Astronomy & Celestial Mechanics
the sky, computed
The observed sky

A clock made of neutron stars

The best atomic clocks are rebuilt every decade and averaged every month. A millisecond pulsar has kept its own time for a billion years. Averaged over years, an ensemble of pulsars is as steady as the world's atomic time scale — steady enough to find its errors — but it can never say how fast that time scale runs.

Assumes Timescales and Pulsars.

16 min read 6 figures The sky as a clockMeasuring the unreachable

The world’s reference time is an average of about four hundred and fifty atomic clocks in some eighty laboratories, weighted by how steady each has been, steered by a handful of primary standards that realise the definition of the second, and recomputed a year after the fact against the best of them. It is the most carefully kept number in science. And it has a weakness that no amount of care can remove: every clock that contributes to it is a laboratory instrument, built within the last few decades, maintained by people, and replaced when a better one comes along. There is no clock in the ensemble that has been running unchanged for longer than a working career, and no independent standard, outside the ensemble, against which its slow wanderings over decades can be checked.

A millisecond pulsar is such a standard. It is a neutron star — a star of about 1.4 solar masses compressed to twenty kilometres across — spinning hundreds of times a second, with a beam of radio emission sweeping past the Earth once a rotation. It was spun up to that rate by accreting matter from a companion hundreds of millions of years ago, and it has been slowing down since at a rate so small that its period changes by one part in 101910^{19} per second. Its flywheel is a moment of inertia of 103810^{38} kilogram-metres squared, and nothing in the Galaxy can apply a torque to it large enough to matter. It is a clock that has been running for a billion years with no maintenance, and the question this essay asks is how good a clock it is — and what, for the time scale kept by atomic clocks, it can and cannot check.

The stability of a pulsar clock against atomic clocks, over averaging times from a day to thirty years. Fractional frequency instability, on a logarithmic axis, against averaging time from a day to thirty years. A single millisecond pulsar timed to 300 nanoseconds every 14 days improves as τ^(−3/2) while its white timing noise averages down, and flattens where its slow spin noise, drawn at 10⁻¹⁵ at ten years, takes over. An ensemble of 20 such pulsars is 4.5 times better if their noises are independent. The atomic curves are representative: a hydrogen maser, excellent at hours and drifting at years, and a steered atomic time scale limited near 3·10⁻¹⁶ at long times. In this representative comparison the pulsar ensemble becomes the steadier of the two beyond about 3 years of averaging. At a day a pulsar is a billion times worse than an atomic clock; over decades it is competitive, because nothing inside a neutron star changes on the timescales on which laboratory clocks are rebuilt.
Fig. 1 Fractional frequency instability against averaging time from a day to thirty years. A single millisecond pulsar timed to 300 nanoseconds every fourteen days improves steeply, as τ−3/2\tau^{-3/2}, while its measurement noise averages down, and flattens where its slow spin noise takes over; an ensemble of twenty such pulsars is 4.5 times better if their noises are independent. The atomic curves are representative: a hydrogen maser, excellent at hours, and a steered atomic time scale limited near 3×10−163\times10^{-16} at long times. In this comparison the pulsar ensemble becomes the steadier of the two beyond about three years of averaging.

Why only some pulsars keep time

Most pulsars are poor clocks, and the reasons are instructive about what makes the good ones good.

An ordinary young pulsar spins a few times a second and slows down quickly, because its magnetic field — around 101210^{12} gauss — radiates away its rotational energy at a high rate; the rate at which it runs down says what it is, its field and its age. Its rotation is also noisy. Young pulsars show timing noise, a random wander of their phase that grows with time, and glitches, sudden increases in their spin rate by up to a part in a million, when the superfluid interior that never slowed down abruptly transfers some of its angular momentum to the crust. Even their long-term spin-down does not follow a clean law: the braking index, which a pure magnetic dipole would make exactly three, comes out as some other number in every pulsar where it has been measured. As clocks they are interesting objects and bad instruments.

