How long a system takes to forget

A galaxy is modelled as a smooth potential with no stars in it. That sounds like a simplification made for arithmetic, and it is not: it is a statement about a timescale, and the timescale is $10^{17}$ years.

A globular cluster is modelled as a collection of stars that exchange energy with each other, segregate by mass, and eventually destroy their own centres. That is a statement about the same timescale, which for a cluster is a billion years.

Both objects are held together by the same force and described by the same equations. What separates them is the number of stars in them, and the way that number enters is not obvious.

One encounter #

Take a star of mass $m$ passing another of the same mass at impact parameter $b$ and relative speed $v$. If the deflection is small, the transverse velocity it picks up can be computed by integrating the perpendicular force along the unperturbed straight line — the impulse approximation, which is exact in the limit of a fast pass and good to a few per cent for anything that is not nearly a collision.

The perpendicular force at time $t$ is $Gm^2b/(b^2+v^2t^2)^{3/2}$, and integrating it over all time gives

$$\delta v_\perp = \frac{2Gm}{bv}.$$

That is the whole of the encounter. It is inversely proportional to the impact parameter and inversely proportional to the speed, and it points perpendicular to the original motion.

Adding them up #

A star crossing a system of radius $R$ containing $N$ others passes each of them at some impact parameter. The deflections point in random directions, so they do not add; their squares do.

The number of encounters with impact parameter between $b$ and $b+\mathrm db$ during one crossing is the number of stars in a cylindrical shell of that radius, which is $N\,(2\pi b\,\mathrm db)/(\pi R^2)$. Multiplying by $\delta v_\perp^2$ and integrating,

$$\Delta v^2 \;=\; \int \left(\frac{2Gm}{bv}\right)^2 \frac{2Nb\,\mathrm db}{R^2} \;=\; \frac{8N G^2m^2}{R^2v^2}\int\frac{\mathrm db}{b}.$$

The integral is $\ln(b_{\max}/b_{\min})$, and that logarithm is the whole character of the result. Every decade of impact parameter contributes equally. A star is deflected as much, in total, by the thousands of stars it passes at a hundred parsecs as by the one it passes at a tenth of a parsec, because the encounters get weaker exactly as fast as they get more numerous.

The limits are physical rather than arbitrary. The upper is the size of the system, beyond which there are no more stars. The lower is the impact parameter at which the deflection stops being small — around $Gm/v^2$ — below which the impulse approximation fails and the encounter is a genuine two-body scattering. The ratio $\Lambda = b_{\max}/b_{\min}$ works out at roughly $N$ for a virialised system, and since it appears only inside a logarithm, being wrong about it by a factor of ten costs a factor of about 1.3 in the answer. That insensitivity is what makes the whole calculation useful.

The timescale #

Setting $\Delta v^2$ equal to $v^2$ — the condition that a star has forgotten its original velocity — and using $v^2 \approx GNm/R$ from the virial theorem gives the number of crossings required:

$$\frac{t_{\rm relax}}{t_{\rm cross}} \approx \frac{N}{8\ln N}.$$

That is the expression the whole subject turns on, and its shape is worth stating in words. The relaxation time is the crossing time multiplied by roughly a tenth of the number of stars. Not by the square root, not by the logarithm, but by $N$ itself, softened only by a logarithm in the denominator.

The consequences follow immediately.

An open cluster: a thousand stars, four parsecs across, half a kilometre a second. The crossing time is eight million years and the relaxation time a hundred million. Open clusters relax within their own lifetimes, and they do not have lifetimes much longer than that — they dissolve.

A globular cluster: a million stars, seven parsecs, eleven kilometres a second. The crossing time is 620,000 years and the relaxation time four billion. Comparable to the age of the cluster, which is why globular clusters show every consequence of relaxation and are the standard laboratory for it.

An elliptical galaxy: $10^{11}$ stars, ten kiloparsecs, two hundred kilometres a second. The crossing time is fifty million years — respectably short — and the relaxation time $2\times10^{16}$ years, a million times the age of the universe. The stars in a galaxy have never interacted with each other individually at all.

Why the equations are different, not just the numbers #

The distinction is not one of degree. It changes what is written down.

A collisionless system obeys the collisionless Boltzmann equation: the distribution function is constant along trajectories in the smooth potential the system generates, and the potential is generated by the distribution function. It is a self-consistent field theory. Nothing in it refers to individual stars, and the number of stars appears only through the total mass.

