Astronomy & Celestial Mechanics
the sky, computed
Concept

Tidal disruption — where it appears

The destruction of a star that passes close enough to a black hole for the hole's tidal field to exceed the star's own gravity. About half the debris escapes and half returns, powering a flare lasting months — unless the hole is heavy enough to swallow the star whole.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

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.

galaxies · Tidal disruption 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.

galaxies · Tidal disruption 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.

galaxies · Tidal disruption How much of a population is driven how close. The fraction of an isotropic population of inner orbits whose Kozai–Lidov cycling brings 1 − e below a given value within 150 secular timescales, against that value, both logarithmic, for octupole strengths ε = 0, 0.01, 0.03; each curve is 128 integrated orbits with random pericentres and nodes. The thin line is the quadrupole closed form, √((3/5)(1 − (1 − q)²)), which the ε = 0 population follows. At 1 − e = 0.01 the fractions are 9.4, 20.3, 40.6 per cent; at 0.001, 3.1, 18.0, 38.3. The octupole matters most where the quadrupole is weakest — at the extreme closeness a planet needs to be captured by tides, or a pair of compact objects needs to merge — because it adds orbits whose quadrupole cycle stopped far short of it.

Eleven in a hundred by geometry, and the rest by waiting

A distant companion drives an inner orbit through cycles of eccentricity, and whether a given system reaches a dangerously close approach depends on its inclination. Counted over a population with random orientations, the answer at quadrupole order is pure geometry — about eleven per cent reach one per cent of their semi-major axis, all at the same inclination. An eccentric outer orbit multiplies that share, spreads the inclinations, and makes the count depend on how long the systems have been waiting.

orbits · Kozai–Lidov An excess the hole cannot afford for long. The mass a 10⁶-solar-mass hole gains from disrupted stars, in units of its own mass, against how long an excess in the disruption rate lasts, for excesses of 30, 100, 300 over the relaxed nucleus's 7.4·10⁻⁵ a year, with half of each Sun-like star accreted. An excess of 100 doubles the hole in 2.7·10⁸ years, an excess of 300 in 9·10⁷. The post-starburst phase that selects the hosts lasts about a billion years, so an excess of a hundred sustained through it would have grown every such hole several times over, and the galaxies that were post-starbursts a few billion years ago would carry holes several times heavier than their stars' motions predict. A binary's spike of 0.1 a year for 3·10⁵ years feeds it only 1.5%.

A flare rate a hole cannot afford for long

Tidal disruption flares are found in galaxies that stopped forming stars within the last billion years far more often than those galaxies' numbers allow — by factors estimated between thirty and two hundred. Each proposed cause can be priced. A dense cluster from the burst would need ten times the hole's mass of new stars inside a parsec; a triaxial nucleus would give twenty times too much; and any excess of a hundred, kept for the length of the phase, would feed a million-solar-mass hole several times its own mass.

galaxies · Tidal disruption Rise, steady state and decline. The mass of magnesium (solid) and iron (dashed) in a white dwarf's mixing layer, each as a fraction of its steady-state value for magnesium, during an accretion episode that delivers both at equal rates for 5 magnesium sinking times and then stops; time in units of the magnesium sinking time. Iron sinks 1.6 times faster. While they rise, both grow together and their ratio is the accreted ratio; once accretion has run for a few sinking times each element saturates at its rate times its own sinking time, so the layer holds 0.63 times the accreted iron-to-magnesium ratio; after accretion stops iron vanishes first, and 7 sinking times later the layer holds 0.009 of the accreted ratio. The same body seen at three moments reads as three compositions.

A planet read in the ashes of its star

A white dwarf's gravity drags every heavy element out of its visible surface, so a white dwarf showing magnesium, iron and silicon is swallowing rock now. Its spectrum is the only place where the bulk composition of a rocky body round another star can be read element by element. But each element sinks at its own rate, so the ratio seen depends on when in the meal the spectrum was taken — and only the hottest hydrogen white dwarfs say.

exoplanets · Planet composition How long a wide pair survives a halo of heavy objects. The time for a binary of one solar mass to be dissolved by passing compact objects, against its separation, both logarithmic, if the whole local dark-matter halo — 0.01 M☉ per cubic parsec, moving past at 300 km/s — is made of objects of 10, 100, 1000 M☉. A pair 0.1 pc apart (about 20,000 AU) lasts 477, 48 and 4.8 billion years respectively; at 1 pc, 47.7, 4.8 and 0.48. The heavier the objects at fixed total density, the fewer and harder the kicks, and the sooner the widest pairs go. The dashed line is 10 billion years, the age of the halo's oldest binaries.

Pairs of stars too loosely bound to survive heavy dark matter

Two stars a tenth of a parsec apart are held together by a force so weak that a passing mass of a few hundred suns, at a distance of a parsec, can loosen them. If the dark matter round the Milky Way were made of heavy compact objects, such pairs would have been unbound long ago. Some have survived ten billion years in the halo. The widest of them sets a ceiling on how heavy the dark matter's pieces can be — and the ceiling has moved twice, both times because of one pair.

gravitation · Impulse approximation

Named alongside it

The objects these essays reach for when they reach for this one.

DiffusionGravitational radiusLoss coneSupermassive black holeTwo-body relaxationWhite dwarfAccretion discApsidal precessionBahcall wolf cuspBinding energyBlack hole growthBlack hole spin

All concepts