Astronomy & Celestial Mechanics
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Stars

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.

Assumes Angular momentum and Eddington limit.

17 min read 8 figures Light is the only messengerTwo bodies, and no more

The most efficient way known of turning mass into light is not fusion. Fusing hydrogen to helium releases 0.7% of the rest energy; matter falling onto a black hole can release eight times that, and the quasars — the brightest sustained sources in the universe — run on the second process rather than the first, which is why a galaxy’s nucleus can outshine the galaxy.

The difficulty is that falling onto a black hole is very hard to arrange. A gas cloud a light-year from a black hole has, by any reasonable standard, an enormous amount of angular momentum; to reach the hole it must lose essentially all of it, and angular momentum is conserved. The energy is easy and the angular momentum is the problem, and a disc is the machine built to solve 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.
Fig. 1 Effective temperature against radius for a disc round a ten-solar-mass black hole accreting 10−810^{-8} solar masses a year. Two features are structural. The profile turns over rather than rising all the way in, because the factor (1−rin/r)(1 - \sqrt{r_{\rm in}/r}) is the statement that no torque acts across the inner edge, so nothing is dissipated there and the peak sits at 49/3649/36 of it. And outside a few inner radii the run is exactly r−3/4r^{-3/4}, which is what makes a disc’s spectrum broad: every decade of radius contributes at a temperature a factor of 5.6 lower. Integrating the whole profile gives GMM˙/2rinGM\dot M/2r_{\rm in} to a per cent — half the binding energy released and no more, because the other half is still going round.

Why it must be a disc

Consider a parcel of gas with specific angular momentum ℓ=GMr0\ell = \sqrt{GMr_0}, in a circular orbit at r0r_0. Its energy is −GM/2r0-GM/2r_0. To reach a smaller radius rr on a circular orbit it needs energy −GM/2r-GM/2r, which is lower, so energy must be removed — that is easy, since gas radiates. It also needs angular momentum GMr\sqrt{GMr}, which is smaller, so angular momentum must be removed too, and there is nothing to radiate it to.

The resolution is that the angular momentum is not destroyed but transported: an inner annulus gives some to an outer one, so material at small radii sinks while a small amount of material at large radii is pushed outwards carrying nearly all the angular momentum of the system. A disc, in this reading, is not a shape that gas happens to settle into. It is a conveyor with the two commodities running in opposite directions.

Where the disc stops, and the spin that follows from it. Two radii against accretion rate, for a neutron star with a 10⁸-gauss field. The falling curve is the magnetospheric radius, where the field's stress on the disc matches the rate at which the flow carries angular momentum inward; it goes as the accretion rate to the minus two sevenths, which is a weak enough dependence that the factor of 1000 in supply drawn here moves the boundary by a factor of 7.2. The horizontal lines are corotation radii for three spin periods — the radius at which the disc orbits as fast as the star turns. Above corotation the field is spinning the gas faster than it wants to go and flings it out; below, the gas is faster and spins the star up. So the crossing is an attractor, and a star accreting steadily walks to the period where the two coincide. That period goes as the field to the six sevenths and the rate to the minus three sevenths — which is why a neutron star that has swallowed a tenth of a solar mass from a companion comes out at a few milliseconds, and why the millisecond pulsars have fields ten thousand times weaker than the young ones.
Fig. 2 Where the disc stops when the central object has a magnetic field. Inside the radius at which the field’s pressure exceeds the accreting gas’s ram pressure the disc is disrupted and the material is channelled along field lines — so the inner edge is not the star’s surface but a magnetospheric radius that depends on the field strength and the accretion rate. Spin the star fast enough and that radius moves outside the corotation point, at which the field flings arriving material away rather than accepting it. The angular momentum still has to go somewhere; this is the case where it goes back out.

The temperature run, from energy alone

The remarkable feature of the standard thin-disc solution is how little it needs. Write down conservation of mass and angular momentum between annuli, assume the disc is thin, steady and Keplerian, and require that the torque vanish at the inner edge. The dissipation per unit area follows:

σT4(r)=3GMM˙8πr3(1−rinr),\sigma T^4(r) = \frac{3GM\dot M}{8\pi r^3}\left(1 - \sqrt{\frac{r_{\rm in}}{r}}\right),

and nothing about the transport mechanism appears in it. Whatever carries the angular momentum — magnetic stresses, turbulence, spiral waves — the local dissipation rate is fixed by the bookkeeping. That is why the theory was useful for twenty years before anybody knew what the mechanism was.

Two checks of the expression are worth having. Its maximum is at r=(49/36) rinr = (49/36)\,r_{\rm in}, which follows from differentiating and is compared in the figure against the peak of the drawn curve. And integrating it over both faces from rinr_{\rm in} to infinity gives GMM˙/2rinGM\dot M/2r_{\rm in}: exactly half the binding energy at the inner edge, with the other half remaining as orbital kinetic energy of the material that has arrived.

