Torques that add because the orbits hold still

Two-body relaxation is a random walk. A star is nudged by each passing neighbour in a direction that has nothing to do with the last nudge, the nudges add in quadrature, and the time to change a star’s orbit by its own size is the number of nudges it takes for their quadrature sum to grow that large. Every estimate of how long a star cluster or a galactic nucleus takes to forget its initial state rests on that assumption of randomness.

Near a black hole that dominates the gravitational field, the assumption fails. The stars there move on orbits that are very nearly fixed Kepler ellipses, because the black hole’s point-mass field produces closed orbits and everything else is a small perturbation. An ellipse that repeats itself for thousands of periods is, as far as its neighbours are concerned, a fixed ring of mass. The torque one such ring exerts on another does not change from orbit to orbit. It points the same way for as long as the two ellipses keep their orientations, and torques that keep pointing the same way add linearly in time instead of in quadrature.

That is resonant relaxation, described by Kevin Rauch and Scott Tremaine in 1996. Its consequence is that angular momentum, and especially the orientation of orbital planes, changes near a massive black hole on timescales much shorter than ordinary relaxation would allow — short enough to matter for stars a few million years old.

A random walk with long steps #

The arithmetic is worth doing explicitly, because the whole effect is a change in how long each step of the walk is.

A star at radius r round a black hole of mass M feels a torque from the other N stars within r. If their orbits were randomly placed fixed ellipses, the net torque would be the residual of N randomly oriented contributions, of order $\sqrt{N}\,Gm/r$ per unit mass for stars of mass m — a small number, since the contributions mostly cancel, but a fixed one. As long as every orbit keeps its shape and orientation, that residual torque acts steadily on the star’s angular momentum, changing it in proportion to time.

The steadiness lasts until the orbits precess. The black hole alone would produce perfectly closed ellipses; the extended mass of the cluster makes each ellipse precess in its own plane, backwards, at a rate set by the ratio of the cluster’s mass within r to the black hole’s; and relativity makes it precess forwards, at a rate set by how close the orbit comes to the black hole. Once an orbit has precessed through a large angle, the torque it exerted has changed direction, and a new steady torque begins. The coherence time is the shorter of the two precession periods.

So the change in angular momentum is a random walk whose steps are coherent stretches of length $t_{\rm coh}$. Each step changes the angular momentum by the torque times $t_{\rm coh}$. The walk needs $(J/\Delta J)^2$ steps to change J by its own size, and the time it takes comes out as

$$t_{\rm s} \approx \left(\frac{M}{m}\right)^2\frac{P^2}{N\,t_{\rm coh}},$$

with P the orbital period. Ordinary relaxation, by comparison, takes about $(M/m)^2 P/(N\ln\Lambda)$: its steps are single orbits, one period long. The resonant time is shorter by the ratio of the coherence time to the period, divided by the Coulomb logarithm — and near a black hole, where the orbits precess once in thousands of revolutions, that ratio is large.

The shape of that figure is the whole idea. Every curve is a sum of torques; the only difference between them is how long each torque keeps pointing the same way before it is replaced by an unrelated one. A slope of one on logarithmic axes is a torque that has not yet changed direction. A slope of a half is a random walk. Ordinary encounters are a random walk from the first orbit. Resonant torques are a straight line until the orbits precess, and that straight stretch is where all their extra effectiveness comes from.

Why a Kepler orbit holds still at all #

The whole effect depends on one property of the inverse-square force that is easy to take for granted. Under most central forces an orbit does not close: the star’s distance from the centre oscillates with one period and its angle advances with another, the two are unrelated, and the ellipse turns steadily round the centre. Only two force laws produce orbits that close exactly — the inverse square and the linear spring — and for the inverse square the closure is guaranteed by a third conserved quantity beyond energy and angular momentum, a vector pointing along the ellipse’s long axis that stays fixed while the star goes round.

Resonant relaxation is what happens when that third conserved quantity is almost, but not quite, conserved. The black hole’s field is very nearly inverse-square, so every orbit very nearly holds its orientation; the small departures — the cluster’s own mass, relativity — make the orientation drift slowly, and the torques from other stars, which would average away over a turning ellipse, instead accumulate over each slow drift. The degeneracy of the Kepler problem, the coincidence that makes the radial and angular periods equal, is what turns a weak torque into a strong one. The same coincidence that let Newton derive Kepler’s ellipses from an inverse square makes a nucleus forget its orbital orientations a thousand times faster than a galaxy does.

Reading the five clocks #

The opening figure is worth reading line by line, because each clock is set by a different property of the nucleus.

Ordinary relaxation is flat: the same eighty billion years at every radius. That is not a coincidence of the numbers chosen. For a cusp whose density falls as the radius to the minus one and a half, the number of stars inside each radius and the orbital period there both grow as the radius to the three halves, and the two cancel in the relaxation time. A cusp of exactly that slope is what stars settle into round a black hole by relaxing among themselves, and a nucleus that has relaxed has the same relaxation time everywhere within it.

