Astronomy & Celestial Mechanics
the sky, computed
Gravitation

Stars lost to a kick that adds no energy

A tidal shock changes each star's energy by two terms. One is the heating every textbook integral computes, always positive and second order in the shock. The other is the product of the star's own velocity and its kick, first order in the shock and exactly zero on average — and for the weak shocks a cluster meets most often it decides who leaves, doubling the losses without adding any energy at all.

Assumes Impulse approximation, Virial theorem and Two-body relaxation.

16 min read 6 figures Small perturbations, large consequencesTwo bodies, and no more

A star cluster crossing the disc of the Galaxy is squeezed. For the few million years of the crossing, the extra mass of the disc pulls every star toward the disc’s mid-plane, and a star above the cluster’s centre is pulled down harder than the centre while a star below it is pulled up. The net effect, once the uniform part that merely moves the cluster is removed, is a change in each star’s vertical velocity proportional to its height above the cluster’s centre: Δvz=−kz\Delta v_z = -kz, with kk set by the disc’s surface density and the cluster’s speed through it.

The energy that change delivers is where the arithmetic has a trap in it. A star moving with vertical velocity vzv_z whose velocity changes by Δvz\Delta v_z changes its kinetic energy per unit mass by

ΔE=vz Δvz+12(Δvz)2=−kzvz+12k2z2.\Delta E = v_z\,\Delta v_z + \tfrac12 (\Delta v_z)^2 = -k z v_z + \tfrac12 k^2 z^2 .

The second term is the tidal heating. It is always positive, it depends only on where the star is, and averaged over the cluster it gives the energy the shock deposits — the integral computed in every treatment of disc shocking. The first term is dropped from that integral, correctly, because it averages to zero: in a cluster in equilibrium, a star at height zz is as likely to be moving up as down. It adds nothing to the cluster’s energy.

It does, however, add to each star’s. And whether a star leaves is a question about that star.

The same energy given to a cluster two ways. The distribution of the specific energies of 20,000 stars in a Plummer cluster (units G = M = a = 1), before and after an impulsive compressive shock of strength k = 0.06, which changes each star's vertical velocity by k times its height. Stars whose energy ends above the potential at the tidal radius, −0.196, marked by the vertical line, are lost. The thin curve is the cluster before. The dashed curve adds only the half-square term, the heating the textbook integral computes: every star gains a little, and 0.8 per cent of the stars cross the line. The solid curve adds the whole change, including the cross term between each star's own velocity and its kick, which averages to zero — the mean energy input is 1.59e-3 against 1.52e-3 — but broadens the distribution: 1.6 per cent of the stars cross. The same energy, 2.2 times as many escapers.
Fig. 1 The loosely bound end of a Plummer cluster’s energy distribution before and after a weak shock, of strength 0.06 in units where the cluster’s central velocity dispersion is about one. Stars whose energy ends above the escape energy, the potential at the tidal radius, are lost. Adding only the heating term moves every star slightly to the right and pushes 0.8 per cent of the cluster over the line. Adding the whole change delivers the same mean energy, to within the sampling, and pushes 1.6 per cent over.

A zero that is not small

The cross term vanishes on average and is the larger of the two terms for almost every star. The reason is visible in the powers of kk. The heating is second order in the shock; the cross term is first order. For a shock weak compared with the cluster’s own velocities — the ratio of the kick kzkz to the star’s speed vv is small — the cross term exceeds the heating by roughly 2v/kz2v/kz, which can be a very large number.