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. 3 What that profile emits. Each annulus is a blackbody at its own temperature, drawn faintly beneath the sum, and each at the area it actually has — the outer rings 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 1/3, which comes out of integrating ν2T(r) r dr\nu^2 T(r)\,r\,dr with T∝r−3/4T\propto r^{-3/4}. That middle section is the observational signature of a disc: no photosphere produces it, and its width rather than its peak says how far in the disc goes.

The efficiency belongs to the inner edge

The energy released per unit mass is GM/2rinGM/2r_{\rm in}, so as a fraction of mc2mc^2 it is GM/2rinc2GM/2r_{\rm in}c^2 — a function of the inner radius in gravitational radii and of nothing else. Not the mass, which cancels; not the accretion rate, which sets the luminosity and not the efficiency.

That single observation sorts the accreting objects.

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 The fraction of rest energy released, against the radius the disc has to stop at. 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 Newtonian curve — the Newtonian expression gives 8.3% at the same radius, 1.46 times too much, and the difference is where this picture has to be handed over to the metric. Hydrogen fusion, 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.

The relativistic value is quoted rather than derived here — the geodesics belong with the physics of the metric — but the structure of the answer is the point. A Schwarzschild black hole’s innermost stable circular orbit is at 6GM/c26GM/c^2 and the binding energy there is 1−8/9=5.72%1 - \sqrt{8/9} = 5.72\%. A maximally rotating one has its last stable orbit at GM/c2GM/c^2 and returns 42%, which is the largest efficiency any process in the universe achieves. The spin of a black hole is worth a factor of seven in how brightly it can shine, and measuring quasar efficiencies is one of the few ways of getting at black-hole spins statistically.

What was actually measured

Four measurements, at four scales, and none of them is a fit to the theory it confirms.

Cataclysmic variables. A white dwarf accreting from a companion shows an eclipse of the disc as the companion passes, and the eclipse’s shape maps the disc’s brightness against radius. Eclipse mapping recovers T(r)T(r) directly, and for systems in a steady state it comes out as r−3/4r^{-3/4} within the errors. That is the temperature law measured rather than assumed.

X-ray binaries. A ten-solar-mass black hole accreting near its limit has a peak disc temperature of a few million kelvin, which is soft X-rays, far outside any of the bands a magnitude is quoted in; the observed spectra are fitted with multicolour disc models, and the fitted inner radius comes out constant as the luminosity varies over a factor of ten — which is what it should do if it is the last stable orbit and not something that moves.

Quasars. The mean radiative efficiency of the whole quasar population can be obtained by comparing the total light they have emitted, integrated over cosmic time, against the total mass now sitting in black holes at galaxy centres. That is Sołtan’s argument, and it gives about 10% — comfortably above the 5.7% of a non-rotating hole, which says the population is on average spinning.

The Eddington limit as a ceiling. No accreting object is observed far above the luminosity at which radiation pressure balances gravity, and that is a prediction with nothing adjustable in it.

What actually moves the angular momentum

The bookkeeping above works without naming a mechanism, and for two decades nobody could name one. That gap is worth describing, because it is a rare case of a theory being quantitatively successful while its central term was openly a placeholder.

Ordinary molecular viscosity in a disc is hopeless. The mean free path of a proton in a disc round a stellar-mass black hole is centimetres and the disc is 10910^9 cm across, so the viscous timescale comes out at 101310^{13} years — longer than the age of the universe by three orders of magnitude, for a process observed to happen in days. The gas must be turbulent, and the question was what makes it turbulent, since a Keplerian disc is linearly stable to hydrodynamic perturbations by the Rayleigh criterion: its specific angular momentum increases outwards, which is exactly the stable case.

The answer, found by Balbus and Hawley in 1991, is that a magnetised Keplerian disc is violently unstable. Two fluid elements joined by a weak field line, one slightly inside the other, are pulled together by magnetic tension; the tension slows the inner one and speeds the outer one, so the inner one falls further in and the outer one moves further out, stretching the field and increasing the tension. The instability’s growth rate is of order the orbital frequency, and it needs only a field weak enough to be dynamically negligible.

A Keplerian shear that is hydrodynamically stable is magnetically unstable, and the reason it took until 1991 is that the result was published in a plasma-physics context in 1959 and nobody in accretion theory read it.

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 The same profile around a neutron star, where the inner edge is a surface rather than an orbit. The disc’s temperature at each radius is fixed by the rate at which orbital energy is being dissipated there, and the innermost radius sets both the peak temperature and the total efficiency — ten kilometres puts the peak in the X-ray and the efficiency near a tenth of the rest mass. Change the central object and the same integral moves band by band; the mechanism that requires angular momentum to be carried outward does not change at all.

