Astronomy & Celestial Mechanics
the sky, computed
Concept

Radiative efficiency — where it appears

The fraction of accreted rest mass emitted as radiation, set by the binding energy at the disc's inner edge. It runs from six per cent for a non-rotating black hole to over forty for a rapidly rotating one, against seven-tenths of one per cent for hydrogen fusion.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

The temperature of a disc around a stellar black hole. Effective temperature against radius, in units of the inner edge, for a stellar black hole of 10 solar masses accreting 10⁻⁸ solar masses a year. Two features are structural. The profile turns over rather than rising all the way in: the factor (1 − √(r_in/r)) is the statement that no torque acts across the inner edge, so nothing is dissipated there and the peak sits at 49/36 of it, measured here at 1.361. And outside a few inner radii the run is exactly r^−3/4, drawn as the dashed line, which is what makes a disc's spectrum broad: every decade of radius contributes at a temperature a factor of 5.6 lower. The peak is 3.46·10⁶ K here, so the disc radiates in X-rays, and integrating the whole profile gives 4.72·10³⁰ W — which is GMṀ/2r_in to a per cent, half the binding energy released and no more, because the other half is still going round.

The disc that has to throw angular momentum away

Matter cannot simply fall onto a compact object. At the energy it arrives with, it has far too much angular momentum, and the only way in is for some of it to be carried outwards — which is what a disc is for.

stars · Accretion A spectrum belonging to no temperature at all. The disc's summed emission, with the individual annuli drawn faintly beneath it. Each ring is a blackbody at its own temperature, and each is drawn at the area it actually has — the outer rings are cool and enormous, the inner ones hot and small. The sum has three parts and only the two ends belong to a temperature: a Rayleigh–Jeans rise of slope 2 from the outermost ring, a Wien cutoff at the hottest, and between them a stretch of slope 0.316, against the 1/3 that comes out of integrating ν²T(r)r dr with T ∝ r^−3/4. That middle section is the observational signature of a disc: no single blackbody produces it, no photosphere produces it, and its width rather than its peak is what says how far in the disc goes. What the figure cannot show is that a real disc's innermost rings are neither thin nor blackbodies, which is where the model's clean edges stop.

A spectrum that is a stack of temperatures

An accretion disc is not hot. Its inner edge is, its outer edge is not, and the temperature runs continuously between them as a power of radius — so what leaves the disc is the sum of a great many blackbodies at different temperatures, which is a spectrum with no temperature in it and a slope no single body can produce.

stars · Accretion 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 thousand times the supply, and eight times the light. The luminosity of an accretion disc, in units of its Eddington luminosity, against the rate at which gas is supplied to it, in units of the rate that would make a thin disc shine at exactly that limit — both on logarithmic axes. Below the limit the luminosity is proportional to the supply, the dashed line, with a tenth of the rest-mass energy radiated. Above it the disc cannot radiate the excess: the luminosity grows only as 1 + ln ṁ, so ten times the supply gives 3.3 times the limit, a hundred times gives 5.6, and a thousand times gives 7.9. The limit is a ceiling that bends rather than breaks — and whatever the supply is doing above it, the light does not say, because the same curve follows whether the surplus is blown away in a wind or carried into the hole.

A ceiling on the light and not on the mass

Pour gas onto a black hole a thousand times faster than the Eddington rate and it shines at only about eight times its limit. The rest of the energy is either blown away or swallowed, and those two fates give the same luminosity while differing by a factor of the supply in how fast the object grows — so the light a super-Eddington source emits cannot say which one it is.

stars · Eddington limit

Named alongside it

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

Accretion discEddington limitBlack hole spinFrame draggingInnermost stable circular orbitInnermost stable orbitIron lineKerr metricMulticolour blackbodyRelativistic beamingSpecific angular momentumAccretion

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