Astronomy & Celestial Mechanics
the sky, computed
Concept

Supermassive black hole — where it appears

A black hole of a million to ten billion solar masses at the centre of a galaxy. Most are not accreting and are detected only through the motions of nearby stars or through the flare produced when a star wanders close enough to be pulled apart.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

S2's orbit, and the mass it implies. The orbit of S2 about the centre of the Milky Way, drawn from its measured elements: a semi-major axis of 0.1251 arcseconds, which at 8.28 kpc is 1036 AU, an eccentricity of 0.8843, and a period of 16.05 years watched round twice. Kepler's third law in solar units — a³/P² — gives 4.31 million solar masses, by exactly the calculation that weighs a planetary system. At periapsis the star is 120 AU from the focus, 1407 Schwarzschild radii, and the mean density inside that radius is 5.3e+15 M☉ per cubic parsec — a million times the densest star cluster known. That density, not the mass, is the argument: there is no configuration of stars that would fit.

The mass at the centre that is not stars

One star at the centre of the Milky Way has been watched round a complete orbit. Its semi-major axis and its period give four million solar masses by Kepler's third law — and the volume that mass occupies is small enough to rule out every alternative to a black hole.

galaxies · Galactic nuclei 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.

galaxies · Tidal disruption 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 often a merged black hole is kicked faster than a given speed. The fraction of black-hole mergers whose remnant recoils faster than the speed on the horizontal axis, for mass ratios spread evenly in logarithm between 0.1 and 1 and spins of 0.9 of the maximum, spins within 10° of the orbit; spins within 30°; spins in random directions, from a fitting formula to numerical relativity over 4000 random mergers each. The median kicks are 113, 234, 534 km/s. Faster than 100 km/s: 62%, 85%, 96%. Faster than 300 km/s: 5%, 40%, 74%. Faster than 1000 km/s: 0%, 2%, 25%. Gas accreted before the merger is expected to drag both spins into line with the orbit, which is why a gas-rich merger keeps its black hole far more often than a gas-poor one.

A black hole thrown out by its own merger

When two black holes merge, the gravitational waves they radiate carry momentum as well as energy, and unless the two are identical and spinning in line the waves leave more in one direction than the other. The remnant recoils, at anything from a few tens of kilometres a second to several thousand — and whether a galaxy keeps its central black hole comes down to how its spins were pointing in the last orbit before the merger.

galaxies · Galactic nuclei The black holes that can be weighed are the heavy ones. 4000 synthetic galaxies spread evenly through a sphere of 120 megaparsecs, with black-hole masses scattered by 0.45 dex about an M–σ relation of slope 4.24 (faint points), and the 511 whose spheres of influence exceed 0.1 arcseconds (filled), which are the ones a dynamical measurement can resolve. At a given dispersion a heavier black hole has a larger sphere and is resolvable further away, in a larger volume, so the resolved sample is dominated by holes above the relation. Fitted to the resolved galaxies alone, the relation comes out with normalisation 8.74 and slope 2.95 (dashed), against 8.11 and 4.20 for all of them (solid) — 0.63 dex too high at 200 km/s, and more at low dispersions.

The black holes that can be weighed are the heavy ones

A black hole's mass is measured from the motions of stars or gas only inside its sphere of influence, the region where its gravity beats the galaxy's, and that region has to be resolved. At a given galaxy, a heavier hole rules a larger sphere and can be weighed further away, in a much larger volume — so the sample of weighed holes is drawn from the top of the distribution, and the relations built from it describe the heavy end of every galaxy rather than the typical one.

galaxies · Galactic nuclei A cusp steepened into a spike by the black hole growing in it. The dark-matter density near the Galaxy's central black hole of 4.3·10⁶ solar masses, in solar masses per cubic parsec on logarithmic axes. The dashed line is the halo's NFW cusp, falling inward as one over the radius. If the black hole grew slowly at the centre of that cusp, the dark matter would have followed it inward on adiabatically shrinking orbits, steepening the cusp inside about a fifth of the black hole's sphere of influence — here 0.4 parsecs, with the sphere of influence at 1.8 — into a spike falling as r to the minus 7/3, measured off the drawn curve. Close in, the dark matter's own annihilation would cap the density at the particle mass over the annihilation rate times the Galaxy's age — for a 100 GeV particle with the thermal cross-section, 2.8·10⁸ solar masses per cubic parsec — the flat plateau. Between the spike's edge and its plateau the density rises by 6 orders of magnitude.

A cusp a black hole would sharpen

A black hole that grows slowly at the centre of a dark-matter cusp drags the dark matter in after it, turning a density that rises as one over the radius into a spike rising as the radius to the minus seven-thirds. Whatever the cusp was, the spike comes out nearly the same — and the annihilation signal it would produce is set less by the particle than by whether the spike survived the stars around it.

galaxies · Halo profiles Where relativity overtakes the margin: 306 Schwarzschild radii for 10⁵ solar masses. Two things against the size of a contracting star, measured in its own Schwarzschild radius, on logarithmic axes. The flat lines are the margin by which the star's pressure-weighted adiabatic exponent exceeds 4/3 — the Newtonian condition for stability — for stars of 1000, 10⁴, 10⁵, 10⁶, 10⁷, 10⁸ solar masses with mean molecular weight 0.61; it does not change as the star shrinks, because it is fixed by the gas fraction of the pressure, and it falls with mass from 0.036 to 1.2·10⁻⁴. The diagonal is what general relativity adds to the requirement, 1.1245 × 2GM/(Rc²), which in these units is the same line for every mass. Contraction moves each star leftward along its flat line, and where the line meets the diagonal the star becomes dynamically unstable: at 31, 97, 306, 966, 3,052, 9,649 Schwarzschild radii respectively. Every one of those crossings lies between 31 and 9,649 Schwarzschild radii out, where the star is not remotely compact — the relativistic correction is tiny, and it decides the matter only because the margin it is compared with is tinier.

A star tipped over at a thousand times its horizon

A star of a million suns is held up almost entirely by its own light, which leaves it stable by about a thousandth. General relativity raises the bar by an amount that is tiny until the star is compact, and a thousandth is tiny too — so the star collapses at nearly a thousand times the radius of the black hole it becomes, at a mean density below that of air, and with no fuel yet burning.

stars · Polytropes 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

Named alongside it

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

M sigma relationSchwarzschild radiusSphere of influenceVelocity dispersionAccretionBlack hole spinEddington limitEvent horizonTidal disruptionTidal radiusAccretion discActive galactic nucleus

All concepts