All three of the phenomena above are absent from the steady solution and all three are what most accreting systems are observed doing, which is a fair summary of what the standard model is for: it fixes the scaling and the energetics, and everything time-dependent has to be added.

Where the model stops

The standard thin disc is a wonderfully productive model and it is wrong in at least four identifiable ways.

The viscosity is parameterised, not derived. Shakura and Sunyaev wrote the stress as α\alpha times the pressure and left α\alpha to be fitted; the physical mechanism, the magnetorotational instability, was not identified until 1991 and the resulting α\alpha is not constant.

The disc is not always thin. At very low accretion rates the gas cannot radiate what it dissipates and puffs into a hot, thick, radiatively inefficient flow — which is what the Galactic centre’s black hole is doing, at a luminosity a hundred million times below its Eddington limit. At very high rates radiation pressure thickens it the other way.

The inner region is not a blackbody. Electron scattering dominates there, so the emergent spectrum is harder than a blackbody at the local temperature, and every inner-radius measurement carries a colour correction of order 1.7 that is estimated rather than measured.

Half the luminosity of a real X-ray binary is not in the disc at all. It is in a hot corona above it, producing a power-law spectrum by Compton scattering the disc’s photons — a component the thin-disc solution says nothing about and which carries the information about the innermost regions. Three assumptions in that list are worth taking further rather than merely noting, because each one turns into a phenomenon of its own.

The disc that is not steady

Everything above assumes a steady state: mass flows in at the outer edge at the same rate it reaches the centre, and the temperature profile is constant. A large fraction of observed accreting systems do nothing of the kind. They sit quiet for months and then brighten by a factor of a hundred in a day.

The mechanism is a thermal instability with a specific cause, and the cause is hydrogen.

Between about six and eight thousand kelvin, hydrogen is partially ionised, and in that range the opacity depends extremely steeply on temperature — as roughly the tenth power. A disc annulus sitting there is unstable: heat it slightly and the opacity rises, trapping more heat, raising the temperature further. There is no equilibrium in the middle of the range, only a cool branch below it and a hot branch above.

So an annulus supplied with mass at a rate that would put it in the unstable range cannot sit there. It accumulates mass on the cool branch, its surface density rising, until it crosses the upper limit of the cool branch and jumps to the hot one — at which point its viscosity rises by two orders of magnitude, it drains rapidly, and the transition propagates through the disc as a heating front.

That is an outburst. The disc dumps a substantial fraction of its stored mass onto the central object in days, then falls back to the cool branch and begins accumulating again.

The observational signature is a limit cycle rather than a variation: a fast rise, a slower decay, and a quiescent interval whose length depends on the supply rate. Dwarf novae do it every few weeks; X-ray transients around black holes do it every few decades and reach a thousand times their quiescent luminosity.

The steady disc is therefore the exception rather than the rule, and the systems that are steady are the ones whose supply rate is high enough to keep the whole disc on the hot branch or low enough to keep it all on the cool one.

The instability is also the reason a great many accreting systems were catalogued as variable stars long before anybody knew what they were.

The same three readings for a disc around a stellar-mass object rather than a supermassive one show which of the conclusions scale and which are absolute.

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. 6 The multi-temperature spectrum for the other central mass. It is a sum of blackbodies and belongs to no temperature at all, and the peak moves with the mass to the minus quarter power — which is why one class of object radiates in X-rays and the other in the ultraviolet from the same physics.
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. 7 And the efficiency, which does not move. The fraction of rest mass released before the inner edge depends only on where the inner edge is in gravitational radii, and that is a property of the metric rather than of the mass — the one quantity in the whole model that is the same for every black hole.

What leaves instead of falling in

The accounting has treated the accretion rate as the rate at which mass reaches the centre. For a substantial fraction of systems it is not, because some of the material leaves.

A magnetised disc launches a wind. Field lines threading the disc and inclined outward act as rigid wires attached to the rotating gas, so material lifted off the surface is flung along them and accelerated centrifugally — the same lever-arm mechanism that brakes a star’s rotation, applied to a disc.

That wind removes mass and it removes angular momentum, and the second is the important one. A wind carrying a small fraction of the mass can carry a large fraction of the angular momentum, because it leaves from a large radius on a long lever. So a disc with a wind does not need internal transport to be as efficient: the angular momentum goes out vertically rather than radially.

The observational evidence is direct. Ultraviolet and X-ray spectra of accreting systems show absorption lines blueshifted by hundreds to thousands of kilometres a second — material between the observer and the source, moving outward. In the most luminous quasars the outflow speeds reach a tenth of the speed of light and the inferred mass loss rates are comparable with the accretion rates.

