Astronomy & Celestial Mechanics
the sky, computed
Concept

Innermost stable circular orbit — where it appears

The smallest radius at which a circular orbit around a compact object is stable, six gravitational radii for a non-rotating hole. Inside it the effective potential has no minimum, so matter does not orbit — it falls.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

The effective potential, for three angular momenta. The radial motion of an orbiting body is one-dimensional motion in an effective potential: the attraction −GM/r plus the centrifugal term L²/2r² that the angular momentum contributes. The barrier at small radius is what stops a body with any angular momentum at all from reaching the centre, and the bottom of each well is the circular orbit.

The wall that angular momentum builds

A body falling toward a star almost never arrives. Sideways motion, which looks like a detail of the initial conditions, turns the attraction into a well with a wall around the middle of it — and the wall is why it costs more fuel to hit the Sun than to leave the solar system.

orbits · Effective potential From six gravitational radii to one. The radius of the innermost stable circular orbit against the dimensionless spin a = Jc/GM², in units of GM/c², for orbits prograde and retrograde with the hole's rotation. Both curves are the Bardeen–Press–Teukolsky expression and are checked at the three places it has exact values: 6 at zero spin, and 1 and 9 at the extremal limit. The separation is the observable consequence of frame dragging — space near the hole is itself circulating, so an orbit going the same way can stay closer before it becomes unstable, and one going the other way cannot come as close as a non-rotating hole allows. The prograde branch is required to fall and the retrograde branch to rise at every step drawn, which is a claim about the direction of the effect rather than about its size. The marked spin of 0.998 is not the extremal value but the equilibrium a hole fed by a thin disc actually reaches, because photons emitted by the disc are preferentially captured on retrograde orbits and spin the hole down again. Nothing here depends on what the hole is made of: two numbers fix the whole geometry, and this figure is the first of them holding still while the second moves.

The second number a black hole has

A black hole in equilibrium is described by its mass and its spin, and nothing else. The mass decides how strongly it pulls. The spin decides how much light a kilogram of infalling matter can emit before it disappears — and between the two extremes that figure changes by a factor of seven.

gravitation · Black hole spin The spin a mismeasured radius reads as. The spin inferred from the disc's inner radius, against the fractional error in that radius, for four true spins. Both spectral methods reduce to one measurement — where the disc stops — and one conversion, the ISCO relation, so this figure is the error propagation both of them share. The behaviour is not uniform and it is the reverse of what the difficulty of the measurement suggests. A ten per cent error reads a hole of true spin 0.3 as anywhere from 0.16 to 0.44, while the same ten per cent moves a hole of true spin 0.99 only between 0.98 and 1.00. The reason is that the ISCO falls from six gravitational radii to one over the whole range of spin and does most of that falling in the last few per cent, so near the extremal limit a small change in radius is a large change in spin and the inversion is stiff. The published spins clustering near 0.9 and above are therefore the ones least sensitive to the systematics, and a published spin of 0.3 carries an error bar the method cannot really support. An overestimated radius always reads slow, so anything that stops the disc outside the last stable orbit biases every measurement the same way.

A spin that is one length in disguise

Both ways of measuring a black hole's spin from its light measure the same thing — where the accretion disc stops — and convert it through the same relation. That conversion is stiff at high spin and slack at low, so the published spins near one are the trustworthy ones and the published spins near a third are barely measurements.

gravitation · Black hole spin A hole's shadow at 17° to its spin axis. The critical curve — the outline of a black hole's shadow on a distant observer's sky — for spins from 0 to 0.998, seen 17 degrees from the spin axis, in gravitational radii. Every curve is the projection of the unstable spherical photon orbits, and nothing about an accretion flow enters it. The sizes are 5.20, 5.12, 4.92, 4.84: a spread of 6.9 per cent across the entire range of spin, which is what makes an image of one of these a mass measurement with almost no model in it. What spin does instead is push the curve sideways and flatten one edge — the fast hole's outline sits 0.70 gravitational radii off centre, because photons co-rotating with the hole are dragged round and escape from closer in than counter-rotating ones can. That displacement is the spin signal, and it is measured against a centre nothing else marks.

A ring whose size is almost only a mass

The dark patch at the centre of an image of a black hole is the projection of the orbits light cannot escape from, and its size changes by seven per cent across the whole range of spin and inclination. That insensitivity is what makes an image a mass measurement with almost no model in it — and what leaves the spin in the part of the picture that barely moves.

gravitation · Black hole spin Two assembly histories, and the sign that separates them. The distribution of effective spin produced by two ways of making a black-hole binary, from 40,000 draws each. A pair that evolved together as two stars keeps its spins near the orbital axis — drawn here aligned to within 25 degrees — and cannot produce a negative projection at all: 0.0 per cent of its mergers fall below zero, and its mean is 0.45. A pair assembled by chance encounters in a dense cluster has no memory of any plane, so its tilts are isotropic, its distribution is symmetric, and exactly half of its mergers have an effective spin below zero. The sign of one number distinguishes two histories — and it does so without any of the individual spins being measured, which is the only reason the question is answerable at all from a catalogue in which every event's spins are individually uncertain. The third curve is a mixture with 50 per cent of its mergers drawn isotropically: its mean is 0.22 and 25 per cent of it lies below zero, which is the shape an observed distribution with a small positive mean and a real negative tail requires. Neither channel alone produces it.

A spin that is measured as an angular momentum

Every other method of measuring a black hole's spin reads it from what the hole does to matter. A merging pair's waveform reads it from the orbit instead — and what it recovers is one mass-weighted projection onto the orbital axis, which is a number that two non-rotating holes and two extremal ones can share.

gravitation · Black hole spin Precession cycles grow as the inverse of the lowest frequency heard. The number of times the orbital plane precesses between a detector's lowest frequency and the last stable orbit, against that lowest frequency, both on logarithmic scales, for pairs of 36 + 29 solar masses, 10 + 8 solar masses, 8 + 1.4 solar masses with spins tilted into the orbital plane. The count grows as the inverse of the lowest frequency — the slopes are −1.07, −1.02, −1.01. Above 20 Hz, the limit of the ground-based detectors that have heard every merger so far, the heaviest pair precesses 0.9 times, the middle 4.4 and the lightest 6.2. Above 5 Hz the counts are 4.9, 18.7, 25.5; above 1 Hz, 26, 95, 129.

Precessions bought one hertz at a time

A merging pair of black holes whose spins are tilted makes its orbital plane wobble, and the wobble is the only part of the signal that separates one hole's spin from the other's. How many wobbles a detector records is a simple formula: two plus three halves of the mass ratio, divided by the total mass and by the lowest frequency the detector can hear. The detectors that have heard every merger so far stop at twenty hertz, where a typical heavy pair wobbles less than once.

gravitation · Black hole spin

Named alongside it

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

Black hole spinKerr metricAngular momentumFrame draggingAccretion discEffective spinEvent horizonIron lineRadiative efficiencyRelativistic beamingSpin precessionApoapsis

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