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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.

Assumes Accretion, Energy transport and Opacity.

16 min read 8 figures Light is the only messengerMeasuring the unreachableTwo bodies, and no more

A star radiates very nearly like a blackbody, because a star has one photosphere at one temperature and everything a distant observer sees comes from it. That single fact underlies most of the machinery in this collection: a colour is a temperature, a spectrum has a peak, and the peak’s wavelength times the temperature is a constant.

An accretion disc does none of that, and the reason is not a complication but a definition. A disc extends from an inner radius to an outer one, it dissipates energy locally at every radius, and it radiates that energy locally too. So it has not one temperature but a continuum of them, and what arrives is their sum.

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.
Fig. 1 The sum. Each thin annulus of the disc contributes a blackbody at its own temperature; the annuli are added, weighted by area; and the result has a middle section that is a power law of slope one-third in flux against frequency — a shape no blackbody at any temperature produces. At the low-frequency end the spectrum reverts to the Rayleigh–Jeans slope of the coolest annulus, and at the high-frequency end it falls exponentially, cut off by the hottest.

The consequence for the observer is immediate and slightly disorienting. Fitting a disc spectrum with a Planck function returns a temperature, and that temperature is not the temperature of anything: it is a weighted average over a profile, and its value depends on which part of the spectrum happened to be observed. Colour is a thermometer for a star and is not one for a disc, and the whole apparatus of magnitudes and colour indices, which works so well on stellar populations, has to be rebuilt before it says anything here.

Where the temperature profile comes from

Two statements produce the whole spectrum, and neither of them is about radiation.

The first is that material spiralling inwards releases gravitational energy. A ring of matter at radius rr moving to r−drr - \mathrm{d}r gives up a well-defined amount, and in a steady disc — one where the same mass per second passes every radius — the energy released per unit area is fixed by the accretion rate and the geometry alone. The standard result is that the dissipation per unit area goes as r−3r^{-3} far from the inner edge.

The second is that the disc is optically thick and radiates locally as a blackbody. Setting the dissipated flux equal to σT4\sigma T^4 then gives

T(r)  ∝  r−3/4.T(r) \;\propto\; r^{-3/4}.

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.
Fig. 2 The profile. Temperature against radius for a disc around a stellar-mass black hole, with the inner edge at the innermost stable circular orbit. It is a power law of index minus three-quarters over most of its range, and it turns over near the inner edge because the disc cannot dissipate energy inside a boundary where there is nothing to hold onto. Note what the profile is not: it is not the temperature of anything falling in, and it is not a temperature the disc equilibrates to. It is what local balance requires at each radius separately.

The dissipation profile is worth one more sentence because it contains a surprise. Naively, half the gravitational energy released between two radii goes into orbital kinetic energy and half is available to radiate — the virial accounting that governs every self-gravitating system. But a disc annulus also does work on its neighbours through the same stress that transports its angular momentum, so energy released at one radius can be radiated at another. The net effect is that the region near the inner boundary radiates roughly three times what it locally liberates, and the outer disc correspondingly less. The r−3r^{-3} law already contains that transfer.

Nothing in that derivation mentions viscosity. The dissipation rate per unit area follows from conservation of energy and angular momentum in a steady flow, so the temperature profile is the same whatever mechanism actually transports angular momentum outwards — which is fortunate, since that mechanism is the least settled part of the subject and the spectrum does not depend on it.

Why the middle is a power law of slope one-third

The one-third exponent is worth deriving, because it is a small piece of arithmetic that explains an observed shape.

Consider a frequency ν\nu in the middle of the spectrum. The annuli whose temperature satisfies kT∼hνkT \sim h\nu radiate most efficiently at that frequency; hotter annuli, further in, are on the Rayleigh–Jeans tail of their own spectra there. Adding up gives Fν∝ν1/3F_\nu \propto \nu^{1/3}, and the exponent is a combination of two things: the Rayleigh–Jeans slope of ν2\nu^2, and the way the emitting area grows as the relevant temperature falls, which is set by the r−3/4r^{-3/4} profile.

Change the temperature profile and the exponent changes with it. A disc truncated inside some radius, or one whose inner region has been evacuated, produces a different slope — so the exponent is a measurement of the profile rather than a decoration on it.

