Astronomy & Celestial Mechanics
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Gravitation

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.

Assumes Black hole spin, Accretion and Quasars.

17 min read 8 figures Measuring the unreachableScales that defeat intuition

The first rung of this anchor established that a black hole’s spin is measurable at all, which is remarkable given that a black hole has no surface, no features and no radius that anything can be laid against. The measurement works because the spin moves the last stable circular orbit: six gravitational radii for a hole that does not rotate, one for a hole at the theoretical limit, and nine for matter going the other way.

That is the whole of the observational content, and it is worth stating in that flat form because both spectral methods rely on it and neither is measuring anything else. Continuum fitting infers where the disc’s hottest emission comes from and calls that radius the last stable orbit. The iron line’s red wing is produced at the innermost radius from which the line is emitted, and that radius is called the last stable orbit. One length, one relation, two routes.

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.
Fig. 1 The spin that follows from a mismeasured inner radius, for four true spins. A ten per cent error reads a hole of true spin 0.3 as anywhere from 0.16 to 0.44, and the same error moves a hole of true spin 0.99 only between 0.98 and 1.00. The conversion is stiff at high spin and slack at low, because the last stable orbit does most of its falling in the final few per cent of the spin range. An overestimated radius always reads slow, so anything that stops the disc outside the last stable orbit biases every measurement the same way.

The consequence is not a caveat. It reverses the usual expectation that an extreme measurement is a fragile one.

The relation that does the work

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 0.95 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.
Fig. 2 The last stable orbit against spin, for orbits prograde and retrograde with the hole’s rotation, drawn to a spin of 0.998. Frame dragging is what separates the two curves: space near the hole is itself circulating, so matter going the same way can stay closer before its orbit becomes unstable, and matter going the other way cannot come as close as it could around a hole that does not rotate at all. The prograde branch falls from six to about 1.24 gravitational radii across the drawn range, and most of that fall happens after a = 0.9.

The bunching is the whole story. Between a = 0 and a = 0.5 the orbit moves from six radii to 4.23 — a change of 1.77 over half the range. Between a = 0.99 and a = 0.999 it moves from 1.454 to 1.182, a change of 0.27 over a range a hundred times narrower. Differentiated, that is a factor of thirty in sensitivity.

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 0.1 as anywhere from 0.00 to 0.27, while the same ten per cent moves a hole of true spin 0.998 only between 0.99 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.
Fig. 3 The same propagation over a narrower range of error and a wider range of true spin. A hole at a = 0.1, measured to ten per cent in radius, is consistent with everything from zero to 0.27; a hole at a = 0.998 is pinned between 0.99 and 1.00. A published spin of 0.1 ± 0.15 is not a measurement of a slowly rotating hole; it is a statement that the disc stopped somewhere near six gravitational radii, which is what it does when there is no spin to detect.

It is worth seeing where the stiffness comes from, since it is not an accident of the algebra. The last stable orbit exists because the effective potential of a circular orbit loses its minimum below a certain radius, and in the extremal limit the coordinate radius of that orbit, the photon sphere and the horizon all coincide at one gravitational radius. Three distinct surfaces converging on one value is a degenerate limit, and quantities are always compressed near a degenerate limit — the proper distance between them stays finite and the coordinate difference does not. So the bunching in the figure is a coordinate artefact in one sense and an entirely real observational fact in another: what a spectrum measures is a coordinate radius, and that is the one that bunches.

What each method has to assume

Both methods measure a radius. They differ entirely in what else they need in order to do it.

Continuum fitting works from the thermal spectrum of the disc. A disc radiating locally as a blackbody at each radius produces a summed spectrum that is a stack of temperatures whose normalisation is proportional to the projected area of the emitting region — that is, to (Rincos⁡i/D)2(R_{\rm in}\cos i / D)^2. Extracting RinR_{\rm in} in kilometres therefore needs the distance and the inclination; converting kilometres into gravitational radii needs the mass. Three external quantities, none of which the X-ray spectrum supplies. It is the method’s strength that its physics is simple and its weakness that it is parasitic on three other measurements.