How much larger the spread is than the heating. The root-mean-square of the cross term, minus k times the star's height times its vertical velocity, divided by the mean of the half-square heating ½k²z², for stars in shells of a Plummer cluster, on a logarithmic scale, for shocks of strength k = 0.05, 0.15, 0.4. The ratio is roughly twice a star's velocity over the velocity change it receives, so it falls as the shock strengthens and falls outward, where stars move slowly and the kick, proportional to height, is large. At the scale radius it is 18 for the weakest shock and 2.2 for the strongest. Only where it approaches one does the textbook heating describe what happens to an individual star.
Fig. 2 The typical size of the cross term, as a root mean square, over the mean heating, in shells of a Plummer cluster for three shock strengths. At the scale radius the cross term is eighteen times the heating for the weakest shock and two times for the strongest. The ratio falls outward, where stars move slowly and the kick, growing with height, is large; only in the outskirts of a strongly shocked cluster does the heating describe what happens to an individual star.

So the energy change of a single star is not a small positive number. It is a large number of either sign, with a small positive mean. A shock does two things to the distribution of energies: it shifts it slightly, and it broadens it considerably. The shift is what the heating integral computes. The broadening is, in the language of transport, a diffusion in energy — each shock is one step of a random walk whose step size is set by the cross term, and the cluster’s energy distribution spreads the way an accumulated rounding error does, as the square root of the number of steps.

That broadening carries no energy, and for most purposes it can be ignored. The one purpose for which it cannot is counting escapers. Escape is a threshold: a star is lost when its energy ends above the potential at the cluster’s tidal boundary. The number of stars crossing a threshold is set by the tail of the distribution near it, and a small shift and a wide spread move a tail by very different amounts.

How many more

The figure below counts the losses both ways over a range of shock strengths, for the same sampled cluster. Every number in it comes from twenty thousand stars drawn from a Plummer model and truncated at the tidal radius, kicked by Δvz=−kz\Delta v_z = -kz, and sorted by whether their new energy exceeds the escape energy.

Stars lost to one shock, with and without the spread. The fraction of a Plummer cluster's stars lost to a single impulsive compressive shock, on a logarithmic scale, against the shock's strength k from 0.01 to 0.5, in units where the cluster's central velocity dispersion is of order one. The dashed curve counts the stars pushed past the escape energy by the half-square heating alone; the solid curve by the whole energy change. For weak shocks the whole change loses many times more: 4.7 times as many at k = 0.03, 2.2 times at k = 0.06, 1.2 at 0.2. For strong shocks the half-square term, second order in k, catches up with the cross term, first order in k, and the ratio falls to 1.04 at k = 0.5, where 21 per cent of the cluster goes.
Fig. 3 The fraction of the cluster lost to one shock, counted with the heating term alone and with the whole energy change, for shock strengths from 0.01 to 0.5. For the weakest shocks the whole change removes several times as many stars: 4.7 times at 0.03, 2.2 times at 0.06. For strong shocks the heating, second order in kk, catches up with the cross term, and the two counts converge — at 0.5 the difference is four per cent and a fifth of the cluster goes either way.

The shape of that result is the argument. At strong shocks the heating dominates everything and the spread is a correction. At weak shocks the heating hardly reaches the threshold and the spread does most of the work, carrying stars that sat just below the escape energy over it by a random amount that averages to nothing. The ratio of the two counts climbs steeply as the shock weakens, because the heating’s contribution falls as k2k^2 and the spread’s as kk.

A real disc crossing is weak. The disc’s vertical pull at the Sun’s radius is about 2πGΣ2\pi G\Sigma with a surface density of about fifty solar masses per square parsec, and a cluster crossing at two hundred kilometres per second receives a velocity change of about 0.014 kilometres per second per parsec of height. For a globular cluster of a hundred thousand solar masses and a three-parsec scale radius, whose internal velocity unit is about twelve kilometres per second, that is a shock strength of a few thousandths in the units of the figure. A more diffuse cluster, or a crossing of the denser inner disc near pericentre, reaches a few hundredths. The regime the figure describes as weak is the regime every globular cluster lives in, and in it the losses are set by the term the heating integral leaves out.

The treatment of shocks in which this was first taken seriously, in the 1990s, called the cross term’s contribution second-order heating, because its effect on the energy appears at second order in kk once the spread is fed back into the distribution. It found the correction comparable to the first-order heating in the cluster’s mean energy over many shocks. What the count above shows is the more direct consequence: well before the spread changes the mean, it changes who escapes.

