a part in $10^6$, followed by exponential recovery — interrupt the timing of the Crab and Vela repeatedly, and measuring $\ddot\nu$ requires modelling around them. The glitches themselves are a probe of the interior superfluid and are the only such probe available.

And the timing model contains everything between here and the star. Dispersion in the interstellar medium delays low frequencies relative to high ones; the solar wind adds a variable term; the Earth’s motion has to be removed to the barycentre, exactly as a stellar radial velocity does, and to greater precision. A pulsar timing solution is a model of the observer’s position and the intervening plasma with a rotating star at the far end of it. And the residuals that say whether any of it has been measured correctly.

Where this ladder goes next #

This rung establishes what is measured, what is derived, and how far the derivations can be trusted.

Above it lies the magnetosphere: the force-free and dissipative models that produce a braking index below 3, the current sheet where the energy is dissipated, and the relation between the spin-down torque and the geometry of the emitting region.

Beside it lies the use of pulsars as clocks rather than as objects. An array of millisecond pulsars timed for decades is a detector for correlated timing residuals, and the correlation pattern expected from a background of gravitational radiation is a specific function of the angle between pairs of pulsars — a measurement made entirely by counting pulses from objects thousands of light years apart.

And below it, as a discipline, lies the habit of separating a catalogue’s measured columns from its derived ones. A table that lists a period, a derivative, an age, a field and a luminosity in five adjacent columns is listing two numbers and three opinions, and knowing which is which is most of what it takes to use it.