The debris that returns fastest lights up last

When a black hole tears a star apart, half the star is flung out and half stays bound, spread evenly in orbital energy, and the bound half returns at a rate that falls as time to the minus five thirds. The first material to come back is the most tightly bound, and the time it takes is the one clock in the problem: 41 days for a Sun-like star round a hole of a million solar masses, shorter for a lighter hole and longer for a heavier one. It is natural to read that clock as the time a flare takes to rise.

It is not, or not reliably, and the reason is that returning is not the same as arriving. The debris comes back on orbits so elongated that almost none of its energy has been released. To shine, it has to fall much deeper, and to fall deeper it has to lose most of its orbital energy somewhere. The place that happens is where the stream of debris crashes into itself — and the returning stream only crosses the stream still falling in because general relativity swings its orbit round at each pass. How far it swings is set by the hole’s mass, and it is smallest for exactly the holes whose clocks run fastest.

Returning is not arriving #

Consider the most bound debris round a hole of a million solar masses. Its pericentre is at the tidal radius, about a hundred solar radii, and its semi-major axis is fifty times larger, so its orbit has an eccentricity of 0.98 and swings out to about a hundred pericentre distances before it comes back. It is a comet’s orbit, not a disc’s.

A disc radiates by letting gas spiral inward through a sequence of nearly circular orbits, releasing energy as it goes, and the efficiency of the whole process is fixed by how deep the disc’s inner edge sits. For debris to join a disc near the pericentre it has to be on a circular orbit there — at twice the pericentre, if it keeps its angular momentum — and the binding energy of that circular orbit is some twenty-five times the binding energy the debris already has. Before the debris can release the energy that makes a flare, it has to get rid of twenty-five times the orbital energy it came back with. Only collisions can do that. A stream of gas moving on a ballistic orbit does not lose energy by itself, and it passes through pericentre, swings out, and returns, on the same ellipse, indefinitely.

The rate at which it returns makes the problem more acute rather than less. Round a hole of a million solar masses the peak return rate is about three solar masses a year, more than a hundred times what the hole could accrete if it were limited by the Eddington luminosity at which radiation pressure balances gravity. The material is arriving far faster than it can be swallowed, and what happens to it is decided by where it first meets itself.

Why an ellipse that turns crosses itself #

In Newtonian gravity the returning debris would retrace its outgoing path exactly. Each piece of the stream follows the same ellipse, with the same orientation, and the returning stream lies precisely on top of the stream still falling in; there is nothing for it to hit.

General relativity adds the same slow turning of the ellipse that accounts for the last 43 arcseconds a century of Mercury’s orbit. Each pass through pericentre advances the orbit’s orientation by $\Delta\omega = 6\pi GM/[c^2 a(1-e^2)]$, and for an orbit as eccentric as the debris’s, $a(1-e^2)$ is very nearly twice the pericentre, so the advance is $3\pi$ times the ratio of the gravitational radius to the pericentre. Mercury’s pericentre is thirty million gravitational radii of the Sun, and it advances by a tenth of an arcsecond an orbit. A star disrupted by a hole of a million solar masses passes at 47 gravitational radii, and its debris advances by 11.6 degrees.

The ratio that sets the angle falls with the hole’s mass as $M^{-2/3}$, because the tidal radius grows as the cube root of the mass and the gravitational radius as the mass itself, so the precession grows as $M^{2/3}$: a hundred times heavier, twenty-one times more swing. After its first pericentre passage the returning stream is on an ellipse rotated by $\Delta\omega$ from the one it arrived on, and two ellipses sharing a focus and rotated relative to each other cross.

Where the crossing is, and what it does #

The crossing point follows from the geometry of two rotated ellipses, and the figure shows it against the debris’s own apocentre, the furthest the stream ever goes.

