Astronomy & Celestial Mechanics
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A flare that puts a ceiling on a mass

A star torn apart by a black hole lights up for a year. The tidal radius grows as the cube root of the hole's mass and the horizon grows as the mass itself, so above about a hundred million suns the star is swallowed whole and nothing is seen — which makes the existence of a flare a measurement.

Assumes Galactic nuclei, Accretion and Tides.

17 min read 8 figures Measuring the unreachableScales that defeat intuition

Most of the black holes at the centres of galaxies are doing nothing. They are not accreting, they emit no light, and the only reason anybody believes they are there is the motion of the stars around them. A quiet hole is the hardest kind of object to study, because the thing that makes it detectable is precisely the thing it is not doing.

Once every ten to a hundred thousand years per galaxy, one of them is handed a star. The star is torn apart, half of it comes back, and for a year or so the nucleus is bright enough to be picked out of a survey — brighter, briefly, than the whole population of ordinary galaxies a survey is counting. What makes these events worth an essay is not the brightness. It is that the existence of the flare is itself a measurement of the hole’s mass, obtained without resolving anything.

Two lengths that cross at 1.1·10⁸ solar masses, above which nothing is seen. The radius at which a star of 1 solar radius and 1 solar mass is pulled apart by a black hole, and the hole's own horizon, both against the hole's mass and both on logarithmic axes. The tidal radius is the star's own radius times the cube root of the mass ratio, so it climbs with a slope of one third; the horizon is proportional to the mass, so it climbs with a slope of one. Two lines of different slope cross once, and this pair crosses at 1.14·10⁸ solar masses. Below that the star is torn apart outside the horizon, half of it is thrown out and half falls back, and the fallback is visible for months. Above it the star crosses the horizon while it is still a star, is swallowed whole, and produces no flare at all. The consequence is the reason these events are worth watching: a flare that is seen is an upper limit on the mass of the hole that made it, obtained without resolving anything, and it is the only such limit available for a hole that is not currently accreting.
Fig. 1 The radius at which a Sun-like star is pulled apart by a black hole, and the hole’s own horizon, both against the hole’s mass. The tidal radius is the star’s radius times the cube root of the mass ratio, so it climbs with a slope of one third; the horizon is proportional to the mass, so it climbs with a slope of one. Two lines of different slope cross once, at about a hundred million solar masses. Below it the star is torn apart outside the horizon and the debris is visible for months. Above it the star crosses the horizon while it is still a star, is swallowed whole, and produces no flare at all.

The crossing is called the Hills mass, and it is the whole argument in one picture. Drawn against the horizon, as here, it is the simplest form of the argument and an upper estimate: a star that passes within four gravitational radii of a hole that does not spin cannot come back out even though it has not crossed the horizon, which lowers this ceiling by a factor of nearly three — and a spinning hole moves it again, in either direction, because the line that actually decides capture is not the horizon.

The same criterion as a moon

The condition for a star to be pulled apart is the condition for anything to be pulled apart, and this collection has already met it. A satellite held together by its own gravity survives until the tide across it exceeds its own surface gravity, and the distance at which that happens depends only on the ratio of the two mean densities. Writing the criterion in terms of densities is what makes the crossing in the hero inevitable. The hole’s mean density inside its horizon falls as the inverse square of its mass, because the radius rises linearly and the volume as the cube. A star’s density is fixed. So there is always a hole so large and so diffuse that a star crosses its horizon before the tide across the star has become interesting, and the only question is where.

For a Sun-like star the answer is around a hundred million solar masses. That is not an exotic value: plenty of galaxies host holes above it. So a survey of these flares is a survey of the lower end of the black hole mass function, and it is blind above a line it can compute. That is an unusual position to be in: most selection effects in astronomy are against the faint and the distant, and this one is against the massive.

Half of it never comes back

Now suppose the star does come apart. It arrives on an orbit that is very nearly parabolic — it fell in from far away, so its centre of mass has almost exactly zero energy — and it crosses the tidal radius in less time than it takes sound to cross the star. Whatever the tidal field has done to the binding energy of each piece by that moment is frozen in.

