Illustrated Physics
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Astrophysics

The mountain a spinning star is allowed

A neutron star spinning thirty times a second with a bump on its surface a few centimetres high is a rotating mass quadrupole. It sends out gravitational waves at twice its spin frequency, as a steady tone lasting millions of years. None has been heard. The silence is a measurement. The Crab pulsar is slowing down, and if all the energy it loses went into gravitational waves they would have been detected many times over. Searches now say the waves carry less than a ten-thousandth of it. The mountain on the Crab is less than about ten centimetres high, on a star twenty-four kilometres across.
15 min read 5 figures Made into an instrumentThe shape decides

Assumes: The orbit that has to shrink · What the instrument actually hears

The wave that stretches one way and squeezes the other drew the pattern a gravitational wave makes in a ring of free masses. The orbit that has to shrink found two orbiting masses radiating such waves and spiralling together as they lose energy. What the instrument actually hears described how an interferometer registers the wave, a few cycles that are only mass and spin followed the ringing of the merged remnant, and the ring that does not come back found the permanent distortion a burst leaves behind.

Every one of those was about a merger: a brief, violent event lasting seconds or less. The quadrupole formula does not require violence. Any mass distribution whose quadrupole moment changes with time radiates, and a single spinning star with a small asymmetry is the most persistent source of all. Its signal is a pure tone at twice its spin frequency, continuing for millions of years. None has been detected. The searches have been sensitive enough, however, that the non-detection is a measurement of the shapes of neutron stars, and for the best-studied pulsar it has reached below the largest mountain its crust is thought able to hold.

A tone from a lumpy star

A rigid body spinning about a principal axis has a quadrupole moment that rotates with it. If the body is symmetric about its spin axis, the rotating quadrupole looks the same at every moment and nothing radiates. If it is not — if it has a mountain, or is slightly elliptical across its equator — the quadrupole changes as the star turns, and the star radiates gravitational waves. The asymmetry is measured by the ellipticity ε\varepsilon, the fractional difference between the star’s moments of inertia about the two equatorial axes. The strain at a distance dd is

h0=4π2GIεf2c4d,h_0 = \frac{4\pi^2 G I \varepsilon f^2}{c^4 d},

with II the moment of inertia, about 103810^{38} kilogram square metres for a neutron star, and ff the wave’s frequency.

The strain a lumpy spinning star sends. The gravitational-wave strain at Earth from a neutron star one kiloparsec away, against the wave's frequency — twice the spin frequency — on logarithmic axes, for ellipticities of 10⁻⁹, 10⁻⁷, 10⁻⁵, 10⁻³, with the most each of three known pulsars could emit if all its observed spin-down were gravitational radiation. Crab: at most 1.4·10⁻²⁴ at 59.2 Hz, needing an ellipticity of 7.7·10⁻⁴; Vela: at most 3.3·10⁻²⁴ at 22.4 Hz, needing an ellipticity of 0.0018; J0437−4715: at most 1.6·10⁻²⁶ at 347.4 Hz, needing an ellipticity of 2·10⁻⁸. Searches have pushed the Crab's strain below about a hundredth of its limit, 1.4·10⁻²⁶. The strain grows as the square of the frequency, so a fast millisecond pulsar with a tiny mountain can send as much as a young pulsar with a large one.
Fig. 1 The strain at Earth from a neutron star 1 kpc away, against the wave’s frequency, for ellipticities 10−910^{-9} to 10−310^{-3} (dashed), with the spin-down limits of three pulsars (dots): the Crab, 1.4×10−241.4\times10^{-24} at 59.2 Hz; Vela, 3.3×10−243.3\times10^{-24} at 22.4 Hz; J0437−4715, 1.6×10−261.6\times10^{-26} at 347.4 Hz. The line below the Crab marks how far searches have reached, 1.4×10−261.4\times10^{-26}.

