An orbit measured to be shrinking

Mercury’s 43 arcseconds a century had to be excavated out of an observed 5,600, and every step of the excavation was a subtraction somebody could get wrong. The system below offers the opposite arrangement: a quantity whose entire observed value is the effect, with nothing large sitting on top of it.

Nothing in the drawing is a correction to something larger. The curve is the whole observed behaviour of the orbit.

Measured, not inferred — and where the mechanism lives #

The claim is narrow and it is worth stating so plainly that it could be wrong: the rate at which this orbit is shrinking is a measured number, on the same footing as a length or an angle, and not a quantity deduced from a theory of why orbits shrink.

The distinction is easy to lose, because the mechanism is famous. That mechanism is not this site’s: the physics of an orbit losing energy to gravitational radiation is developed as physics at illustrated-physics.com, whose arguments stand with nothing whatever observed in the sky. What is argued here is the reverse — a timing residual accumulated over four decades, and the chain of assumptions between an arrival time recorded at a telescope and a number quoted as a period derivative. No radiated power is computed anywhere below.

Sixty-eight nanoseconds a turn #

Take the period derivative as given and ask what it does. In one revolution of 27,907 seconds the period shortens by $\dot P_b P_b = 6.8\times10^{-8}$ s — 68 nanoseconds, against an orbit measured to microseconds. No single revolution could show it.

What makes it visible is that the shortening does not reset. Each orbit is a little shorter than the last, so the count of orbits since some epoch runs ahead of the count a fixed period would predict, and the discrepancy grows quadratically:

$$\Delta t(T) = \tfrac12 \frac{\dot P_b}{P_b}\,T^2 .$$

The period shortens by 76 microseconds a year and the orbit comes round 1,131 times a year, and neither number is what is measured. What is measured is the parabola: after forty-five years and some fifty thousand revolutions, periastron arrives 87.5 seconds earlier than a constant period would put it — three parts in a thousand of one orbit, out of a per-orbit change of two parts in $10^{12}$.

The curvature of that parabola is $\dot P_b$, and nothing else in it matters: a wrong epoch tilts the curve, a wrong period offsets it, and neither changes the second difference. That is worth holding onto, because a period is how nearly every mass in the sky is weighed, and here the quantity of interest is not the period but the rate at which the period is failing to be constant.

What an arrival time is, and what has to be taken out of it #

The instrument is a radio telescope — Arecibo’s 305 m dish for most of this record — and the recorded quantity is a time of arrival. The pulsar spins once every 59 milliseconds, and a single pulse is too weak and too variable to time; several minutes of data are folded at the current spin period into an average profile, which is cross-correlated with a template to give one number: the epoch at which a fiducial point of the pulse crossed the observatory’s clock. Residual scatter on such a number is tens of microseconds — a part in a few thousand of one turn of the star.

Between that and an orbit sit four removals, and each is an assumption.

The star’s own clock drifts. A pulsar spins down, so the spin frequency and its derivative are fitted alongside everything else, and the fit assumes the star does not glitch or wander in ways the model cannot absorb.

The telescope is not an inertial platform. Arrival times must be referred to the solar system’s barycentre, which means removing up to 500 seconds of light-travel across the Earth’s own orbit — a position quoted in the wrong frame is a different quantity — and that removal needs a planetary ephemeris, a model of the Sun’s own delay, and a conversion from an atomic clock at a particular place to a barycentric time scale.

The interstellar medium delays low frequencies more than high ones. The delay goes as the inverse square of the observing frequency, so a dispersion measure is fitted too, and its slow drift over decades is a systematic nobody can switch off.

And what remains is an orbit of the pulsar, not of the pair. The periodic delay gives the projected semi-major axis of one star’s path about the barycentre, in light-seconds, with the inclination folded inside it — the same degeneracy that makes a spectroscopic mass a lower bound rather than a mass, and for the same reason. Neither body is still, and only one of the two is pulsing.

Where the shrinking rate is separable from everything else #

The orbit is therefore not observed; it is fitted. Some seventeen parameters — spin, spin-down, position, proper motion, dispersion measure, five Keplerian elements, and a handful of post-Keplerian terms — are adjusted at once until the model reproduces every arrival time ever recorded. $\dot P_b$ is one coefficient in that model.