Millisecond pulsars are different in every one of those respects. Their magnetic fields are ten thousand times weaker, buried or decayed during the long accretion that spun them up, so they slow down ten million times more slowly. They are old and cold, and their interiors have long since settled. Glitches are almost unknown among them — two small ones have been seen in all the millisecond pulsars ever timed — and their timing noise is far smaller, often undetectable above the measurement noise over decades. The quality of a pulsar as a clock is almost entirely a matter of how weak its magnetic field is, because the field is what drives both the spin-down and much of the irregularity, and the recycling that spun these pulsars up is also what quietened them.

How a pulsar is read as a clock

A pulsar’s pulses are too faint to count individually except in the brightest few, so a pulsar is timed by averaging: the radio signal is folded at the known period for tens of minutes, producing an average pulse profile, and that profile is compared with a template to find its arrival time to a small fraction of the pulse width. For the best millisecond pulsars, whose pulses are narrow and bright, a single observation gives an arrival time to about a hundred nanoseconds; for most it is between half a microsecond and a few.

Each arrival time is then compared with a prediction. The prediction is a model of everything between the pulsar and the telescope: the pulsar’s spin frequency and its slow decline; its position, proper motion and parallax; its orbit if it has a companion; the delay from the interstellar plasma, which varies with the observing frequency; the telescope’s motion round the Solar System’s barycentre, which requires a planetary ephemeris; the relativistic delays in the Sun’s gravitational field. And the observatory’s clock, which timestamps every arrival — traced through GPS to UTC and from there to Terrestrial Time as realised by the atomic ensemble.

The difference between the measured arrival times and the prediction — the timing residuals — is where the clock comparison lives. If the reference clock wandered by a microsecond over a decade, every pulsar’s residuals would show the same microsecond wander, because every pulsar is being timed against the same clock. That shared signal, extracted from many pulsars at once, is a pulsar time scale.

How steady a pulsar is

The stability of a clock is conventionally described by how much its frequency varies from one averaging interval to the next, as a function of the length of the interval — the Allan deviation. Every clock has its own curve, and no clock is best at every interval.

How steady each kind of clock is, by how long it is averaged. The Allan deviation — the typical fractional difference in frequency between successive averages of length τ — against τ, for representative noise models of caesium beam, hydrogen maser, caesium fountain, optical lattice clock. Each falls as τ^−½ while random frequency noise dominates, then levels off at a floor set by slower, correlated noise, and a clock with a systematic drift eventually rises again. A caesium beam is still improving at the longest averaging drawn, reaching 10⁻¹⁴; a hydrogen maser reaches its best, 7.3·10⁻¹⁶, after about 27 hours; a caesium fountain is still improving at the longest averaging drawn, reaching 1.2·10⁻¹⁶; an optical lattice clock is still improving at the longest averaging drawn, reaching 10⁻¹⁸. No clock is best at every averaging time — the maser is steadiest over hours and wanders over weeks, the fountain is noisier over minutes and more accurate over months — and a time scale is built by using each where it is best. The curves are typical shapes, not the specification of any particular instrument.
Fig. 2 The Allan deviation of representative caesium beam, hydrogen maser, caesium fountain and optical lattice clocks against averaging time. Each improves as random frequency noise averages down and levels off where slower, correlated noise takes over. A hydrogen maser reaches its best after about a day and wanders beyond it; a caesium fountain keeps improving; an optical lattice clock is a hundred times steadier than either. No laboratory clock has a curve that extends, measured, beyond a few years, because none has run unchanged for longer.

A pulsar’s curve has the opposite shape. Its measurement noise is enormous by atomic standards — a hundred nanoseconds in a single observation, when a hydrogen maser keeps time to a fraction of a nanosecond over a day — so over short intervals it is useless, a billion times worse than an atomic clock at a day, with an uncertainty set by counting photons from a faint radio source. But the noise is in the measurement, not in the clock, and it averages down steeply: the fractional frequency error over an interval τ\tau falls as τ−3/2\tau^{-3/2}, because both the number of measurements and the length of the baseline grow with τ\tau. Over a decade a single good pulsar reaches a stability of about 10−1510^{-15}, and an ensemble of twenty, if their noises are independent, a few parts in 101610^{16}.