Where the extra escapers come from

The heating term grows as the square of height. It reaches the outskirts of the cluster, where the kick is large and the stars are already loosely bound, and almost nowhere else. The cross term depends on the product of height and velocity, and stars deeper in the cluster move faster. It can reach further in.

Where in the cluster the escapers started. The fraction of stars lost from each radial shell of a Plummer cluster (radii in units of the scale radius a) after one impulsive shock of strength k = 0.1. The pale bars count only stars pushed out by the half-square heating, which grows as the square of height and so reaches only the outskirts: inside 1.5 scale radii it removes 0.0 per cent of the stars. The dark bars count the whole energy change: inside 1.5 scale radii it removes 0.3 per cent, because a star deep in the cluster moving fast and near the top of its own orbit's energy range can be carried over the line by the cross term even though the average kick there is tiny. In the outermost shell the two nearly agree — 57 per cent against 63 per cent — and there the half-square term alone even loses slightly more, because the cross term takes energy from as many stars as it gives it to.
Fig. 4 The fraction of each radial shell lost to one shock of strength 0.1, counted both ways. The heating term alone removes stars almost only from beyond three scale radii. The whole change removes a few per cent of the stars between one and a half and three scale radii as well, stars moving fast near the top of their own orbit’s energy range. In the outermost shell the two nearly agree, and the heating term alone loses slightly more there, because the cross term takes energy from as many stars as it gives it to.

The outermost shell shows something the averages never could. There, the cross term’s losses and gains nearly balance: a star near the edge is as likely to be pushed away from the threshold as toward it, and some stars the heating alone would have lost are saved. The extra escapers come from inside, from the shells where the heating is too small to matter and the spread is not. So the spread does not simply scale up the loss the heating predicts. It moves the source of the loss inward.

That matters for the cluster’s structure afterwards. A shock that removes only the outermost stars trims the cluster’s edge. One that removes stars from a range of radii changes its profile, and it changes the energy of what remains in a way that depends on which stars went.

Afterwards: the cluster re-settles

A shock lasts a few million years and a cluster’s crossing time is a few million to ten million, so a cluster is left out of equilibrium: the stars that remain have new energies, and the cluster’s size no longer matches its binding. Over a few crossing times it re-virialises, by the same violent relaxation in a changing potential that sets up an equilibrium in the first place. The virial theorem then fixes its new size from its new mass and energy.

The escapers carry away the least bound stars, so the survivors are on average more tightly bound than the cluster was. A cluster that lost its loosely bound stars has a larger binding energy per unit mass and settles smaller. This is the counter-intuitive thermodynamics of self-gravitating systems acting in the direction that protects the cluster: the next shock, whose kick grows with height, finds a smaller target.

A cluster through 80 shocks of strength 0.06. The mass remaining in a cluster after each of 80 identical impulsive shocks of strength k = 0.06, with the survivors re-virialised between shocks as a Plummer sphere of the mass and energy they hold, and the tidal radius shrinking as the cube root of the mass. The dashed curve lets only the half-square heating decide who escapes; the solid curve uses the whole energy change. After 40 shocks the first has lost 22 per cent of the cluster and the second 34 per cent; after 80, 35 per cent and 46 per cent. The survivors are the more tightly bound stars, so each re-virialised cluster is smaller — the scale radius falls to 0.38 of its starting value with the spread and 0.58 without — and a smaller cluster is heated less by the same shock, since the kick grows with height. The first shock removes 1.8 per cent; the last, 0.6 per cent. The cluster is peeled and hardens as it goes.
Fig. 5 Eighty identical shocks of strength 0.06, with the survivors re-virialised between each as a Plummer sphere of the mass and energy they hold, and the tidal radius shrinking as the cube root of the mass. After forty shocks the heating term alone has removed 22 per cent of the cluster and the whole change 34 per cent; after eighty, 35 and 46 per cent. The cluster shrinks to 0.38 of its starting scale radius with the spread included, and each shock then removes less than the one before it.

