The gas a hole catches from far away

The target that grows as the traveller slows found that a black hole is a strange target for anything moving slowly past it. Its horizon is tiny, but what decides capture is angular momentum rather than aim at the horizon itself: a particle is swallowed if its angular momentum is small enough, and a slow particle aimed some distance off has very little. The capture cross-section therefore grows as the inverse square of the speed, $4(c/v)^2$ times the area of the horizon’s silhouette, and a hole drifting through cold, slow particles catches them from much further out than its size suggests.

Much further, but still not far. A hole of one solar mass moving at ten kilometres a second through collisionless particles captures those whose paths would have passed within about 180,000 kilometres of it — half the distance to the Moon, from an object whose horizon is three kilometres across. Every particle that passes outside that distance swings round the hole on a hyperbola and leaves, with its speed unchanged.

Interstellar gas does not behave like that, and the reason is collisions. This essay is about what changes when the particles can hit each other, and the answer, worked out by Hermann Bondi in 1952, is that the effective target becomes enormously larger — by a factor of about two hundred and fifty million in the example above — and that the rate at which the hole grows then depends on its mass in a way that, unchecked, ends in a runaway.

Why gas cannot swing past #

Picture the particles that miss. Those passing above the hole are deflected downward, those passing below are deflected upward, and behind the hole the two streams cross. Collisionless particles pass through one another at the crossing and carry on. Gas cannot. The two streams meet head-on in their sideways motion, the collision converts that motion into heat, and what is left is gas moving straight away from the hole at less than escape speed, which turns round and falls in. The angular momentum that would have saved each particle is cancelled by the other side’s.

So for a gas the question is not whether each atom’s orbit misses, but whether the gas as a whole can resist being pulled in, and the only thing resisting is pressure. Gravity pulls with a strength measured by $GM/r$, the depth of the potential at distance $r$; pressure pushes with a strength measured by $c_s^2$, the square of the sound speed. They balance at

$$r_B = \frac{GM}{c_s^2},$$

the Bondi radius. Inside it, gravity wins and the gas falls; outside it, the gas barely notices the hole. It is the same competition that the disturbance that grows instead of travelling found deciding whether a lump in a self-gravitating gas oscillates as sound or collapses — there the gravity was the gas’s own, and the length at which it beat pressure was Jeans’s; here the gravity belongs to a point at the centre, and the length is Bondi’s. In both, the comparison is of a gravitational potential with the square of a sound speed, and nothing else enters. For a solar-mass hole in warm interstellar gas, with a sound speed of ten kilometres a second, the Bondi radius is 1.3 × 10¹² metres, about nine times the distance from the Earth to the Sun. Every bit of gas inside a sphere the size of Saturn’s orbit is drawn in.

The one flow that works #

Bondi asked for the steady flow: a hole at rest, gas far away at rest with density $\rho_\infty$ and sound speed $c_\infty$, and gas falling in at a constant rate $\dot M$, with the same density and speed at each radius for ever. Two equations decide it. Mass is conserved, so the flux through every sphere is the same,

$$4\pi r^2 \rho v = \dot M,$$

and energy along each stream is conserved, which for a gas held at constant temperature — one that radiates away the heat of being squeezed — is Bernoulli’s equation with gravity,

$$\frac{v^2}{2} + c_s^2 \ln \rho - \frac{GM}{r} = \text{constant}.$$

Eliminating the density between them and measuring distance in units of $r_c = GM/2c_s^2$ and speed by the Mach number $m = v/c_s$ leaves a single relation that every steady flow obeys,

$$\frac{m^2}{2} - \ln m - 2\ln x - \frac{2}{x} = \text{constant}, \qquad x = \frac{r}{r_c}.$$

Each value of the constant is a curve in the plane of distance and Mach number, and each curve is a possible steady flow.

The picture is a landscape with one saddle. Most curves are useless. Some loop round on the outside and never get close to the hole: they describe gas that slows to a halt before reaching it, which cannot carry a steady inflow inward. Some exist only close in and turn back before reaching large distances: they describe flows that are supersonic everywhere and cannot start from rest. Two curves pass through the saddle at $x = 1$, $m = 1$, crossing each other there.

One of them starts subsonic far away, with the gas nearly at rest, accelerates as it falls, passes the speed of sound exactly at the sonic radius, and continues supersonic into the hole. That is accretion, and it is the only solution that does what the problem asks. The other crossing curve is the same flow run backwards — a gas starting slow near the mass and accelerating outward to supersonic speed far away. Eugene Parker found that one in 1958, and it is the solar wind.

The sonic point is a horizon for sound #

The crossing has a second reading, and it connects the gas back to the hole. Inside the sonic radius the gas is falling faster than sound travels through it. A pressure wave sent outward from there is carried inward by the flow faster than it can move against it, exactly as the river that sound cannot swim up found for sound trying to travel upstream through a nozzle’s supersonic throat. Nothing that happens inside the sonic radius can send a signal through the gas to the region outside it.