A binding energy that runs from bound to unbound across one stellar diameter. The specific orbital energy of the debris, plotted against where in the star it came from, for a 1 solar-mass star disrupted by a hole of 10⁶ solar masses. The star arrives on a nearly parabolic orbit, so its centre has almost exactly zero energy; the tidal field across its diameter adds and subtracts the same amount either side, 1.91·10¹³ joules per kilogram. That is a small number beside the 61767 kilometres a second the escape speed at the tidal radius corresponds to, and it is the entire structure of the event. The near half of the star is left bound and comes back; the far half is unbound and leaves, at speeds of thousands of kilometres a second, never to be seen again. Because the star crosses the tidal radius far faster than it can readjust, the distribution across it is frozen at that instant and is very nearly flat in energy — equal masses at equal intervals. The most bound material, at the extreme left, is the first to return, 41 days later, and everything after that is the same distribution read as a clock.
Fig. 2 The specific orbital energy of the debris, against where in the star it came from, for a solar star disrupted by a hole of a million solar masses. The star’s centre has almost exactly zero energy; the tidal field across its diameter adds and subtracts the same amount either side. The near half is left bound and comes back; the far half is unbound and leaves at thousands of kilometres a second, never to be seen again. Because the star crosses the tidal radius far faster than it can readjust, the distribution across it is frozen at that instant and is very nearly flat in energy — equal masses at equal intervals.

Two consequences follow immediately and neither is obvious. First, half the star is lost. It leaves the galaxy entirely, and it is not a small amount of mass moving at a small speed: a solar mass at several thousand kilometres a second is a considerable amount of energy going nowhere anybody can see it. Second, the returning half arrives with every binding energy between zero and the maximum, not with one.

That flatness is what makes the light curve computable. The distribution of debris in energy is very close to uniform, and the return time of an orbit is fixed by its energy alone.

Two lengths that cross at 1.1·10⁸ solar masses, above which nothing is seen. The radius at which a star of 1 solar radius and 1 solar mass is pulled apart by a black hole, and the hole's own horizon, both against the hole's mass and both on logarithmic axes. The tidal radius is the star's own radius times the cube root of the mass ratio, so it climbs with a slope of one third; the horizon is proportional to the mass, so it climbs with a slope of one. Two lines of different slope cross once, and this pair crosses at 1.14·10⁸ solar masses. Below that the star is torn apart outside the horizon, half of it is thrown out and half falls back, and the fallback is visible for months. Above it the star crosses the horizon while it is still a star, is swallowed whole, and produces no flare at all. The consequence is the reason these events are worth watching: a flare that is seen is an upper limit on the mass of the hole that made it, obtained without resolving anything, and it is the only such limit available for a hole that is not currently accreting.
Fig. 3 The two lengths again, and the only thing that has changed is which hole is marked on them. The crossing is where it was — at 1.1×1081.1\times10^8 solar masses, because it is set by the star and not by the hole — and the mark now sits at 10710^7, a decade below it. The ceiling is a property of the crossing and the mark is a choice of subject, which is worth separating: every flare ever seen comes from a hole somewhere to the left of that crossing, and the crossing itself is a statement about the star that was eaten.

An exponent that is a change of variable

Put those two facts together. If mass is spread evenly in binding energy, and if the period of an orbit goes as the binding energy to the power minus three halves, then the mass returning per unit time is fixed by nothing but the chain rule.