The iron line works from the shape of a fluorescent line emitted by the disc’s surface. The line is broadened by the orbital motion, beamed by it, and redshifted by the hole’s gravity, and the red wing extends further the closer the emitting gas gets. Its enormous advantage is that a line profile is a shape: the distance divides out, the mass divides out, and the inclination is fitted from the profile’s own blue edge.

A line with a wing where the last orbit is. The profile of an iron Kα line at a rest energy of 6.4 keV, emitted from a disc seen at 45 degrees with emissivity falling as the 5th power of radius, for a hole of spin 0 and 0.5 and 0.9. Each photon's energy is shifted by g = √(1 − 3/r) ÷ (1 + sin i sin φ/√r) — gravitational redshift and orbital time dilation in the numerator, the Doppler shift of the orbital motion in the denominator — and the profile is the histogram of that shift weighted by the emissivity and by g³. The blue horn comes from material on the approaching side of the outer disc and is at 7.01 keV for both spins, because the spin does not move the outer disc: the two edges are required to agree to within a hundredth. The red wing is where the spins separate. A non-rotating hole's disc stops at six gravitational radii and its line reaches 3.52 keV; a hole at 0.9 has material orbiting at 3.02 radii and its line reaches 0.47 keV. Measuring a spin therefore means measuring the extent of the faintest part of a line, which is why it is contested. The shift is computed in the Schwarzschild metric and the spin enters only through the inner radius, which is where nearly all of the difference lies but not quite all of it.
Fig. 4 The line profile at three spins for a disc inclined at 45 degrees with a steep emissivity, falling as the fifth power of radius. The blue horn is the approaching side, beamed and boosted; the red wing is the innermost material, which the spin governs. A steep emissivity weights the innermost radii heavily and is what makes the wing detectable — and the emissivity index is fitted from the same profile it is needed to interpret, which is the method’s own version of the problem the other one has with distance.

The methods are therefore not independent checks of one another in the way it is usually put. They share the physical assumption that carries all the risk — that the disc’s inner edge is at the last stable orbit and the emission stops there — and differ only in the auxiliary data each needs.

What could stop the disc somewhere else

That shared assumption is worth taking seriously, because there are several reasons a disc might not reach the last stable orbit.

At low accretion rate the inner disc can evaporate into a hot, thick, radiatively inefficient flow, leaving the thin disc truncated at tens or hundreds of gravitational radii. This is not a subtlety: the truncation radius in that state moves with the accretion rate, and it is why spin measurements are attempted only in the high-luminosity states where the thin disc is believed to extend inward.

At high accretion rate the disc thickens for the opposite reason, radiation pressure inflates it, and matter can begin to plunge from outside the last stable orbit. Both errors push the inferred radius outward, and by the figure above both therefore read as a slower hole.

There is a third possibility that is neither of those and is the most awkward, which is that the inner disc is not flat. A disc around a hole whose spin axis is tilted with respect to the orbital plane is torqued by frame dragging into alignment near the centre and left tilted further out, so the inner region warps. A warped disc presents a different projected area at every radius, which corrupts the continuum normalisation, and it emits its line from a surface that is not the plane the profile calculation assumes.

There is also emission from inside the last stable orbit. Matter crossing it does not vanish; it plunges, and it continues to radiate for the fraction of an orbit the crossing takes. That contribution is small in a thin disc and is not zero, and it pushes the inferred radius the other way.

Every one of these effects has the same character: it is a departure from a geometry that the measurement assumes rather than tests, and none of them shows up as a poor fit. A truncated disc fitted with an untruncated model returns a good fit and a lower spin. That is the signature of a systematic rather than a statistical error, and it is why the error bars quoted on individual spins are so much smaller than the disagreements between authors analysing the same data.

A spin measured with no spectrum at all

There is a third route to spin that shares none of these assumptions, and it does not measure a hole. It measures all of them at once.