What keeps it from soaking up other terms’ errors is its shape. Every Keplerian term in the fit is periodic in the orbit, the geometric ones are periodic in the year, and $\dot P_b$ alone grows without bound. A secular term is separable from a forest of periodic ones not because it is large but because it is the only thing in the model that never comes back. Forty-five years of data contain forty-five cycles of the annual terms and fifty thousand of the orbital ones, and exactly one parabola.

One term does mimic it, and it is the honest weak point of the measurement. The pulsar and the solar system sit in different parts of the Galaxy and accelerate differently, so the line-of-sight component of that relative acceleration makes every clock in the system appear to run slow at a constant rate — indistinguishable, in the timing model, from a shrinking orbit. Removing it requires the pulsar’s distance and a model of how the Galaxy rotates, built from inside it. The correction is about four parts in a thousand of $\dot P_b$: it moves the ratio of observed to predicted decay from 0.997 ± 0.002 to 1.0013 ± 0.0021, and the uncertainty grows when it is applied, because the distance is poorly known. That growth is the signature of a model-dependent step, and it is why the agreement is now often quoted the other way round — with the theory taken as given and the residual read as a distance.

The same orbit’s other secular term, thirty-five thousand times Mercury’s #

The fit returns a second unbounded term, and it is the one that makes this rung and Mercury’s one ladder rather than two subjects. Periastron advances, at 4.226598 degrees per year.

Two factors build that number out of Mercury’s 43 arcseconds a century, and both are geometrical rather than exotic. Per revolution the advance is 13.5 arcseconds against Mercury’s 0.104 — a factor of 130, from a far smaller and far more eccentric orbit. And the revolutions arrive 272 times more often, because the period is 7.75 hours rather than 88 days. The product is about 35,400, and the whole of it is that the same effect has been given a tighter orbit and more turns of it. The eccentricity is doing more work here than the formula shows. A periastron is a measurable direction only if the orbit has a long axis worth speaking of, and an orbit at low eccentricity can look exactly like a circle while its apsidal line wanders unmeasurably; at 0.617 the two stars are four times further apart at apoapsis than at periapsis.

A prediction nobody fitted, and a second pair that does it better #

The advance of periastron and the amplitude of the orbit’s time dilation are each functions of the two masses. Two measured quantities, two unknowns: the masses come out at 1.4398 and 1.3886 solar masses, with no use made of the decay whatever. The predicted $\dot P_b$ follows from those masses, so comparing it with the measured one sets a computed number against an observation that had no part in computing it. Three post-Keplerian terms, two unknowns, one test — the same over-determination that makes an eclipsing binary the only kind of star whose mass is known outright, in a system where nothing is eclipsed and one star has never been seen at all. Russell Hulse and Joseph Taylor found the 59-millisecond pulsar at Arecibo in 1974 and published in 1975 with its period already visibly Doppler-shifted by an unseen companion; the decay was detected in 1979, on four years of timing.

PSR J0737−3039, found at Parkes in 2003, does the same job thirty times better, for a reason about visibility rather than about physics. Both neutron stars pulse, so the two projected orbits are separately measured and their ratio gives the mass ratio directly, with no post-Keplerian term and no theory involved. The orbit is 2.45 hours, the pair is near enough for a parallax measured by radio interferometry — so the galactic correction that limits PSR B1913+16 is itself well determined — and the ratio of observed to predicted decay comes out at 0.999963 ± 0.000063.

The end the rate implies, and the part of it nobody measured #

A rate that is measured now says something about a future that is not.

What the picture cannot show #

The residuals themselves. The points in the hero figure are the published prediction scaled by the published ratio, not the archival arrival times. A real timing plot is thousands of measurements over four decades, unevenly spaced, with receivers and back-ends changing under them; the smoothness of the parabola is a property of the model rather than of the data.

The mechanism. Nothing above says why the period shortens. The figures compute no radiated power and quote no formula for one; the two closed forms that appear are quoted with their sources, and the decay rate enters as a measurement.

The correction that is not on the axes. The one systematic that matters — the galactic acceleration term — depends on a distance the timing cannot supply. It is four parts in a thousand of the effect, larger than the quoted uncertainty, and it appears nowhere in the drawing. A figure whose error bars are invisible invites the conclusion that nothing limits the measurement, and something does.

The final approach. The inspiral curve is a closed form for an orbit changing slowly compared with its own period. In the last seconds that ceases to be true and the two objects stop being points. The line is honest across nine decades of separation and silent about the only part of the history anybody could watch.