The stability of a pulsar clock against atomic clocks, over averaging times from a day to thirty years. Fractional frequency instability, on a logarithmic axis, against averaging time from a day to thirty years. A single millisecond pulsar timed to 100 nanoseconds every 14 days improves as τ^(−3/2) while its white timing noise averages down, and flattens where its slow spin noise, drawn at 10⁻¹⁵ at ten years, takes over. An ensemble of 20 such pulsars is 4.5 times better if their noises are independent. The atomic curves are representative: a hydrogen maser, excellent at hours and drifting at years, and a steered atomic time scale limited near 3·10⁻¹⁶ at long times. In this representative comparison the pulsar ensemble becomes the steadier of the two beyond about 1.4 years of averaging. At a day a pulsar is a billion times worse than an atomic clock; over decades it is competitive, because nothing inside a neutron star changes on the timescales on which laboratory clocks are rebuilt.
Fig. 3 The same comparison with each pulsar timed to 100 nanoseconds, the precision of the best-timed millisecond pulsars rather than a typical one. The white part of each pulsar’s curve drops threefold, and the ensemble overtakes the atomic scale after about 1.4 years instead of three. Better timing moves the crossing earlier; it does not move the floor at long times, which is set by the pulsars’ spin noise and by any noise they share.

That is the regime where the atomic time scale is weakest. Over years, the atomic ensemble’s stability is limited by the slow drifts of its hydrogen masers, the changes in its composition as clocks are added and retired, and the accuracy with which the primary standards realise the second; its long-term stability is estimated, not measured, because there is nothing steadier to measure it against. Beyond a few years of averaging, a pulsar ensemble is the only independent clock the atomic time scale has.

The blind spot the timing model creates

A pulsar is not a perfect clock in one specific respect, and it is the respect that matters most for checking an atomic time scale.

Every pulsar’s spin frequency and its rate of change are free parameters of its timing model. They are not known in advance to the required precision — a pulsar’s period is measured, not predicted — so they are fitted, pulsar by pulsar, from the same arrival times that are being used to check the clock. A reference clock that runs slightly fast makes every pulsar appear to spin slightly slower, and the fit absorbs that into the pulsar’s fitted frequency; a clock whose rate drifts makes every pulsar appear to spin down slightly faster, and the fit absorbs that into the fitted spin-down.

A clock error, and the part of it a pulsar can see. A simulated error in a reference clock over 20 years (upper curve, microseconds): a steady rate offset, a slow change of rate, and a random wander. Timing a pulsar against that clock requires fitting the pulsar's own spin frequency and its rate of slowing down, and those two parameters absorb any clock error that is constant, linear or quadratic in time — the pulsar cannot tell a clock running fast from a pulsar spinning slowly. What remains after the fit (lower curve) is 0.23 microseconds rms against 1.53 in the original: the wander survives and the drift does not. A pulsar time scale can therefore never measure the rate of an atomic clock, only the irregularities of it, and it is exactly those irregularities that a time scale built from a few hundred atomic clocks is least able to check against itself.
Fig. 4 A simulated error in a reference clock over twenty years — a rate offset, a slow change of rate and a random wander — and what remains after fitting each pulsar’s spin frequency and spin-down. The fit removes any clock error that is constant, linear or quadratic in time, leaving 0.23 microseconds rms of the original 1.53: the wander survives and the drift does not. A pulsar time scale cannot measure the rate of an atomic clock, only its irregularities.

That is a fundamental limit, not a technical one. A pulsar cannot tell a clock that runs fast from a pulsar that spins slowly, and since the pulsar’s true frequency is not known independently, there is no way to break the degeneracy. Pulsars cannot realise the second; they cannot say whether an atomic clock’s second is the right length; they cannot measure a steady drift of atomic time against some ideal. What they can do is measure the parts of the atomic clock’s error that are not smooth — the wanders on timescales of years that a quadratic does not fit — and those are exactly the errors the atomic ensemble is least able to see in itself, because they are shared across its clocks through their common steering.

A measurement with no atomic clock in it

The atomic time scale exists in two versions. TT(TAI) is Terrestrial Time as realised in real time, from International Atomic Time as it is published each month. TT(BIPM) is Terrestrial Time recomputed afterwards, each year, by the International Bureau of Weights and Measures, using the full record of the primary frequency standards to correct the ensemble’s slow errors. The two differ by a slowly varying amount, reaching a few microseconds over the decades since the recomputation began, and the difference is the Bureau’s own best estimate of how wrong the real-time scale was.