A mean spin of 0.69, from the light of every quasar that ever shone. The radiative efficiency of a thin accretion disc against black-hole spin — the binding energy per unit rest mass at the last stable orbit, which is 5.72 per cent for a hole that does not rotate and 42.3 per cent in the extremal limit — with the value the quasar population requires marked on it. The argument is Sołtan's and it uses no spectrum of any individual object. Every solar mass that a black hole swallowed released a fraction η of its rest energy as light and kept the remaining 1 − η, so the total energy quasars have emitted per unit volume and the mass in black holes now are two measurements of the same accretion history, and their ratio is η averaged over all of it. Taking 4.8e+4 solar masses per cubic megaparsec of emitted energy against 4.2e+5 of surviving mass gives η = 10.3 per cent, which the curve converts to a mean spin of 0.69, with the quoted uncertainties spanning 0.25 to 0.90. The strength of it is that no distance, no inclination and no disc model enters. The weakness is that the curve it is read against assumes a thin disc radiating locally, which the objects doing most of the swallowing may not have had, and the whole inference is an average over a population rather than a measurement of anything.
Fig. 5 The radiative efficiency of a thin disc against spin — 5.72 per cent for a hole that does not rotate, rising to 42.3 per cent in the extremal limit — with the value the quasar population requires marked on it. Every solar mass a hole swallowed released a fraction of its rest energy as light and kept the rest, so the total light quasars have emitted per unit volume and the mass in black holes today are two measurements of one accretion history. Their ratio is the efficiency averaged over all of it: 10.3 per cent here, which the curve converts to a mean spin of 0.69.

The arithmetic is one line and it is worth writing out. If a mass density ρacc\rho_{\rm acc} fell into black holes and each unit of it radiated a fraction η\eta of its rest energy, then the emitted energy density is ηρaccc2\eta\rho_{\rm acc}c^2 and the mass left behind is (1−η)ρacc(1-\eta)\rho_{\rm acc}. Dividing one by the other,

η1−η  =  UquasarρBH c2\frac{\eta}{1-\eta} \;=\; \frac{U_{\rm quasar}}{\rho_{\rm BH}\,c^{2}}

and both quantities on the right are measured. Nothing about any individual object appears.

Sołtan’s argument dates from 1982 and it is one of the cleanest things in the subject. It needs the quasar luminosity function integrated over redshift and a bolometric correction, and it needs the local black-hole mass density — which comes from the correlation between black-hole mass and bulge dispersion applied to a galaxy luminosity function. Neither ingredient is a spectrum of a black hole. Neither involves a distance to a particular object, an inclination, or a disc model beyond the assumption that the accretion was radiatively efficient.

A mean spin of 0.90, from the light of every quasar that ever shone. The radiative efficiency of a thin accretion disc against black-hole spin — the binding energy per unit rest mass at the last stable orbit, which is 5.72 per cent for a hole that does not rotate and 42.3 per cent in the extremal limit — with the value the quasar population requires marked on it. The argument is Sołtan's and it uses no spectrum of any individual object. Every solar mass that a black hole swallowed released a fraction η of its rest energy as light and kept the remaining 1 − η, so the total energy quasars have emitted per unit volume and the mass in black holes now are two measurements of the same accretion history, and their ratio is η averaged over all of it. Taking 6.0e+4 solar masses per cubic megaparsec of emitted energy against 3.2e+5 of surviving mass gives η = 15.8 per cent, which the curve converts to a mean spin of 0.90, with the quoted uncertainties spanning 0.70 to 0.99. The strength of it is that no distance, no inclination and no disc model enters. The weakness is that the curve it is read against assumes a thin disc radiating locally, which the objects doing most of the swallowing may not have had, and the whole inference is an average over a population rather than a measurement of anything.
Fig. 6 The same argument with a lower local mass density and a higher emitted energy density, which is within the range different authors have adopted. The efficiency rises to 15.8 per cent and the mean spin to 0.90. The lever is short: the efficiency runs from 6 to 42 per cent across the whole of spin, so a factor of two in the input ratio moves the answer across most of the available range. That is the argument’s weakness, and it is a different weakness from the spectral methods’ — it is an error in a population average rather than in an object.