Round a hole of 10⁵ solar masses the streams cross at 45 pericentre distances, and the apocentre is also 45: the two streams meet at the very far end of their orbit, where each is moving at a small fraction of its pericentre speed and the angle between them is only a few degrees. A collision there dissipates very little of the energy that has to be lost. Round a hole of 10⁶ solar masses the crossing moves in, to 79 of the 99 pericentre distances the orbit spans. Round a hole of 10⁷ solar masses it is at 17 of 214, a twelfth of the way out, where the streams are fast and meet at a large angle, and a single collision can remove a large share of the orbital energy at once.

The shape of that dependence carries the argument. Round a light hole the streams barely cross, and they cross where crossing does least. Material that is not stopped at the first crossing goes round again, precesses again, and meets the next arriving stream a little further in, so circularisation proceeds over many returns rather than one; the energy is released gradually, and much of it far from the hole, where it emits at lower temperatures. Round a heavy hole the first crossing is violent and close in, and the gas can settle into a compact hot disc within roughly one return.

The fastest clock and the weakest collision #

That puts the two timescales of a tidal disruption in opposition.

The return clock runs as the square root of the hole’s mass: 13 days for 10⁵ solar masses, 41 days for 10⁶, 130 days for 10⁷. The rate at the peak falls as the mass rises, from 9.4 solar masses a year for the lightest of the three to three and then to under one. Measured against each hole’s own Eddington rate the contrast is extreme — the lightest hole receives about four thousand times what it can accrete, the heaviest about four times.

So a light hole is fed first, fastest and in the greatest excess, and it is the hole around which that material has the least means of becoming a disc. A heavy hole is fed later and more gently, and its debris can circularise almost at once. If a flare’s rise is timed by the formation of a disc rather than by the first return, the flares of light holes should rise later than their fallback clocks suggest, and those of heavy holes close to on time.

Put beside the line at which a star is swallowed rather than torn apart, that leaves a band. Below about a million solar masses the swing is too small for a prompt collision; above about four times ten to the seven, a Sun-like star round a hole that does not spin is not disrupted at all. The flares that can be expected to light up close to their fallback clock come from the holes in between — a factor of forty in mass — and outside it, on the light side, a flare is expected but not promptly, and on the heavy side it is not expected.

A finer view of the turnover #

The crossing point in units of the pericentre is not monotonic in mass, and the reason is two competing trends.

As the hole gets heavier, the debris’s orbit becomes more elongated relative to its pericentre, because the tidal radius grows more slowly than the spread in orbital energy it produces — the apocentre climbs from 66 to 310 pericentre distances across these three masses. That alone would push the crossing outward. At the same time the precession angle grows, which pulls the crossing inward along each ellipse. At the lightest masses the orbit’s growth wins and the crossing sits at the apocentre; somewhere near a million solar masses the swing takes over; and by 3·10⁷ solar masses the streams cross at five pericentre distances, almost as soon as the returning stream has turned the corner.

The share of the orbit at which the crossing falls tells the story more directly than the distance does. It is 97 per cent of the way to apocentre at 3·10⁵ solar masses, 39 per cent at 3·10⁶, and under 2 per cent at 3·10⁷. Across two orders of magnitude in the hole’s mass, the collision moves from the slowest point on the orbit to nearly the fastest.

A denser star, or a closer pass, swings further #

The hole’s mass is not the only thing that sets the pericentre in gravitational radii. The star and the depth of the encounter set it too.

A small dense star has a smaller tidal radius, so it has to come closer to be torn apart, and its debris precesses more: 18.3 degrees rather than 11.6 at a million solar masses. The commonest disrupted stars are exactly these low-mass dwarfs, so the typical flare round a light hole is a little more favourable to prompt circularisation than the Sun-like case suggests — though a factor of 1.6 in angle does not move a 10⁵ solar-mass hole out of the regime in which the streams meet near apocentre.

A star can also be sent in deeper than the tidal radius. The penetration factor β is the ratio of the tidal radius to the actual pericentre, and a star arriving with β = 2 passes at half the distance at which it would first be disrupted.

