A drag that is strongest in the middle

A massive body moving through a sea of stars is slowed by the wake it builds behind itself, and that first rung established the mechanism: no gas, no collisions, and a drag anyway. What it did not establish is how the drag depends on how fast the body is going, and the answer is the opposite of the intuition every fluid gives.

The peak is at 0.97, which is to say at very nearly the point where the body’s speed equals the root-mean-square speed of the field. Either side of it the drag falls, and at both ends it goes to zero.

Why only the slower stars count #

Chandrasekhar’s derivation sums the impulses from encounters at every impact parameter, which is the calculation that turns a single deflection into a rate, and then averages over the velocity distribution of the field.

The averaging is where the surprise is. Consider the body moving at some velocity through stars with their own velocities. Transform into the frame of a field star: the body approaches it at the relative velocity, is deflected along a hyperbola, and the star is deflected the other way. What the body loses is proportional to the component of the relative velocity along its own motion, and that component changes sign depending on whether the field star is going faster or slower.

Integrating over an isotropic distribution, the contributions from stars faster than the body cancel exactly. Only the ones slower are left, and their number is the fraction of the distribution inside a sphere of radius equal to the body’s speed.

That cancellation is exact only for an isotropic distribution. In a rotating system — a disc, say — the field is not isotropic in the frame of a body moving through it, and the drag acquires a component perpendicular to the motion. That is a real effect and it is why satellites sinking through discs do not sink radially.

The two limits, and why both are zero #

A body much faster than the field sees almost all of it as slow, so the numerator saturates at one, and the drag falls as the inverse square of the speed. It is not zero but it is small, and the time to slow down grows as the cube of the speed. That is the regime a spacecraft crossing a stellar system is in, and it is why a flyby’s energy bookkeeping can ignore the field stars entirely.

A body much slower than the field is the interesting limit. Almost none of the field is slower than it, and the fraction goes as the cube of the speed for small speeds — the volume of a small sphere in velocity space. Divided by the square of the speed, the drag goes to zero linearly. So a body at rest with respect to the field feels no drag at all, which is obvious in hindsight: there is no wake without motion.

The consequence is that dynamical friction is a decelerating force that switches itself off as it succeeds. A satellite spiralling into a galaxy does not asymptotically approach rest; it slows until the drag is comparable to the other terms in its equation of motion and then behaves differently.

Sinking, and the times it takes #

The classic application is a satellite on a circular orbit in a host with a flat rotation curve. There the field’s dispersion is roughly the circular speed, so the satellite sits near the peak of the drag curve for its whole descent, and the sinking time comes out proportional to the square of the starting radius divided by the satellite’s mass.

The number that keeps this from being a clean prediction is the Coulomb logarithm — the ratio of the largest to the smallest useful impact parameter, which enters as a multiplicative factor and is not determined by the theory. Taking the outer cutoff to be the satellite’s orbital radius or the host’s size changes the answer by tens of per cent, and there is no principle that settles it.

Where the velocity dependence matters most #

Three cases turn on the shape of the hero curve rather than on the magnitude of the drag.

The first is a galaxy falling into a cluster. Cluster velocity dispersions are around a thousand kilometres a second and infall speeds are comparable, so a galaxy is near the peak — but the peak’s height is inversely proportional to the square of the dispersion, so the drag is weak in absolute terms. Galaxies in clusters therefore do not sink; they orbit for a Hubble time, which is why clusters have galaxies in them at all rather than one enormous central object. The second is a pair of black holes at the centre of a merged galaxy. They sink rapidly at first, because they are heavy and the stellar dispersion is low; then they form a bound binary, and the drag on the binary as a whole becomes irrelevant because what matters is the ejection of individual stars by the pair. That handover — from dynamical friction to three-body scattering — happens at a separation of about a parsec, and it is exactly where the process is known to stall.

The third is mass segregation inside a cluster. Heavy stars are dragged by light ones, sink to the core, and speed up as they sink because a self-gravitating system gets hotter as it loses energy. The equilibrium they head for is equipartition, and the drag curve says how long the approach takes.

The same wake in a gas #

There is a version of this calculation for a gaseous background rather than a stellar one, and comparing the two is the clearest way to see what the collisionless case is actually doing.