The comparison between the two families is the useful thing. Individual spectral spins cluster high, at 0.9 and above, in a sample selected for being measurable. Sołtan’s average sits nearer 0.7 and is an average over everything that ever accreted, weighted by mass. Those are consistent with each other and with a picture in which most mass is swallowed at moderate spin while the objects bright enough to have their discs fitted are the fast ones — and they are also consistent with the spectral sample being biased, since the systematics all push the same way.

Between six and forty-two per cent. The fraction of the rest mass of infalling matter that can be released as light before it crosses the horizon, against the hole's spin, for prograde and retrograde discs. The efficiency is one minus the specific energy of the last stable orbit, so it is fixed entirely by where that orbit is: 5.72 per cent for a non-rotating hole — exactly 1 − √(8/9), which is checked rather than quoted — rising to 34.0 per cent at the largest spin drawn and to 42.3 per cent in the extremal limit the curve is approaching. A retrograde disc round a rapidly spinning hole is worse than no spin at all, at 3.79 per cent. The horizontal line is hydrogen fusion, which converts 0.7 per cent of rest mass: accretion onto even a static hole beats it by a factor of eight, and onto a fast one by fifty. At the 0.998 spin a thin disc actually settles at, the efficiency is 32 per cent. This is the number behind every argument that compares the light quasars have emitted with the mass in black holes today, and the comparison is a measurement of the average spin rather than an assumption about it.
Fig. 7 The efficiency curve drawn out to 0.999, with the retrograde branch for comparison and hydrogen fusion marked for scale. A prograde disc around a hole at the equilibrium spin releases 32 per cent of the rest mass it swallows; fusion releases 0.7 per cent. That factor of forty-five is why accretion onto a rotating hole is the most efficient process in the universe that is not annihilation, and why the same amount of swallowed mass produces wildly different amounts of light depending on a number that has nothing to do with how much was swallowed.

Why the holes are not all extremal

If accretion spins a hole up, and holes have been accreting for ten billion years, the natural expectation is that every one of them sits at the limit. It is not what is inferred, and the reason is a piece of physics with no free parameters.

A hole is spun to its limit by swallowing √6 of itself. The spin of a black hole against its mass, as it accretes from a thin disc whose material arrives at the innermost stable orbit and carries that orbit's specific energy and specific angular momentum. Nothing is assumed about the accretion rate or the time it takes: the track is a relation between two of the hole's own numbers. Starting from no spin at all, the hole reaches a = 0.9999 after its mass has grown by a factor of 2.359, and the integrated track is checked against Bardeen's closed relation between spin and mass at every point along it rather than only at its end. That distinction is the figure's own arithmetic lesson: the extremal limit a = 1 is reached at √6 = 2.449, so the last thousandth of the spin costs as much swallowed mass as the first nine hundred and ninety-nine. The spin-up is fast at first and slow at the end, because the specific angular momentum of the last stable orbit falls as that orbit moves inward — the hole becomes harder to spin the faster it turns. The mark at a = 0.998 is reached after a growth of 2.202, and it is where a real hole stops: photons emitted by the disc are captured preferentially onto retrograde orbits, which removes spin at exactly the rate accretion adds it. The consequence for the collection is that a hole that has grown by more than a factor of two by accretion should be spinning near that limit, and a hole assembled by mergers of randomly oriented pairs should not.
Fig. 8 Spin against accreted mass for a hole fed by a prograde thin disc. Reaching the extremal limit takes swallowing √6 times the starting mass — a factor of 2.45, which is not much — and the track passes 0.998 at a factor of 2.20. That value is Thorne’s limit and it is where the spin-up stops: photons emitted by the disc are preferentially captured onto retrograde orbits by the hole’s own frame dragging, and the counter-torque exactly balances the accretion torque there. A hole cannot be spun past it by a disc, whatever it swallows.