The figure plots that strain against frequency for a star at one kiloparsec, for ellipticities from 10−910^{-9} to 10−310^{-3}. The strain grows as the square of the frequency, because the quadrupole moment’s second time derivative does, so a star spinning ten times faster sends a hundred times stronger waves for the same mountain. The numbers are tiny. An ellipticity of 10−510^{-5} at 100 hertz from a kiloparsec gives a strain of about 10−2510^{-25}, a change in a four-kilometre detector arm of a hundred-thousandth of a proton’s radius.

Two cycles for every turn

The frequency of the wave is fixed by the star’s rotation, and it is twice the rotation frequency.

Two wave cycles for every turn. The two polarisations of the gravitational wave from a spinning star with a single mountain, viewed at 60° to the spin axis, in units of the peak strain, over two rotations. A spinning ellipsoid returns to the same shape every half turn, so the wave has twice the spin frequency: the Crab, turning 29.6 times a second, would radiate at 59.2 Hz. At this inclination the plus polarisation has amplitude 0.625 and the cross 0.500, a quarter-cycle apart. The signal is a pure tone lasting for years, so weak that it can only be found by adding up its phase coherently over months of data, which is possible only because its frequency is known from radio timing.
Fig. 2 The two polarisations of the wave from a spinning star with one mountain, viewed at 60° to the spin axis, over two rotations. The wave completes two cycles per turn: the Crab, turning 29.6 times a second, would radiate at 59.2 Hz. At this inclination the plus polarisation has amplitude 0.625 and the cross 0.500, a quarter-cycle apart.

The figure draws the two polarisations of the wave over two rotations, for a star viewed at 60 degrees to its spin axis. A spinning ellipsoid returns to the same shape after half a turn, because an ellipsoid looks the same from opposite ends, so the quadrupole and the wave repeat twice per rotation. The Crab, spinning 29.6 times a second, would radiate at 59.2 hertz. The two polarisations have amplitudes that depend on the viewing angle, 0.625 and 0.500 of the peak at 60 degrees, and they are a quarter-cycle apart, so the combination traces an ellipse in the plane of polarisation — the gravitational analogue of elliptically polarised light.

The doubling is the signature of quadrupole radiation, the same doubling that makes a binary’s waves come at twice its orbital frequency. It is also what makes a known pulsar a precise target. Radio astronomers time pulsars to a fraction of a millisecond over years, so the wave’s expected frequency and its slow decline are known in advance to many decimal places, and a search can look for exactly that tone and nothing else.

The spin-down limit

Every pulsar loses rotational energy: its spin slows by a measured amount each second. Most of that energy drives a wind of magnetised particles, and for the Crab the wind lights up the whole Crab Nebula. Some of it could go into gravitational waves. If all of it did, the ellipticity would have to be large enough for the waves to carry away the whole spin-down power, and that fixes the largest strain the pulsar can possibly send, its spin-down limit.

For the Crab, which loses about 4.3×10314.3\times10^{31} watts, the spin-down limit is a strain of 1.4×10−241.4\times10^{-24} at 59.2 hertz, requiring an ellipticity of 7.7×10−47.7\times10^{-4}. For Vela, younger and closer, it is 3.3×10−243.3\times10^{-24} at 22.4 hertz. For the millisecond pulsar J0437−4715, the nearest of its kind, it is 1.6×10−261.6\times10^{-26} at 347 hertz, needing an ellipticity of only 2×10−82\times10^{-8}, because a millisecond pulsar spins down very slowly and a small mountain at such a high frequency would drain it quickly.

The same argument works for any spinning source whose energy loss is measured, and it has a pleasing symmetry with the binary pulsars that first proved gravitational waves exist. There the orbit was seen to shrink at exactly the rate the quadrupole formula demands, so the waves were inferred from the energy they took. Here the spin is seen to slow at a rate that sets a ceiling, and the waves are inferred to be smaller than the ceiling because the detectors do not hear them. The spin-down limit is a ceiling set by energy conservation, independent of any theory of neutron-star structure. A search that reaches below it measures something new, because it excludes shapes that energy alone would allow.

