A drag computed with a logarithm nobody can pin down

A massive body moving through a sea of lighter ones is slowed by them. It is slowed not by hitting them — nothing hits anything — but by pulling them into a wake behind itself and then being pulled back by the wake it made. The mechanism has no contact in it anywhere, and Chandrasekhar wrote the force down in 1943 by adding up the deflections from every encounter at every impact parameter.

The result is one of the most-used expressions in galactic dynamics. It decides whether a globular cluster reaches the centre of its galaxy, whether a satellite galaxy survives as a satellite or is swallowed, how quickly a massive black hole settles after a merger, and how long a stellar-mass black hole takes to sink into a nuclear cluster. Every one of those is a timescale, and every one of those timescales is proportional to the same quantity.

The addition diverges. At small impact parameters the deflection per encounter grows without bound; at large ones the number of encounters grows faster than their individual effect falls. Both ends have to be cut off by hand, and the answer is proportional to the logarithm of the ratio of the two cuts.

This is a common enough situation in physics that it has a standard shape and a standard remedy. A quantity diverges logarithmically at both ends of an integration; the divergence is cut off by physics the calculation does not contain; and because the dependence on the cut-offs is only logarithmic, the answer is insensitive to getting them approximately right. The same structure appears in the Coulomb scattering the name is borrowed from, in radiative corrections in field theory, and in the drag on a body moving through a plasma. What makes the astronomical case awkward is not the structure but the numbers.

Two divergences with different characters #

The integral is over impact parameter $b$, and the two ends fail for different reasons, which is worth separating because only one of them is a real feature of the physics.

The small-$b$ divergence is an artefact of the approximation. Chandrasekhar’s derivation treats every encounter as a small deflection and adds the deflections linearly. That is exactly right for distant encounters and wrong for close ones, where a single passage can reverse the direction of motion. The formal integral of the small-angle expression diverges logarithmically as $b\to0$; the true physics does not, because a close encounter deflects by at most 180 degrees. The correct cut-off is the impact parameter at which the small-angle approximation fails, which is the 90-degree deflection radius $b_{90} = GM/v^2$.

The large-$b$ divergence is real and is cut off by geometry. Distant encounters are individually feeble and enormously numerous, and the two effects nearly cancel: the contribution per logarithmic interval of $b$ is constant, which is why the answer is a logarithm at all. The integral is stopped not by physics but by the medium running out — the host system has a size, and beyond that there is nothing to drag against.

The first cut-off is defensible and nearly unambiguous. The second is a choice, and it is the one that varies between authors: the size of the host, the current orbital radius, the half-mass radius, the distance at which the density falls by some factor.

There is a third thing that is sometimes cut off and should not be, and it is worth naming because it is a common confusion. The formula’s derivation also assumes the field particles are much lighter than the perturber and are not themselves deflected appreciably. That is not a divergence and cannot be fixed by a cut-off; it is a condition on when the formula applies at all. Papers that extend the integration to encounters with objects of comparable mass are not choosing a cut-off badly — they are outside the theory.

Why the flatness does not save the calculation #

The standard reassurance is that a logarithm is insensitive: getting $b_{\rm max}/b_{\rm min}$ wrong by a factor of ten changes $\ln\Lambda$ by 2.3, which sounds small.

It is not small, because $\ln\Lambda$ itself is not large. In practice the conventions give values between about 3 and 10 for the astrophysically interesting cases, so an uncertainty of 2.3 is an uncertainty of a factor of two in the answer. The drag force is proportional to $\ln\Lambda$, not to its logarithm.

That is the honest situation and it is worth stating without hedging. For a substantial class of problems — globular clusters sinking to a galaxy’s centre, dwarf satellites spiralling into a halo, a massive black hole settling after a merger — the answer to “has this happened yet” depends on a number nobody derives.

The three regimes where it matters differently #

The sensitivity is not uniform across the problems the formula is applied to, and the differences follow from how the two cut-offs are set in each case.

Point mass in a much larger system. Here the ratio is genuinely enormous — the host’s size over the 90-degree radius can be $10^6$ or more — so $\ln\Lambda$ is 12 to 15 and a factor of two in the ratio changes it by five per cent. This is the regime the formula was derived for, and it is the regime in which the logarithm is genuinely a harmless constant. It applies to a star sinking in a globular cluster.