Minus five thirds, and the 41 days before it starts. The rate at which disrupted material returns to the hole, against time since the disruption, both axes logarithmic, for a 1 solar-mass star and a hole of 10⁶ solar masses. Nothing arrives for 41 days, which is the orbital period of the most bound debris and is the one timescale in the problem; then the rate peaks at 2.97 solar masses a year and falls with a logarithmic slope of −1.667. That exponent is not fitted and not assumed. It follows from how the debris is spread in orbital energy by one change of variable: the mass is spread evenly in binding energy, the return time of an orbit goes as the binding energy to the minus three halves, and a flat distribution in energy read as a distribution in time is exactly a minus five thirds power. The horizontal line is the rate at which this hole could radiate at its Eddington limit if a tenth of the rest mass came out as light. The return rate is above it for the first part of the flare, so what is actually seen depends on what happens to material that arrives faster than it can be radiated away — which is why the light curves of real events follow this slope less often than the mass return does.
Fig. 4 The rate at which disrupted material returns to the hole, against time since the disruption. Nothing arrives for about forty days, which is the orbital period of the most bound debris and is the one timescale in the problem; then the rate peaks and falls with a logarithmic slope of exactly minus five thirds. That exponent is not fitted and not assumed: it is what a flat distribution in energy becomes when energy is read as a return time. The horizontal line is the rate at which this hole could radiate at its Eddington limit — the return outruns it for the first part of the flare.

The minus five thirds is one of the few genuinely parameter-free predictions in the subject, and it is worth being clear about what it predicts. It is the rate at which mass comes back, not the rate at which light comes out. Those are the same thing only if everything that returns is promptly converted into radiation at a fixed efficiency, and the figure shows why that cannot hold at early times: the return rate exceeds the Eddington rate by a large factor for the first months.

A binding energy that runs from bound to unbound across one stellar diameter. The specific orbital energy of the debris, plotted against where in the star it came from, for a 1 solar-mass star disrupted by a hole of 10⁷ solar masses. The star arrives on a nearly parabolic orbit, so its centre has almost exactly zero energy; the tidal field across its diameter adds and subtracts the same amount either side, 4.11·10¹³ joules per kilogram. That is a small number beside the 133074 kilometres a second the escape speed at the tidal radius corresponds to, and it is the entire structure of the event. The near half of the star is left bound and comes back; the far half is unbound and leaves, at speeds of thousands of kilometres a second, never to be seen again. Because the star crosses the tidal radius far faster than it can readjust, the distribution across it is frozen at that instant and is very nearly flat in energy — equal masses at equal intervals. The most bound material, at the extreme left, is the first to return, 130 days later, and everything after that is the same distribution read as a clock.
Fig. 5 The binding-energy spread for a hole ten times lighter than the standard case. The picture is the same shape — half the debris bound, half unbound, and the division running through the star’s centre — and the scale of the energy spread is what has changed, because the tidal field across the star at the disruption radius depends on the hole. A smaller spread means a longer period for the most bound material, and the next figure is that statement turned into a time.

What the returning material has to do

Debris on a highly eccentric orbit does not simply fall in. It comes back to pericentre, swings around, and goes out again on almost the same orbit. Turning that into an accretion flow requires the streams to intersect and dissipate, and turning the dissipated energy into light requires it to reach a radius from which it can escape. Above a certain rate it cannot: radiation pressure on the infalling material balances gravity and the excess is driven back out, which is the limit the previous figure’s horizontal line marks. The disc has to export angular momentum outward in order to let mass move inward, which takes time, and the time it takes is not the fallback time. If the viscous time is longer than the fallback time the light curve is smeared and the minus five thirds is not seen at all; if shorter, the light tracks the return rate. This is where the honest account has to stop being tidy. The observed flares do follow power laws, and the fitted exponents cluster near minus five thirds often enough to be convincing as a class, but individual events depart from it by amounts that no single reprocessing model has explained.

Why anybody would want this measurement

There is a well-established way to weigh a black hole in a galaxy that is not accreting: measure the velocity dispersion of the stars in the bulge and use the empirical relation between the two. That relation works well at the top end and badly at the bottom, and the bottom is where the question is. Whether every galaxy has a central hole, and how small the smallest ones are, is a question about objects too faint and too distant to weigh by stellar motions. A disruption flare gives a different handle. Its existence bounds the mass from above through the Hills crossing; the timing of its rise bounds it from below, because the return time of the most bound debris scales as the square root of the hole’s mass; and neither requires resolving anything.