The two curves differ by a factor that is not constant. Early on the spread loses about twice as many stars per shock. Later, the cluster that has lost more has also contracted more, and a smaller cluster is harder to shock, so the per-shock losses of the two histories come closer together while their totals stay apart. The spread’s extra losses are front-loaded. They are paid while the cluster still has a loosely bound envelope to lose.

A stronger series shows the other side of the same result: the spread matters most when the shock is weak.

A cluster through 60 shocks of strength 0.1. The mass remaining in a cluster after each of 60 identical impulsive shocks of strength k = 0.1, with the survivors re-virialised between shocks as a Plummer sphere of the mass and energy they hold, and the tidal radius shrinking as the cube root of the mass. The dashed curve lets only the half-square heating decide who escapes; the solid curve uses the whole energy change. After 30 shocks the first has lost 34 per cent of the cluster and the second 41 per cent; after 60, 47 per cent and 53 per cent. The survivors are the more tightly bound stars, so each re-virialised cluster is smaller — the scale radius falls to 0.32 of its starting value with the spread and 0.43 without — and a smaller cluster is heated less by the same shock, since the kick grows with height. The first shock removes 3.1 per cent; the last, 0.7 per cent. The cluster is peeled and hardens as it goes.
Fig. 6 Sixty shocks of strength 0.1, the same model. After thirty the heating term alone has removed 34 per cent of the cluster and the whole change 41; after sixty, 47 and 53. The gap is narrower than for the weaker series, as the single-shock count predicts, and both clusters end at about a third of their starting scale radius. The first shock removes three per cent of the stars and the last less than one.

That is the step that turns an energy input into a number of stars. The heating integral says how much energy a shock deposits, and that number is right. It does not say how many stars leave, because escape is decided star by star at a threshold, and the answer depends on a term whose average is zero. The re-virialisation then sets what the next shock meets, and it sets it by the mass that left rather than the energy that arrived.

Why the averages were never going to see it

It is worth asking why a term this large could be set aside for so long, and the answer is that it was set aside for a good reason in a calculation that was asking a different question. The heating integral was written to answer how much energy a cluster gains from a shock, and for that question the cross term is exactly zero. Every star that is moving up when it is kicked down loses energy, every star moving down gains it, and in an equilibrium cluster the two populations are equal. The energy budget is right.

The question it was then used to answer was how long a cluster survives, and the route from one to the other went through an assumption that is easy to miss: that a cluster which has gained a given energy has lost a given number of stars. That would be true if every star gained the same energy, or if the energies were far from the threshold. Neither holds. The energies near the threshold are exactly the ones that decide escape, and a star’s gain is dominated by a term whose sign is a coin toss.

The same structure appears in a drag computed from the same kind of integral, where the mean deceleration of a massive body is the quantity wanted and the fluctuating part is discarded because it cancels. There the discard is safe, because a massive body moving through many stars has its own fluctuations averaged away. A star in a shocked cluster has no such averaging. It receives one kick per crossing, and the kick either carries it over the threshold or it does not.

There is also a sense in which the spread is the more robust of the two effects. The heating depends on the square of the shock strength and on the mean square height of the stars, both of which are uncertain — a thirty per cent error in a cluster’s size is a factor of 1.7 in its heating. The spread depends on the first power of each, so the same uncertainty moves it by only thirty per cent. For weak shocks, where the spread dominates the losses, the escape rate is less sensitive to the poorly known size of a cluster than the heating calculation would suggest.

What the model leaves out

The model behind these figures is deliberately simple, and three of its simplifications should be stated because each moves the numbers.