That is a horizon, for sound rather than light, and it sits much further out than the hole’s own: at $GM/2c_s^2$ instead of $2GM/c^2$, larger by the factor $(c/c_s)^2/4$. It is why the rate can be fixed from outside. Whatever the gas does after it passes the sonic point — however it is heated, shocked or swallowed — cannot reach back and change how much gas enters, in the same way that the surface that only lets things in shields everything outside it from what happens within. The steady flow is decided entirely by conditions on the subsonic side, and the transonic condition is the statement that the gas has handed the problem over to the hole.

The analogy has been taken seriously in the laboratory. Flows of water, of light in a moving medium and of Bose–Einstein condensates driven past their own sound speed have been used as analogue black holes, to look for the counterpart of Hawking’s radiation in the sound waves at a sonic horizon. Bondi’s flow is the original example, found decades before anyone looked for it, and in it the sonic horizon is a sphere.

The requirement that the flow pass through the saddle is what fixes the rate. The constant on the transonic curve is $-3/2$, and evaluated far from the hole it sets how much mass crosses each sphere:

$$\dot M = 4\pi\lambda \frac{(GM)^2 \rho_\infty}{c_\infty^3}, \qquad \lambda = \frac{e^{3/2}}{4} = 1.12.$$

A smaller rate would put the gas on a curve that never becomes supersonic and so piles up against the hole; a larger one would demand a flow that does not exist. Pressure, in other words, does not slow the inflow down; it selects it.

What the gas does on the way in #

Followed inward, the accretion flow has three regions. Far outside the Bondi radius the gas is barely moving and its density is the ambient one — the hole’s influence falls off quickly once gravity is weaker than pressure. Near the Bondi radius the gas begins to move and to pile up: 2.45 times denser than ambient at $r_B$, 4.48 times at the sonic radius, where it is moving at exactly the sound speed. Inside, the inflow approaches free fall from rest, its speed rising as $r^{-1/2}$, and continuity then forces the density up as $r^{-3/2}$.

The last region is the one that matters for the light. Gas falling freely into a small volume is dense and fast, and if anything stops it — a surface, a shock of the kind the front that steepens until it cannot found every compressive wave heading for, or the collisions that circularise any small residual rotation into a disc — it heats enormously and radiates. Bondi’s calculation stops at the edge of that region, where the gas is a long way from the hole in units of its horizon, and says nothing about what happens next. But it says how much gas arrives, and the rate is set out where the gas is cool and slow, not where it is hot.

The same structure holds for a gas that keeps its heat of compression, with a different number in front.

A gas that cannot radiate its compressional heat gets stiffer as it falls, and resists more. For a monatomic gas, with an adiabatic index of $5/3$, the rate falls to a quarter of $4\pi(GM)^2\rho_\infty/c_\infty^3$, against 1.12 for an isothermal one. The sonic point moves inward as the index rises, reaching the centre at $5/3$. But every $\lambda$ is of order one. The physics is in the powers: twice of the mass, once of the density, and minus three of the sound speed.

A rate that grows as the square of the mass #

The power of two on the mass has a consequence that no exponential process shares. A population of bacteria, a sum at compound interest, a chain reaction — anything whose growth rate is proportional to its size — grows exponentially, doubling at equal intervals for ever. A rate proportional to the square of the size does something more extreme. Write $dM/dt = kM^2$ and integrate:

$$M(t) = \frac{M_0}{1 - t/t_B}, \qquad t_B = \frac{1}{kM_0} = \frac{M_0}{\dot M_0}.$$

The mass doubles at half of $t_B$, doubles again in a quarter more, and again in an eighth, and reaches infinity at exactly $t_B$ — the time the initial rate alone would take to double it.

Real holes do not run away, because the gas does not fall in quietly. It is heated as it falls and radiates, and the radiation pushes back on the infalling gas. Light carries momentum, as light has a pressure measured, and the brightness a mass cannot exceed computed the ceiling it sets: a luminosity at which radiation pressure on electrons balances gravity on protons, proportional to the mass. If a fixed fraction of the infalling rest energy is radiated, that ceiling is a maximum accretion rate proportional to $M$, and growth at it is exponential, with an e-folding time of a few tens of millions of years for a radiative efficiency of a tenth.

The figure puts the two together. While the hole is small, its Bondi rate, proportional to $M^2$, is below its Eddington rate, proportional to $M$, and Bondi controls the growth, which is slow at first. As the hole grows the Bondi rate catches up — in the figure at ten times the starting mass — and from then on the hole’s own light is the limit. A small hole is starved by the gas; a large one is starved by itself. Which regime a given hole is in depends on the density and temperature around it, through $t_B$, and both have changed enormously over the history of the universe.