The debris’s spread in energy is set where the star is torn apart and does not depend much on how deep it then goes, so a deeper pass leaves the orbits’ size nearly unchanged while halving the pericentre. The apocentre in pericentre units doubles, the precession angle doubles, and the two effects move the crossing in opposite directions just as mass did. The balance is unchanged at the light end — the streams still meet near apocentre at 10⁵ solar masses — and at 10⁷ solar masses the crossing is five pericentre distances out, where the approximation of a slowly turning ellipse no longer holds: a pericentre of five gravitational radii is deep in the strong field, and the true orbit there is not an ellipse at all but a rosette whose shape the effective potential decides.

What the geometry leaves out #

The streams have thickness. A stream of stellar debris is narrow but not a line, and two streams that cross in projection may pass above and below each other. Round a spinning hole the orbit’s plane itself precesses at each pass, by an effect of the hole’s rotation, and a stream pushed out of its original plane can miss the stream it would otherwise hit for several returns. That delays circularisation further, and it does so most for the holes whose spin — the second number a black hole has — is largest.

The collision is not the only dissipation. As the debris passes through pericentre it is squeezed vertically and shocked in a nozzle, which removes some energy even without a crossing, and the heating from the first collision can inflate the stream and make later collisions easier.

A stream is not a single orbit. The debris spans a range of energies, so the returning material is a whole family of ellipses with different sizes, not the orbit it came from. The most bound part defines the first crossing; later arrivals cross further out, where their orbits are larger.

And the formula is first order in the ratio of gravitational radius to pericentre. At tens of gravitational radii it is accurate to a few per cent; at ten it is indicative; at five it is a sketch.

How far out the light comes from #

The crossing distances in the figures are in units of the pericentre, and in ordinary units they are large. The tidal radius of a Sun-like star round a hole of a million solar masses is a hundred solar radii, about half an astronomical unit, so a crossing at 79 pericentre distances is some 37 astronomical units from the hole — further out than Neptune is from the Sun. Round 10⁵ solar masses the crossing is about 10 astronomical units out, and round 10⁷ about 17.

That scale has an observational counterpart. Most tidal disruption flares have been found at optical wavelengths, and their light is close to that of a blackbody at a temperature of a few tens of thousands of degrees. A luminosity and a temperature together fix the size of the surface that is radiating, and for these flares it comes out at tens of astronomical units: a surface radiating $10^{37}$ watts from a radius of 37 astronomical units has a temperature of about 26,000 degrees. A disc near the hole, a few gravitational radii across, would be far hotter and far smaller and would emit mainly X-rays. The optical flares are too large and too cool to be discs, and their size is about the size of the region in which these streams cross — one of the reasons stream collisions, rather than an accretion disc, are thought to power much of the optical light.

What a delayed flare would look like #

The prediction has observable consequences, and some have been seen. A flare in which circularisation is slow should first appear at optical and ultraviolet wavelengths, from the energy released in shocks far from the hole, with the hot X-ray-emitting inner disc forming only later, if at all. Several tidal disruption flares have shown just that sequence: optical emission peaking first, and X-rays that appeared or brightened months to a year afterwards. Not every flare does, and the mass of the hole is only one of the things that decides — the spin, the star, the depth of the pass and the direction it arrived from all enter — but the geometry here is the reason a delay of that kind is expected at all.

For the lightest holes it sharpens a problem. Holes of 10⁵ solar masses and less are the hardest to find and weigh, because the galaxies they live in are small and the stars that orbit them are too few and too faint to resolve. Tidal disruption flares are one of the few ways to see them at all, and they are the flares most likely to be slow, dim at first, and emitted far from the hole at wavelengths that are not those of a compact disc.

Still open: how many stars are sent in #

Everything here assumes a star has arrived. The rate at which a nucleus supplies stars on orbits that reach the tidal radius is set by the slow random walk of their angular momenta under encounters with other stars, and it decides both how often flares happen and which radii the disrupted stars come from — a question with an answer that depends on the hole’s mass in the opposite sense to the one that decides how promptly each flare lights up.