The difference between two versions of Terrestrial Time, as a pulsar array would see it. A modelled slow difference between two realisations of Terrestrial Time — the one broadcast at the time and the one recomputed afterwards — over 18 years, in microseconds, with its best-fitting quadratic removed as any pulsar timing fit removes it (curve), and yearly measurements of the same difference from an ensemble of pulsars with errors of 60 nanoseconds (points). The measurements follow the curve with a reduced chi-squared of 1.0, and depart from a flat line — no difference at all — with a reduced chi-squared of 12. This is the form of the published result: pulsar timing arrays have detected the difference between the atomic time scale as first realised and as later corrected, at the level of a few hundred nanoseconds over two decades, with no atomic clock involved in the comparison. The curve drawn is a model of that difference, not the published one.
Fig. 5 A modelled slow difference between two realisations of Terrestrial Time over eighteen years, with its best-fitting quadratic removed as any pulsar timing fit removes it (curve), and yearly measurements of the same difference from a pulsar ensemble with errors of 60 nanoseconds (points). The measurements follow the curve with a reduced chi-squared of 1.0, and depart from a flat line with a reduced chi-squared of 12. This is the form of the published result, though the curve here is a model of the difference, not the measured one.

The Parkes Pulsar Timing Array published a pulsar-based time scale in 2012, built from nineteen millisecond pulsars timed over nearly two decades, and it detected the difference between TT(TAI) and TT(BIPM) — the wander that the Bureau’s annual recomputation corrects. The pulsars saw the correction without any access to the primary frequency standards that produced it. Later analyses by the International Pulsar Timing Array, with more pulsars and longer baselines, confirmed the result and began to probe differences between successive versions of TT(BIPM) itself.

It is a remarkable closure. The Bureau computes its correction from the laboratory standards that define the second; the pulsar arrays find the same correction from neutron stars thousands of light-years away that have never been near a laboratory. Two instruments with nothing in common agree on how wrong the world’s clock was — to the extent the pulsar fit allows them to disagree, which is in everything except the smooth part.

How many pulsars, and the noise they share

An ensemble improves on its members only as far as their errors are independent, and for pulsars the most interesting noise is not independent at all.

A pulsar ensemble's stability against the number of pulsars in it. The fractional frequency instability over 10 years of an average of N millisecond pulsars, each timed to 100 nanoseconds every 14 days with spin noise of 10⁻¹⁵, against N on a logarithmic axis. Independent noises average down as one over the square root of N (dashed); a noise common to all pulsars does not, and the solid curve includes one at 3·10⁻¹⁶. One pulsar gives 10⁻¹⁵; twenty give 3.7·10⁻¹⁶; a hundred and twenty give 3.1·10⁻¹⁶, within 5 per cent of the common floor. The gravitational-wave background the pulsar timing arrays detected is exactly such a common noise, correlated between pulsars in a known way — so the same arrays that could keep time are limited, as clocks, by the signal they were built to find.
Fig. 6 The instability over ten years of an average of NN millisecond pulsars, each timed to 100 nanoseconds every fourteen days, against NN. Independent noises average down as one over the square root of NN (dashed); a noise common to all pulsars does not, and the solid curve includes one at 3×10−163\times10^{-16}. One pulsar gives 10−1510^{-15}; twenty give 3.7×10−163.7\times10^{-16}; a hundred and twenty come within 5 per cent of the common floor. An ensemble cannot average away a noise all its members share.

Three noises are shared. The reference clock’s error is shared by construction, and it is the signal a pulsar time scale looks for. Errors in the planetary ephemeris, which moves the telescope round the barycentre wrongly, are shared too, with a characteristic pattern: a pulsar’s residuals from an ephemeris error depend on the direction to the pulsar through a dipole on the sky. And the gravitational-wave background — the combined signal of supermassive black-hole binaries throughout the universe, which pulsar timing arrays detected in 2023 — is shared with a quadrupolar pattern, the Hellings–Downs curve, that depends on the angle between each pair of pulsars.

The three can be separated by their patterns on the sky. A clock error is the same in every direction — a monopole; an ephemeris error is a dipole; a gravitational-wave background is a quadrupole. The separation needs many pulsars spread over the sky, and it is one of the main reasons the timing arrays have grown to more than a hundred pulsars. The same arrays that could keep time are limited, as clocks, by the signal they were built to find: the gravitational-wave background adds a common red noise to every pulsar, correlated in its own way, and a pulsar time scale has to fit it out before it can see the clock.