Where a 5772 K spectrum meets five filters. A single Planck curve at 5772 K, scaled to its own peak at 502 nm, with the five Johnson–Cousins responses shaded beneath it. Each shaded hump is the spectrum multiplied by that filter's response, so its area is the light the band actually collects, and the percentage beside each name is that area as a fraction of everything the star radiates at all wavelengths: U — 6.9%, B — 12.3%, V — 11.9%, R — 16.3%, I — 13.3%, 60.7% between them. The bands overlap, so those figures do not partition the light; and they cannot add to all of it, because most of a hot star's output is ultraviolet and most of a cool one's is infrared, where none of these filters looks. The responses are idealised as Gaussians at the published effective wavelengths and widths — a real filter's blue edge is steep and its red tail is the detector's, which this cannot show.
Fig. 3 What a single temperature looks like for comparison: one Planck curve, one peak, and a Wien displacement relating the two. Everything about a star’s continuum can be read off a picture like this one, and nothing about a disc’s can. A disc spectrum fitted with a single blackbody returns a temperature that belongs to no part of the disc.

The inner edge, and what it is a measurement of

The hottest annulus sets the high-frequency cutoff, so the cutoff is a measurement of the inner radius — and the inner radius is a statement about the object at the centre.

For a black hole the disc cannot extend inwards indefinitely: inside the innermost stable circular orbit there are no circular orbits, and material there falls in on a dynamical time without radiating much. The radius of that orbit is six gravitational radii for a non-rotating hole and shrinks to about one for a maximally rotating one, so the inner edge — and therefore the peak temperature at a given accretion rate — is a probe of the spin.

How much of a mass can be turned into light, and what decides it. The fraction of rest energy an accreting object releases, against the radius its disc has to stop at. The curve is the Newtonian half-binding-energy GM/2rc², and the whole content of the plot is that the answer is set by the inner edge and by nothing else — not by the mass, which cancels when the radius is measured in GM/c², and not by the accretion rate, which sets the luminosity and not the efficiency. A white dwarf's surface is far out and returns 0.015%; a neutron star's is at a few gravitational radii and returns 11%; a black hole has no surface, so the disc stops at the last stable orbit instead and returns 5.72%. That last number is not on the curve: the Newtonian expression gives 8.3% at the same radius, 1.46 times too much, and the difference is the point at which this picture has to be handed over to the metric. Hydrogen fusion, drawn as the flat line, returns 0.7% — an accreting black hole is an order of magnitude better at converting mass into light than a star is, which is why quasars outshine the galaxies they sit in.
Fig. 4 How much of a mass can be turned into light, and what decides it. The binding energy at the innermost stable orbit is the efficiency, and it runs from about six per cent for a non-rotating black hole to over forty for a rapidly rotating one — against the seven-tenths of one per cent that hydrogen fusion manages. Accretion onto a compact object is by a wide margin the most efficient way the universe has of converting mass into radiation, and the number is a property of geometry rather than of any nuclear physics.

For a neutron star or a white dwarf the disc stops at the surface instead, and the leftover kinetic energy is dissipated in a boundary layer — a distinct component with its own temperature, sitting on top of the disc spectrum, and one of the observational signatures that distinguishes an accreting neutron star from an accreting black hole.

For a magnetised star it stops further out still, where the magnetic pressure exceeds the ram pressure of the inflow, and the material is channelled along field lines onto the poles. That is the geometry that makes an accreting pulsar rather than an accreting disc.

Putting numbers on it

The scalings are more useful than the derivation, and there are two.

The peak temperature of a disc accreting at a fixed fraction of its Eddington rate goes as M−1/4M^{-1/4}. A ten-solar-mass black hole peaks near a kilovolt — soft X-rays. A hundred-million-solar-mass one peaks a factor of about sixty cooler, in the far ultraviolet, which is the “big blue bump” seen in every quasar spectrum. The same equations, the same accretion physics, and two phenomena that were studied separately for decades under different names.

The size scalings run the other way. The gravitational radius is proportional to the mass, so the disc around the supermassive object is ten million times larger and its light-crossing time is ten million times longer. That is why a stellar-mass X-ray binary varies on milliseconds and a quasar on months, and why the two are studied with completely different instruments: the physics is identical and the observing strategy cannot be.

