A pulsar that relativity is turning out of sight

The orbit of Mercury precesses by forty-three arcseconds a century beyond what Newton’s law provides, and the orbit of a binary pulsar shrinks by exactly the amount its gravitational radiation demands. Both of those are statements about an orbit. Einstein’s theory makes another statement, about something the orbit carries with it: a spinning body taken round a closed path in curved space comes back with its axis pointing in a new direction. Nothing has pushed it, no torque is at work, and the axis has done the nearest thing to staying straight that the geometry permits.

Willem de Sitter worked this out in 1916, within a year of the theory, and applied it to the Earth and Moon going round the Sun. The result was an angle of 19 thousandths of an arcsecond a year, far below anything then measurable. It took until the 1980s for lunar laser ranging to detect it, and until 2011 for Gravity Probe B, four quartz spheres in a polar orbit round the Earth, to measure the same effect directly.

The pulsars did not wait. In a binary with a short period, the same term amounts to degrees per year. An instrument that sits on the turning object and points along its axis would record that as a slowly moving direction. A pulsar is such an instrument, and a peculiar one, because it is observed only when its beam crosses the Earth. So the visible consequence of this piece of curved-space geometry is that certain stars change shape on a timescale of years and then disappear, on dates that can be written down.

One formula at four strengths #

For a test gyroscope on a circular orbit the rate is simple: the fraction $\tfrac{3}{2}GM/c^2a$ of the orbital angular speed. That makes it the same kind of quantity as the advance of Mercury’s perihelion, a first post-Newtonian correction proportional to the depth of the potential well. When the gyroscope’s own mass is not negligible, as in a pair of neutron stars, the formula of Barker and O’Connell applies:

$$\Omega = \left(\frac{2\pi}{P_b}\right)^{5/3} T_\odot^{2/3}\,\frac{m_c(4m_p + 3m_c)}{2(m_p + m_c)^{4/3}}\,\frac{1}{1-e^2},$$

with $T_\odot = GM_\odot/c^3$ about 4.9 microseconds. There is nothing in that expression to adjust. Measure the orbit’s period and eccentricity, and the two masses by other means, and the rate is fixed.

The comparison shows why pulsars can do what they do. Gravity Probe B needed sapphire and superconducting readouts, gyroscopes that were the roundest objects ever made, and a telescope locked on a guide star, to detect an angle about the size of a coin seen from a kilometre away after a year. The same term at the double pulsar is about 2,800 times larger than that, and its signature can be seen in the pulse shape without any special instrument at all. Reaching those strengths needs two things that a neutron-star binary provides naturally: a compact orbit, for the depth of the potential, and a short period, since the effect accumulates once per orbit.

The rate falls gently with the orbital period, as the five-thirds power, so only a few of the known relativistic binaries are placed to show it quickly.

Those rates are predictions, and they serve as a test. Every quantity on the right of the formula comes from the orbit’s relativistic timing parameters. The precession measures something independent of the timing, and so a measured rate that matches the prediction is a test of the same theory in a different sector, involving spin rather than orbit. For the double pulsar, the rate was measured through an ingenious accident. The two stars eclipse each other, and the way the magnetosphere of one blocks the light of the other depends on its orientation, so the pattern of the eclipse changes as the axis moves. René Breton and collaborators read it off in 2008 as $4.77^{+0.66}_{-0.65}$ degrees a year. The prediction is 5.07.

A beam, a cone, and a calendar #

A pulsar’s radio emission comes from near the magnetic pole. It rotates with the star, and the observer sees a pulse only when the beam sweeps across the line of sight. Spin and magnetic axis are locked together, and so they precess together. If the spin lies at an angle $\delta$ from the total angular momentum, it traces out a cone of half-angle $\delta$ over one precession period. The observer’s angle to the spin axis $\zeta$ then changes as

$$\cos\zeta = \cos\delta\cos i + \sin\delta\sin i\cos\Omega t,$$

where $i$ is the inclination of the orbit, which a Shapiro delay can give independently. Then the pole, an angle $\alpha$ from the spin, is seen only while $|\zeta - \alpha|$ is less than the beam’s half-width. Precession therefore moves the line of sight in latitude across the beam.

