Energy

Feeding a black hole

Gas releases gravitational energy as it spirals into a black hole, producing light before it crosses the horizon.

How does gas falling into a black hole turn into light?

Gas approaching a hole generally has some sideways motion. As it falls inward, it spins faster, the way water speeds up as it circles a drain. So the gas piles into a flat, whirling pancake round the hole, a disk, and goes round and round for a long time before it gets in.

A disk seen from above, and how hot each ring is. The hottest ring sits just outside the last stable orbit, the innermost circle the gas can hold. Farther out it is cooler, and the fall in temperature follows a simple rule.

The rings orbit at different speeds. Their friction moves angular momentum outward so inner gas can fall farther; it also heats the gas until it glows and radiates before crossing the horizon.

As gas spirals from far away to the last stable orbit, 60,000 metres from Vesper's centre, it releases gravitational energy. A gram is a fortune calculates the efficiency: 10.4 percent of its rest-mass energy is radiated. Stellar fusion releases less than one percent of its fuel's rest-mass energy, so accretion releases more energy per kilogram of fuel.

Guess first Feed a disk ten thousand times more gas. How much hotter does it glow?

Only ten times hotter. A surface's glow goes as the fourth power of its temperature (the rule behind Cooling is the economy), and the disk has the same area as before. Ten thousand times the power needs ten times the temperature, because ten to the fourth is ten thousand.

That rule decides what colour the Vigil's light is. In the 2020s astronomers watched stellar black holes in binary stars shine as X-ray sources, hot because they were fed hard, near the ceiling this page comes to. The Vigil feeds Vesper a trillionth of that or less. At First Light in 2540, on 0.11 kilograms a second, the Newtonian thin-disk reference for a black surface peaks at about 550 kelvin, the heat of a hot oven, and shines in infrared. At the plan's goal of 1 × 10²⁰ watts, its hottest ring would be near 9,800 kelvin, in ultraviolet. At either temperature the blackbody tail in the X-ray band is nonzero but negligible; the sim boosts its soft-X-ray view for visibility.

A feed this thin does not make a disk like that by itself. Glowing black takes gas too thick to see through, and the First Light feed, spread over its disk, is a wisp you could see straight through. Left alone, gas that cannot shine its heat away keeps it, grows far hotter, and falls in as a hot flow that glows in X-rays and radio but carries most of its energy through the horizon unseen. So the Vigil does not leave it alone. It slows the gas's drift inward until the gas piles up thick enough to glow black, and the light it gathers is infrared and ultraviolet. Holding a disk that way is engineering no one in the 2020s had tried. challenge

There is a limit on the other side too. Light pushes on gas. Make the glow bright enough and its push outward matches the hole's pull inward, and no more gas can be fed through that light. That brightness is the hole's ceiling, and the Vigil lives far below it.

How much gas does a petawatt take?

The rule from A gram is a fortune turns around into a feeding rule. Gas arrives at a rate ṁ, and a share η of its rest-mass energy is radiated:

radiated power = η × ṁ × c²

η is the share of the feed's rest-mass energy emitted as light. Spin raises it, because it lets the gas orbit closer in before it plunges.
ṁ is the feed, in kilograms a second, and c is the speed of light.
To run a power P you need ṁ = P ÷ (η c²).

For the Vigil's first petawatt, 1 × 10¹⁵ watts:

Hole	Rest-mass energy radiated	Feed for a petawatt
No spin	5.7 %	0.19 kg a second
Vesper (spin 0.7)	10.4 %	0.11 kg a second
Spin 0.998, the most feeding can give	32 %	0.035 kg a second

Spin changes the required feed by a factor of 5.6 between the top and bottom rows. At the plan's goal of 1 × 10²⁰ watts, Vesper would take 10,700 kilograms a second, where a hole with no spin would take 19,500 and one at the limit 3,470.

How hard can a hole be fed?