It is impulsive. Every star is kicked instantaneously, including stars deep in the core whose orbital periods are shorter than a real disc crossing. The adiabatic protection of the core — the core’s stars following a slow perturbation and returning what they took — is absent. That protection applies to both terms and suppresses the cross term’s inward reach most of all, so the extra escapers from one and a half to three scale radii are an upper bound. The factor of two at weak shocks survives in the outskirts, where orbital periods are long and the shock is impulsive in fact.

Its escape criterion is an energy. A star above the escape energy is counted as lost at once. A real star above the energy threshold of a cluster in a galactic tide escapes through two narrow gaps near the Lagrange points of the cluster–galaxy system, and it can take many orbits to find one. Stars with energies just over the threshold linger, and some are scattered back below it by later encounters. The count is the number of stars made able to leave, not the number that have left.

And its re-virialisation is homologous. Each settled cluster is drawn as a fresh Plummer sphere of the right mass and energy, which erases whatever the spread did to the shape of the energy distribution between shocks. In a real cluster the broadening accumulates over successive crossings, the diffusion continues between them through two-body relaxation, and the envelope is refilled from inside. A full treatment follows the distribution of energies through every shock and every interval, which is what the modern Monte Carlo and N-body calculations of cluster evolution do.

What is actually measured

No individual shock has ever been observed. What is observed is the result: globular clusters with mass functions depleted of their low-mass stars, clusters with extended tidal tails along their orbits, and a population whose mass distribution has lost its low-mass end. The tails are the most direct witness. A cluster that has recently crossed the disc should carry an excess of stars beyond its tidal radius, released within the last orbit and not yet sheared into a stream, and several do.

The size of that excess is a number of stars, and the number of stars is exactly the quantity this essay is about. A calculation that counts escapers from the heating integral predicts fewer extra-tidal stars than one that follows the spread, by the factor of two or more drawn above, and predicts them from further out. Comparing the observed excess with the prediction for clusters whose orbits are measured — from the space astrometry that now gives every globular cluster a three-dimensional velocity — is how the escape rate is tested. The factor-of-two agreement usually quoted for that comparison is about the size of the effect this whole essay describes, which is a reason to state which calculation is being compared.

A term that is zero and decisive

The general shape is one that recurs wherever a threshold meets a random process. A mean is the right description of what a population gains; a distribution is the right description of what happens to its members; and a threshold samples the distribution only at its edge. A term that contributes nothing to the mean can dominate the edge. Relaxation in a cluster is the same statement on a slower clock: the encounters that change no star’s average energy are what drive stars over the escape threshold, one by one, until the cluster evaporates.

What is special about the shock is that the zero-mean term is not a small correction to the heating but the larger of the two, by a factor that grows as the shock weakens. The weaker the perturbation, the less its average effect describes what it does. For the weak, repeated shocks of disc crossings, the heating integral computes the energy correctly and the escapers wrongly, and the cluster keeps count of the escapers.

The same distinction runs through the whole of the impulse approximation. A deflection computed along the path a body would have taken gives each passing star a definite kick; summed over a cluster, the translation cancels and leaves the tidal part; summed over the cluster’s stars, the tidal part’s mean leaves the heating. At each step something is removed because its average vanishes, and at each step the removal is correct for the average and incomplete for the member. The escapers are members.

Still open: shocks that are neither impulsive nor slow

Every calculation here, and every closed-form treatment of shocks, sits at one of two limits: the shock is instantaneous and every star is kicked, or it is slow and stars follow it adiabatically and give the energy back. Real disc crossings sit between the two for a large part of every cluster. A star whose orbital period is comparable to the crossing time is kicked partly, and its cross term — the product of its velocity and a kick that is changing while the velocity is — has no closed form.

How much of a cluster’s energy diffusion comes from that intermediate regime, and whether the core’s protection against the heating also protects it against the spread, is decided only by integrating the stars through the shock. The answer sets how deep into a cluster a disc crossing reaches, and so whether the population of globular clusters seen today was carved mainly from outside or partly from within.

