A black hole thrown out by its own merger

Two galaxies merge — if their encounter is slow enough for them to capture each other — their central black holes sink towards the middle of the new one, and if they get past the last parsec — where friction has stopped working and gravitational radiation has not yet started — they spiral together and merge within a few million years. The final orbits and the merger itself convert several per cent of the pair’s mass into gravitational waves, at a peak power greater than that of every star in the observable universe combined.

Energy is not all the waves carry. They carry momentum too, and the momentum radiated in different directions does not have to cancel. If the two holes differ in mass, or spin at different rates or in different directions, the radiation is beamed: more goes one way than the other during the last orbits, and the remnant recoils in the opposite direction like a rifle firing. The recoil is set entirely by the geometry of the final orbits, and the numbers are large. For the configurations that maximise it, the merged hole leaves at several thousand kilometres a second — faster than the escape speed of any galaxy.

A galaxy’s central black hole therefore cannot be taken for granted after a merger. Whether it stays depends on its galaxy’s depth and on a property of the two holes no telescope can see directly: which way they were spinning.

Why a merger kicks at all #

A pair of black holes on a circular orbit radiates gravitational waves in a pattern that rotates with the orbit. If the two holes are identical, the pattern has a symmetry under swapping them — rotating it by half an orbit — and the momentum radiated in any direction during one half-orbit is radiated in the opposite direction during the next. Nothing accumulates. The remnant stays put.

Break the symmetry and the cancellation fails. With unequal masses the lighter hole moves faster on its larger orbit, and the radiation is beamed slightly in its direction of motion. At any moment there is a net momentum flux, pointing along the lighter hole’s velocity. On a circular orbit that direction rotates, so over one orbit the flux nearly cancels — but the orbit is shrinking, the radiation is growing, and each orbit’s flux is larger than the last. The incomplete cancellation leaves a residue, most of it from the final plunge, where the orbit ends abruptly and the last unbalanced burst has nothing after it to cancel against.

For non-spinning holes this mass-asymmetry kick is modest. It vanishes for equal masses, vanishes for very unequal ones, where the radiation is weak, and peaks at about 175 kilometres a second for a mass ratio near one third. That number was among the first produced when numerical relativity became able to follow a merger in 2005, and it was about what earlier approximations had estimated.

The surprise came in 2007, when spins were added.

The superkick #

A spinning black hole drags space around with it — the spin is the second number a black hole has, after its mass — and two spinning holes in orbit interact through their spins as well as their masses. A spin component perpendicular to the orbital plane, along the orbit’s axis, changes how fast the holes plunge and adds a kick in the orbital plane of a few hundred kilometres a second. A spin component lying in the orbital plane does something stranger: it makes the whole orbit bob up and down along the axis, once per orbit, and the bobbing beams radiation preferentially up or down.

When the two holes have equal masses and their in-plane spins point in opposite directions, the bobbing adds coherently over the final orbits, and the recoil is along the orbital axis — out of the plane, perpendicular to everything the mass-asymmetry kick does. The fitting formula to the numerical results, drawn here, gives up to about 3,400 kilometres a second for spins at nine-tenths of the maximum lying flat in the plane; later simulations with more extreme configurations found kicks approaching 5,000. These are the superkicks, and their size made the problem astrophysical overnight. No galaxy has a central escape speed above about 3,000 kilometres a second.

The figure shows how sharply the kick depends on the spin directions. With both spins along the orbital axis there is no in-plane component, the bobbing does not happen, and equal holes do not recoil at all. As the spins lean, the kick grows roughly as the sine of the lean, and it is already above a thousand kilometres a second at thirty degrees. Unequal masses reduce the spin kick, because the formula weights the spins by the mass ratio and the lighter hole’s spin counts for less, and add the modest mass-asymmetry floor.

The final orientation of the in-plane spins relative to the line between the holes at merger is effectively random, because the spins precess during the inspiral and nobody can predict where they will be pointing at the last orbit. The recoil therefore has a distribution even for fixed spin magnitudes and tilts, and the averaged curve in the figure is the root-mean-square over that unknown phase.

Whether spins are aligned is a question about gas #

The kick distribution depends almost entirely on how the spins are oriented relative to the orbit, and the opening figure shows the three cases. With spins within ten degrees of the orbital axis, the median kick is about a hundred kilometres a second and essentially none exceeds a thousand. Within thirty degrees, the median roughly doubles. With spins in random directions, the median is over five hundred and a quarter of mergers exceed a thousand.

What decides the orientation is how the holes grew. A black hole that accretes most of its mass from a gas disc has its spin dragged into line with the disc, because the disc’s angular momentum is much larger than the hole’s and the disc’s inner region warps until the two agree — the Bardeen–Petterson effect. If both holes in a merger accreted from gas that settled into the merged galaxy’s central disc before the final inspiral, their spins are aligned with each other and with the orbit, which also lies in that disc. Gas-rich mergers should therefore produce small kicks.