Light carries momentum, so it pushes on whatever scatters it, and the free electrons in hot, ionized gas scatter it well. The brighter the glow, the harder the push outward, while the hole's pull stays the same. Where the two match is the ceiling, named for Arthur Eddington:

ceiling = 4π × G × M × (proton mass) × c ÷ (electron scattering cross-section)

M is the mass of the hole, and the ceiling grows in proportion to it: 1.26 × 10³¹ watts for each solar mass.
Vesper weighs 12 solar masses, so its ceiling is 1.51 × 10³² watts. That is Kardashev 2.62, and 2,417,844 times the light of Ember, its star.
The feed that lights Vesper to its ceiling at spin 0.7 is 1.6 × 10¹⁶ kilograms a second: 0.086 of an Earth's mass every year.

First Light runs at 6.6 × 10⁻¹⁸ of the ceiling, and the plan's goal would run at 6.6 × 10⁻¹³ of it. The ceiling is no limit on the Vigil: Vesper could shine 1.5 × 10¹² times brighter than even the goal asks.

What colour is the glow?

How hot the disk gets depends on how much power it sheds from its area. Here is the hottest ring for four feeds on Vesper, if the disk glows perfectly black:

Feed	Light	Hottest ring	Where it glows
First Light	1 × 10¹⁵ W	550 K	infrared, peak near 5.2 micrometres
The plan's goal	1 × 10²⁰ W	9,800 K	ultraviolet, peak near 0.29 micrometres
A hundredth of the ceiling	1.5 × 10³⁰ W	3.5 × 10⁶ K	X-rays, peak near 0.84 nanometres
The ceiling	1.5 × 10³² W	1.1 × 10⁷ K	X-rays, peak near 0.27 nanometres

Nikolai Shakura and Rashid Sunyaev worked this out in 1973, and it is why stellar-mass holes fed hard are X-ray sources. Bigger holes at the same share of their ceiling run cooler, because a larger disk spreads the power over more area: a hole of a billion solar masses at its ceiling has a hottest ring near 1.1 × 10⁵ K, in the ultraviolet.

With JavaScript on, this is a calculator: choose a spin and a feed and it shows the light, how close that is to the ceiling, and how hot the disk's hottest ring runs. The tables above give the same numbers for the Vigil's own feeds.

These are the temperatures of a perfectly black disk, the coolest a disk of that power can be. Gas that glows less well has to run hotter, and First Light's feed is far too thin to glow black. The last fold takes up what that does to the light and to the power bill.

Open everything

Why does a disk glow this way, and where does the ceiling come from?Energy balance ring by ring, and light against gravity

The temperature of one ring

Take a ring of the disk at radius $r$ , and let gas flow inward through it at a rate $\dot m$ . On a circular orbit around a hole of mass $M$ , a unit of mass has energy $-GM/2r$ in Newton's picture. Gas that moves from $r + dr$ to $r$ gives up the difference, so the ring releases

$dL = \frac{GM\dot m}{2r^2}\,dr$

The ring leaves the disk as light from both faces. Its area is $2\pi r\,dr$ on each face, and a black surface sheds $\sigma T^4$ from each square metre (the rule on Cooling is the economy). Setting what is released equal to what is shed:

$2 \cdot 2\pi r\,dr\,\sigma T^4 = \frac{GM\dot m}{2r^2}\,dr \quad\Longrightarrow\quad T^4 = \frac{GM\dot m}{8\pi\sigma r^3}$

Take the fourth root and the disk's temperature falls as $r^{-3/4}$ . That is the line in the diagram: ten times farther out is $10^{-3/4}$ = 0.18 of the temperature. The energy balance makes no mention of the rubbing, so the rule holds whatever does the rubbing.

The full result is a little different. Gas can only move inward by passing its spin outward, and the torque that does it also carries energy outward along the disk. Shakura and Sunyaev's bookkeeping of that adds a factor 3, and puts in a bracket that makes the heating vanish at the inner edge $r_\text{in}$ , where there is no torque:

$T^4 = \frac{3GM\dot m}{8\pi\sigma r^3}\left[1 - \sqrt{\frac{r_\text{in}}{r}}\right]$

Far from the inner edge the bracket is nearly 1, and the $r^{-3/4}$ stays. Add up the light from every ring in the Newtonian version and you get $L = GM\dot m / 2r_\text{in}$ : half of the fall's energy, which is the 8.3 percent that A gram is a fortune gets at $r_\text{in} = 6GM/c^2$ . The other half is still in the gas's orbital motion when it plunges.