What the flare actually measures, and what it assumes

The chain from a light curve to a mass has three links and the middle one is the weak one.

The first link is solid. The peak time is the orbital period of the most bound debris, and that period is fixed by the hole’s mass, the star’s mass and the star’s radius through the geometry above. Doubling the hole’s mass lengthens it by a factor of the square root of two.

The second link is the assumption that the star was Sun-like. A disrupted red giant and a disrupted main-sequence star of the same mass give timescales differing by a factor of tens, because the radius enters to the power three halves — and where a star sits on the main sequence fixes its radius only while it is on it. Nothing in the flare says which was disrupted, and the only defence is a population argument: giants are rare and their disruption produces a flare so slow that a survey would classify it as something else.

The third link is that the light is tracking the mass. That is the one the previous section undermined.

Minus five thirds, and the 13 days before it starts. The rate at which disrupted material returns to the hole, against time since the disruption, both axes logarithmic, for a 1 solar-mass star and a hole of 10⁵ solar masses. Nothing arrives for 13 days, which is the orbital period of the most bound debris and is the one timescale in the problem; then the rate peaks at 9.40 solar masses a year and falls with a logarithmic slope of −1.667. That exponent is not fitted and not assumed. It follows from how the debris is spread in orbital energy by one change of variable: the mass is spread evenly in binding energy, the return time of an orbit goes as the binding energy to the minus three halves, and a flat distribution in energy read as a distribution in time is exactly a minus five thirds power. The horizontal line is the rate at which this hole could radiate at its Eddington limit if a tenth of the rest mass came out as light. The return rate is above it for the first part of the flare, so what is actually seen depends on what happens to material that arrives faster than it can be radiated away — which is why the light curves of real events follow this slope less often than the mass return does.
Fig. 6 The fallback rate for a hole of 10510^5 solar masses — an intermediate-mass hole rather than a nuclear one. Nothing arrives for 13 days against the 130 of the standard case: the delay is the orbital period of the most bound debris, and it scales as the square root of the hole’s mass. A lighter hole gives a faster flare, which is the observational handle on the mass and also the reason the lightest holes are the hardest to catch: the whole event is over before most surveys revisit the field.

The population, and what it is a census of

There is a further reason to care about the rate at which these flares occur. A hole that has been fed by disruptions throughout the history of a galaxy has grown by a computable amount, and comparing that with how much it must have grown by ordinary accretion is a constraint on both. The arithmetic is worth doing because it settles a question that sounds harder than it is. At one solar mass per ten thousand years, a hole gains a hundred thousand solar masses over the age of the universe. For a hole of a hundred million that is a rounding error. So disruption is not how these objects were built; it is how the ones that were built quietly can be found.

Two lengths that cross at 2.6·10⁹ solar masses, above which nothing is seen. The radius at which a star of 8 solar radii and 1 solar mass is pulled apart by a black hole, and the hole's own horizon, both against the hole's mass and both on logarithmic axes. The tidal radius is the star's own radius times the cube root of the mass ratio, so it climbs with a slope of one third; the horizon is proportional to the mass, so it climbs with a slope of one. Two lines of different slope cross once, and this pair crosses at 2.59·10⁹ solar masses. Below that the star is torn apart outside the horizon, half of it is thrown out and half falls back, and the fallback is visible for months. Above it the star crosses the horizon while it is still a star, is swallowed whole, and produces no flare at all. The consequence is the reason these events are worth watching: a flare that is seen is an upper limit on the mass of the hole that made it, obtained without resolving anything, and it is the only such limit available for a hole that is not currently accreting.
Fig. 7 The same crossing for a star eight times the Sun’s radius and the same mass — an evolved star on the subgiant branch. The tidal radius moves up by a factor of eight while the horizon does not move at all, so the crossing shifts upward by a factor of about twenty-two: giants can be disrupted by holes that would swallow a main-sequence star whole. This is the loophole in the ceiling, and it is why a flare from a nucleus known to be above the Hills mass is evidence about what was disrupted rather than evidence against the argument.

