A flare rate a hole cannot afford for long

A star that wanders too close to a massive black hole is torn apart, and about half of its gas falls back onto the hole over the following months in a flare as bright as a supernova. Some hundred of these flares have now been found in optical surveys, and their host galaxies are an odd sample. A disproportionate share are post-starburst galaxies: systems whose spectra show the strong hydrogen absorption of stars a few hundred million years old and no sign of present star formation — galaxies that formed stars vigorously and then stopped within the last billion years.

The overrepresentation is large. The strongest post-starbursts are about two in a thousand galaxies and have hosted something like a third of the well-studied flares; broader definitions of the class, which take in galaxies with weaker absorption, find excesses of thirty or so. The essay on the narrow band of stars a hole eats worked out how often a relaxed nucleus feeds its hole — about one star in ten thousand years round a hole of a million solar masses — and ended with the observation that the rate belongs to the nucleus rather than the hole, so a recently disturbed nucleus could flare far more often. It did not say how much more, or what it would cost.

This essay prices the explanations. Each proposed cause of the excess is a change to the nucleus that the same loss-cone calculation can follow, and each comes with a bill — in stars that must have formed, in rates that must have been sustained, and in mass the hole must have swallowed. The bills turn out to rule out more than the rates do.

A share that implies an excess #

The measurement is a share: how many of the flares found sit in galaxies of a particular class. Turning a share into an excess per galaxy is simple arithmetic, and the arithmetic has a property worth seeing before anything else.

A class holding a fraction f of galaxies, each flaring E times as often as an ordinary galaxy, hosts a share $fE/(1 + f(E - 1))$ of the flares. For the rarest class, a third of the flares needs an excess of about two hundred and fifty; for a class ten times commoner, about twenty. The curves flatten as the share rises, so a share measured from a few dozen flares — with its uncertainty of a factor of two — bounds the excess from below much better than from above. The estimates in the literature, from thirty to two hundred, are this figure read with different definitions of the class.

There is a selection effect inside the share too. A flare is easier to find and to classify in a galaxy with little dust and no active nucleus, and post-starbursts are both. Some of the excess may be detection rather than disruption. But even generous allowances for that leave a factor of ten or more to explain, and the rest of this essay asks what could produce excesses of that size.

Buying flares with density #

In a relaxed nucleus round a hole of a million solar masses, most disruptions come from stars within about a tenth of the hole’s sphere of influence, where the loss cone — the narrow set of orbits that reach the tidal radius — is emptied faster than encounters between stars refill it. The rate there is the number of stars divided by the time it takes relaxation to change their angular momenta, and both depend on the density. Doubling the density doubles the stars and halves the relaxation time.

So the rate rises almost as the square of the density, and a modest overdensity buys a large excess: seven times the density gives thirty times the flares, fifteen times gives a hundred. That is the appeal of the explanation. A starburst concentrated at the centre of a galaxy leaves a dense cluster of new stars there, and observations of post-starburst nuclei with the Hubble Space Telescope have found some with stellar densities rising more steeply towards the centre than ordinary galaxies show.

The price is in the stars. A relaxed nucleus holds about 0.6 hole masses of stars inside half the influence radius — about three quarters of a parsec for a million-solar-mass hole. Raising that density fifteen times means adding about nine hole masses of new stars inside three quarters of a parsec: a nuclear star cluster ten million solar masses heavy, packed into a volume the size of the Sun’s nearest neighbours’ separation. Nuclear star clusters of that mass exist, but they are typically a few parsecs across, and building one ten times more concentrated in a single burst asks for a great deal of gas to have reached the centre and formed stars there without forming them anywhere else first.

Where the extra flares come from #

Where in the nucleus the extra flares originate depends on how the nucleus was changed.

A burst cluster adds its flares where the cluster is, inside the sphere of influence, and they come from orbits of the same kind as before — stars diffusing slowly into an empty cone. The distribution of their orbital energies is little changed, and so are the flares’ properties: their timescales and peak luminosities, which depend on how deeply bound the debris is, should look like those of flares in ordinary galaxies.

A loss cone kept full is different. Stars on orbits far outside the sphere of influence, which in a relaxed nucleus hardly ever reach the cone because relaxation there is slow, now arrive at the rate their orbits carry them in. The flux from those radii is enormous, and the flares they produce come from stars on more loosely bound orbits — which affects how quickly the debris returns and how the flare rises — since the debris that returns fastest lights up last, a star arriving on a looser orbit changes the order in which the stream comes back. In principle the two explanations could be separated by the flares themselves. In practice the dependence of a flare’s shape on the star’s orbit is weak compared with its dependence on the star’s own mass and structure, and nobody has yet made the separation.