The hottest ring

Let $x = r / r_\text{in}$ . The temperature obeys $T^4 \propto x^{-3}(1 - x^{-1/2}) = x^{-3} - x^{-7/2}$ . Its slope is zero when

$-3x^{-4} + \tfrac{7}{2}x^{-9/2} = 0 \quad\Longrightarrow\quad \sqrt{x} = \frac{7}{6}$

so the hottest ring sits at $x = 49/36 = 1.36$ , just outside the inner edge. Put that back in:

$T_\text{max} = 0.488 \left(\frac{3GM\dot m}{8\pi\sigma r_\text{in}^3}\right)^{1/4}$

For Vesper at First Light, $M$ = 2.39 × 10³¹ kg, $\dot m$ = 0.107 kg/s and $r_\text{in}$ = 3.39 $GM/c^2$ = 60,100 m. The result is 550 K.

Two things follow. The temperature goes as the fourth root of the feed, so each factor of ten thousand in the feed is a factor of ten in temperature, as in the guess earlier. And because $r_\text{in}$ is a fixed number of $GM/c^2$ , it grows with the mass of the hole, so $T^4 \propto \dot m / M^2$ . A hole at a fixed share of its ceiling is fed in proportion to its mass, which makes $T \propto M^{-1/4}$ . A hole of a billion solar masses runs 96 times cooler than one of 12.

The estimate uses Newton's profile with the inner edge moved to the relativistic last stable orbit. Don Page and Kip Thorne worked out the fully relativistic flux in 1974. For Vesper's spin it moves the hottest ring out from 4.62 to 5.22 $GM/c^2$ and makes it 15 percent cooler, the same share at every feed and every mass, so the tables and the toy keep the simpler numbers: First Light's hottest ring would be nearer 470 K.

Where the ceiling comes from

Light of luminosity $L$ at distance $r$ carries a momentum flow $L / 4\pi r^2 c$ through each square metre. A free electron scatters it with the Thomson cross-section $\sigma_T$ = 6.6525 × 10⁻²⁹ m², and so feels a push outward of

$F_\text{light} = \frac{\sigma_T L}{4\pi r^2 c}$

The electron is tied by electric attraction to a proton, so the pair is pushed together, and the pull of the hole on the pair is mostly on the proton, of mass $m_p$ = 1.6726 × 10⁻²⁷ kg:

$F_\text{gravity} = \frac{GMm_p}{r^2}$

Both forces go as $1/r^2$ , so their ratio does not depend on the distance. They match when

$L_\text{Edd} = \frac{4\pi G M m_p c}{\sigma_T}$

For $M$ = 1 solar mass that is 1.26 × 10³¹ W. Divide by $\eta c^2$ to turn it into a feed, and for Vesper you get 1.62 × 10¹⁶ kg/s, which is $1.5 \times 10^{17}$ times the feed that gave First Light its petawatt.

What happens when the feed is faint, and can anyone catch the light?Hot flows, magnetic turbulence, and the limits on catching and converting light

The faint end

The temperature model assumes a thin disk, gas dense enough to glow like a black surface. Ramesh Narayan and Insu Yi showed in 1994 that gas can fall another way. If it cannot radiate its heat as fast as it falls, the heat stays in the gas, which swells into a hot flow and carries the energy through the horizon unseen. In 1995 they found these flows exist only below a critical feed, where a thin disk and a hot flow are both possible solutions. Feng Yuan and Narayan's review of 2014 puts that feed at a few tenths of $\alpha^2$ of the ceiling feed, with $\alpha$ the viscosity parameter: a few thousandths for $\alpha$ = 0.1. A brighter kind of hot flow carries on to just under a hundredth, where the feed dial draws its line. Their review uses Sgr A* as its faint-flow prototype and finds lower feeds radiate a smaller share of the gas's energy. A 2012 fit by Xie and Yuan gives 1.4 percent at a thousandth of the ceiling feed. At First Light's feed, far outside its range, it gives 1.3 × 10⁻¹⁰ of the rest-mass energy, not the thin disk's 10.4 percent. A flow that shines like a thin disk needs 1.0 × 10¹⁵ times that feed, so the minidisk is engineered.