A collisional system needs a collision term — a Fokker–Planck operator describing the diffusion of stars through energy space at the rate computed above. That turns a conservation law into a diffusion equation, and diffusion equations have irreversible solutions: heat flows, gradients smooth, and the system evolves towards a state it cannot leave.

The two descriptions are not approximations to one another. Which one applies is decided by $N$, and the boundary between them runs straight through the sample above.

There is a way to make the point sharper. Suppose a galaxy’s mass were carried by a hundred million objects of a thousand solar masses each rather than by $10^{11}$ stars. The total mass is the same, the smooth potential is the same, every orbit computed from that potential is identical — and the relaxation time is shorter by a factor of a thousand, which brings it within a Hubble time. Discs would be thickened, cold streams would be heated, and the galaxy would look different. That is a real observational constraint and it has been used as one: the survival of thin discs and of cold stellar streams places an upper bound on the mass of whatever dark matter is made of, at a few hundred solar masses for objects in the halo.

What relaxation does when it happens #

Three consequences, and each is observed.

Equipartition and mass segregation. Encounters transfer energy preferentially from the faster body to the slower, which drives the system towards equal kinetic energies rather than equal speeds. A star twice the average mass ends up with $1/\sqrt2$ of the average speed, sinks in the potential, and settles towards the centre. The timescale is the relaxation time divided by the mass ratio, so the heaviest objects segregate first. In a globular cluster the observable is that the blue stragglers and the binaries are concentrated towards the core while the low-mass stars are spread through the halo, and it is one of the cleanest confirmations that the mechanism operates.

Evaporation. The velocity distribution that relaxation drives towards has a tail, and a star in the tail above the escape speed leaves. Removing it lowers the cluster’s energy, the remaining stars re-relax, the tail refills, and another leaves. The fraction escaping per relaxation time works out at about 0.0074, so a cluster empties in roughly 136 relaxation times — half a trillion years for the cluster above, which sounds safe until the tidal field of the host galaxy is included, and that cuts it by more than an order of magnitude.

Core collapse. The most dramatic, and the one that makes relaxation more than a slow smoothing.

The one place where a galaxy relaxes #

The million-Hubble-time answer above is for a galaxy as a whole. It is not the answer everywhere in one.

Relaxation time scales as $N/\ln N$ times $R/\sigma$, and both factors run the wrong way near a galactic centre: the density is enormous, so the crossing time is short, and the relevant $N$ is the number of stars within the region rather than in the whole galaxy. Within a parsec of Sagittarius A* the relaxation time falls to a few billion years — inside a Hubble time, and therefore relevant.

That has consequences that are observed. The stars there should be mass-segregated, with stellar-mass black holes concentrated into the innermost region; the density profile should approach the cusp that relaxation drives towards; and the rate at which stars are scattered onto orbits that pass close enough to the central black hole to be torn apart is set by the same diffusion coefficient. A tidal disruption event is a relaxation calculation with an observable outcome, and the observed rate of such events is one of the few direct tests of the theory outside star clusters.

The logarithm, and what it means #

It is worth returning to $\ln\Lambda$, because it is the least intuitive part of the result and the most consequential.

The statement that every decade of impact parameter contributes equally means the relaxation is not dominated by close encounters. That is the opposite of the intuition the word “collision” invites. A star is not knocked about by occasional near-misses; it is nudged continuously by the whole system, and the nudges from a hundred parsecs away matter as much as the ones from a tenth of a parsec.

That is why the answer depends on $N$ almost linearly and on the structure of the system hardly at all. Two clusters with the same number of stars, the same size and quite different density profiles have relaxation times within a factor of two of each other, because the integral that produces the answer is dominated by no particular radius.

It is also why the calculation transfers. The identical logarithm appears in the Coulomb collision rate of a plasma, for the identical reason — an inverse-square force, encounters weakening as $1/b$ and multiplying as $b$ — and the two subjects have borrowed the name from each other for a century.

Where the model stops #

Four assumptions, and each is violated somewhere that matters.

The encounters are two-body, and in a dense cluster core they are not. Three-body encounters involving a binary are qualitatively different: a hard binary hardens further and gives up energy to the passing star, which makes binaries an energy source that can halt core collapse and reverse it. The post-collapse evolution of a globular cluster is driven entirely by an effect this calculation does not contain.

The stars are all the same mass, and a real mass spectrum changes the answer. Relaxation is driven mainly by the heaviest objects present, because $\delta v \propto m$ and the diffusion carries $m^2$; a cluster containing black holes relaxes considerably faster than its mean mass suggests.