The rate is wrong, and the hosts are peculiar

Two facts about the population sit awkwardly beside the tidy argument above, and both are informative.

The first is the rate. Calculating how often a star is scattered onto an orbit that reaches inside the tidal radius is a well-posed problem: stars in a nucleus diffuse in angular momentum through two-body encounters, the orbits that lead to disruption occupy a narrow cone in velocity space, and the rate is the rate at which that cone is refilled. Doing the calculation for a typical galaxy gives something like one disruption per ten thousand years.

What surveys find is closer to one per hundred thousand — a factor of several to ten below. The discrepancy has survived every improvement in the surveys, so it is not a matter of missing faint events, and the candidate explanations divide into the astrophysical and the observational. Either something removes stars from the loss cone before they reach it, or a substantial fraction of disruptions produce something a survey does not classify as a disruption. Both are plausible and neither is established.

The second fact is stranger and is a genuine clue. Disruption flares are strongly overrepresented in a particular kind of galaxy: post-starburst systems, whose spectra show strong absorption from a large population of A stars with no ongoing star formation — a galaxy that formed stars vigorously and stopped within the last billion years. Those galaxies are a small fraction of a per cent of the local population and they host a substantial fraction of the observed flares, an overrepresentation of order thirty.

That is far too large to be a coincidence and far too large to be a selection effect. The leading reading is dynamical: a post-starburst galaxy is the remnant of a recent merger, and a merger leaves a nucleus that is dense, centrally concentrated and possibly still hosting a black-hole binary. Any of those raises the rate at which stars are delivered to the loss cone — a dense nucleus by shortening the relaxation time, a binary by scattering stars into it directly.

So the population statistics of these flares have turned into a probe of the dynamical state of galactic nuclei, which is a quantity nothing else measures. That was not what the technique was for.

The ones that come back

The account so far assumes the star is destroyed once. A growing number of events do not fit that, and they are reshaping what the light curves are read as.

A star on an orbit whose pericentre lies just outside the tidal radius is not disrupted but is stripped: the outer layers exceed the binding limit while the core survives, a fraction of a solar mass is lost, and the remnant returns on its original orbit to be stripped again. The resulting light curve is a series of flares with a fixed period, decaying in amplitude as the star loses mass — and several sources are now known to flare on periods of months to years with exactly that character.

Repeaters solve one problem and create another. They are attractive because a partial disruption is far more common than a full one — the cross-section for grazing a boundary exceeds that for crossing it — so a population of repeaters could account for some of the rate discrepancy above while producing flares that a survey classifies as variability rather than as a disruption. They are awkward because the period of a repeater is the orbital period of a bound star, which has nothing to do with the fallback time, so the timing argument that converts a rise time into a black hole mass does not apply to them at all.

Distinguishing the two classes therefore matters for every mass in the catalogue. A single flare observed once could be the first of a series, and reading its rise time as a fallback time would give a mass that is simply a different quantity. The only reliable discriminator is time: watch the nucleus for another decade and see whether it does it again.

Two lengths that cross at 5.7·10⁷ solar masses, above which nothing is seen. The radius at which a star of 0.5 solar radii and 0.5 solar masses is pulled apart by a black hole, and the hole's own horizon, both against the hole's mass and both on logarithmic axes. The tidal radius is the star's own radius times the cube root of the mass ratio, so it climbs with a slope of one third; the horizon is proportional to the mass, so it climbs with a slope of one. Two lines of different slope cross once, and this pair crosses at 5.72·10⁷ solar masses. Below that the star is torn apart outside the horizon, half of it is thrown out and half falls back, and the fallback is visible for months. Above it the star crosses the horizon while it is still a star, is swallowed whole, and produces no flare at all. The consequence is the reason these events are worth watching: a flare that is seen is an upper limit on the mass of the hole that made it, obtained without resolving anything, and it is the only such limit available for a hole that is not currently accreting.
Fig. 8 And the ceiling for a smaller star. A half-solar-mass main-sequence star of half the Sun’s radius is disrupted outside the horizon only up to 5.7×1075.7\times10^7 solar masses, against 1.1×1081.1\times10^8 for a solar twin: the tidal radius goes as R⋆M⋆−1/3R_\star M_\star^{-1/3} and the horizon does not care about the star at all. The ceiling is not one number for a galaxy; it is one number per kind of star, so a survey’s non-detection of flares above some hole mass is a statement about which stars were available to be eaten as much as about the holes.

