The wake and the meal are one calculation

Two earlier rungs of this anchor took accretion for granted and asked what it produces: a disc, because the material has too much angular momentum to fall straight in, and a spectrum that is a stack of temperatures rather than one. Neither asked where the material came from or at what rate it arrives. That question has an answer with one length in it.

The formula is Bondi’s, from 1952, with Hoyle and Lyttleton’s earlier treatment of the supersonic case folded in. It is the simplest statement in the subject and it is wrong by orders of magnitude wherever it has been tested, which is the more interesting half of the essay.

One radius, arrived at twice #

Consider a body of mass $M$ sitting in a gas with sound speed $c_s$, at rest. Gas thermal motion resists being captured; gravity pulls it in. The radius at which the two balance is where the escape speed equals the sound speed,

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

and gas well inside that radius is bound while gas well outside is not. The mass flux through a sphere of that radius, at the sound speed, is the accretion rate to within a factor of order unity.

Now let the body move at speed $v$. In the frame of the body the gas streams past at $v$, so the criterion becomes the escape speed matching the total relative speed rather than the thermal one, and the accretion radius shrinks to $2GM/(c_s^2 + v^2)$. The flux through it is the area times the speed times the density, so the rate carries the square of the radius and one power of the speed, giving the shape drawn above. The same radius appears in the drag calculation, because the material that is deflected but not captured is exactly the material that forms the wake. That is why the two answers share a geometry: what is captured falls inside the focusing radius and what pulls back was focused just outside it.

Two limits that behave differently #

At Mach numbers well below one the motion is irrelevant, the rate is Bondi’s original spherical answer, and the flow is close to hydrostatic out to the accretion radius.

Above Mach one the picture changes qualitatively rather than quantitatively. The flow is supersonic, so a bow shock forms upstream; gas passes through it, loses its transverse momentum, and collects along the axis behind the body in a dense column. Material in that column with too little momentum to escape falls back onto the body. The accretion is therefore not spherical at all: it happens along a line.

The distinction matters because the two cases fail differently. Spherical Bondi accretion is unstable to almost nothing and is a decent description of the flow it models. Supersonic accretion along a column is violently unstable — the column flaps, the shock oscillates, and the instantaneous rate varies by factors of several — so the formula describes a time average of something that is never in that state.

Why the rate is almost always an overestimate #

The formula assumes the gas has no angular momentum with respect to the body, no magnetic field, and no way of getting rid of the energy it gains falling in. All three assumptions fail, and each failure reduces the rate.

Angular momentum is the largest. Gas with even a small amount of rotation cannot reach the body; it settles into a disc at the radius where its angular momentum is Keplerian, and from there it has to be transported outward before anything can fall in.

Second, radiation. Material falling onto a compact object releases a large fraction of its rest mass, and that radiation pushes back on the material still arriving. Above the luminosity at which radiation pressure balances gravity the flow is stopped, so the accretion rate is capped regardless of what the supply is. Third, the gas has to cool. Material compressed as it falls heats up; if it cannot radiate that heat away faster than it falls, its pressure rises and it stops falling — the same competition between compression and cooling that decides whether a cloud of interstellar gas can collapse at all. Hot, tenuous flows around low-luminosity nuclei are in exactly that regime, and their accretion is radiatively inefficient — most of the energy is advected inward or blown outward rather than radiated.

The case where the numbers can be checked #

The Galactic centre is the one place where the supply, the rate and the output can all be measured independently, and they do not agree. The gas around it is observable too: X-ray observations resolve the hot medium out to the accretion radius, giving a density of about a hundred particles per cubic centimetre and a temperature of a keV. Putting those into the Bondi formula gives about ten to the minus five solar masses a year.

The observed luminosity implies a rate several hundred times smaller. So of the material that crosses the accretion radius, the overwhelming majority does not reach the hole — it is turned around, blown back out, or is radiating so inefficiently that the luminosity is not a measure of the rate at all. Distinguishing those is an active question and the observations that bear on it are polarisation measurements of the inflowing gas.

Where the formula is the right one #

The failures above are all cases where something intervenes between the supply and the delivery. There are cases where nothing does.

A star moving through a molecular cloud accretes at very nearly the Bondi–Hoyle rate, because the amount it gathers is far too small to affect the flow, the gas has essentially no angular momentum on the relevant scale, and no radiation pressure is involved. The rate is also negligible — a solar-mass star crossing a dense cloud picks up perhaps a millionth of its mass — which is why the case is a good test and not an important process.