Mergers of gas-poor galaxies — the giant ellipticals that formed by merging smaller ellipticals — have no such disc. The holes’ spins point wherever their own earlier histories left them, the orientation is close to random, and kicks of over a thousand kilometres a second are common. The physics of the last orbit, a few hundred gravitational radii across, is therefore set by the gas supply of a galaxy a hundred thousand light-years across.

Whether alignment works as well as the simple argument says is still debated. The alignment takes time, the gas may not settle into a single plane, and the spins may precess out of alignment during the inspiral. The aligned case is the optimistic limit.

Kicks against the depth of the well #

A recoiling black hole escapes its galaxy if its speed exceeds the escape speed from the centre. That speed depends on how much mass is in the galaxy and how concentrated it is, and it scales roughly with the galaxy’s velocity dispersion.

The estimate drawn treats each galaxy as an isothermal halo with the same dispersion as its stars, truncated at its virial radius, with the black hole starting from its own sphere of influence. That gives an escape speed of about 360 kilometres a second for a dwarf with a dispersion of 50 and about 1,340 for a large galaxy with a dispersion of 200 — roughly six or seven times the dispersion in each case, the factor growing slowly because the potential well’s depth depends logarithmically on its extent. Real galaxies, with their stars concentrated towards the centre, are deeper than this by tens of per cent.

The bands are where the kicks lie. Aligned spins give kicks below the escape speed of every galaxy except the smallest dwarfs. Random spins give kicks that overlap the escape speeds of galaxies with dispersions up to a hundred and fifty kilometres a second — which includes the Milky Way’s bulge and most galaxies in the universe.

Which galaxies keep their black holes #

The fraction of mergers that leave a black hole behind follows directly.

A galaxy with a dispersion of two hundred kilometres a second keeps its merged black hole in nearly every merger, even with random spins: 86 per cent. A dwarf with a dispersion of fifty keeps 98 per cent with aligned spins and a third with random ones; at thirty, 86 per cent and 17 per cent. And a galaxy that has had several mergers must survive each of them, so the fraction compounds.

That has a consequence for the smallest galaxies that no other process produces. If dwarfs grew partly by mergers of smaller dwarfs that each had a black hole, and the spins were not aligned, many of them should have lost their black holes — and the fraction of dwarfs that host one should fall towards low masses faster than the black holes’ formation alone would make it. Whether dwarfs host central black holes at all is a measurement made galaxy by galaxy, with difficulty, because a black hole of a hundred thousand solar masses in a galaxy forty million light-years away has a sphere of influence far too small to resolve. The occupation fraction is not yet known well enough to say whether recoils have emptied any.

The same process works in the opposite sense in the early universe. The first black holes grew in small haloes with shallow wells, and a recoil large enough to eject a seed from its halo would stop its growth. The difficulty of growing black holes of a billion solar masses within a billion years is made worse if seeds were being ejected from their haloes every time two of them merged.

How far a kicked hole goes #

A kick below the escape speed does not leave the hole at the centre. It sends it out on a radial orbit through the galaxy, and the excursion can be long.

In an isothermal potential, the depth grows only logarithmically with distance, so the distance a given speed can reach grows exponentially with the square of that speed. Half the escape speed takes a hole a couple of hundred parsecs out — still within the galaxy’s core. Seven-tenths takes it to a few kiloparsecs, comparable to the size of the visible galaxy. Nine-tenths sends it well over a hundred kiloparsecs, into the halo, from which it returns on an orbit of hundreds of millions of years.

Dynamical friction, which drags on any massive body moving through a background of stars and dark matter, shortens the excursions of slow holes in dense centres and barely touches fast ones far out, because friction falls steeply with speed and with the density of the medium. A hole kicked at nine-tenths of the escape speed can spend a significant fraction of a Hubble time wandering, carrying with it whatever gas and stars were bound within its own sphere of influence. A wandering hole with a cluster of bound stars around it still disrupts the occasional star, and a tidal-disruption flare far from a galaxy’s centre is one of the few ways such a hole could announce itself.

The same arithmetic in a star cluster #

The recoil does not care how massive the holes are; it depends only on their mass ratio and spins, and a merger of two ten-solar-mass black holes kicks its remnant exactly as hard as a merger of two billion-solar-mass ones. What changes is the well it has to climb out of.

A globular cluster has an escape speed of a few tens of kilometres a second, and a young star cluster less. Against those, almost every recoil in every spin case is fatal. Even with aligned spins, the median kick of a hundred kilometres a second is several times the escape speed of a large globular cluster. A black hole formed by a merger inside a cluster is almost always ejected from it.