The √6 is exact and is worth a line. The angular momentum a hole gains per unit mass swallowed is the specific angular momentum at the last stable orbit, and integrating dJ=LISCO dM\mathrm{d}J = L_{\rm ISCO}\,\mathrm{d}M from a non-rotating hole to the extremal limit gives Mfinal/Minitial=6M_{\rm final}/M_{\rm initial} = \sqrt{6} — Bardeen’s result, from 1970, with no free parameter and no disc model beyond the assumption that matter arrives on circular orbits and crosses the last stable one. A factor of 2.45 in mass is a modest amount of accretion by the standards of quasar growth, which is why the expectation of high spin is a strong one.

Two further mechanisms keep real holes below even that. Accretion delivered in randomly oriented episodes spins a hole up and down alternately, and the random walk settles near a modest spin rather than at the ceiling; and a merger between two holes of comparable mass produces a remnant whose spin is set by the orbital angular momentum at the last stable orbit, which for random orientations averages around 0.7 — a process whose rate depends on whether the last parsec can be crossed at all.

That the observations, the Sołtan average and both theoretical arguments all land in the region of 0.7 to 0.9 is the strongest thing that can be said about black-hole spin at present. It is a convergence of four weak measurements rather than a strong one.

There is a useful asymmetry between those two. Chaotic accretion holds the spin down because each episode is small compared with the hole and randomly oriented, so the walk is unbiased; coherent accretion from a single large disc drives the spin up. The observed spins therefore say something about how the mass arrived rather than only how much, and a population of fast-spinning holes is evidence for prolonged accretion in a fixed plane — which is what a disc that has to throw its angular momentum outwards does over many orbits.

What is actually measured

Setting the chain out plainly is sobering in the same way it was for the virial factor.

For a continuum fit: an X-ray spectrum, a distance from a parallax or a kinematic model, an inclination from an optical light curve of the binary, a mass from the same, a disc atmosphere model that converts the observed colour temperature into an effective temperature through a hardening factor, and the assumption that the emission stops at the last stable orbit. The hardening factor alone is a fitted number between 1.5 and 1.9 that enters the fourth power of the temperature.

For an iron line: an X-ray spectrum with enough counts in the 3–10 keV band to distinguish a broad wing from the continuum underneath it, a model for that continuum, a model for absorption along the line of sight, an emissivity profile, and the same assumption about the inner edge. The most disputed spins in the literature are disputed over the continuum model rather than over the line.

The hardening factor deserves its own sentence because it is the least visible assumption in the whole business. A real disc atmosphere is scattering-dominated, so photons emerge having been scattered many times and carry a spectrum harder than a blackbody at the local temperature; the ratio of the observed colour temperature to the effective one is the hardening factor, it is computed from an atmosphere model, and the inferred radius goes as its square. A value of 1.7 rather than 1.6 moves the inferred radius by twelve per cent, which by this essay’s first figure is a substantial move in spin for a slow hole and almost none for a fast one.

Neither list contains a measurement of a black hole. Both contain a measurement of a spectrum and a chain of models, and the spin is the last link. That is not a criticism of the technique — nothing else is available for an object with no surface — but it is why the strongest constraint in the field remains the one that measures nothing in particular.

The spins that are not measured this way

Two other kinds of spin measurement exist and are worth setting beside these, because they fail differently.

A merging pair of black holes radiates a gravitational-wave signal whose phase depends on the components’ spins, and whose amplitude and polarisation precess if those spins are misaligned with the orbit. That is a measurement of an angular momentum as an angular momentum, with no accretion disc anywhere in it, and it comes from the same signal that gives a distance with no ladder under it. Its weakness is precision: the spin combination the waveform constrains best is a mass-weighted projection onto the orbital axis, and for most detected mergers it is consistent with zero. What that has established is not a set of spins but an absence — the holes that merge are not, in general, rapidly spinning in the orbital plane’s direction, which is itself informative about how they were assembled.

