The pulse that shows the back of the star

The radius of a neutron star is the number that would settle what matter does at several times the density of an atomic nucleus. Every proposed description of that matter predicts a relation between a neutron star’s mass and its radius, and a single measured mass above a description’s maximum deletes it. Masses are the easy half. They come from orbits, and for the best cases from the delay of a pulsar’s signal as it passes its companion, measured to a few per cent. Radii are the hard half: a neutron star twelve kilometres across at a distance of a thousand light years subtends an angle a hundred million times smaller than any telescope can resolve, and its thermal glow gives a radius only through a distance and an atmosphere model, each uncertain by tens of per cent.

The route that has worked comes from the star’s gravity itself. Some neutron stars have hot spots — the patches at the magnetic poles where the pulsar’s own currents return to the surface and heat it to a few million kelvin — and as the star turns, the spots swing into and out of view and the X-ray flux pulses. The shape of that pulse depends on how strongly the star’s gravity bends the light leaving its surface, and that bending depends on the ratio of mass to radius. A pulse is a measurement of the star’s compactness, taken without resolving it.

Why the old, fast stars are the ones to use #

The hot spots worth fitting are not on young pulsars. A young neutron star is hot all over from its formation, its magnetic field is strong enough to distort its atmosphere and its X-rays are dominated by emission from its magnetosphere, which pulses for reasons unrelated to the surface. The clean cases are millisecond pulsars: old neutron stars that were born spinning slowly, spun down for hundreds of millions of years, and were then spun up again by gas transferred from a companion, which also buried or reduced their magnetic fields. What remains is a cool surface, a weak field, and two small patches at the magnetic poles heated by the particles the pulsar’s electric field accelerates back onto the star. Their X-rays are almost entirely thermal, from the surface itself.

The same stars are the steadiest clocks in astronomy, and that matters here twice. Their rotation frequencies are known well enough that X-ray photons collected over years can be folded on the rotation with no loss of phase, which is how a pulse profile is built from a few photons per turn. And some of them sit in binaries whose timing gives their masses independently, which is what turns a compactness into a radius at the end of the argument.

A star that shows its own back #

On a star whose gravity did not bend light, an observer would see exactly half the surface: the hemisphere facing them, out to a limb at ninety degrees from the point directly below the line of sight. A ray that leaves the surface horizontally, grazing it, comes from that limb, and nothing behind it is visible. A neutron star’s gravity changes that. Light leaving the surface curves towards the star as it climbs out, so a ray that reaches a distant observer can have left from a point well beyond the geometric limb, heading partly away from the observer and turned round by the star’s gravity on the way out.

The bending is computed by following the ray through the empty space outside the star, where the geometry is the one Schwarzschild found in 1916 and nothing about the star’s interior enters. For a star of 1.4 solar masses and twelve kilometres, the grazing ray reaches the observer from 121 degrees round the star — thirty-one degrees past the limb. For a two-solar-mass star of the same size it comes from 146 degrees. The dashed curves are an approximation Andrei Beloborodov found in 2002, $1 - \cos\alpha = (1 - \cos\psi)(1 - 2GM/Rc^2)$, which is accurate to a few degrees here and makes the physics transparent: the cosine of the emission angle is a linear function of the cosine of the position angle, and gravity changes only the slope.

The same bending that lets the observer see round the star also concentrates the light. A bundle of rays leaving a patch near the limb is spread over a wider range of positions on the star, so each patch is seen at a different brightness than its projected area alone would give. Both effects are in the figures that follow, and both come from the one integral.

The consequence is striking. A typical neutron star shows three-quarters of its surface at once, and the heaviest known show more than nine-tenths. At a compactness of 0.568 — a radius of about three and a half times $GM/c^2$ — the grazing ray comes from the point directly behind the star, and the observer sees the whole surface simultaneously, the back of the star wrapped round its edges in a ring. No neutron star measured is that compact, but the most massive come within about a sixth of it. The bending that weighs a galaxy cluster from the distortion of the galaxies behind it is here a correction of order one to what a single star looks like.