That has a measurable consequence for gravitational-wave astronomy. In a dense cluster, black holes sink to the centre, form pairs by three-body encounters, and merge, and a remnant that stayed could merge again — a second-generation merger, with a heavier, faster-spinning component that carries the signature of having been built from two black holes rather than from a star. Recoils make such repeat mergers rare in globular clusters and possible only in the deepest wells: nuclear star clusters at the centres of galaxies, with escape speeds of a hundred kilometres a second or more, or gas discs around active nuclei. A merger with a component of unusual mass and high spin, of the kind the detectors have reported several times, is therefore evidence about the depth of the well it happened in.

It is also evidence about how the binaries were assembled. A pair formed by encounters in a cluster arrives with random spin orientations, because nothing aligned the two holes’ spins with each other or with the orbit; a pair formed from two stars that evolved together arrives with spins roughly aligned. The recoil distribution therefore differs between the two channels for the same reason it differs between gas-rich and gas-poor galaxy mergers, and the retention of merger products in clusters depends on which channel built them.

What was actually measured #

No recoil has been measured unambiguously, and the search shows how hard a clean signature is to find.

Displaced nuclei. A recoiling hole that is still accreting appears as an active nucleus offset from its galaxy’s centre, by up to several kiloparsecs. Candidates exist, found in images as point sources displaced from the light-weighted centre. Each has an alternative explanation: a second, unrelated nucleus from the merger; a dusty centre that displaces the light; a chance superposition.

Velocity offsets. A hole moving at a thousand kilometres a second carries its broad-line region with it, so the broad emission lines from the gas bound to the hole are shifted relative to the narrow lines from the host galaxy. Several quasars show such offsets, some of thousands of kilometres a second. The trouble is that a binary black hole, still orbiting, produces exactly the same signature: one hole moving relative to the system’s centre. The two interpretations predict different behaviour over years — a binary’s offset changes as it orbits, a recoiling hole’s does not — and monitoring has shifted some candidates each way.

Gravitational waves. The recoil imprints itself on the waveform of the merger, as a Doppler shift of the ringdown and as asymmetries in the final cycles. For the stellar-mass mergers detected by ground-based instruments, which already measure a combination of the two masses to a fraction of a per cent, the signal-to-noise is just becoming sufficient to constrain the kick in individual events, and at least one merger has been reported with a kick of over a thousand kilometres a second inferred from the waveform. For supermassive holes, the space-based detector planned for the next decade would measure kicks directly in many mergers, and the pulsar-timing arrays that use the Galaxy itself as a detector are already sensitive to the population of binaries whose mergers produce them.

Where the estimates stop #

One fitting formula. The kicks are computed from a 2007 fit to numerical relativity that captures the mass-asymmetry kick, the in-plane spin kick and the superkick. Later fits add higher-order spin terms and a dependence on the spins’ orientation relative to the line of merger that reach larger extremes, near five thousand kilometres a second, and shift the distribution’s tail. The median and the dependence on spin alignment are robust.

An isothermal galaxy. The escape speed and the excursion use a single isothermal sphere truncated at the virial radius. A real galaxy has a stellar bulge concentrated within a few kiloparsecs that deepens the central well, and the escape speed from the centre is higher than drawn — for giant ellipticals, by a few hundred kilometres a second.

No friction, no gas. The excursion figure ignores the drag that brings slower holes back sooner, and ignores the gas at the centre of a merger remnant, which can both deepen the potential and slow the hole.

Distributions, not populations. The retention fractions assume mass ratios spread evenly in logarithm and spins at nine-tenths of maximal. Real merger populations have mass-ratio distributions from cosmological simulations and spin distributions from accretion histories, and both change the numbers.

A symmetry that holds only on average #

The general statement is about conservation laws and symmetric systems. A pair of bodies radiating symmetrically conserves its momentum in the radiation’s average, and the remnant stays at rest. Break the symmetry — unequal masses, misaligned spins — and the radiation’s momentum no longer cancels, and the source must move to compensate. Nothing about this is specific to black holes; an asymmetric supernova kicks its neutron star by the same accounting, at hundreds of kilometres a second.

A conserved quantity that sums to zero only because of a symmetry is free to sum to something else when the symmetry fails, and the failure is usually in the detail nobody controls. Here the detail is spin orientation, set by accretion histories in galaxies the size of the Milky Way, and it decides whether a hole of a billion solar masses stays where it is or leaves at a hundredth of the speed of light.

Still open: the black holes that ought to be resolvable and are not #

Every argument about which galaxies host black holes depends on measuring black holes in galaxies, and the measurement has a selection of its own. A hole’s mass is measured dynamically only where its sphere of influence — the region where its gravity dominates the stars’ — can be resolved, and that sphere is larger for heavier holes at fixed galaxy properties. The sample of measured holes is therefore weighted towards the heavy ones. How strongly that selection has shaped the relations every merger argument relies on, and how many of the holes in small galaxies are simply below the resolution rather than absent, is the question the sphere of influence itself poses.