The Vigil operates far below those feed rates. The faintest flows, where even the electrons cannot radiate, set in below about 1 × 10⁻⁵ of the ceiling feed. Vesper's feed is 1.6 × 10⁷ times below it at the plan's goal and 1.6 × 10¹² times below at First Light. Spread by the rubbing, First Light's feed lays only 0.014 grams of gas on each square metre of its hottest ring (taking $\alpha$ = 0.1, and gas at the black-disk temperature, the thickest case). Even fully ionised, that layer would scatter only 3 × 10⁻⁷ of the light crossing it. It cannot glow black, and left to itself it would heat up and fall hot.

The ledger counts the full 10.4 percent of the feed's rest-mass energy as radiation at every feed because the Vigil keeps the gas in a cool, black disk. The handle is the rubbing: a disk passes gas inward at a rate set by how hard its rings rub, so a disk that rubs less holds more gas at the same feed. Slow the drift enough and the wisp above thickens into a layer that glows black. challenge The design is a disk of dust grains, which cool by their own surfaces: 0.008 kg on each square metre, 2.2 × 10⁹ kg in orbit before First Light (the Engine) design. The torque that drives it is open frontier. Left to fall hot, the gas would radiate a smaller share of its rest-mass energy, as X-rays and radio. frontier

What rubs the disk?

Ordinary friction in gas is far too weak. Shakura and Sunyaev in 1973 represented the unresolved transport with one number, the viscosity parameter $\alpha$ . In 1991 Steven Balbus and John Hawley showed that a weak magnetic field makes a disk whose spin slows outward unstable, growing about as fast as the disk turns. Their 1998 review traces what followed: the instability drives turbulence, which carries spin outward and is the rubbing.

A cool disk has a second problem. The instability needs gas ionized enough for the field to grip it, and gas at First Light's 550 K is not ionized by its own heat. Disks round young stars have the same trouble: in Charles Gammie's 1996 model, turbulence runs only close to the star and in a surface layer lit by cosmic rays, with a quiet layer between. A quiet disk holds its gas, which is half the answer; the other half is moving gas inward at the ledger's rate with fields of the Vigil's own. Whether a disk can be run that way is open. frontier

The bright end

Past the ceiling the disk is so dense that light cannot get out faster than the gas falls in, and some is carried into the hole. Marek Abramowicz and colleagues described these "slim" disks in 1988. The efficiency drops, so feeding a hole harder than the ceiling does not light it proportionally harder. The simple ceiling also assumes light spreads evenly; a disk can run past it by letting light out through its faces while gas falls through its plane.

Can the light be caught and used?

Two limits bound the light the ledger captures and delivers.

Catching. A disk sheds light from both faces in every direction, a collector covers part of the sky, and some light bends back into the hole. A ring of 1,000 dishes near the disk delivers 4 × 10⁷ W of the 1 × 10¹⁵ watts (the Engine) design.
Converting. Light from a surface at temperature $T_s$ can be turned into work, against surroundings at $T_c$ , with at most the efficiency

$\eta_\text{max} = 1 - \frac{4}{3}\frac{T_c}{T_s} + \frac{1}{3}\left(\frac{T_c}{T_s}\right)^4$

This is lower than the Carnot value $1 - T_c/T_s$ , because light has entropy of its own to get rid of. Landsberg and Tonge reviewed limits of this kind in 1980. With $T_c$ = 30 K, the Vigil's radiator temperature, black-disk light at First Light's 550 K can give at most 93 percent of its energy as work, light at the goal's 9,800 K can give 99.6 percent, and X-rays from a hole at its ceiling nearly all of it. The limit is for light at one temperature, so a disk that shines at many needs an integral over its rings.