A ratio that does not scale is the one to watch. The efficiency — the fraction of rest mass emitted — depends only on the spin, not on the mass or the rate. So the total light emitted by all the accretion in the universe, integrated over cosmic time, divided by the total mass now locked in black holes, is a measurement of the average spin. That comparison has been made, it gives an efficiency around ten per cent, and it is one of the few statements about black holes that comes from a budget rather than from an object.

How bright it is allowed to be

The accretion rate cannot be arbitrary, and the ceiling is one this collection has already met.

The temperature of a disc around a neutron star. Effective temperature against radius, in units of the inner edge, for a neutron star of 1.4 solar masses accreting 10⁻⁹ 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 5.33·10⁶ K here, so the disc radiates in X-rays, and integrating the whole profile gives 4.88·10²⁹ 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.
Fig. 5 And around a neutron star, which sits between the two. The inner edge is at ten kilometres rather than at a white dwarf’s ten thousand, so the peak moves into the X-ray and the total efficiency climbs toward a tenth of the rest mass — the number that makes accretion the most efficient energy source in the universe short of annihilation. The three panels are one formula read at three central masses, and the observable band is what changes.

The Eddington limit is what makes accretion discs bright enough to be the most luminous persistent objects in the universe and what stops them being brighter. It also sets a maximum temperature, since the peak temperature at the Eddington rate scales as the inverse fourth root of the mass — so a stellar-mass black hole peaks in X-rays and a billion-solar-mass one in the ultraviolet, which is why accreting stellar remnants and quasars look like completely different phenomena and are the same physics.

What the accretion rate is inferred from

The spectrum has two free parameters in the simplest version — the mass and the accretion rate — and they enter the observables differently, which is what makes both recoverable.

The peak temperature depends on the rate to the quarter power and on the mass to the minus quarter. The total luminosity depends on the rate linearly and on the mass not at all. So a measured peak and a measured luminosity are two equations in two unknowns, and the system closes. In practice the mass is usually known from something else — a companion’s velocity curve, or the orbits of stars around a galactic nucleus — and the spectrum is used to get the rate alone.

The result is invariably that the rate is small compared with what is available. A supermassive black hole in a typical nearby galaxy is accreting at something like a ten-thousandth of its Eddington rate, which raises the question of why, and the answer is that the material has to lose essentially all of its angular momentum to arrive. Gas at a kiloparsec has ten million times too much of it. The bottleneck in feeding a black hole is never gravity; it is always transport.

Measuring the disc’s size without resolving it

None of these discs is resolved. A quasar’s accretion disc subtends about a microarcsecond, which is four orders of magnitude below anything achievable.

A 20-day delay, recovered at 20 days from two curves that share no wavelength. Above: a quasar's continuum, drawn as a damped random walk with a 90-day damping time, and the broad emission line responding to it. The line curve is the continuum convolved with a top-hat response of half-width 20 days, so it is later and smoother — it varies only 51 per cent as much, because at any instant it is an average of the continuum over a range of light-travel times. Below: the cross-correlation of the two, which peaks at 20 days. That number is a length: 20 light-days is 5.2·10¹⁴ metres, or 3463 astronomical units, and it has been measured for an object that subtends 3.5·10⁻⁵ arcseconds at a hundred megaparsecs — some thirty times finer than the best optical interferometry has ever resolved anything, and reached here with a photometer and a clock. The irregularity of the continuum is what makes this work. A periodic source would give a cross-correlation with many equal peaks and no way to choose; a random one gives a single peak, and the whole method rests on active nuclei being erratic.
Fig. 6 The measurement that gets round it. The central source varies; the material further out sees the variation later, by the light travel time; so cross-correlating two light curves at different wavelengths returns a delay, and the delay is a distance. A size measured from a delay is the whole technique, and applied across wavelengths it maps the temperature profile directly — since longer wavelengths come from cooler and therefore more distant annuli, the lag should grow as the four-thirds power of wavelength if the profile is the one derived above.

The measured lags are consistently a factor of two or three larger than the standard profile predicts, which is one of the sharpest quantitative disagreements in the field. Nobody is quite sure whether the discs are genuinely bigger, whether the emission is contaminated by reprocessed light from further out, or whether the naive lag-to-radius conversion is too naive.

