The wobble that radiating makes larger

A few cycles that are only mass and spin followed a black hole that has just formed in a merger as it rings down: its distortions oscillate and die away within milliseconds, because each oscillation radiates gravitational waves and the waves carry off its energy. The orbit that has to shrink found the same loss draining a binary. The mountain a spinning star is allowed turned the loss into a measurement, bounding how lumpy a neutron star can be from the fact that no search has heard it.

In every one of those, radiation is friction. It takes energy out, and whatever is radiating settles down. That is so familiar — a bell, a plucked string, an antenna, an atom in an excited state — that it reads like a law. It is not. In 1970 Subrahmanyan Chandrasekhar found that a rapidly rotating fluid body could be made unstable by the emission of gravitational waves: the radiation, instead of damping a certain oscillation, fed it. In 1978 John Friedman and Bernard Schutz showed that this was generic and gave the reason, and in 1998 Nils Andersson showed that one family of oscillations present in every rotating star satisfies their condition at any rate of spin. The effect is named after all three, and it is one of the few places in physics where radiating energy away makes something larger.

One crest, running two ways #

Consider a pattern on the surface of a spinning star — a set of bumps and hollows arranged round the equator, two of each for the simplest case. The star turns at a rate $\Omega$. The pattern is not fixed to the star’s material; it is a wave, and it moves through the material at a speed of its own.

For the family of oscillations called r-modes the speed is set by the rotation itself. They are driven by the Coriolis force: a parcel of fluid displaced sideways on a rotating sphere is turned by it, and the result is a slow, swirling wave whose restoring force exists only because the star spins. The wave is the stellar version of the planetary wave that can only travel west — the same Rossby wave that sets the meanders of the jet stream — and like it, it runs against the rotation. For the simplest r-mode, with two crests round the equator, the crest moves backward through the star’s material at a third of the spin rate.

The figure plots three angles against time. A fixed point on the star turns forward one full turn per rotation period. The crest, measured against the star’s own surface, falls behind by a third of a turn per period. The same crest, measured against the distant stars, is carried forward by the rotation faster than it runs back, and advances at two-thirds of a turn per period. Both statements are true of the same crest at the same time. The pattern runs backward to an observer standing on the star and forward to one watching from far away.

Why the sign of the angular momentum is the one that matters #

That double motion decides the sign of the pattern’s angular momentum, and the sign decides everything.

A wave on a rotating star is a rearrangement of the star’s material, and it changes the star’s total angular momentum by some amount. For a pattern running forward through the material the change is positive: the wave’s crests carry fluid forward faster than the rotation does. For a pattern running backward through the material the change is negative. The star with the backward wave on it has less angular momentum than the same star without it, and less energy in the rotating frame — the wave’s presence is a deficit rather than a surplus.

Now let the pattern radiate. Gravitational waves from a rotating lumpy pattern carry away angular momentum in the sense of the pattern’s motion as seen by the distant observer who receives them — forward, for the r-mode. The star’s total angular momentum must go down by the amount the waves carry. But the part of the star’s angular momentum that can pay is the wave’s, and the wave’s is negative, so paying makes it more negative: the wave grows. Radiation that drains a forward-moving vibration feeds a backward-moving one.

The same accounting runs through several other places in this collection, and recognising it is most of the understanding. The speed below which nothing can be made found that a superfluid flowing past a wall creates excitations only when it flows faster than a certain speed, because above that speed an excitation moving backward relative to the fluid has negative energy in the wall’s frame, and making one lowers the total. A boat faster than the waves on a lake makes a wake for the same reason, and a charge moving through glass faster than light moves in the glass makes a cone of light behind it. In each case something moving through a medium outruns the medium’s own waves, and the waves become a way for it to lose energy by creating them. A rotating star is a medium moving past the distant observer, and a backward wave slow enough to be dragged forward is a wave the star can make by radiating.

The band in which radiation feeds a mode #

The condition can be drawn as a band.

The vertical axis is the speed of a pattern as the distant observer sees it, divided by the star’s spin. A pattern at zero stands still in the sky, a pattern at one turns with the star, and the shaded band between them holds every pattern that runs forward in the sky and backward on the star. Any mode inside the band is driven by radiation; any outside is damped.

Most of a star’s vibrations are ordinary sound and surface waves whose speed through the material has nothing to do with the rotation. The dashed curves are three schematic modes of that kind, whose crests run backward through the star at a fixed fraction of the break-up rate. At slow spin the rotation cannot carry them forward, they run backward in the sky as well, and radiation damps them. Only when the star spins faster than the crest runs is the pattern dragged forward, into the band. For the fundamental oscillations of a real star that requires a spin close to break-up, which is why Chandrasekhar’s original instability was thought to matter only for the most extreme objects.