The half that leaves

Everything above concerns the bound debris. The unbound half is not merely uninteresting leftovers, and following it makes a connection to a population that would otherwise look unrelated.

The far side of the star is left with positive energy of order the same magnitude as the bound side’s negative energy, which for a solar star at a million-solar-mass hole is a few thousand kilometres a second. That is well above the escape speed of any galaxy. So each disruption launches roughly half a solar mass of stellar debris out of its host at speeds that carry it into intergalactic space, and it does so at every disruption, everywhere, for the age of the universe.

The total is not negligible on the scale of a galaxy’s own enrichment, and it is essentially undetectable: half a solar mass of hydrogen and helium spread through a growing sphere thousands of parsecs across has no emission measure worth mentioning. It is one of the few genuinely unobservable outputs of a well-understood process.

The same energetics, applied to a different object, produce something that is observable. Replace the single star with a binary, and the tidal field pulls the pair apart rather than pulling a star apart — the binary’s own binding is far weaker, so this happens much further out, at a radius where nothing else interesting occurs. One component is captured onto a tight orbit around the hole; the other is ejected with the binary’s orbital velocity boosted by the exchange, at up to a few thousand kilometres a second.

That is the Hills mechanism, and its products are the hypervelocity stars: individual stars observed moving fast enough to leave the Galaxy, whose trajectories, traced back, point at the Galactic centre. A few dozen are known. They are the only direct evidence anyone has that a specific dynamical process is operating in a nucleus, since the process leaves the object it acted on intact and moving in a direction that records where it happened.

The captured component is the more consequential half, and it is invisible. A star left on a tight orbit around the hole is a candidate for later disruption, and repeated captures build up a population of stars bound to the hole on orbits far tighter than relaxation alone would supply — which feeds back into the rate problem of the previous section. The same encounter that produces the one observable star also produces the unobservable one that raises the disruption rate, and the two are not independent measurements of the nucleus but two ends of the same event.

Where the ladder goes

Two directions open from here and they are different in kind.

One is observational: what happens between the debris returning and the light coming out. That is a question about shocks, about how streams intersect when general relativity makes their orbits precess, and about where the reprocessing photosphere sits — and it is being answered by finding more events rather than by thinking harder about the ones already found.

The other is dynamical, and it is the one this collection is better placed to follow. A star is disrupted only if it arrives on an orbit whose pericentre is inside the tidal radius, and orbits like that are a tiny fraction of the phase space available. What puts stars onto them is two-body relaxation among the stars of the nucleus, acting on angular momentum rather than on energy — a diffusion into a narrow cone that empties as fast as it fills. The rate of these flares is therefore a measurement of a relaxation process nobody can watch, in a region nobody can resolve, which is the same trick this essay has already played once.

Where this goes next

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. The tide across a moon, against the moon's own gravity. The tidal acceleration across a satellite and the satellite's own surface gravity, both in units of that surface gravity, against distance from the primary in planet radii. The tide falls as the inverse cube and the self-gravity does not fall at all, so they cross once — at 2.23 radii for the density ratio drawn. Inside the crossing the tide wins and a body held together only by its own weight comes apart. The distance at which a moon stops holding together The tide across a body falls as the inverse cube; the body's own gravity does not fall at all. There is therefore exactly one crossing, and Saturn's rings end within a few per cent of it. 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.

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

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.

AccretionAccretion discEddington limitEvent horizonLight curveM sigma relationOrbital energySchwarzschild radiusSupermassive black holeTidal forceTidal radius