In gas the perturber launches a sound wave rather than merely deflecting particles, and the geometry of the resulting overdensity depends on whether it is moving faster or slower than the sound speed. Subsonically, the disturbance propagates ahead of the body as well as behind it, and in a steady state a perfectly symmetric distribution would exert no force — the drag comes entirely from the fact that the perturbation has only had a finite time to develop, so the front–back symmetry is incomplete.

Supersonically, the body outruns its own sound waves and the disturbance is confined to a Mach cone trailing behind it. That is an unambiguous overdensity in the rear hemisphere, and the drag is correspondingly strong.

The result is a drag curve that peaks near Mach one and falls on both sides — the same shape as the collisionless curve, arrived at through completely different physics, with the sound speed playing the role that the velocity dispersion plays in the stellar case. The two are not the same expression and their peaks do not sit at the same place, but the qualitative statement is identical: a wake requires relative motion to form and requires time to form, and a body that moves too fast leaves it behind.

The difference matters wherever both backgrounds are present. A satellite galaxy sinking through a host contains stars and gas and moves through stars and gas, and the gaseous drag on the gaseous component can substantially exceed the collisionless drag on the whole — which is one of the mechanisms that separates a satellite’s gas from its stars during infall and leaves the two on different orbits. The stripped gas trails behind and the stars carry on, which is what a galaxy caught mid-infall looks like.

There is a second place the comparison earns its keep, and it is the one that decides whether a black-hole pair merges. Two holes at a parsec have exhausted the stars available to scatter, so the collisionless drag has nothing left to act on; a gaseous disc around them has not been exhausted, because gas is a fluid and refills. Whether the final parsec is crossed at all may therefore be a question about which of the two backgrounds is present, and the gaseous version of this calculation is what decides it. The awkwardness is that a disc massive enough to do the job is also massive enough to be unstable and to fragment into stars, at which point it becomes a stellar background with the same exhaustion problem.

It is worth noticing what the two derivations have in common at the level of method, because it is the transferable part. Both compute a density perturbation caused by the perturber, and both then compute the gravitational force that perturbation exerts back on it. Neither contains a drag coefficient, a viscosity or a collision cross-section. The force is gravity acting on the arrangement gravity has made, and the medium’s properties enter only in deciding what arrangement it makes.

What was actually measured #

Dynamical friction has never been measured directly, in the sense of watching a body slow down. Every test is a population argument, and they are worth separating by how much they assume.

The strongest is the Magellanic Clouds. The Large Cloud is massive enough — of order a tenth of the Milky Way’s mass within its own radius — that the drag on it is substantial, and the Stream of gas trailing it records where it has been. Reconstructing its orbit backwards requires the friction term, and the reconstruction that fits the Stream’s geometry also fits its measured proper motion, which is a consistency check rather than a measurement of the friction coefficient. The Stream is itself a tidal tail of the kind a disrupting satellite leaves, so the same object constrains two processes at once.

The second is the absence of massive satellites. A galaxy like the Milky Way should have accreted many satellites above ten to the ten solar masses over its history, and those should all have sunk. The observed satellite population is dominated by objects two decades lighter, which is what the sinking-time curve requires — though a survey’s own selection decides what is counted, and the faintest satellites were being discovered at a rate that made the comparison a moving target for two decades.

The third is globular cluster populations. Clusters in a galaxy’s halo sink at rates set by their masses, so the cluster mass function should be truncated at the bottom by evaporation and at the top by friction. Both edges are seen; the top edge is at about the mass the sinking-time argument predicts, given a Coulomb logarithm that has to be chosen.

What a rotating background does #

The cancellation that leaves only the slower stars was derived for an isotropic field, and the qualification made there deserves to be taken further, because it changes the geometry of the answer rather than its size.

In a rotating system the field’s velocity distribution is not isotropic in the frame of a body moving through it. A satellite on a prograde orbit inside a disc is being overtaken by material on the inside of its orbit and is overtaking material on the outside, and the two populations do not cancel: the drag acquires a component perpendicular to the motion as well as one along it.