The solid lines are the r-modes. Their backward speed through the star is proportional to the spin, so their speed in the sky is a fixed fraction of it — two-thirds for the simplest, five-sixths and nine-tenths for the next two — and they sit inside the band at every spin, however slow. Every rotating star is unstable to them. What decides whether the instability matters is how fast it grows compared with how fast the star’s internal friction damps it.

Racing friction #

Both rates can be estimated for a model neutron star, and a widely used one — a ball of fluid of 1.4 solar masses and 12.53 kilometres radius, with no solid crust, analysed by Lee Lindblom, Benjamin Owen and Sharon Morsink in 1998 — gives the numbers in the next figure.

The growth rate by radiation rises steeply with spin. The power a pattern of fixed size radiates rises steeply with how fast it turns, and the mode’s size in the relevant sense grows with the spin too; together they make the growth time fall as the sixth power of the spin: nineteen million seconds at 100 hertz, and 160 seconds at 700. Shear viscosity, the ordinary friction of the fluid, damps the mode in a time that does not depend on the spin and grows longer as the star gets hotter; at a billion kelvin it is 250 million seconds. Bulk viscosity — the friction of a fluid that is compressed faster than its nuclear reactions can keep up with — damps it in a time that grows very rapidly shorter as the star gets hotter, and is negligible at a billion kelvin.

Where the growth time dips below the shortest damping time, the mode is unstable. At a billion kelvin that happens at 65 hertz. Anything spinning faster, in this model, cannot keep the r-mode quiet.

The window a star has to pass through #

Doing the same comparison at every temperature gives the instability window.

The window is the region above the curve and below break-up. Its floor is 40 hertz at about five billion kelvin, where both kinds of friction are weakest. Toward lower temperatures the shear viscosity grows and the threshold rises, but only as the cube root of the cooling: 140 hertz at a hundred million kelvin, 649 at a million. Above about ten billion kelvin the bulk viscosity takes over and closes the window quickly.

A neutron star is born at over a hundred billion kelvin and cools through ten billion within about a minute and through a billion within a year. If it is born spinning rapidly, it enters the window almost at once and stays in it for a long time, and the r-mode grows. The original estimates in 1998 supposed that the mode would grow until its amplitude was of order one and then saturate, and found that a newborn star would radiate most of its angular momentum within a year and spin down to about a hundred hertz. That was a striking prediction, because it would explain why young pulsars are all found spinning slowly, and why none has been caught spinning near break-up.

Later work made the picture less dramatic. A growing r-mode transfers energy to other modes through the nonlinear coupling between them, and calculations of that coupling suggest it saturates at amplitudes of a thousandth or less, which spins a newborn star down over thousands of years rather than one. And a solid crust, which forms within minutes of birth, changes the damping: the fluid of the core rubbing against the crust creates a thin viscous boundary layer that damps the mode far more strongly than the bulk fluid, raising the floor of the window by a factor that depends on how rigidly the crust is attached and is still argued about.

Accreting stars, and the ceiling at seven hundred hertz #

The window has a second population to explain, and it is the more pressing one. Neutron stars in close binaries accrete matter from their companions, and the matter arrives with the angular momentum of the orbit, so accretion spins the star up. Over hundreds of millions of years a star can be spun to many hundreds of turns a second, and these are the millisecond pulsars. Break-up for a neutron star is above about 1,000 hertz for most models of its matter, and above 1,500 for many. Yet the fastest pulsar known turns at 716 hertz, and the accreting stars whose spins are measured from flickers in their X-rays cluster below about 620. Statistical studies of the measured spins suggest a cutoff well below break-up rather than a gradual thinning.

Something is stopping them. Three explanations have been proposed. One is magnetic: the star’s field couples to the inflowing disc and holds the star at a spin where the torque from accretion balances the torque from the field. A second is a mountain on the crust: accreted matter piled unevenly on the crust makes a mountain, whose radiation grows as the fifth power of the spin and eventually balances the accretion torque. The third is the r-mode: a star spun up into the window starts to radiate, the radiation removes angular momentum as fast as accretion delivers it, and the spin stalls at the window’s edge.