Where this goes next

The kick falls as b⁻² and the heating as b⁻⁴. Two quantities delivered by the same distant encounter, against impact parameter in units of the target's half-mass radius, both logarithmic and both scaled to cross near one. The upper line is the impulse itself, 2Gm/vb, which every star in a bound system receives almost equally — so it moves the system and changes nothing inside it. The lower line is what is left after the common part is subtracted: the difference of the impulse across the system, which is its gradient multiplied by the system's own size, one power of b smaller and squared in the energy. The slopes are −2 and −4 exactly, and they are measured off the drawn curves rather than quoted. What follows is the point. Encounters at impact parameter b arrive at a rate proportional to b db, so summing the impulse over all of them gives ∫b⁻¹ db, which diverges logarithmically and is the origin of the Coulomb logarithm that appears in every treatment of relaxation. Summing the heating gives ∫b⁻³ db, which converges: extending the population from ten half-mass radii out to 300 multiplies the summed impulse by 2.48 and the summed heating by 1.010. Distant encounters diffuse velocities and heat nothing, and a tidal-heating calculation therefore needs no cutoff at large impact parameter, where a relaxation calculation cannot proceed without one. The part of a kick that heats nothing Most of the impulse a passing mass delivers to a star cluster is delivered equally to every star in it, so the cluster moves and nothing inside it changes. What heats it is the difference across it — one power of the impact parameter smaller, squared in the energy — and that one distinction decides which encounters matter and which cannot. 7 decades of relaxation time, and a Hubble time between them. Crossing time and two-body relaxation time for 5 self-gravitating systems, against the number of bodies in each. The lower marks are the crossing time R/σ, which spans 3 decades; the upper ones are t_relax ≈ 0.1N/ln N times it, which spans 8. The horizontal rule is a Hubble time, and the same expression puts these systems on opposite sides of it. Two of them relax — an open cluster and a globular cluster — so their stars have exchanged energy, their heavy stars have sunk, and their present structure is not the one they were born with. The rest do not: a dwarf spheroidal, an elliptical galaxy, a cluster of galaxies, the elliptical missing it by a factor of 10⁶, which is why a galaxy is modelled as a smooth potential with no stars in it at all. The dependence is on N and hardly at all on anything else, because the Coulomb logarithm makes every decade of impact parameter contribute equally. A cluster that relaxes also evaporates: about a stellar mass in a hundred and forty leaves per relaxation time, so the globular cluster empties in 6.1·10¹¹ years. A cluster that boils itself away A star cluster has no thermostat. Encounters between its members push a few of them above escape speed, the cluster loses them, and losing them makes it contract — which makes it hotter, which makes more of them escape. A self-gravitating system heats up as it loses energy, and the process ends by destroying the system. A body that heats up because it is losing energy. A uniform self-gravitating sphere of 1.0 solar masses radiating at 1.0 solar luminosities, with no nuclear source at all, from 3.0 solar radii. Everything is in units of the starting energy, and the two curves that matter run in opposite directions: the total energy falls, and the temperature rises. That is not a paradox and it is not a special case. The virial theorem makes 2K = −U for any self-gravitating gas in equilibrium, so E = U + K = −K, and −dE/dt = +dK/dt: energy leaving as light is energy arriving as heat. The bookkeeping is exact and is measured here rather than quoted — over the run 5.465·10⁴⁰ J of gravitational energy is released and 2.733·10⁴⁰ J is radiated, a ratio of 2.0000. Half the release is spent on the star's own heat and only half escapes. The consequence is a body with a negative heat capacity, which is why a contracting protostar gets hotter until it ignites, why a globular cluster's core runs away instead of settling, and why nothing self-gravitating ever comes to thermal equilibrium. The system that gets hotter as it loses energy The virial theorem makes a self-gravitating body's total energy equal to minus its kinetic energy. Radiating heat away therefore raises the temperature, there is no equilibrium to settle into, and every star and every cluster is running away from one.

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Disc shockingEnergy diffusionEscape energyGlobular clusterImpulse approximationPlummer sphereTidal shockVirial theorem