The consequence is that the satellite’s orbit does not simply shrink. The tangential component removes angular momentum at a different rate from the way the radial component removes energy, so the eccentricity evolves — and which way it evolves depends on whether the satellite is prograde or retrograde with respect to the field. A prograde satellite in a rotating halo circularises as it sinks; a retrograde one does not, and can become more eccentric on the way in.

That asymmetry is measurable in principle and it matters for what a satellite looks like when it arrives. A satellite that circularises spends its final orbits at a nearly constant radius, being tidally stripped over many passages, and disperses into a shell. One that keeps its eccentricity plunges through the centre repeatedly, is stripped in violent episodes at each pericentre, and leaves a stream rather than a shell. The morphology of a galaxy’s accreted halo therefore carries a record of the sense in which its satellites arrived.

There is a further wrinkle, and it is the reason simulations were needed before anybody trusted this. The rotating background is not merely anisotropic; it responds. A satellite moving through a disc excites the disc’s own collective modes — spiral arms and a bar — and the torque from those can exceed the local wake’s by a large factor. In that regime the language of a drag against a background stops being appropriate, and the exchange is better described as angular momentum being carried away at resonances. The classical formula is then not a poor approximation to the right answer; it is an answer to a different question that happens to be numerically close over some range.

The assumption that does most of the work #

Every number above comes from a formula derived for a body moving in a straight line through an infinite, homogeneous, isotropic Maxwellian field. None of those conditions holds anywhere it is applied.

A real satellite is on a curved orbit through a field whose density and dispersion change along it, and the wake it builds does not have time to reach the steady state the formula assumes. Simulations that measure the drag directly find the formula right to tens of per cent when the Coulomb logarithm is tuned, and wrong in a specific direction in cored density profiles — where a satellite can stall at a finite radius instead of sinking to the centre, because the field inside a core has no gradient to build a wake against.

A body being dragged is also being fed #

There is a second consequence of the same wake, and it belongs here because it uses the identical geometry.

The material that converges behind the perturber does not all escape. Whatever passes inside the radius at which the perturber’s escape speed exceeds the local relative speed is captured, and the rate at which that happens is the same focusing calculation read as a flux rather than as a force.

Comparing the two curves is instructive. The drag peaks at the dispersion and dies either side; the accretion rate is flat below it and falls steeply above. So there is no speed at which a body both grows fastest and is slowed hardest, and the two effects have different consequences for the same orbit: a body moving supersonically through gas is slowed efficiently and fed badly, while one moving with the flow is fed well and not slowed at all.

For a star it hardly matters — the mass a star gains from the interstellar medium over its life is negligible. For a black hole wandering through a galaxy’s gas it is the whole question of whether such objects can be detected at all, and the answer depends on where on these two curves it sits.

The last thing worth stating is what the shape of the curve implies for a body that starts fast and slows down. It enters the figure from the right, where the drag is weak, and the drag increases as it decelerates — so the deceleration accelerates. Then it crosses the peak and the drag begins to fall, so the deceleration slows again. The result is a sinking history that is neither exponential nor a power law but has an inflection in it, with most of the energy lost in a fairly narrow band of speeds around the dispersion. That concentration is why the sinking-time formula is as robust as it is despite everything the last two sections raised: the answer is dominated by the part of the trajectory where the body’s speed matches the field’s, and the details of what happens at the two ends contribute little to the total. It is also why a body that is somehow prevented from entering that band — by a core, by a resonance, by a tidal field — does not merely sink more slowly. It does not sink.

Both of the quantities the drag is built from are worth reading at a second value, since one of them is a shape and the other is only a scale.

Where the ladder goes #

The natural third rung is the same integral read as a rate of accretion rather than as a force: the material that ends up inside the perturber rather than merely behind it, which is the Bondi–Hoyle–Lyttleton problem and has the same focusing at its heart.

The other direction is the failure this rung has flagged twice. A formula derived for an infinite homogeneous medium is being applied to bodies in resonance with structures — bars, spiral arms, cores — and in those cases the drag is better described as a torque exchanged at particular resonant radii than as a friction. That reformulation is the modern one, and it recovers the classical answer in the limit where the resonances are dense.