The picture offers a sobering thought about discovery. The Hulse–Taylor pulsar was found in 1974, within a window of about eighty years out of a cycle of three centuries. Had Russell Hulse been surveying two centuries sooner or later, the pulsar would not have been there to find, by this reckoning, and neither would the first evidence for gravitational radiation. A radio survey can only find what is currently pointed at it, and a precessing pulsar is pointed at any particular observer only part of the time.

The profile change has been followed since the late 1980s, when Joel Weisberg, Roger Romani and Joseph Taylor noticed the first trace of change in the shape of this pulsar’s pulse. By 1998 Michael Kramer could fit the change to a model of the moving beam, and from that he forecast that the pulsar would be lost to view about 2025. A forecast of that kind depends on the unknown shape of the beam, and so its date has real uncertainty, and the pulsar’s actual behaviour has been the subject of continued monitoring. The structure of the forecast is what matters, a date on the calendar computed from orbital geometry.

A pulse that narrows as the edge comes #

The line of sight’s path across the beam is a chord, and its length is what the observer records as a pulse width. As the line of sight approaches the beam’s edge, the chord shortens.

This is a good case of a figure doing what no timing residual can. The long flat stretch of pulse width means the line of sight has been passing near the beam’s centre, and little has appeared to change. As it nears the edge, the rate of change climbs, and the last few years show the fastest shrinking of all. The behaviour resembles the last moments of a total eclipse. Observers see nothing dramatic until shortly before the end. This has an uncomfortable implication: a stable pulse profile over decades is no evidence that a pulsar will stay in view, and any pulsar whose spin is misaligned with its orbit could be near its own edge.

The beam itself is what is being mapped. Each year the line of sight takes a new slice through it, and the sequence of slices, combined with the geometry, gives a two-dimensional picture of the beam. No other kind of pulsar has this available. A solitary pulsar shows its observer one chord only, for ever, and all that is known about the shape of its beam has to be inferred statistically from many different stars. A precessing pulsar scans its own beam. For Hulse–Taylor, this scan revealed a structure more like an hourglass than a filled circle. For J1906+0746, published work in 2019 went further, tracking the pulse from both magnetic poles for over a decade. It confirmed the precession rate predicted by general relativity at 2.17 degrees a year, within errors of five per cent, and it showed that the star’s magnetic geometry was close to what the classical model expects. That is a test of relativity and a map of a neutron star’s emission, from the same data. The pulsar itself has faded from view since.

The one that vanished on schedule #

The double pulsar’s second star had the most dramatic history. Discovered in 2004 alongside its companion, pulsar B was bright only in two short stretches of its orbit, as the wind of pulsar A disturbed its magnetosphere. Over the next four years its profile changed quickly, and its brightest stretches faded. In March 2008 it dropped below detection. It had been observed for just about four and a third years.

It did not die. At five degrees per year the cone takes seventy-one years to go round, and in those four years the axis had moved through about twenty degrees of it, enough to turn the beam away. Models of the emission geometry predict that the beam will come back, and at least one model places that return as early as the mid-2020s, while others place it about a decade later. A measurement of when it does return would fix the geometry, and it would give a separate check on the rate. There is a small thrill in writing a statement about an object not currently visible, whose return date is set by a term in a post-Newtonian expansion.

Pulsar A, in the same system, has shown no profile change at all, and the reason is informative. The rate at which it should precess is almost the same as B’s. What differs is the angle of the cone. A’s spin must be very nearly aligned with the orbit, to within a few degrees according to fits, so its axis is turning on the spot, and its appearance does not change. So the same physics can be invisible for one star of a pair and decisive for the other, depending on an angle.

What set the angle #

That angle is a piece of evolutionary history. A pulsar with a measured misalignment has inherited its orientation from the event that formed the system. The older star of a neutron-star binary was spun up by accreting from its companion, and mass transfer leaves its spin aligned with the orbit. Then the companion explodes as a supernova, forming the second neutron star, and any asymmetry in that explosion gives it a kick. The kick changes the plane of the orbit, while the first star’s spin stays pointing the way it was.