Why the flash is weaker on a denser star #

A hot spot on a star with no bending disappears behind the limb for part of each turn, and during that part the flux falls to nothing. The pulse is then as deep as a pulse can be: its pulsed fraction, the difference between the brightest and faintest flux divided by their sum, is one. On a neutron star the spot stays in view for longer, and for a compact enough star it never sets at all.

With no bending, a spot sixty degrees from the pole seen from sixty degrees above the equator is hidden for thirty-nine per cent of each turn. At a compactness of 0.2 it is hidden for twenty-seven per cent; for the typical neutron star it only just grazes the limb; and for the two-solar-mass star it never sets, and the flux at the faintest point of the turn is more than a fifth of the flux at the brightest. The pulse is shallower and rounder, and the reason is simply that the star’s gravity is holding the spot in view.

That is the measurement in principle: the more compact the star, the smaller the pulsed fraction a spot of given geometry produces. It depends on the ratio of mass to radius and on nothing else about the star — not on its composition, not on its interior, and not on its distance, because the pulsed fraction is a ratio of fluxes. The bending is the one that shifted the positions of stars seen near the Sun in 1919, produced by the same geometry round a mass that accounts for forty-three arcseconds of Mercury’s orbit, grown from a correction of parts per million at the Sun’s surface to one of order unity: general relativity is being used here not as the thing tested but as the ruler.

The same flash from a different star #

The difficulty is the phrase a spot of given geometry. The pulsed fraction depends on where the spot is and how the star is tilted towards the observer as much as on the compactness.

A spot near the pole, seen from near the pole, never sets on any star and gives a shallow pulse whatever the compactness; a spot on the equator seen from the equator sets on all but the most compact stars and gives a pulse of full depth. Between those extremes, every curve falls as the compactness rises, but they fall from different places, and any given pulsed fraction is produced by one geometry on a star of one compactness and by another geometry on a star of another. The amplitude of the pulse is a single number, and it has been asked to report three.

Neither angle is known in advance. The inclination of a pulsar’s spin axis to the line of sight can sometimes be estimated from the swing of its radio polarisation, with errors of ten or twenty degrees, and the position of the hot spots on the surface is not known at all: the magnetic field that heats them need not be a centred dipole, and the fits find that in some stars it is not.

A sinusoid has nothing more to say #

It would be natural to hope that the rest of the pulse, beyond its amplitude, carries the missing information. For a spot that never sets, it does not.

The three curves lie on top of each other. One is from a star of compactness 0.2 tilted at forty degrees, one from a star of 0.3 at forty-three, one from a star of 0.45 at fifty-one, and once the amplitudes are matched the shapes agree to a tenth of a per cent. The reason is Beloborodov’s formula. The flux from a small spot is proportional to the cosine of its emission angle times the focusing of its bundle of rays, and in the approximation both are linear in the cosine of the angle round the star — which, for a spot turning on a tilted star, is a constant plus a constant times the cosine of the rotational phase. The flux is a pure sinusoid. A sinusoid has a mean, an amplitude and a phase, and the phase says only when the spot faced the observer. Everything the pulse could report about compactness is in the amplitude, and the amplitude is shared with the geometry.

The degeneracy is not an accident of a poor fit or too few photons; it is exact in the approximation and nearly exact in the full calculation. It is the same situation as a weak-lensing survey that measures two cosmological parameters only as one combination of them, or as the inspiral of a binary that reports only one combination of two masses: the data are good, and the model maps a whole line of parameter values onto the same observation.

A spot that sets does better. The flat bottom of an eclipsed pulse is a feature a sinusoid cannot have, and its length depends on how far past the limb the star lets the observer see, which puts harmonics into the pulse that carry information the amplitude does not. That helps only for stars and geometries in which the spot sets, and for compact stars seen at moderate angles it often does not.

The spin writes the radius into the pulse #

What breaks the degeneracy for the stars that matter most is the spin. The stars with the best pulse profiles are millisecond pulsars, spinning a few hundred times a second, and at that rate a spot on a twelve-kilometre star is moving at five to ten per cent of the speed of light.

A source moving towards the observer is brightened and one moving away is dimmed, and for a spot on a spinning star the approach comes before the moment of closest alignment and the recession after it. The pulse leans: it rises steeply and falls slowly, and its peak moves earlier. The size of the lean is set by the speed of the surface, which is $2\pi R f$ times the sine of the spot’s colatitude — and the frequency $f$ is known, from timing the pulses, to a dozen decimal places. So the lean measures the radius, in kilometres, and not only the ratio of mass to radius.