And the shadow. An image of the emission around a hole shows a dark region whose diameter is set almost entirely by the mass and whose shape is very weakly dependent on spin. The asymmetry a fast spin produces is a few per cent of the diameter, which is below what the two published images resolve. The measurement that image made cleanly is a mass; the spin is inferred from the surrounding structure and the polarisation, with the same modelling chain as everything else in this essay.

The generalisation

The structure worth extracting is that a stiff conversion is a good conversion, and the intuition runs the other way.

Where the relation between the observable and the quantity is steep, an error in the observable produces a small error in the quantity. Every spin measurement in this essay is one length converted through the ISCO relation, and that relation is steep exactly where the spins are extreme. The consequence — that the most extreme claims are the best supported — is the opposite of the usual heuristic that an extraordinary result needs extraordinary evidence, and it is a case where the heuristic is wrong for a computable reason.

There is a second reading, less comfortable, which is about what a sample of measurements means when the systematics all point one way. Every effect in this essay that could stop the disc outside the last stable orbit reads as a slower hole, and none reads as a faster one. A distribution of measured spins that piles up near the extremal limit is therefore exactly what a biased-low method applied to genuinely extremal holes would produce, and also exactly what an unbiased method applied to moderately spinning holes could not produce. The pile-up is evidence, and the Sołtan average sitting lower than the individual measurements is the sort of tension that a one-sided systematic predicts.

The same shape appears wherever a measurement is made near a singular limit. The mass a cold star cannot exceed is approached along a radius–mass relation that steepens to vertical, so a white dwarf near the limit has its mass better determined by its radius than one far below it. Look at the derivative before deciding which end of a range is the safe one.

Where the ladder goes next

The next rung leaves the spectrum for the shadow. A hole’s photon sphere and the ring of light around it are geometric consequences of the same two numbers, and the shape of that ring — its diameter, and its asymmetry — depends on spin and inclination in a way an image can constrain directly. The measurement has been made for two objects, and what it gives is a mass with almost no model in it and a spin with a great deal.

Further rungs on this anchor: the quasi-periodic oscillations whose frequencies would be a spin measurement if anybody knew which resonance produced them; the spins of the holes that merge, read off the gravitational-wave signal’s precession, which is the only method that observes spin as an angular momentum rather than through its effect on matter; jet power against spin, which is the Blandford–Znajek prediction and the least conclusive of all the tests; and the spin of the hole at the centre of the Galaxy, which is measured by four techniques that do not agree.

Where this goes next

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. 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. A single-epoch mass, good to a factor of 2.6. The error budget of a black-hole mass obtained from one spectrum, as probability densities in the logarithm of the ratio of the estimate to the truth. Four independent widths go in. The radius comes from the luminosity through a relation with 0.19 dex of scatter about it. The line width has to be measured on a profile that is blended with narrow lines and with iron, and it enters the mass squared, so 0.12 dex. The luminosity was not measured at the epoch the relation was calibrated for and the object has varied since, which is 0.053 dex. And the term this figure exists for is orientation: the virial factor is one number fitted to a sample, every individual object has its own according to how its broad-line region is inclined, and the spread of that is 0.35 dex — the largest of the four by some margin. In quadrature the total is 0.42 dex — a factor of 2.6 — and the drawn total's half-width is checked against that combination rather than assumed. The bracket at the foot is a different kind of error and is not in the quadrature: the absolute calibration of the virial factor, uncertain by about a factor of 3, which moves every quasar mass ever published by the same 0.24 dex and widens nothing. The factor that multiplies every quasar mass A reverberation lag and a line width give a length times a velocity squared, which is a mass multiplied by an unknown number of order one. That number is fixed by making a few dozen nearby active galaxies lie on a correlation measured in quiescent ones — so every black-hole mass at redshift two rests on a fit performed at redshift zero.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Accretion discBlack hole spinContinuum fittingFrame draggingInnermost stable circular orbitIron lineKerr metricRadiative efficiencyRelativistic beamingSoltan argument