Why collisions make the target so large #

The ratio between the two capture problems can now be written down. For collisionless particles arriving slowly at speed $v$, the cross-section is $4(c/v)^2$ in units of the horizon’s silhouette, $\pi r_s^2$. For gas with sound speed $c_s$, dividing Bondi’s rate by the flux $\rho c_s$ gives an effective cross-section of $\lambda(c/c_s)^4$ in the same units — the fourth power where the particles had the second.

The two extra powers have a plain reading. A particle is captured if it comes within a distance of order $GM/cv$, set by the relativistic capture of the circle light cannot leave and its cousins for massive particles. Gas is captured if it comes within $GM/c_s^2$, the Newtonian radius at which gravity first matters at all. The ratio of those distances is $c/v$, and the ratio of areas is its square. At ten kilometres a second it is a few hundred million.

Nothing in the gas calculation knows about relativity. The horizon enters Bondi’s problem only as a sink at the centre, small enough that the flow does not care what it is: a neutron star, or a normal star whose surface is far inside the Bondi radius, would draw in gas at the same rate, as long as something at the centre absorbs what arrives. What makes a black hole special is only that the gas cannot pile up on it, so the flow can stay steady.

What Bondi’s flow leaves out #

The flow drawn here is an idealisation in at least four ways, and each matters for real holes.

The hole is moving. A hole drifting through gas at speed $V$ captures it within $GM/(c_s^2 + V^2)$ rather than $GM/c_s^2$: once $V$ exceeds the sound speed, the gas swept up in a wake behind the hole is what falls in, as Hoyle and Lyttleton found for a star moving through a cloud in 1939. Most black holes in the Galaxy move at tens of kilometres a second relative to the gas around them, and their accretion is reduced accordingly.

The gas has angular momentum. Interstellar gas is turbulent, and even a small rotation, conserved as the gas falls inward, eventually becomes large enough to stop radial infall and spread the gas into a disc. Bondi’s rate then measures the supply to the disc, not the rate at which the disc feeds the hole, which is set by the disc’s internal friction.

The gas is heated. Bondi assumed the gas follows a fixed relation between pressure and density, as if its own thermal history did not matter — the opposite of the self-gravitating cloud in the ball of gas that heats up as it cools, where the history is everything. Real gas near a hole is heated by the radiation from further in, which can reverse the flow, and is cooled by its own emission. The heated case can drive an outflow from the same gas that would otherwise fall in.

The surroundings are not uniform. The gas at the Bondi radius of the hole at the centre of the Galaxy — about a tenth of a parsec for gas at ten million kelvin — is not a uniform medium at rest but the colliding winds of the young massive stars orbiting there.

What the calculation does survive is its structure: a single transonic solution, a rate fixed by conditions far away, the square of the mass, and the inverse cube of the sound speed. Those are the parts that appear in every more elaborate treatment, with corrections multiplying them.

Still open: how far below Bondi’s rate real holes accrete #

For the hole at the centre of the Galaxy the inputs to Bondi’s rate can be estimated: X-ray telescopes resolve hot gas near its Bondi radius, and its density and temperature give an expected rate of about $10^{-5}$ solar masses a year. Measurements of the gas much closer in — from the way it rotates the polarisation of the radio emission passing through it — show that far less than that reaches the hole, by a factor of a hundred or more. Most of the gas that crosses the Bondi radius never arrives.

Where it goes is disputed. In one picture the gas, unable to radiate its heat, becomes so hot that it is barely bound and most of it flows out again in a wind. In another, convection inside the flow carries energy outward and stalls the inflow. In a third the gas’s rotation and magnetic fields dominate everything inside the Bondi radius, and the rate is set by how fast the field can let angular momentum out. Simulations that include all three effects at once give accretion rates spread over two orders of magnitude, depending on how the gas is fed at the outer boundary.

The same question governs the early universe, where the first black holes had to grow from a few hundred solar masses to a billion within the first billion years. Growth at the Eddington limit is barely fast enough for that, if it is uninterrupted, and Bondi’s $M^2$ makes small seeds grow slowly at first. Whether some holes grew faster than the Eddington rate — in dense gas, where the radiation is trapped and carried inward with the flow — or began much larger than a few hundred solar masses has not been settled.

The part of the picture nobody doubts is the one drawn here. A collisionless particle is saved by its own angular momentum, which a slow particle aimed a long way off has very little of; gas is not saved even by that, because the streams on either side cancel each other’s. What decides how much a hole swallows is not how small its horizon is, nor how strong its gravity is near it, but how far out its gravity first beats the motion of what is around it — a distance set by the sound speed, and as large, for a hole of one solar mass in warm gas, as the orbit of Saturn.