What the disc is not doing

The clean picture above accounts for the thermal component and for essentially nothing else in a real spectrum. Accreting systems switch between states in which the relative strength of the disc and the corona changes dramatically, on timescales of days for stellar-mass objects, and the transitions come with jets appearing and disappearing. What decides the state is not settled. The leading picture is that the disc is truncated at a large radius in the hard state and reaches the innermost stable orbit in the soft one, so that the states are a statement about the inner radius — which brings the argument back to the cutoff, and to reading a spectrum for a geometry.

The temperature of a disc around a white dwarf. Effective temperature against radius, in units of the inner edge, for a white dwarf of 0.8 solar masses accreting 10⁻⁹ 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 4.38·10⁴ K here, so the disc radiates in the ultraviolet, and integrating the whole profile gives 5.58·10²⁶ 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.
Fig. 7 The same construction around a white dwarf, three orders of magnitude lower in central mass. The disc temperature at a given radius goes as the mass over the radius cubed, and the inner edge sits at the star’s own surface rather than at an innermost stable orbit — so the whole profile is shifted down and out, peaks in the ultraviolet rather than the X-ray, and the stack of blackbodies it sums to is a stack of much cooler ones. Same integral, same shape, a different band to look in.

Where the disc stops on the outside

The inner edge was given a section because it says what the central object is. The outer edge is less discussed and it is what decides whether a disc exists at all.

For a disc in a binary system the boundary is imposed from outside: the accreting object’s Roche lobe, beyond which material belongs to the companion rather than to the disc. Tidal torques from the companion truncate the disc at something like ninety per cent of that radius, so the outer edge is set by the orbital separation and the mass ratio, both of which are measurable.

For a disc around a supermassive black hole there is no companion, and the boundary is set by the disc’s own gravity. A disc is held vertically thin by the central mass, but its own surface density grows outwards relative to that support, and beyond some radius the disc’s self-gravity dominates locally. There it fragments — into clumps, and then into stars.

The radius at which that happens is of order a tenth of a parsec for a typical quasar, which is thousands of gravitational radii and is very much smaller than the region material has to be fed from. So the standard disc model does not describe a quasar’s outer regions at all: whatever brings gas from a kiloparsec to a tenth of a parsec is not a thin accretion disc, because a thin accretion disc that large would not survive.

There is direct evidence that the fragmentation happens. The centre of the Milky Way contains a disc of young massive stars, a few tenths of a parsec across, orbiting the central black hole in a coherent plane — an unlikely place for stars to have formed by any ordinary route and exactly what a self-gravitating accretion disc would leave behind.

A disc that becomes unstable turns into stars, and the stars are then a record of a disc that no longer exists.

Where the picture stops

The disc is assumed to be steady, and observed discs are not. The whole temperature profile follows from the same mass per second passing every radius, which is true only on timescales longer than the viscous time at the outer edge — which for a quasar disc is thousands of years. The variability that reverberation mapping exploits is direct evidence that the assumption fails.

Irradiation has been ignored. The inner disc shines on the outer disc, and for a flared geometry the reprocessed flux can exceed the locally dissipated one at large radii. That flattens the outer temperature profile, changes the optical slope, and is the most likely reason the measured sizes come out large.

The inner boundary condition is a choice. The standard profile assumes zero torque at the innermost stable orbit, which is what makes the dissipation vanish there and produces the turnover in the temperature curve. If the magnetic field threading the plunging region can exert a torque back on the disc — and there is no obvious reason it cannot — the inner disc is hotter and more luminous than the standard picture allows, and every spin measured from a continuum fit moves.

And relativity has been left out of the emission, not merely the orbit. Light from the inner annuli is gravitationally redshifted, Doppler-boosted on the approaching side, and bent; a line emitted at a single wavelength there arrives as a broad asymmetric profile. Fitting that profile is currently the main way black-hole spin is measured, and it is a measurement of the inner radius by an entirely different route from the continuum cutoff. The two do not always agree.

The parameter the spectrum cannot see

It was noted above that the temperature profile does not depend on how angular momentum is transported. That is worth taking further, because it explains why the standard model has been so durable and so uninformative at the same time.