The other clean case is the wind-fed X-ray binary. A compact object in orbit around a massive star with a strong wind is moving supersonically through that wind, and the geometry is exactly the Hoyle–Lyttleton one — a bow shock, an accretion column, and a rate that varies as the orbit carries the compact object through denser and thinner parts of the flow. The observed X-ray luminosities of those systems track the formula to within the factor of a few that the instability of the column would predict.

The transition radius, which is where the two pictures meet #

There is a way to see the angular-momentum problem as a length rather than as an objection, and it makes the comparison between spherical and disc accretion concrete.

Gas at the accretion radius has some specific angular momentum, whatever the flow at large scales gave it. That angular momentum defines a circularisation radius — the radius at which a circular orbit would have exactly that much — and inside it the gas cannot go without shedding some. So the flow is spherical from the accretion radius down to the circularisation radius, and a disc from there in.

The ratio of the two is the whole question, and it is enormous for a supermassive black hole fed by a galaxy and modest for a compact object fed by a companion’s wind. For the Galactic centre the accretion radius is about a tenth of a parsec and the circularisation radius is estimated at a few hundred gravitational radii — a factor of ten thousand — so essentially the entire flow is a disc, and the Bondi calculation describes only the outermost boundary condition.

For the wind-fed binary the two radii are comparable, because the wind’s own velocity gradient across the accretion radius supplies only a little angular momentum. That is why those systems are the ones where the formula works: they are the systems in which there is barely a disc.

A rate that is not steady #

One more feature of the supersonic case deserves stating, because it is the reason observed sources vary.

The accretion column behind a supersonically moving body is not stable. Small asymmetries in the flow give the column a net angular momentum that switches sign as the instability develops, so the material arriving at the body alternates between a disc rotating one way and a disc rotating the other. The instantaneous rate swings by factors of several on the timescale of the flow crossing the accretion radius. The timescale is worth putting a number to. For a compact object moving at a thousand kilometres a second through a wind, the accretion radius is of order ten to the ten centimetres and the crossing time is a few minutes — and wind-fed X-ray binaries do vary on exactly that timescale, by exactly that amount.

What was actually measured #

Three quantities, and only two of them well.

The density and temperature of the ambient gas come from X-ray spectroscopy: the emission measure gives the square of the density integrated along the line of sight, and the shape of the continuum gives the temperature. Both require assuming the gas is in collisional ionisation equilibrium, which for a hot diffuse plasma is safe.

The body’s mass comes from orbits where there are orbits, and from a scaling relation where there are not. For the Galactic centre it is the best-measured black hole mass anywhere, and everywhere else it comes from a relation calibrated on the few that could be weighed directly.

The rate is never measured. It is inferred, either from a luminosity through an assumed efficiency, or from a rotation measure through an assumed field geometry, or from a formula whose assumptions the same observations show to be violated. The honest statement is that the supply is measured, the delivery is not, and the ratio between them is the physics.

One question the essay has not asked is what happens to the momentum. A body accreting from a medium it is moving through gains mass, and the material it gains was moving relative to it, so its velocity changes even before any drag is considered. That effect is the accretion of momentum rather than a force, and it always slows the body, because the material it captures was on average moving backwards in its frame. For a body growing slowly the term is negligible beside the gravitational drag; for one whose mass is changing on the same timescale as its velocity, it is not, and the two have to be treated together. That regime is reached by protoplanetary cores in a gas disc, which is why the calculation this essay describes for compact objects reappears, in almost the same algebra, in a subject about how planets are built. The identical geometry produces a growth rate there and a migration rate here, and the two are the same integral read against different denominators.

The ceiling the supply can run into #

The formula computes what gravity can gather, and there is a separate question of what the accretor can accept.

Material falling in releases energy, and the energy leaves as radiation that has to travel outward through the material still arriving. Radiation carries momentum, so it pushes back — and above a certain luminosity the push exceeds the pull. That luminosity depends only on the accretor’s mass, since both the pull and the push scale with it in the same way, and it corresponds to a limiting accretion rate once an efficiency is assumed.

The two rates are computed from entirely different things. The supply rate depends on the ambient gas: its density, its temperature, and how fast the body is moving through it. The limiting rate depends on the accretor: its mass, and how efficiently the infall radiates. Nothing connects them, so either can be the larger.

Where the supply is smaller, the accretor takes everything on offer and the formula in this essay is the answer. Where the supply is larger, the excess has to go somewhere, and what happens then is genuinely unsettled: the flow may be regulated down to the limit, with the surplus driven off as a wind, or it may become so optically thick that the photons are carried inward with the gas faster than they can diffuse out — trapped, and delivered to the accretor along with the material, which allows rates far above the nominal limit.