A clock checked by something older than itself

The comparison between pulsars and atomic clocks has a curious asymmetry. The atomic clocks are better at almost everything: at short times, at realising the second, at being read in real time, at being understood physically. The pulsars are better at one thing, which is being old and unchanged. And that one thing is what a reference clock cannot provide for itself.

A laboratory time scale is maintained by continual comparison among its members. If every clock in it drifts in the same way — because they share a design, or a steering algorithm, or an environmental sensitivity — the comparison does not reveal it, and the ensemble drifts with them. The primary standards guard against a drift in the length of the second; nothing inside the ensemble guards against a common wander on timescales of years that the primary standards are too noisy or too intermittent to see. A pulsar time scale is an outside witness, and the value of an outside witness is not that it is more accurate than the clocks it checks but that its errors have nothing in common with theirs.

The same logic runs through much of metrology. The second itself is about to be redefined in terms of an optical transition, and the new clocks are a hundred times steadier than caesium at every averaging time a laboratory can reach. The pulsars will not compete with them at short times, and will not have to. What they offer — a record of the same rotating objects, timed continuously for decades — becomes more valuable the longer it runs, and it is the one kind of clock whose quality improves with age.

A clock that can also say where

A pulsar’s arrival time depends on where the receiver is as well as on when it is, and the same timing that makes a pulsar a clock makes it a beacon. A receiver displaced by a light-microsecond — three hundred metres — along the direction to a pulsar sees its pulses arrive a microsecond early. Timing several pulsars in different directions therefore fixes a position in space, with no transmitter on the ground, in the way a satellite navigation receiver fixes one from several satellites.

In 2017 an experiment on the International Space Station, using the NICER X-ray telescope to time millisecond pulsars that emit X-ray pulses, determined the station’s orbit autonomously to within about five kilometres, without any help from the ground. That is far worse than tracking a spacecraft from Earth by the Doppler shift of its radio signal, which reaches metres, but it works anywhere in the Solar System and needs no antenna on Earth. A network of neutron stars is a positioning system that was built before there was anyone to use it, and the precision of its clocks is what limits the precision of its fixes.

What the curves assume

The pulsar noise model is representative. Real millisecond pulsars have white measurement noise from a few tens of nanoseconds to several microseconds, red spin noise of different strengths and spectral shapes, and noise from variations in the interstellar plasma along the line of sight, which is itself red and has to be modelled from multi-frequency observations. The curves drawn show the shape of the comparison; the crossing time depends on which pulsars are used.

The atomic time scale’s long-term instability is an estimate. Its curve beyond a year cannot be measured against anything steadier, which is the whole point of the comparison; the value drawn is a representative floor.

The clock-error simulation is a single random realisation, and the fraction of it absorbed by the quadratic fit depends on how much of the error is smooth. A clock whose error is mostly drift would lose almost all of it to the fit; one whose error is mostly wander would lose little.

And the pulsar-measured difference in Terrestrial Time is drawn from a model. The published measurements have larger error bars at the ends of the data span, where the quadratic fit is least constrained, and the published difference is not the sinusoidal combination drawn.

Still open: a time scale that improves with age

The pulsar time scales published so far have used baselines of twenty to thirty years and reached stabilities comparable with the atomic scale at the longest intervals. Their stability should keep improving as the baselines lengthen, since their white noise keeps averaging down, until it reaches the floor set by the pulsars’ own spin noise and by the gravitational-wave background they share. Where that floor lies is not known. If it is low — if the best millisecond pulsars are as steady on timescales of decades as their short-term behaviour suggests — then within a few more decades a pulsar ensemble will be the most stable clock available for intervals longer than about ten years, and it will be able to check not only the atomic time scale’s past but the new optical time scale’s future. If it is high, the pulsars will have found the limit of what a spinning neutron star can keep, which is itself a measurement of how quiet the interior of a neutron star is. Either result depends on nothing but continuing to point radio telescopes at the same small set of stars, once a fortnight, for the rest of the century.