The precession driven by the cluster’s own mass is flat as well, for the same reason: it depends on the ratio of the stellar mass inside an orbit to the black hole’s mass, times the period, and those factors also cancel for this cusp. Relativistic precession, by contrast, rises steeply with radius, because it depends on how deep in the black hole’s potential the orbit lies — the same advance that Mercury’s perihelion shows, enormously magnified. The two precession lines cross at about six thousandths of a parsec; inside that radius relativity sets the coherence time, outside it the cluster does.

The two resonant clocks then follow. Scalar resonant relaxation has a kink where the precession that limits it changes hands, falling inward while relativity governs and rising outward while the cluster does. Vector resonant relaxation has no kink at all, because neither precession limits it, and it lies below every other relaxation clock at every radius drawn. The young stars’ age, six million years, crosses it at about a hundredth of a parsec for a cluster of solar-mass stars, and at about four hundredths if the torques come from ten-solar-mass remnants — the radius at which the observed young stars change from a coherent disc to a randomised swarm.

Two torques, one of which precession cannot stop #

The torque between two fixed ellipses has two effects. Part of it changes the size of the star’s angular momentum — which, for an orbit of fixed energy, means its eccentricity. Part of it changes the direction — which means the orientation of the orbital plane. The two are called scalar and vector resonant relaxation, and they behave very differently once precession is included.

Precession turns an ellipse within its own plane. That scrambles the part of the torque that depends on where the ellipse’s long axis points, which is the part that changes the eccentricity. It does nothing to the plane. Seen from far away, a precessing ellipse sweeps out an annulus — a flat, axisymmetric ring of mass in a fixed plane — and the torque between two such annuli in different planes is perfectly steady, and tends to rotate each plane about the other. Vector resonant relaxation is therefore not limited by precession at all. Its coherence is limited only by the planes themselves turning, and it is the fastest of all the processes near the black hole.

Its timescale is the time for the residual torque to turn a plane through a large angle in one coherent stretch, about $(M/m)\,P/\sqrt N$, and at 0.1 parsec from the Galactic centre’s black hole that is a few tens of millions of years. Two-body relaxation at the same place takes about eighty billion. Orbital planes near the black hole are turned thousands of times faster than the encounters between stars could turn them.

A cluster made of heavier things #

The timescales depend on the mass of the bodies doing the torquing, in a way that again rewards weighing them rather than counting them.

The question is not academic. Stellar-mass black holes are heavier than ordinary stars and sink towards the centre of any cluster by the drag of the lighter stars around them; a nucleus several billion years old should have concentrated thousands of them within its inner parsec. Their total mass may be small compared with the stars’, but because the resonant torque grows with the mass of each body, a minority of heavy remnants can dominate the turning of orbital planes. The resonant timescales are therefore a probe of a population that cannot otherwise be seen — the dark cusp of remnants at the Galactic centre.

The barrier relativity raises #

Relativistic precession has a peculiarity that matters for the black hole’s diet. Its rate grows sharply with eccentricity: an orbit whose closest approach is ten times nearer the black hole precesses much faster than a circular orbit of the same size, because most of the precession happens at periapsis. A star whose eccentricity is being driven up by scalar resonant torques therefore finds its coherence time shrinking as it goes. At some eccentricity the precession becomes so fast that the torque can no longer change the angular momentum coherently at all, and the star is stranded.

The band between the two lines was named the Schwarzschild barrier when it was found in simulations in 2011: resonant relaxation drives stars towards high eccentricity, and then the relativistic precession it does not control throws them back. If the barrier were absolute, resonant relaxation could not feed the black hole directly. Stars would have to cross the band by ordinary encounters, which work regardless of precession, and the rate of disruptions and captures would be set by the slow process rather than the fast one.

Later work showed that the barrier is leaky. The random part of the encounters, small as it is, keeps nudging stars across the band, and a careful statistical treatment finds the flux through it reduced rather than stopped. The barrier is a real feature of the dynamics and not a wall. What it does is make the black hole’s feeding rate insensitive to resonant relaxation, so that the rate of tidal disruptions predicted from ordinary encounters is roughly right even in a nucleus where resonant torques dominate everything else.

It also bears on the sources that gravitational-wave detectors in space are being built to find. A compact remnant spiralling slowly into a supermassive black hole, emitting gravitational waves for years, must first be placed on an orbit eccentric enough to begin the inspiral. The barrier lies across exactly the path such an object takes, and the rate of these inspirals depends on how it is crossed.