A silence below the ceiling

For the Crab, that point was passed in 2008, when a search of data from the LIGO detectors’ fifth science run found no signal at a strain four times below the spin-down limit. Each later run pushed further. By the observing runs of the late 2010s and early 2020s the Crab’s strain was constrained to about a hundredth of its spin-down limit.

The share of a pulsar's spin-down a mountain could carry. For the Crab pulsar, the fraction of the rotational energy it loses each second that a mountain of each ellipticity would radiate as gravitational waves — the square of its ellipticity over the ellipticity that would account for all of the spin-down — on logarithmic axes; the ellipticity that would account for all of it is 7.7·10⁻⁴. The searches' limit on the strain, about a hundredth of the spin-down value, is a limit of 7.7·10⁻⁶ on the ellipticity and of 10⁻⁴ on the share: gravitational waves carry less than a ten-thousandth of the energy the Crab loses. The rest drives the nebula around it, through the pulsar's magnetised wind — which is what had been assumed, and is now measured.
Fig. 3 For the Crab, the share of its spin-down power a mountain of each ellipticity would radiate as gravitational waves, on logarithmic axes; all of it at 7.7×10−47.7\times10^{-4}. The searches’ strain limit, about a hundredth of the spin-down value, limits the ellipticity to 7.7×10−67.7\times10^{-6} and the share to 10−410^{-4} (shaded region excluded).

The figure turns that into an energy budget. The power a mountain radiates goes as the square of its ellipticity, so the share of the Crab’s spin-down carried by gravitational waves is the square of its ellipticity over the ellipticity that would account for all of it. A strain limit of a hundredth of the spin-down value is an ellipticity limit of 7.7×10−67.7\times10^{-6} and a limit of 10−410^{-4} on the share. Gravitational waves carry less than a ten-thousandth of the energy the Crab loses. The rest goes into the magnetised wind that powers the nebula, which is what astrophysicists had assumed; the searches have turned the assumption into a measurement.

How tall a mountain can be

An ellipticity is an abstract number. As a height, roughly the ellipticity times the star’s radius, it becomes vivid.

How tall a mountain a neutron star is allowed. Mountain heights on a neutron star 12 km in radius, taken as the ellipticity times the radius, on a logarithmic scale. Crab, if all spin-down were waves: ellipticity 7.7·10⁻⁴, about 9.2 m; Crab, largest allowed by searches: ellipticity 7.7·10⁻⁶, about 9.2 cm; largest a crust can hold, estimated: ellipticity 10⁻⁵, about 12.0 cm; J0437−4715, if all spin-down were waves: ellipticity 2·10⁻⁸, about 237 μm; floor suggested by millisecond pulsars: ellipticity 10⁻⁹, about 12 μm. A neutron star's gravity is so strong that a bump of about ten centimetres is estimated to be the most its crust could hold before breaking, and the gravitational-wave searches have now reached below that for the Crab. The millisecond pulsars' slow spin-down caps their mountains at tens to hundreds of micrometres, and suggests a floor near ten.
Fig. 4 Mountain heights on a 12 km neutron star, as ellipticity times radius. The Crab if all its spin-down were waves: 9.2 m. The Crab as allowed by searches: 9.2 cm. The most a crust can hold, estimated: about 12 cm. J0437−4715 if all its spin-down were waves: 240 μm. A floor suggested by millisecond pulsars: 12 μm.

The figure lists five heights. If all the Crab’s spin-down went into gravitational waves, its mountain would be about nine metres high. The searches have excluded that, and everything above about nine centimetres. The largest mountain a neutron star’s crust could support is estimated at an ellipticity of about 10−510^{-5}, some twelve centimetres, so the searches have reached below the most a crust could plausibly hold. For the millisecond pulsar J0437−4715 the spin-down limit alone caps the mountain at about 240 micrometres, the width of a few hairs.