The transport is conventionally summarised by one dimensionless number, defined so that the stress is that number times the pressure. Everything about the disc’s structure — its thickness, its density, its surface temperature — depends on it, and yet the emergent spectrum does not, because the spectrum is fixed by the dissipation rate and the dissipation rate is fixed by conservation laws.

So a spectrum measures the accretion rate and says nothing about the mechanism producing it. That is convenient — the rate can be read without settling an unsolved problem — and it means the observation most easily made is the one least able to discriminate.

What does depend on the parameter is time. The radial drift speed of the material scales with it, so the viscous timescale — the time for a change at one radius to propagate — is inversely proportional to it. That makes the variability the diagnostic: the rise and decay times of dwarf-nova outbursts, the propagation of a disturbance through an X-ray binary’s disc, and the timescale on which a quasar’s brightness can change all measure it.

The numbers that come out of those are around a tenth. The numbers that come out of numerical simulations of magnetically driven turbulence, which is the favoured mechanism, are around a hundredth. That is an order-of-magnitude disagreement between the best measurement and the best calculation of the same quantity, it has stood for two decades, and it is the reason the transport problem is still described as open.

A parameter that the easiest observable is blind to and the hardest one measures is a parameter that stays unsettled, and the accretion disc’s is the standing example.

One more decade of radius shows how much of the spectrum comes from how far out.

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.
Fig. 8 The same summed spectrum built from four decades of disc rather than three. The extra decade adds only to the long-wavelength tail, because each annulus contributes near its own temperature and the outer annuli are cold — the peak is set by the inner edge and nothing else.

Where this ladder goes next

Later rungs on this anchor: the corona and the hard X-ray power law, and what Comptonisation of a known seed spectrum can be made to give up; the relativistically broadened iron line as a spin measurement; the thermal-viscous instability and the outburst cycles it produces; the states of X-ray binaries and the jets that accompany them; and the boundary layer, which is the component that exists when the central object has a surface and is absent when it does not.

Where this goes next

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. The main sequence against its own ceiling. Luminosity against mass, in solar units, with the Eddington limit drawn on the same axes. The limit is exactly linear in mass. The main sequence is not — until it is: above about 55 solar masses the relation flattens to a straight proportionality as well, and the two lines then run parallel at a fixed ratio of 83 per cent. They never cross on this range, which is the more interesting outcome: the most massive stars do not approach their limit gradually, they are built at a fixed fraction of it. The Sun, by contrast, radiates 0.0026 per cent of its own ceiling. A brightness that would blow the star apart Light pushes. For a star of a given mass there is a luminosity at which the outward push on its own outer layers equals the inward pull of gravity, and it depends on nothing but the mass and the opacity — not on the star's structure, its composition or its age. A 20-day delay, recovered at 20 days from two curves that share no wavelength. Above: a quasar's continuum, drawn as a damped random walk with a 90-day damping time, and the broad emission line responding to it. The line curve is the continuum convolved with a top-hat response of half-width 20 days, so it is later and smoother — it varies only 51 per cent as much, because at any instant it is an average of the continuum over a range of light-travel times. Below: the cross-correlation of the two, which peaks at 20 days. That number is a length: 20 light-days is 5.2·10¹⁴ metres, or 3463 astronomical units, and it has been measured for an object that subtends 3.5·10⁻⁵ arcseconds at a hundred megaparsecs — some thirty times finer than the best optical interferometry has ever resolved anything, and reached here with a photometer and a clock. The irregularity of the continuum is what makes this work. A periodic source would give a cross-correlation with many equal peaks and no way to choose; a random one gives a single peak, and the whole method rests on active nuclei being erratic. A size measured from a delay An active nucleus at redshift two is a point source in every telescope ever built, and its central mass is measured anyway. The continuum varies, the broad lines follow days later, and the lag is a light-travel time — which is a length, recovered from two light curves that share no wavelength.

About the same objects

Not linked from either essay — found by the objects both name.

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 discAccretion rateDisc truncationEddington limitEffective temperatureInnermost stable orbitMulticolour blackbodyRadiative efficiencyReverberation mappingSpectral stateThermal spectrumViscous dissipation