A disc that should have been turned #

The Galactic centre has provided a test of vector resonant relaxation that no theorist designed. Within half a parsec of the black hole there are about a hundred massive young stars, a few million years old, and many of them orbit in a single disc, all in the same sense, clockwise as seen from the Earth. They must have formed together, probably from a disc of gas that fell in and fragmented — the black hole’s tidal field forbids ordinary star formation that close, but a massive enough gas disc could collapse under its own weight. Inside about 0.04 parsecs, by contrast, the stars — mostly younger-looking B stars with highly eccentric orbits — move in random orientations.

The young stars are a clock whose zero is known — they were all born at once, in one plane — and the disc’s present state is a measurement of how far their planes have been turned since. The disc is still a disc, with a thickness of some tens of degrees and a noticeable warp, coherent in its inner parts and less so in its outer. Vector resonant relaxation, with the order-one coefficients set to one, would have turned planes by about that much in six million years. That is close enough to be interesting and too close to be decisive. If the coefficients are a little larger, or the cluster’s heavy remnants a little more numerous, the disc should not have survived as a disc; if they are smaller, the disc’s thickness needs another explanation, such as the warp the infalling gas brought with it.

Seen the other way, the disc’s survival is a limit. It bounds the number of heavy remnants that can be lurking at the centre, because more of them would have turned the planes faster than the disc’s coherence allows. The isotropic stars further in tell the same story from the other side: their orbits have been randomised, which is what a few million years of vector resonant relaxation should do at those radii, and which the orbits’ tightness — their periods of decades — makes plausible on any reckoning.

Why the precession has to be computed and not assumed #

The timescales in these figures depend on the precession rates, and the precession rates depend on the density of the cluster in a way that makes the whole calculation self-referential. The mass precession is set by how much stellar mass lies inside each orbit; that mass is exactly what is uncertain at the Galactic centre, where the cusp of stars the theory of relaxation round a black hole predicts has been hard to find in the observable old giants. If the cusp is shallower than assumed, the mass precession is slower, the coherence time longer, and scalar resonant relaxation faster.

The relativistic precession is the more secure of the two. It depends only on the black hole’s mass and the orbit’s size and shape, both known for the stars that matter; its value for the star with the best-measured orbit near the Galactic centre has itself been detected, a twelve-arcminute advance of the periapsis per orbit, exactly as general relativity predicts. That measurement fixes one of the clocks in the opening figure directly. The rest are inferred from it and from the cluster’s density, and the inferences are only as good as the density.

What the figures leave out #

The coefficients of order one, which the figures set to one, are not negligible; detailed calculations give values between a few tenths and a few, and they differ between the scalar and vector cases. The cluster is drawn as a single power law in density with a single stellar mass, where a real nucleus has a spectrum of masses, a cusp whose slope differs between stars and remnants, and a young disc that is itself a significant mass concentration in one plane and exerts its own coherent torque. And the treatment of resonant relaxation as a random walk with coherent steps is a simplification of a process that is better described as a stochastic diffusion with a specific correlation time, which matters near the barrier and for the details of how orbital planes wander.

What is robust is the ordering: in a nucleus dominated by a black hole, orbital planes turn fastest, eccentricities change more slowly, and orbital energies — which neither resonant process touches, because a steady torque does no work on a Keplerian orbit’s energy — change slowest of all, by ordinary encounters alone.

Order out of a fixed potential #

The general point concerns what makes relaxation fast. Relaxation is slow in a galaxy because the deflections are uncorrelated: each encounter is independent of the last, and the walk has steps one encounter long. It is fast near a black hole because the potential is nearly integrable — every orbit repeats — and the deflections are correlated over thousands of orbits. The same mass, arranged in the same numbers, relaxes differently depending only on how regular the motion it produces is.

The same structure appears in the secular evolution of planetary systems, where planets torque one another’s orbits as fixed ellipses averaged over their periods, and the eccentricities and inclinations exchange on timescales of tens of thousands of years while the orbital periods hold steady. Resonant relaxation is that secular theory applied to a cluster of millions of bodies whose ellipses are not fixed but precess and are scattered, with the statistics of the torques replacing the deterministic coupling of a few planets. A planetary system is a nucleus with four members; a nucleus is a planetary system with four million.

Still open: how much of the centre’s mass is in remnants #

The resonant timescales near the Galactic centre are dominated by the heaviest bodies present, and the heaviest bodies are the ones that cannot be seen: stellar black holes, sunk to the centre over billions of years. Their number determines how fast the young stars’ planes are being turned, how quickly the eccentric B stars were randomised, and how effectively the black hole is fed. Counting them directly is impossible. Inferring them from the stars they affect — the disc’s thickness, the distribution of the inner stars’ orientations, the eccentricities of the stars on the tightest orbits — is a measurement with several unknowns, and the next decade of orbit measurements, extending the census to fainter stars on longer periods, is what will constrain it.