That second possibility matters for one specific question. The most distant quasars are already a billion solar masses when the universe is under a billion years old, and growing a stellar-mass seed to that in the time available at the limiting rate is arithmetically tight. Every proposed resolution either starts with a heavier seed or exceeds the limit, and the trapped-photon regime is the mechanism by which the limit is exceeded.

The wake, photographed #

The essay’s central claim is that the drag and the meal are one calculation, and there are objects where the wake is not a calculation at all but an image.

A star moving supersonically through the interstellar medium and losing mass in a wind produces a bow shock: the wind pushes outward, the ambient medium pushes back, and the two balance along a surface ahead of the star. Its standoff distance is where the wind’s ram pressure equals the ambient ram pressure, so measuring the distance and knowing the speeds gives the mass-loss rate — a quantity otherwise obtainable only from spectra and with far more modelling.

Dozens have been imaged, in the infrared where the shocked dust radiates, around runaway massive stars that were ejected from clusters and are crossing the medium at tens of kilometres a second. The shape is a paraboloid whose opening angle is set by the ratio of the wind’s speed to the star’s motion.

The other half of the picture is behind. A red giant losing mass slowly while moving quickly leaves the material behind it rather than around it, and the result is a tail — the clearest case extends for several parsecs behind a star of the kind, containing the mass it has shed over hundreds of thousands of years, laid out as a record of its own mass-loss history in the order it happened.

So the wake is observable, and the same geometry that produces the drag produces the picture. What is not observable in these systems is the accretion, which is negligible for a mass-losing giant — the two halves of the calculation are separated by which one happens to be large.

It is the usual arrangement in this subject: one geometry, two effects of very different magnitudes, and which of them is observable decided by the parameters of the particular object rather than by anything in the physics.

The bow shocks are the clearest instance: the drag they represent is utterly negligible for the star’s motion, and the image is the best measurement of its wind that exists.

Which is worth remembering when a calculation is described as unified: what is unified is the derivation, and the two consequences can still be separated by twenty orders of magnitude in whether anybody can see them.

One difference between the two media the same formula is applied to is worth stating, because it is the place where the analogy stops being exact. A collisionless medium — a field of stars around a sinking satellite — has no pressure and no sound speed, so its wake is purely gravitational and the drag depends on the velocity distribution of the stars rather than on any property of a fluid. A gas has both, and above Mach one its wake is not a smooth overdensity but a shock: the disturbance cannot propagate ahead of the body, the enhancement piles into a Mach cone, and the drag acquires a term that the collisionless calculation has no counterpart for. Near the peak of the curve above, where the body moves at about the sound speed, the two accounts differ by tens of per cent and the gaseous one is not a small correction to the other. Which formula applies is therefore a question about the medium and not about the perturber, and a satellite sinking through a galaxy’s stars and its gas at once is subject to both at different strengths.

The drag curve has a shape that is worth seeing twice, because the Coulomb logarithm changes its size and not its shape at all.

That separation is what makes the formula useful despite its worst feature. The Coulomb logarithm is the ratio of the largest to the smallest impact parameter that contributes, both ends are chosen rather than derived, and the answer depends on the choice — but only linearly, and only in the overall rate. The location of the peak, the fact that a very fast body feels almost nothing, and the fact that drag vanishes as the speed goes to zero are all independent of it.

Those three facts are what the astrophysics actually rests on. A satellite galaxy sinking into a halo slows until it is moving at about the halo’s dispersion and then feels the maximum drag, which is why sinking accelerates rather than stalls. A star moving much faster than its neighbours passes through them almost freely, which is why the fastest stars in a cluster are the last to be thermalised. And a body at rest with respect to its surroundings feels nothing, which is why dynamical friction cannot pull a system tighter than its own dispersion allows.

None of that depends on knowing the logarithm to better than a factor of two, and nobody does know it better than that. The formula’s reputation for imprecision is earned and misplaced: it is imprecise in the one quantity that is easiest to calibrate against a simulation, and precise in the three that decide what happens.

Where the ladder goes #

The next rung of this anchor has to be the transport itself: what makes a disc viscous, why the answer is a magnetic instability rather than molecular viscosity, and how a dimensionless parameter introduced as a placeholder in 1973 turned out to have a value simulations agree on.

The other direction leads to the radiatively inefficient flows this essay kept meeting. A flow that cannot cool is a flow whose energy has to go somewhere, and the two candidates — advection through the horizon and an outflow — make very different predictions about what a low-luminosity nucleus does to the galaxy around it.