The system is isolated, and none are. A cluster orbiting in a galaxy’s tidal field has a truncation radius, stars beyond it are stripped, and the loss rate from tidal stripping exceeds the evaporation rate for most of the Galactic globulars.

And the impulse approximation assumes the deflections are small, which fails for the closest encounters. Those are rare and their contribution is inside the logarithm, so the error is bounded — but in a core dense enough for physical stellar collisions the whole framework is being used past its range, and the blue stragglers are the evidence that something outside it is happening. The one number the calculation cannot fix is the Coulomb logarithm, so the honest presentation is to draw the two conclusions it enters at a very different value of it.

The relaxation that is not this one #

A galaxy has never relaxed by the mechanism in this essay and yet it looks relaxed: smooth, centrally concentrated, with a density profile close to what a settled system would have. Something arranged it, and it was not two-body encounters.

The mechanism is collisionless and it operates on a crossing time rather than on a relaxation time. During the collapse of a protogalaxy the potential is changing rapidly — matter is falling in, the mass distribution is rearranging, and the potential a star moves in is not the one it started in. A star’s energy is not conserved in a time-varying potential, so stars exchange energy with the collective field rather than with each other.

The exchange is chaotic and its outcome is a broad redistribution of energies, achieved in a few crossing times. It has been called violent relaxation ever since it was described in 1967, and the name is apt: it is fast, it is irreversible, and it happens once.

Three features distinguish it from the two-body case and are worth holding separately.

It does not depend on the number of bodies at all. A system of a hundred and a system of a hundred billion violently relax at the same rate in units of their own crossing time, because the mechanism is the collective potential rather than the granularity of it.

It does not drive equipartition. Energy per unit mass is what is redistributed, not energy per particle, so violent relaxation produces no mass segregation. A galaxy’s heavy stars are not concentrated in its centre for this reason, and a globular cluster’s are — which is a direct observational separation of the two mechanisms.

And it is incomplete. The process switches itself off as soon as the potential stops changing, which is after a few crossing times, so it never reaches a fully thermal state. The end product is a particular non-thermal distribution, and the fact that observed elliptical galaxies resemble it is one of the standard arguments that they were assembled this way.

A system can look relaxed without ever having relaxed, and separating the two is what makes the timescale in this essay worth computing.

The experiment that checks it #

Every number in this essay comes from an analytic estimate with an approximation in it, and the way they are checked is to integrate the equations of motion directly with no approximation at all.

That is harder than it sounds, for two reasons that have shaped the whole subject. The force calculation is a sum over every pair, so the cost per step grows as the square of the number of bodies, and a simulation with a million bodies does $10^{12}$ force evaluations per step. And the close encounters — the ones inside the logarithm’s lower limit — require arbitrarily small timesteps, so a naive integrator spends all of its effort on a handful of pairs.

Both have standard solutions. The pairwise cost is reduced by treating distant groups of particles as single masses, which converts the scaling from $N^2$ to $N\log N$ — at the price of introducing exactly the smoothing that relaxation is about, so a tree code is the wrong tool for measuring a relaxation rate and the right one for a collisionless system. And the close encounters are handled by regularisation: a change of variables that removes the singularity in the two-body problem analytically, so that a near-collision becomes a smooth passage in the transformed coordinates.

Direct integrations of that kind have been run for the number of stars in a real globular cluster, which took special-purpose hardware in the 1990s and graphics processors since. What they confirm is the scaling — the relaxation time really does go as $N/\ln N$ times the crossing time — and what they correct is the coefficient, which the analytic estimate gets right to within a factor of about two.

The analytic result supplies the scaling and the simulation supplies the constant, which is the same division of labour the Coulomb logarithm’s cutoffs required, and for the same reason: the approximation is controlled everywhere except at its boundaries.

And the sinking time from a larger starting radius, since the whole point of the timescale is that it is compared with the age of something.

One more reading of the drag’s velocity dependence closes the comparison between the two timescales.

Where this ladder goes next #

This rung establishes the timescale and the boundary it draws.

Above it lies the Fokker–Planck description proper: not “how long until a star has forgotten” but the full diffusion of the distribution function through energy space, which is what produces the collapsed-core profile and the evaporation rate as solutions rather than as estimates.

Beside it lies the regime where the description fails on purpose. A cluster core dense enough for binaries to dominate, or for stars to physically touch, needs direct $N$-body integration, and the results of those integrations are what the analytic timescale is calibrated against.

And below it, in a sense, lies the whole of galactic dynamics, which exists as a separate subject only because the number in this essay came out so large. A galaxy is not a gas of stars, and the reason is arithmetic.