Extended satellite in a host of comparable size. Here the satellite is not a point and has no well-defined $b_{\rm min}$: its own half-mass radius is larger than the deflection radius, so encounters closer than that are with part of the satellite rather than with all of it. And $b_{\rm max}$ is not much larger than $b_{\rm min}$. The ratio may be only 30 or 100, so $\ln\Lambda$ is 3 to 5 and every choice matters. This is the regime of dwarf-galaxy infall, and it is where most of the astrophysics is.

Comparable masses. Here the derivation’s central assumption — that the perturber is much heavier than the field particles and that its own motion is not deflected — has failed, and the formula does not apply at all. Whether two galaxies merge is a ratio of two times computed with a version of this formula that has been calibrated rather than derived.

It is worth being explicit about why the third regime is the interesting one, since a formula that fails there might seem simply inapplicable. Almost every case a galaxy-formation calculation cares about is in it. Satellites accreted by a halo have masses between a thousandth and a tenth of their host’s; the mergers that build galaxies are between comparable objects; the globular clusters that sink to a galactic nucleus are extended and are stripped as they go. The regime the derivation was written for — a heavy point moving through a sea of much lighter ones — describes a star in a cluster, and it is not what a modern simulation is being asked about.

What was actually measured #

Nothing here is measured in the sky. The Coulomb logarithm is calibrated against numerical experiments, and the experiments are the closest thing to a measurement the subject has.

The method is direct. Set up a satellite of known mass on a known orbit in a known host, integrate every particle, and watch the orbit decay. Then ask what value of $\ln\Lambda$ the Chandrasekhar formula would have needed to reproduce that decay. The answer is a number, and the interesting result is what it depends on.

It depends on the mass ratio. Simulations find that the effective logarithm is well fitted by something like $\ln(1 + M_{\rm host}/M_{\rm sat})$ rather than by a constant, which is the statement that $b_{\rm max}$ should be interpreted as the radius enclosing a mass comparable to the satellite’s rather than as the host’s size.

It depends on the satellite’s structure. An extended, loosely bound satellite is stripped as it falls, so its mass is a decreasing function of time and the drag falls with it. A compact one is not. Fitting a single logarithm to both produces different answers, and the difference is a property of the satellite rather than of the drag.

And it depends on the orbit’s eccentricity, though weakly. The formula is derived for straight-line motion through a uniform medium, and a real orbit passes through a density gradient; the effective logarithm that reproduces an eccentric decay is a few tens of per cent different from the one that reproduces a circular one.

Where the drag comes from, drawn rather than integrated #

Return to the mechanism, because the logarithm’s origin is visible in it rather than merely derivable from it. The formula’s awkward feature is not an artefact of the algebra; it is a direct statement about which material is doing the pulling, and a picture of the wake makes that statement in one glance.

The balance is not approximate. The deflection per encounter falls as $1/b$, the number of encounters at a given $b$ grows as $b$, and the momentum transfer goes as the square of the deflection times the number — which leaves $1/b$ per unit $b$, or a constant per unit $\ln b$. A sum whose contributions are equal in every decade is a sum that depends only on how many decades there are, and that count is exactly what nobody can supply.

One further result of the calibration programme deserves a mention because it is a genuine physical discovery rather than a fitting exercise. The effective logarithm measured in simulations is not the same for the drag on the orbit and for the heating of the satellite, even though both come from the same encounters. The first is dominated by distant, resonant interactions and the second by close ones, so they sample different parts of the integral and stop at different places. Two quantities that a single formula treats as proportional turn out not to be, and the discrepancy is a signature that the local approximation is being used past its range rather than merely imprecisely. The same warning applies to the heating that unbinds a cluster during an encounter, where the impulse and the drag are again two accounts of one interaction that a single parameter cannot cover.

Where the picture stops #

Three limits stand out, and the third is the one that suggests the eventual replacement.

