Astronomy & Celestial Mechanics
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Tidal disruption — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Two lengths that cross at 1.1·10⁸ solar masses, above which nothing is seen. The radius at which a star of 1 solar radius and 1 solar mass is pulled apart by a black hole, and the hole's own horizon, both against the hole's mass and both on logarithmic axes. The tidal radius is the star's own radius times the cube root of the mass ratio, so it climbs with a slope of one third; the horizon is proportional to the mass, so it climbs with a slope of one. Two lines of different slope cross once, and this pair crosses at 1.14·10⁸ solar masses. Below that the star is torn apart outside the horizon, half of it is thrown out and half falls back, and the fallback is visible for months. Above it the star crosses the horizon while it is still a star, is swallowed whole, and produces no flare at all. The consequence is the reason these events are worth watching: a flare that is seen is an upper limit on the mass of the hole that made it, obtained without resolving anything, and it is the only such limit available for a hole that is not currently accreting.

    A flare that puts a ceiling on a mass

    A star torn apart by a black hole lights up for a year. The tidal radius grows as the cube root of the hole's mass and the horizon grows as the mass itself, so above about a hundred million suns the star is swallowed whole and nothing is seen — which makes the existence of a flare a measurement.

    part 1 · galaxies
  2. The horizon and the orbit that cannot come back, against the hole's spin. Two radii round a rotating black hole, in gravitational radii GM/c², against its spin a from −1 (an orbit against the rotation) to +1 (with it), for orbits in the equatorial plane. The lower curve is the horizon, 1 + √(1 − a²), which is the same whichever way the orbit goes. The upper one is the marginally bound orbit, 2 − a + 2√(1 − a): the closest a body falling in from far away can pass and still escape back out. For a hole with no spin the horizon is at 2 and the marginally bound orbit at 4 — twice as far out. With maximal spin the marginally bound orbit comes in to 1.09 for a prograde orbit (a = 0.998) and moves out to 5.83 for a retrograde one. A star whose tidal radius lies inside this curve is swallowed whole, so the curve, not the horizon, is the line a disruption flare is measured against — and it moves by a factor of 5.3 with the spin.

    The line a star is swallowed at is not the horizon

    A star torn apart by a black hole makes a flare, and a star swallowed whole makes nothing, so the heaviest hole that can produce a flare is a measurement. That ceiling is set not at the horizon but at the closest orbit from which infalling matter can still come back out — twice as far out for a hole that does not spin, and moved by a factor of five by the hole's rotation.

    part 2 · galaxies
  3. How far the returning debris swings round, against the hole's mass. The relativistic advance of pericentre, per orbit, of the most bound debris from a disrupted star of 1 solar radius and 1 solar mass, passing at the tidal radius, against the hole's mass. Δω = 6π GM / c² a(1 − e²), and for these nearly parabolic orbits a(1 − e²) is about twice the pericentre, so the angle is set by how many gravitational radii the pericentre is — and that falls as the mass to the minus two thirds, because the tidal radius grows as the cube root of the mass while the gravitational radius grows as the mass. At 10⁵ solar masses the pericentre is 219 gravitational radii and the stream swings round by 2.5°; at 10⁶ solar masses the pericentre is 47 gravitational radii and the stream swings round by 11.6°; at 10⁷ solar masses the pericentre is 10 gravitational radii and the stream swings round by 53.4°. A light hole barely bends its debris's orbit, and a heavy one bends it by tens of degrees.

    The debris that returns fastest lights up last

    The debris of a star torn apart by a light black hole comes back within two weeks, in greater excess of what the hole can swallow than for any heavier hole. It should make the promptest flare, and it may make the slowest — because to shine, the returning stream has to crash into itself, and where it does so is decided by how far relativity swings its orbit round. Round a light hole the swing is a few degrees, and the streams meet only near the far end of their orbit, moving slowly.

    part 3 · galaxies
  4. Where the stars that are torn apart come from, round a 10⁶ solar-mass hole. The rate at which stars are delivered onto orbits reaching the tidal radius, per logarithmic interval of their orbital radius, for a hole of 10⁶ solar masses at the centre of a nucleus that is isothermal outside the hole's influence and a Bahcall–Wolf cusp within it, with a velocity dispersion of 54 km/s set by the black hole mass–dispersion relation. The radius is in units of the hole's influence radius, GM/σ² = 1.47 pc. Close in, a star's orbit is so short and its angular momentum changes so slowly that the loss cone empties every orbit, and the rate is limited by relaxation diffusing stars into it; far out, the angular momentum wanders across the whole cone in one orbit, the cone is full, and the rate is limited by how many stars there are and how long they take to come in. The flux peaks at 0.27 influence radii, near where the two regimes meet (0.19), and the total is 7.4·10⁻⁵ disruptions a year. The nucleus model is the simplest there is and real nuclei differ from it by factors of several; the shape — a narrow band of radii supplying most of the flares — is what survives.

    The stars a black hole eats come from a narrow band

    A star is torn apart only if its orbit happens to point almost exactly at the hole — within a millionth of the possible directions of its angular momentum. Stars on those orbits are gone within one orbit, and the only way new ones arrive is by the slow random walk of encounters with other stars. Close to the hole that walk is too slow; far from it the orbits take too long; and most of the stars a hole destroys come from a narrow band between.

    part 4 · galaxies

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