The heights are small because the gravity is enormous. The size at which a body becomes round found that the tallest mountain a body can hold falls as one over its surface gravity. A neutron star’s surface gravity is about 101110^{11} times the Earth’s, and its crust, though far stronger than any terrestrial material, supports only centimetres before it breaks. The crust is a lattice of neutron-rich nuclei in a sea of electrons, the matter whose composition the protons a star cannot afford traced, compressed by the weight of the star above it, and its breaking strain is estimated from molecular-dynamics simulations of that lattice at about a tenth, far beyond what any terrestrial material survives. Even so, twelve centimetres is the limit.

The millisecond pulsars suggest something further. Their spin-down rates, plotted against their spin frequencies, show a lower edge, as though no millisecond pulsar spins down more slowly than an ellipticity of about 10−910^{-9} would force. Woan and colleagues proposed in 2018 that this edge is set by a minimum mountain, some twelve micrometres on a twelve-kilometre star — perhaps raised by the star’s own buried magnetic field, which strains it slightly out of round. If so, every millisecond pulsar emits continuous gravitational waves at a strain the next generation of detectors might reach.

What could raise a mountain

A neutron star settles into a nearly perfect spheroid within seconds of forming, so a mountain needs something to hold it up against the star’s gravity. Three mechanisms are taken seriously, and each predicts a different size.

Strain frozen into the crust. A young neutron star spins fast and its crust solidifies while the star is flattened by rotation. As the star spins down, it wants to become rounder, and the solid crust resists. The stress it stores can be released in sudden cracks, starquakes, and whatever asymmetry the crust keeps is a mountain. This is what the crust’s breaking strain limits, and it is the source of the twelve-centimetre estimate.

Magnetic stress. A neutron star’s interior field, which may be much stronger than the field measured at its surface, distorts the star as a stiff spring would. A field of 10810^{8} tesla wound round inside the star could raise an ellipticity of the order of 10−910^{-9} to 10−710^{-7}, depending on how it is arranged. That is the mechanism usually proposed for the millisecond-pulsar floor, and it would make every magnetised neutron star slightly lumpy, whether or not its crust were strained.

Accretion. A neutron star gathering matter from a companion builds its crust from the top down. The fresh matter is compressed as it is buried, and at each of the densities where a nucleus captures electrons — the thresholds the protons a star cannot afford listed for iron and its lighter neighbours — the composition changes and heat is released. Bildsten pointed out in 1998 that if the crust is slightly hotter on one side than the other, those capture layers sit at slightly different depths on the two sides, and the density jumps they make form a buried mountain. It is a mechanism that turns a chemical-potential threshold into a gravitational-wave source, and it may be why the fastest accreting neutron stars stop spinning up near 700 hertz.

A crust that cracks, and a superfluid that slips

The Crab and Vela do not spin down smoothly. Every few years Vela suddenly spins up, by about a part in a million of its rotation rate, and then relaxes over days to weeks. These glitches are the clearest evidence that a neutron star is not a rigid body. The favoured explanation involves the neutron superfluid inside the crust, which rotates by forming an array of quantised vortices, the same objects the whirlpool that comes in one size described in helium. The vortices pin to the nuclei of the crust. As the crust slows, the superfluid cannot, until the lag grows large enough to unpin many vortices at once. Their angular momentum is dumped into the crust in a moment, and the star spins up.

A glitch rearranges a neutron star’s interior in seconds, and it may briefly excite oscillations that radiate gravitational waves. Searches for such transient signals after observed glitches have found none so far. They also help to calibrate the continuous searches, because a glitch changes the pulsar’s frequency and would shift the phase of any continuous tone, and the searches must follow the radio timing through it. Nothing is allowed to be rigid found that no body can transmit a push faster than its sound speed; a neutron star’s crust, with a shear-wave speed of about a thousand kilometres a second, takes some tens of milliseconds to feel a glitch across its whole surface, which is fast by any human standard and slow compared with a millisecond pulsar’s rotation.

Why the search takes a year

A continuous signal is weak, and it can only be found by adding it up coherently over a long time. That is harder than it sounds, because the detector is moving.