Where this goes next

How steady each kind of clock is, by how long it is averaged. The Allan deviation — the typical fractional difference in frequency between successive averages of length τ — against τ, for representative noise models of caesium beam, hydrogen maser, caesium fountain, optical lattice clock. Each falls as τ^−½ while random frequency noise dominates, then levels off at a floor set by slower, correlated noise, and a clock with a systematic drift eventually rises again. A caesium beam is still improving at the longest averaging drawn, reaching 10⁻¹⁴; a hydrogen maser reaches its best, 7.3·10⁻¹⁶, after about 27 hours; a caesium fountain is still improving at the longest averaging drawn, reaching 1.2·10⁻¹⁶; an optical lattice clock is still improving at the longest averaging drawn, reaching 10⁻¹⁸. No clock is best at every averaging time — the maser is steadiest over hours and wanders over weeks, the fountain is noisier over minutes and more accurate over months — and a time scale is built by using each where it is best. The curves are typical shapes, not the specification of any particular instrument. A time scale that is finished after it is used The world's reference time is not kept by any clock. It is an average of about four hundred and fifty, weighted by how steady each has been, published a month late, and then recomputed a year later against the handful of clocks that define the second — so the best available time for any moment is known only long after the moment has passed. The best clocks are now so good that averaging them requires knowing each one's height to a centimetre. Three ways for a timing model to be wrong, and three shapes that say which. Timing residuals over 6 years for the Crab pulsar, one curve per kind of error in the model, in microseconds. A position error of 1.2 mas leaves a sinusoid of period exactly one year — measured off the drawn curve as 1.000 — with amplitude (a/c)·δθ·cos β = 2.7 μs, because the error is being projected onto a baseline that is the Earth's own orbit and nothing about the pulsar. An unmodelled proper motion of 0.9 mas/yr leaves the same sinusoid with an envelope growing linearly: twice as large at 6 years as at 3. An error of one part in 10⁹ in Ṗ leaves a parabola, the second integral of a frequency drift, whose second derivative is constant to 4e-12 across the span — and that fractional error is deliberately minute, because anything larger produces a residual thousands of times the other two and draws them as flat lines. The shapes do not resemble each other, which is the whole reason a pulsar is an instrument rather than a clock: fitting them simultaneously delivers a position, a proper motion and — from the annual curvature term, not drawn here — a parallax, all from the arrival times of pulses and no image of anything. What is left when every known shape has been removed is the science: glitches, red noise, and the correlated residual between pairs of pulsars that a timing array exists to find. A clock read against a model of everything in between A pulsar delivers arrival times and nothing else. Everything else is a model, and what is measured is the difference between the model and the arrivals — a residual whose shape says which term is wrong, and whose annual sinusoid is a position measured from pulses rather than from an image. Nothing visible in any pulsar, and a quadrupole in the angle between them. Above: 4 millisecond pulsars' timing residuals over 15 years, at the few hundred nanoseconds a good one reaches. Each wanders, and none of them shows anything a reader could call a signal; a gravitational-wave background of amplitude 2.4·10⁻¹⁵ at one cycle per year contributes a common part to all of them that is smaller than each pulsar's own red noise. Below: the correlation between pairs, against the angle on the sky between them. 2211 pairs out of 67 pulsars, binned into 15 angles, against three curves with no free parameters between them. A quadrupolar background gives the Hellings–Downs shape — positive for nearby pulsars, negative near 83°, and back up to exactly half its zero-separation value at 180° because a background looks the same in opposite directions. An error in the observatory clock would give a flat line, because it shifts every pulsar identically. An error in the solar-system ephemeris would give a cosine, because it moves the barycentre in one direction. The drawn points prefer the quadrupole over the flat line by Δχ² = 358. That is the detection: not a waveform, not an event, not a moment — a shape in an angle, accumulated over fifteen years, on data taken for another purpose entirely. A detector the size of the galaxy At a nanohertz no instrument can be built, so the clocks already in the sky are used instead. The signal is in no single pulsar's data — it is in the correlation between pairs as a function of the angle between them, and that curve has no free parameters at all.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Allan deviationClock ensembleGravitational-wave backgroundInternational Atomic TimeMillisecond pulsarPulsar time scaleTerrestrial timeTiming model