A full cone is worth too much #

The second explanation is that a recent merger has left the nucleus triaxial — elongated along three different axes rather than round — and that the torques of a non-spherical potential change stars’ angular momenta far faster than encounters between individual stars do. Many orbits in a triaxial potential are chaotic and wander through a wide range of angular momenta in a few orbital periods, passing repeatedly through the loss cone.

The upper limit is a cone kept completely full at every radius, and it is worth a factor of two thousand round a million-solar-mass hole. That is not an explanation of an excess of a hundred; it is twenty times too much. A triaxial nucleus that refilled the cone at a twentieth of the full rate — because only some of its orbits are chaotic, or because the chaotic ones wander slowly — would give the right excess, and simulations of non-spherical nuclei find refilling well below the full value, in the general range required. So the triaxial explanation needs no extreme assumption about the nucleus. Its difficulty is the opposite of the burst cluster’s: it can easily give too much, and what it predicts depends on a refilling fraction that is hard to compute and harder to measure.

The heavier holes gain more from a full cone, because relaxation is slowest in their nuclei and an empty cone starves them most. A merger remnant hosting a hundred-million-solar-mass hole could see its rate rise by tens of thousands — but for holes that heavy a Sun-like star falls through the horizon before it is torn apart, and the extra stars are swallowed without a flare.

The bill, in hole masses #

Every star disrupted feeds the hole. Roughly half of each star’s mass stays bound and falls back, and whatever fraction of that is accreted rather than blown away adds to the hole. In a relaxed nucleus, a disruption every ten thousand years delivers a few tenths of a solar mass each time, and over the age of the universe that is comparable with the hole’s own mass — which is one reason tidal disruptions are counted among the ways light holes grow.

An excess multiplies the bill.

An excess of a hundred doubles the hole in 270 million years. An excess of three hundred does it in ninety. The post-starburst phase — the interval in which a galaxy’s spectrum still shows the A stars that define the class — lasts about a billion years, and an excess of a hundred sustained throughout it would feed the hole almost four times its own mass.

That is the surprising connection the flares make. Their statistics, which were collected to understand how stars are delivered to black holes, turn out to constrain how black holes grow. If the excess in post-starbursts were large and long-lived, every galaxy that passed through a starburst would have its central hole multiplied several times by eating stars, and the galaxies that were post-starbursts a few billion years ago — now ordinary early-type galaxies — would carry holes several times heavier than the relation between a hole’s mass and its galaxy’s stellar motions predicts. A starburst is common enough in a galaxy’s history that such growth would not be a curiosity; it would be a channel.

The arithmetic does not forbid the excess. It forbids the excess from being both large and long. Either it lasts a few hundred million years and not the whole phase, or it is closer to thirty than to two hundred, or the accreted fraction is much smaller than a half — most of the debris blown away in an outflow rather than swallowed.

A cluster that outlasts its own excess #

The budget is hardest on the burst cluster, because a dense cluster does not go away quickly.

A cluster dense enough to raise the rate a hundredfold has its own relaxation time of about two and a half billion years at the radius that matters — the time over which encounters among its stars let a cluster boil itself away — and nothing faster acts on it unless the cluster is disturbed from outside. It spreads only on that timescale, so the excess it causes fades only on that timescale — halving after more than a billion years. It would cross the line where the hole has doubled within a few hundred million years and keep flaring above it for billions more, long after the galaxy’s spectrum had stopped looking like a post-starburst.

That makes a prediction the burst-cluster explanation cannot avoid. If dense central clusters are the cause, the excess should not be confined to post-starbursts; it should continue into the older early-type galaxies that were post-starbursts one or two billion years before, at a level only a few times lower. The surveys find the excess strongest in the strongest post-starbursts and weaker in the broader class whose hydrogen absorption has begun to fade — a decline with the strength of the absorption, which is itself a clock of the burst’s age, rather than a plateau lasting billions of years. That decline is evidence against a long-lived cause of any kind.