A steady tone that the Earth's orbit keeps retuning. The frequency shift, over a year, of a 59.2 Hz gravitational wave from a source on the ecliptic, caused by the Earth's orbital motion towards and away from it at up to 29.8 km/s. The shift swings by ±5.88 mHz, a part in ten thousand. A search that adds up a year of data coherently resolves frequencies to about 3.2·10⁻⁸ Hz, so the shift spans some 370,000 resolution bins; unless it is removed exactly, the signal's power is smeared across all of them. That is why a search is fastest for a known pulsar, whose position, frequency and spin-down are measured by radio astronomers, and slowest for a neutron star nobody has seen.
Fig. 5 The frequency shift over a year of a 59.2 Hz tone from a source on the ecliptic, caused by the Earth’s orbital motion at up to 29.8 km/s. It swings by ±5.88 mHz, a part in ten thousand. A search adding up a year coherently resolves about 3.2×10−83.2\times10^{-8} Hz, so the shift spans some 370,000 resolution bins.

The figure plots the frequency of a 59.2-hertz source on the ecliptic as received at the Earth over a year. The Earth’s orbital motion towards and away from the source Doppler-shifts it by up to a part in ten thousand, ±5.88\pm5.88 millihertz, the same shift the shift a mirror gives twice measures from a moving target. A search that adds up a year of data coherently can distinguish frequencies about 3×10−83\times10^{-8} hertz apart, so over the year the signal wanders across some 370,000 frequency bins. Unless the shift is removed exactly, the signal’s power is smeared across all of them and lost in the noise. The Earth’s daily rotation adds a smaller shift, and the detector’s sensitivity to the wave’s direction changes through the day as well.

For a known pulsar all of this can be corrected: its position fixes the Doppler shift, and radio timing fixes its frequency and spin-down. The search then follows one precise template through months of data. For a neutron star nobody has seen — and there are expected to be a hundred million in the galaxy — the position, frequency and spin-down are unknown, and the number of templates needed to cover the possibilities grows so fast with the length of data that the most sensitive all-sky searches are limited by computing power. Some have been run as distributed computing projects, on hundreds of thousands of volunteers’ computers.

What the search assumes

The limits drawn above rest on three assumptions, each stated in the searches themselves.

The moment of inertia. Every conversion from strain to ellipticity uses I=1038I = 10^{38} kilogram square metres. Real neutron stars may have moments up to about three times larger, depending on their mass and on the equation of state of their interiors, and the ellipticity limits scale inversely with it.

The distance. The Crab’s distance is about two kiloparsecs, uncertain by some tens of per cent, and the spin-down limit scales inversely with it. The ratio of the search limit to the spin-down limit does not depend on the distance, which is why the energy statement is more robust than the height.

The emission mechanism. The searches assume the wave comes from a mountain rotating rigidly with the star, at exactly twice the radio frequency. A neutron star could also radiate at other frequencies, through oscillation modes of its fluid interior or through a slight wobble of its spin axis. Searches for those use wider frequency bands and are less sensitive.

Still open: whether any spinning star is heard

No continuous gravitational wave has yet been detected from any source. The most promising targets are young pulsars whose spin-down limits are within a factor of a few of current sensitivity, accreting neutron stars in X-ray binaries — where the infalling matter may build and sustain mountains and whose spin frequencies seem to stop rising at about 700 hertz, perhaps because gravitational waves carry away the angular momentum the accretion delivers — and the millisecond pulsars, if their minimum ellipticity is real. Detectors now being planned would be about ten times more sensitive, which would reach the proposed millisecond-pulsar floor for the nearest of them. Whether neutron stars carry mountains at the level the crust allows, at the floor the pulsars hint at, or at neither, and what the first detected tone will say about the matter in their crusts, is one of the open questions gravitational-wave astronomy is built to answer.