It assumes an infinite homogeneous medium. A real host has a density gradient and a finite extent, and the drag on a satellite at a given radius depends on material at other radii in a way the local formula cannot express. This is the origin of the drag being strongest somewhere in the middle rather than at the centre, which the local formula gets qualitatively right and quantitatively wrong.

It assumes the perturber is a point that does not change. Real satellites are stripped, and a satellite that has lost ninety per cent of its mass exerts a hundredth of the drag. The stripping and the drag are coupled and the formula treats neither.

And the whole framework is a local approximation to a global problem. The modern alternative is to compute the response of the host’s distribution function to the perturber directly — a linear-response calculation in which the drag emerges as a resonance between the satellite’s orbital frequency and the frequencies of the host’s own orbits. That calculation has no free logarithm in it. It is much harder, it agrees with the calibrated Chandrasekhar formula where both apply, and it explains why the effective logarithm depends on the mass ratio: the resonances that dominate are set by the perturber’s own influence radius.

There is a fourth worth stating separately because it changes what the formula is for. In a real host the material being dragged against is not a smooth medium but a set of orbits, and a satellite on a circular orbit at a given radius is in resonance with some of them and not others. That means the drag depends on the satellite’s orbit in ways a local formula cannot express — a stream’s track is not the orbit its progenitor was on for a related reason, and both are consequences of treating a structured system as a medium.

Why a fudge factor is worth writing an essay about #

There is a temptation to treat this as an embarrassment to be handled quietly, and it is worth resisting for two reasons.

The first is that the formula is right. It is derived rather than fitted, its dependence on the mass, the velocity and the density are all exact, and it makes correct predictions across twelve decades in the number of particles — the same expression describes a star being deflected in a globular cluster and a galaxy being deflected in a cluster of galaxies. A theory that is exact except for one logarithm is a very good theory.

The second is that the logarithm is honest about what it is. It is written down explicitly, its origin is stated, and every paper that uses the formula says which convention it adopted. Compare that with the many places in astrophysics where a similar ambiguity has been absorbed into a coefficient somebody fitted once and nobody has revisited. A visible fudge factor is a research programme; an invisible one is a systematic.

The programme in this case has been productive. Asking what value of $\ln\Lambda$ the simulations require is what revealed the mass-ratio dependence, which is what pointed at the resonance calculation, which is the theory that will eventually make the logarithm unnecessary. The uncertainty was the route to the improvement rather than an obstacle to it.

There is one more reason to keep the formula rather than to wait for its replacement, and it is practical. A galaxy-formation model that follows millions of satellites cannot afford a linear-response calculation for each of them, and it can afford one evaluation of an algebraic expression. So the calibrated Chandrasekhar formula is what actually runs, in every semi-analytic model of galaxy assembly currently in use, with a mass-ratio-dependent logarithm fitted to simulations that themselves used the full N-body dynamics. The chain from first principles to production code passes through a fitted constant, and it does so knowingly. That is a normal state of affairs in computational astrophysics and it is worth seeing clearly at least once, in a case where every step of the chain is visible — an integrator that is right about the energy and wrong about the position is the same compromise in a different currency.

It is worth naming the one situation in which the ambiguity genuinely disappears, because it shows what the logarithm is standing in for. When the perturber’s mass is small enough that its influence radius is well inside the system and the background is smooth over that scale, both limits are unambiguous — the inner one is the impact parameter at which the deflection becomes large, and the outer one is the scale over which the density changes. The logarithm is then a computable number rather than a convention. The trouble arises when the perturber is a substantial fraction of the system’s own mass, which is precisely the interesting case: a satellite galaxy sinking into a halo, a black hole sinking into a merger remnant. There the outer limit is comparable to the system and the inner one is comparable to the perturber, so the ratio is small, the logarithm is of order unity, and the whole expansion the derivation rests on is being used outside its range. The uncertainty is therefore not a missing constant but a signal that the formula is being applied where it was not derived.

Where the ladder goes next #

The obvious next rung is the linear-response calculation itself: how a drag emerges from resonances between a perturber’s orbit and its host’s, and why that formulation has no cut-off to choose. The rung after that is the case the local formula cannot reach at all — the last parsec of a black-hole pair, where the stars that would supply the drag have already been ejected by it.