The third explanation has a difficulty the window figure makes plain. The cores of accreting neutron stars are heated by the accretion to a few hundred million kelvin, where the fluid model’s threshold is about 100 to 140 hertz. Most of the observed stars spin much faster than that and are therefore, in the model, deep inside the window — unstable now, with modes that should be growing. Either real neutron stars damp r-modes far more strongly than a ball of fluid does, which is possible given the crust, superfluid friction and exotic matter in their cores, or the modes saturate at amplitudes so low that they limit nothing. The quiet stars themselves put a bound on the second possibility: an r-mode that radiates also heats the star by viscous friction, and some accreting stars are so cool in their quiet phases that the amplitude of any mode in them must be below something like one part in ten million. This mismatch is known as the r-mode puzzle, and it has not been resolved.

The tone that would say which it is #

A detection would settle more than whether the mode exists, because an r-mode and a mountain radiate at different frequencies.

A mountain is fixed to the crust and turns with it, and a pattern with two bumps round the equator returns to the same shape twice a turn, so it radiates at twice the spin frequency, as the argument about mountains found. The r-mode’s pattern turns, as seen from outside, at two-thirds of the spin, and it too has two crests, so it radiates at four-thirds of the spin. Relativity and the star’s internal structure shift that ratio, and calculations for plausible neutron stars put it between 1.39 and 1.57. A signal at a frequency between those multiples of a known pulsar’s spin would be an r-mode, and where in the band it fell would say something about the star’s compactness.

Searches have been made. The young pulsar J0537−6910, in the Large Magellanic Cloud, spins at 62 hertz and suffers frequent glitches whose pattern of recovery was suggested to be the fingerprint of an r-mode taking angular momentum out of it; the detectors listened at 1.4 to 1.6 times its spin frequency and heard nothing, which bounds the mode’s amplitude in that star. No r-mode signal has been detected from any source.

Where the fluid model stops #

The numbers in the figures belong to one model, and every simplification in it is one the argument is sensitive to. The star is a single uniform-composition fluid in Newtonian gravity, rotating slowly enough that its shape is nearly spherical. Real neutron stars have a crust, whose boundary layer can damp the mode by orders of magnitude; superfluid cores, in which the neutron fluid and the proton fluid move partly independently and rub against each other through vortices; possibly hyperons or quark matter in the centre, whose bulk viscosity can be enormous at low temperature; and magnetic fields, which the mode’s drift winds up, draining its energy into the field. Each of these pushes the damping up and the window down, and none is known well enough to compute the window of a real star to better than a large factor.

The instability mechanism itself is independent of all of that. It needs only rotation, a mode that runs backward through the rotating material and forward in the sky, and radiation that carries angular momentum in the sky’s frame. General relativity supplies the radiation; Newtonian fluid dynamics supplies the mode; the bookkeeping of angular momentum between two frames supplies the sign. A close relative — the amplification of waves scattered off a rotating body, which Yakov Zel’dovich predicted in 1971 — has been demonstrated in the laboratory with water waves round a draining vortex and with sound reflected from a spinning absorber.

What the pictures cannot show #

The figures show a mode’s linear growth, its pattern speed and a threshold. They do not show what the mode does once it is large: its nonlinear coupling to other modes, the differential rotation it drives in the star, the heat it deposits, the spin-down it causes, or whether it ever reaches a steady amplitude. They show one mode of one symmetry; the other r-modes are also unstable but grow more slowly, and a real star rings with many. And the window is drawn as a sharp boundary between growth and decay for a star with a single uniform temperature, when a real star has a hot centre and a cold surface and the viscosity varies across it. The figures are a map of where the mechanism can operate, not a prediction of what any star does.

Still open: why no neutron star spins faster than about seven hundred hertz #

Millisecond pulsars are among the best-measured objects in astronomy, and their spins are known to many decimal places. The distribution of those spins, and of the spins of the accreting stars that become them, stops well short of what the stars’ own structure would allow. Whether the ceiling is set by gravitational waves — from mountains, from r-modes, or from both — or by the balance between accretion and the star’s magnetic field, is not known. The r-mode explanation is the one most sensitive to the physics of the neutron star’s interior, because the window’s position is set by the star’s friction, and it is also the one least comfortable with the observations, because the simple model puts the observed stars where they should not be. A gravitational-wave detection from an accreting star, at 4/3 of its spin or at twice it, would decide between them directly, and the next generation of detectors is designed to reach the relevant amplitudes for the nearest.

The habit worth carrying away concerns the frame in which a loss is counted. Radiation takes away energy and angular momentum as measured by whoever receives it, and a mode that runs backward through a rotating body but forward across the sky has negative angular momentum in exactly that account — so losing some makes it larger. Every rotating star has such modes, because the Coriolis force makes waves that run against the rotation; whether they matter is a race between the radiation and the star’s friction, and the race is run at a rate that rises as the sixth power of the spin.