The habit worth carrying away is to treat a well-bounded silence as a measurement. When the energy a source loses is known, the largest signal it could send is known too, and a search that reaches below that ceiling measures the source even if it hears nothing. For the Crab the silence says that its mountain is shorter than about nine centimetres and that its gravitational waves carry less than a ten-thousandth of its lost energy, and the searches reached that far without detecting a single cycle.

Part 6 of 6

This essay is one argument about Gravitational waves. The others:

The ring does not come back. A ring of freely floating masses at four phases of a passing gravitational wave and once after it has gone. The first four are the familiar picture: stretched one way, then the other, with the area unchanged. The fifth is the one the standard picture does not draw — the ring is permanently deformed, by 25 per cent of the largest deformation the wave itself produced, and nothing brings it back. The masses are not oscillating about a new centre; they are at rest, at new separations. Both the oscillation and the offset are exaggerated enormously: the real strain at 440 megaparsecs is 9.9e-22 and the real permanent offset is 2.5e-22, so the drawing magnifies both by about 2e+20. What is honest in the picture is the ratio between them. The ring that does not come back Part 5 — Every picture of a passing gravitational wave shows a ring of free masses stretched, squeezed and let go. The last frame is wrong. The ring ends a different shape — permanently, with the masses at rest at new separations — by about a fifth of the largest distortion the wave itself produced. What sources the offset is the energy the wave carried away, so the wave is remembering itself, and nobody has measured it.

Where this goes next

How long an orbit has before the waves take it. The time a circular orbit has left before gravitational radiation brings it together, against its separation, for four pairs of masses. Both axes are logarithmic and every curve has the same measured slope, 4.00: the lifetime goes as the fourth power of the separation, so halving an orbit shortens its remaining life by a factor of sixteen. The Earth's orbit is 13 decades above the age of the universe and the neutron-star binary is below it, which is the whole difference between a system that is losing energy and a system that is going to merge. The orbit that has to shrink Two masses in orbit radiate gravitational waves and lose energy, so the orbit tightens, so they go faster and radiate harder. The runaway takes 10²³ years for the Earth and the Sun and eight minutes for the last thousand kilometres of a black-hole pair — and the same one-line formula gives both. Where the instrument can hear. The response of an L-shaped interferometer to the plus polarisation, over the whole sky: azimuth across, cosine of the polar angle up, so that equal areas of the picture are equal areas of the sky. The plus pattern is largest directly overhead and underfoot and along the arms, and vanishes on four lines where a wave stretches both arms equally and the interferometer has nothing to compare. The average of the square over the whole sky is 0.2333, summed over a hundred and sixty thousand directions, and the two polarisations' averages add to 0.4000 — two fifths exactly, whatever the polarisation angle. Averaged over that angle as well, each polarisation contributes a fifth, and that fifth is what turns a strain sensitivity into a range: it is why a detector's quoted reach is substantially less than what it would manage for a source overhead. There is no direction in which the instrument is deaf to both polarisations at once, and none in which it is fully sensitive either. What the instrument actually hears A gravitational-wave detector is not a ruler laid against a stretching space. It is a clock comparison, its response falls to nothing at frequencies where the wave turns over while the light is still in the arm, and there are directions in the sky where it is deaf. Where a body can no longer be any shape it likes. The tallest mountain a body can carry, against the body's radius, on logarithmic axes, beside the line on which a mountain would be as tall as the body. The first falls as 1/R and the second rises as R, so they cross exactly once — here at 282 km, for rock of 200 MPa strength and density 3000 kg/m³. Below that radius a body's own gravity cannot enforce anything and it stays whatever shape it was made; above it, the shape is decided by gravity and the answer is a sphere. The crossing moves as the square root of the strength, so it is an order of magnitude and not a boundary. The size at which a body becomes round A mountain can be no taller than the height at which the rock beneath it begins to crush, and that height falls as the body gets bigger — so there is a size above which a mountain would have to be taller than the world it stands on. Above it, nothing can be any shape but a sphere.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Coherent integrationDoppler effectEllipticityGravitational wavesNeutron starPulsarQuadrupole radiationSpin down