That is the second, independent handle. The bending measures $M/R$; the lean measures $R$ times a geometric factor that the pulse’s other features also constrain. Together with the amplitude and the harmonics that the lean introduces — a leaning pulse is no longer a sinusoid — they are enough to separate the compactness from the geometry. The real fits add two things the figure leaves out, aberration of the emitting area and the difference in light-travel time between the near and far sides of the star, which is a few tens of microseconds across a star that turns in a few milliseconds and shifts features by a per cent of a turn. They also fit the spectrum of the spot as well as its brightness, since the Doppler shift moves photons between energy bands as well as brightening them.

A radius from a compactness and a mass #

Even with the degeneracy broken, what a pulse fit measures most tightly is the compactness. On a plane of mass against radius, a measured compactness is not a point but a band through the origin.

The first star analysed this way, the millisecond pulsar J0030+0451, was fitted by two independent teams in 2019 using X-ray data from an instrument on the International Space Station built for this measurement. They found its compactness $GM/Rc^2$ to about six per cent, and its mass and radius individually only to about ten per cent, the two errors correlated along the band: a lighter star with a smaller radius fitted about as well as a heavier one with a larger radius. The mass and radius they quoted, about 1.4 solar masses and thirteen kilometres, are where the band was crossed by the lean and the harmonics, and those crossings are looser than the band itself.

The second star supplied what the first lacked. J0740+6620 is the most massive neutron star with a precise mass, measured by the Shapiro delay of its pulses as they pass its white-dwarf companion in an almost edge-on orbit. With the mass fixed from timing, the compactness from the pulse fit becomes a radius, and the radius came out at about twelve and a half kilometres, uncertain by about a kilometre either way. That single number has done more to constrain the equation of state of dense matter than any other: a two-solar-mass star that is not much smaller than a 1.4-solar-mass one requires matter whose pressure keeps rising steeply with density to the centres of the heaviest stars, and it rules out the descriptions in which the matter softens.

Where the method is exposed #

The pulse is a clean probe of gravity and a less clean probe of everything else. The spots are not small: they have sizes and shapes the fit must include, and the analyses of J0030 found two spots, both in the southern hemisphere, neither a circle — one a crescent — which says the star’s magnetic field is far from the centred dipole the textbooks draw. The dipole formula every pulsar field strength comes from was already suspect for other reasons, and here the surface itself shows the departure.

The spots’ emission is not a black body. Their radiation passes through a thin atmosphere of hydrogen or helium whose opacity depends on energy and angle, so a spot is limb-darkened and its spectrum is harder than its temperature suggests. The atmosphere changes how brightness depends on emission angle, which is exactly the dependence the bending acts on, and the choice of atmosphere moves the inferred radius by several per cent. The two teams that analysed J0030 used different atmospheres, different spot models and different treatments of the instrument’s background, and they differed in the radius of J0740 by more than a kilometre, which is larger than either team’s statistical error.

And the fits depend on the instrument’s response. How efficiently it records photons of each energy is known only to a few per cent, and the pulse fits that combine its data with another telescope’s, to calibrate the background, inherit the relative calibration of both. None of these limits touches the principle, which is that gravity bends light from the back of the star into view by an amount set by its compactness. They are what stands between that principle and a radius to a few per cent.

Still open: what the next radius will change #

Only a handful of neutron stars have had their radii measured this way, and all of them sit between about eleven and fourteen kilometres. The radius that would change most is that of a star of known mass well below 1.4 solar masses, or one close to the maximum: two radii at very different masses fix the slope of the mass–radius relation, which is where different descriptions of dense matter diverge most, and the present measurements are nearly all at one end. A second handle is emerging from the gravitational waves of merging neutron stars, whose tidal distortion just before contact depends on their radii, and whose systematic errors have nothing in common with a hot spot’s atmosphere. Where the two methods agree, the radius of a neutron star — the least accessible length in astronomy — will have been measured twice, once by the light a star’s gravity bends round it and once by the way its shape yields to a companion’s pull.