What a template is, and the assumption inside it #

An arrival time was described above as the epoch at which a fiducial point of the pulse crossed the observatory’s clock, and that phrase conceals the step where most of the practical difficulty lives.

A single pulse from this star is not reproducible. Individual pulses vary in shape, in strength and in the phase at which they peak, and many are absent altogether. What is stable is the average of several thousand of them, and the timing procedure is to fold a few minutes of data at the current spin period, producing an average profile, and then to cross-correlate that profile with a standard template.

The output of the correlation is a phase offset, which is converted to a time. So an arrival time is not a measurement of when anything happened; it is a measurement of how far a smoothed shape has slid relative to a reference shape.

That makes the stability of the template an assumption of the same standing as the ephemeris or the dispersion model. If the average profile changes — if the star’s beam sweeps across the line of sight differently — the correlation returns a shifted phase and the fit reads it as a timing residual.

The profile of this pulsar does change, slowly, over decades, and the change is not a nuisance: it is the geodetic precession named at the end of this essay, the pulsar’s spin axis being carried around in the curved space it orbits in. So the same effect that is one of the system’s most interesting measurements is also a slow corruption of the instrument used to measure everything else, and disentangling the two requires modelling the beam’s geometry rather than treating the template as fixed.

An instrument whose calibration is one of its own results is a familiar situation in this subject, and the usual resolution — measure the drift and carry it as a parameter — is what is done here too.

Why the record cannot be restarted #

The parabola grows as the square of the elapsed time, which makes the length of the campaign the single most valuable thing about it. What is less obvious is that the value is in its continuity rather than in its total span.

Pulsar timing is coherent: the model does not fit the arrival times as a set of independent measurements but counts pulses, and every pulse between the first observation and the last is accounted for as an integer. That is what delivers the precision — a phase-connected solution over decades knows the total number of rotations exactly, and an error of one would show up immediately.

A long enough gap breaks the count. If the accumulated uncertainty in the model’s prediction grows past half a rotation before the next observation, the number of pulses across the gap is ambiguous and the phase connection is lost. What remains is two separate coherent stretches whose relative phase has to be re-established, and some of the precision of the combined span is gone.

The same problem appears when instruments change. A new receiver or a new back-end introduces an unknown constant offset in the recorded arrival times, so each combination of telescope, receiver and processing has to carry its own fitted offset — parameters that absorb some of the signal and that must be constrained by overlapping observations taken with both systems before the old one is retired.

That is why the collapse of the Arecibo dish in 2020 mattered beyond the loss of collecting area. Most of the record for this system was taken there, and continuing it elsewhere means establishing the offset between the old instrument and the new one using whatever overlap exists — which, for an instrument that failed rather than being decommissioned on a schedule, is whatever happened to have been recorded already.

A measurement that accumulates rewards not being interrupted, and that is a property of the observing programme rather than of the physics.

The same parabola, drawn by friction instead #

The algebra of this measurement is not about relativity at all, and the clearest proof is a system in which nothing relativistic happens.

The Earth’s rotation is slowing under ocean tides, so a day is longer than it was, and the discrepancy between an event’s predicted and observed time accumulates as the square of the elapsed interval — the same $\tfrac12(\dot P/P)T^2$, with the sign reversed because a lengthening period makes things late rather than early. Over 2,700 years of Babylonian, Chinese and Arab eclipse records that parabola reaches some four hours, and running the measured rate backwards refutes the assumption that it was constant. Two effects with nothing in common — gravitational radiation from two neutron stars, friction in shallow seas — are read out of the sky by the same quadratic, one over 45 years of microsecond timing and the other over three millennia of naked-eye reports.

The same trick is what lets a transit arriving late weigh an unseen planet: a repeating event is a clock, and a clock’s slow drift is easier to see than its instantaneous rate. The sky offers few instruments and a great many periodic phenomena, and a period derivative is the cheapest way to convert patience into precision.

The measurement is a parabola in accumulated phase and the inspiral is its extrapolation, and both are worth drawing over a longer span.

Where the ladder goes next #

This anchor’s first rung ended by pointing here, and this one owes at least two more. An orbit seen nearly edge-on delays its own pulses as they pass the companion, which turns the inclination and the companion’s mass from assumptions into measurements. And the pulsar’s spin axis precesses in the curved space it moves through, so the pulse profile’s shape changes over decades — a secular term in a quantity that is not an orbital element at all, read off the same data by asking a different question of it.