The consequence for the energy budget is that a luminosity measured at the centre underestimates what the disc has done. Some of the released energy went into lifting and accelerating the wind, and that fraction is not radiated.

There is a further consequence at galaxy scale. A wind carrying a per cent of the accretion luminosity, deposited in the surrounding gas, is enough to disturb the whole galaxy — which is the standard explanation for why the mass of a central black hole correlates with the properties of a galaxy that its gravity does not reach.

The edge that is set by a magnetic field

The efficiency argument turns on where the disc stops, and for a black hole that is a property of the metric. For an accreting object with a magnetic field it is not, and the inner edge is set by a competition.

A magnetised central object threads its field through the inner disc. Inside some radius the magnetic stress exceeds the material stress, the gas is forced to co-rotate with the star rather than orbit Keplerian, and the disc is truncated. That radius depends on the field strength and on the accretion rate: a stronger field pushes it out, a higher rate pushes it in.

The consequences depend on how that radius compares with the radius at which the star’s own rotation is Keplerian.

If the truncation radius is inside the co-rotation radius, the field lines connecting star and disc are being wound in the sense that transfers angular momentum to the star, and material flows along them onto the poles. That is the ordinary accreting state, and the accretion is channelled onto small polar caps rather than spread over the surface.

If the truncation radius is outside, the star’s field is sweeping the inner disc faster than the disc orbits, so the torque is in the other direction: the field spins the material up and flings it out. That is the propeller regime, and accretion is inhibited.

The transition between the two is observable. An accreting pulsar whose accretion rate falls will move its truncation radius outward, and when it crosses the co-rotation radius the source turns off far more sharply than the declining supply alone would explain.

A magnetised star therefore sets its own inner boundary and adjusts it, and the same argument applied to a young star with a kilogauss field puts the truncation at several stellar radii — which is where the accretion columns and the jets of a forming star originate.

And the case where the inner edge is set by something other than the metric, which is the one configuration the model has to be told about rather than deriving.

Where the disc stops, and the spin that follows from it. Two radii against accretion rate, for a neutron star with a 10⁹-gauss field. The falling curve is the magnetospheric radius, where the field's stress on the disc matches the rate at which the flow carries angular momentum inward; it goes as the accretion rate to the minus two sevenths, which is a weak enough dependence that the factor of 1000 in supply drawn here moves the boundary by a factor of 7.2. The horizontal lines are corotation radii for three spin periods — the radius at which the disc orbits as fast as the star turns. Above corotation the field is spinning the gas faster than it wants to go and flings it out; below, the gas is faster and spins the star up. So the crossing is an attractor, and a star accreting steadily walks to the period where the two coincide. That period goes as the field to the six sevenths and the rate to the minus three sevenths — which is why a neutron star that has swallowed a tenth of a solar mass from a companion comes out at a few milliseconds, and why the millisecond pulsars have fields ten thousand times weaker than the young ones.
Fig. 8 The truncation radius for a magnetic field ten times stronger. The disc stops where the magnetic pressure matches the accreting material’s ram pressure, so a stronger field pushes the inner edge outward — and the accretion then arrives along field lines at the poles rather than through a boundary layer.

Where this ladder goes next

This rung is the steady, thin, Newtonian disc and its energy accounting. The rungs above go outward and inward.

Outward: the same physics with the compact object replaced by a young star gives a protoplanetary disc, where the temperature run sets the snow line and therefore where a giant planet can form, and where the same angular-momentum transport is what lets a star assemble at all.

Inward: what the last stable orbit means, why it exists, and what the 5.72% is actually a statement about. That belongs with the metric rather than with the disc, and this essay has deliberately taken it as an input — but the observational half, of measuring an inner radius from a spectrum and reading a spin off it, is a rung this ladder can climb.

And across: the variability. A disc is not steady; it flickers on every timescale from milliseconds to years, and the flickering is how the innermost regions are probed, because a light curve resolves in time what no telescope resolves in angle.

Where this goes next

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. 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. Equal areas in equal times, at eccentricity 0.65. Positions of an orbiting body at equal intervals of time, obtained by solving Kepler's equation. The two shaded sectors span the same interval and enclose the same area — a long thin one at the far end, a short fat one at the close approach. Equal areas in equal times, which is angular momentum in disguise Kepler's second law is a statement about the area a radius line sweeps. It looks like an odd thing to have noticed, and it turns out to be a conservation law arriving eighty years early.

What this makes readable

Essays that name this one as a prerequisite.

About the same objects

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

What links here

The 8 of 20 essays linking to this one that name the most of the same objects.

The objects this essay names

Each one links to every other essay that touches it.

Accretion discAngular momentum transportEddington limitInnermost stable orbitMagnetorotational instabilityMulticolour blackbodyQuasarRadiative efficiencySpecific angular momentumViscosity