The misalignment therefore records the kick. The large angle of about twenty degrees in Hulse–Taylor implies a substantial one, of the order of two hundred kilometres per second. The small angle in the double pulsar implies a very small kick, and with its other unusual features this has been taken to suggest a low-mass progenitor, possibly a stripped star collapsing through electron capture, a quieter kind of explosion. So an observable motion of a beam now tells how a supernova went off a long time ago. This connects the most precise tests of gravity to the much messier physics of core collapse, by way of an angle.

The result is not fully determined by that one measurement, as the kick has three components and the mass lost in the explosion also changes the orbit. Only the combination of the tilt, the eccentricity, the period and the system’s motion through the Galaxy can pin it down. Together, though, these make these systems among the very few objects in which a supernova kick has been constrained for an individual star, rather than statistically.

What it does to the census #

A rate of neutron-star mergers can be estimated by counting the binaries found in pulsar surveys and asking how many more there must be. The measurement of gravitational waves from those mergers has made that calculation important. The count depends on what fraction of all such pulsars is beamed towards the Earth, and precession might appear to change it.

The flatness is what the figure is meant to show. At any moment the same fraction is visible, whatever the precession, because an observer’s chance of lying in the swept region is set by the area of the beam, and that does not change when the beam moves. Averaged over all the binary pulsars in the Galaxy, geodetic precession does not hide any more of them. So the correction to the census remains the one that has always been in it, the beaming fraction. What precession does change is the survey’s interpretation: a given pulsar has a limited lifetime of visibility, the same pulsar seen again decades later may be invisible, and a pulsar known to exist may no longer be findable by any future survey. There is also a subtler effect. Because a beam that passes across the observer gives a wider pulse at some moments than others, the survey’s detection threshold will favour some phases. This is a minor correction next to the uncertainties in pulsar luminosities that dominate these rate calculations.

The surprise in the term #

The most surprising connection here is between a mechanical instrument and a celestial one. Gravity Probe B was forty years in the making and was among the most difficult experiments ever flown, while a pulsar in a tight orbit performs the same measurement on its own, in a stronger field, at a larger signal, without having been built. Its readout is the shape of a radio pulse. The two now agree with the same prediction: a geometric rotation of a vector carried along a curve.

A second connection joins this to the older essays on the Earth’s own rotation axis. The tilt that is not a constant includes a small part of this same effect, as de Sitter realised: the Earth’s axis is carried round the Sun and turns by about two arcseconds a century, included in the precession constant that astronomers use every day. It is hard to find a term with a wider reach, as it affects the calendar of the Earth’s equinoxes, and it also shuts off the signal of a pulsar a few thousand light years away.

What the picture cannot show #

The geometry in these figures is a single set of numbers. Real fits give several solutions, because the sign of the inclination and the pole’s position can usually be swapped with little change to a pulse-width history, and each gives a different forecast date. So the 1943 to 2025 window is one illustration. The beam has been drawn as a filled circle with a sharp edge. Actual beams are patchy, and what looks like the edge can be one of their dark patches, so that a pulsar may fade and then return in a way the chord model does not capture. The formula neglects the spin of the companion and higher post-Newtonian terms, all of which are small here.

The kick figure neglects mass loss and treats the kick as purely perpendicular, while real kicks have components in the plane that change the orbit’s size and shape. The visibility figure uses random inclinations and random magnetic angles, whereas real pulsar geometries are not random, in that older pulsars tend to align their fields, and recycled pulsars have narrower beams. Such pulsar beams also depend on the observed frequency. So the numbers in it are only about the general behaviour.

Still open: whether the return of the second pulsar can be timed #

The double pulsar’s lost star is the cleanest available test of the whole account, because its orbit is known to extraordinary precision, and so the geometry is limited only by the beam model. A detection of pulsar B’s reappearance would fix a date on a precession cycle, and an extended observation of it would turn its profile into a slice through a beam already partly mapped. The models differ by a decade in when that should occur, which itself measures how little is known about a beam seen for only four years. The largest radio arrays now coming into operation are able to detect a far fainter trace of it, so its return could be noticed early, while it is still faint. Whether the beam that comes back resembles the one that left is a question about whether a pulsar’s beam is a fixed property of the star, or whether it, too, has been changing in the meantime, and that is what the timing of its return would begin to say.