The third explanation, a pair of black holes left by the merger, has the opposite problem. A hole binary at the moment it becomes tightly bound can scatter stars into its loss cone at around a tenth of a star a year — a thousand times the relaxed rate — but only for a few hundred thousand years. Averaged over a billion-year phase, that is an excess of forty per cent. A binary makes a brief spectacle and contributes almost nothing to a population’s statistics, unless the binaries in post-starbursts become bound at a strongly preferred time and the surveys happen to catch them then. The binary’s own history makes that unlikely: a pair of holes stalls at the last parsec for a time that depends on how quickly the nucleus resupplies it with stars, which varies by orders of magnitude between nuclei, and the slower Kozai-driven delivery of stars by a wide binary raises the rate by less and for longer.

A clock already on the spectrum #

The class that hosts the excess is defined by a clock. A post-starburst spectrum is dominated by A stars, which live a few hundred million years to a billion; the strength of their hydrogen absorption rises for a few hundred million years after star formation stops, as the hotter O and B stars die and stop diluting it, and then falls as the A stars die in turn. The time a galaxy spends crossing between the blue and red populations is read from the same stars, and a galaxy’s age since its burst can be fitted from its spectrum to tens of per cent.

That makes the post-starburst selection unusual among the host classes of any transient. It is not a type of galaxy; it is a stage, with a start and an end fixed by stellar evolution rather than by the galaxy’s structure. An excess that lives in that stage is an excess with a measurable time dependence, and the loss-cone arithmetic gives each explanation a characteristic shape in time: flat for billions of years for a dense cluster, a few hundred million years for a relaxing triaxial shape, a few hundred thousand for a binary. The spectrum supplies the time axis for free.

What survives the bill #

Put together, the pricing narrows the possibilities. A cause that persists for billions of years is excluded by the concentration of the excess in the post-starburst phase and by the mass it would feed the holes. A cause that lasts a few hundred thousand years is excluded by the small fraction of galaxies it could affect at once. What is left is a cause with a lifetime of a few hundred million years and a strength of tens: a triaxial remnant that refills the loss cone at a few per cent of the full rate and settles back towards symmetry within a fraction of a billion years, or a modest central concentration of new stars that is disrupted or dispersed faster than relaxation alone would do.

That lifetime is itself a test. The time since the burst can be read from a post-starburst galaxy’s spectrum, from the balance of its young and old starlight, to a precision of tens of per cent. If the flares’ hosts are drawn from the young end of the post-starburst population — those whose bursts ended within the last two or three hundred million years — the cause is short-lived, as the budget requires. If they are spread evenly across the phase, the excess is weaker than the share suggests, and part of it is selection. The host ages measured so far are too few, and the definitions of the class too unsettled, to say yet which end of the phase they favour.

What the model leaves out #

One nucleus model stands for all. The relaxed nucleus is an isothermal sphere joined to a Bahcall–Wolf cusp, with the velocity dispersion set by the hole’s mass. Real nuclei vary by factors of several in density at fixed hole mass, and the base rate the excess is measured against varies with them.

The burst cluster has a sharp edge. The overdensity is drawn as a step at half the influence radius. A real cluster has a smooth profile, which spreads its extra flares over a wider range of radii without changing their total much.

The stars are all Sun-like. A post-starburst nucleus has an excess of A stars, which are larger and more massive than the Sun. They are disrupted further from the hole and their debris is more massive, which changes both the rate and the brightness of the flares, and possibly the classification: a brighter flare is easier to find.

The heaviest hosts swallow rather than flare. Above about a hundred million solar masses for a Sun-like star, the line a star is swallowed at lies outside the one it is torn apart at, and an excess in such a nucleus feeds the hole without producing flares. The bill then grows without any sign in the statistics, and the constraint drawn from the flares applies only to the lighter holes.

The accreted fraction is uncertain. The bill assumes half of each star is swallowed. Simulations of the debris’s return show much of it may be expelled in outflows, which would lower the bill by a factor of a few and relax every constraint drawn from it.

Still open: how long the excess lasts #

The overrepresentation of post-starburst galaxies among the hosts of tidal disruptions is too large to be a coincidence and too large, if it were sustained, for the holes to afford. Its explanation has to be strong for a short time, and the candidates differ chiefly in how short.

The surveys now running will find flares by the thousand rather than the hundred, with hosts characterised spectroscopically. Measuring the distribution of time since the burst among those hosts will say how long the excess lasts, and the lifetime will say which kind of nucleus produces it: a shape relaxing within a few hundred million years, a cluster dispersing within that time, or something in the selection of the flares themselves. The answer bears on more than the flares. It decides whether starbursts double the black holes at the centres of the galaxies they happen in, or merely make them conspicuous for a while.