A slow shock heats the halo and robs the core

A star cluster crossing the disc of the Galaxy is squeezed by the disc’s gravity for the million or so years it takes to pass through, and the squeeze does work on its stars. If the crossing were instantaneous the result would be simple: every star receives a kick proportional to its height above the cluster’s own mid-plane, and the cluster as a whole gains energy in proportion to the square of the kick. If the crossing were infinitely slow, every star would follow the changing force smoothly and return to its original orbit when the force went away, and nothing would be gained at all.

Real crossings are neither. A star deep in a cluster’s core completes an orbit in a few hundred thousand years, faster than the crossing; a star in the outskirts takes ten million, far slower. The standard way of handling the gap is to compute the instantaneous answer and multiply by a correction factor that depends on how many orbits a star completes during the shock — near one for slow stars, falling steeply for fast ones. The factor is computed for an idealised star, and its derivation for a harmonic oscillator gives an exponential fall that is both clean and severe.

Nothing guarantees that a cluster behaves like a collection of oscillators, and the question left open by the treatment of the cross term was exactly this: what happens in the intermediate regime, where no closed form applies? The answer can only be computed, by putting stars through a shock one at a time.

The experiment #

The cluster is a Plummer sphere, the standard model of a globular cluster’s density, with twelve hundred stars sampled from its equilibrium distribution of positions and velocities. The shock is a compressive tidal force along one axis, growing and decaying as a Gaussian in time, with its duration varied while its total impulse — the time integral of its strength — is held fixed, so that every shock would deliver the same energy if it were instantaneous.

Each star is placed at its sampled position at the moment the shock peaks, integrated backwards without the shock to well before it begins, and then forwards through it to well after it ends, with a symplectic integrator and a step much shorter than both the shock and the fastest orbits. Its energy before and after gives its energy change.

One trick makes the measurement possible with so few stars. A star’s energy change has a first-order part, the product of its velocity and its kick, which is large and averages to zero over the cluster, and a second-order part, the heating, which is small and positive on average in the impulsive limit. In any finite sample the first-order part swamps the second. Running each star through the shock twice, once with the force reversed, and averaging the two cancels every odd power of the force exactly, leaving the second-order part star by star. That is the quantity drawn. Halving the step and changing the shock’s strength by a factor of two and a half move the totals by less than a per cent, which says both the integration and the second-order approximation are sound.

A slow shock still heats the cluster #

The opening figure is the total. For a shock lasting a twentieth of the half-mass orbital time divided by two pi, the cluster gains exactly the instantaneous energy, as it should. As the shock lengthens the gain falls, but slowly: to 87 per cent when the duration times the half-mass orbital frequency reaches a half, to 70 per cent when it reaches one, to half at two, and still to over a quarter at four.

Against that are three prescriptions, each applied to the same twelve hundred stars. The first treats every star as a harmonic oscillator at its own circular frequency, which suppresses its heating exponentially in the square of the duration times the frequency. The second uses a power-law suppression of the kind derived for real orbits in the 1990s, also star by star. The third uses a single power-law factor for the whole cluster, evaluated at the half-mass radius — a shortcut in which a cluster’s heating is characterised by one frequency.

The star-by-star prescriptions bracket the integration closely: at the longest shock, 0.24 for the oscillators and 0.36 for the power law, against 0.28 integrated. The single-factor shortcut gives 0.017, seventeen times too little. The cluster-wide suppression is not a property of the cluster’s typical star at all. It is set by the stars that carry the energy.

The energy goes where the lever arm is #

The reason is in where the instantaneous energy would go. A tidal squeeze along one axis kicks each star in proportion to its distance from the mid-plane, and the energy goes as the kick squared — as the square of that distance. A Plummer cluster’s outermost tenth of its mass, beyond about four scale radii, is so much further out than the rest that it would receive nearly sixty per cent of the instantaneous energy on its own. The next fifth of the mass would receive a quarter, and the inner half of the cluster less than a tenth.

Those outer stars have orbital periods many times longer than the half-mass radius’s. For them, any shock short enough to matter for the inner cluster is instantaneous, and they take their full share whatever the duration. The total therefore stays high as long as the outer stars stay impulsive, and falls only when the shock becomes long compared even with their orbits — and those are the orbits most easily stripped by the Galaxy’s tide anyway.

The shortcut fails because it applies the half-mass radius’s frequency to energy that is mostly delivered somewhere else. The star-by-star prescriptions succeed, for the total, because they get the outskirts right, and the outskirts are most of the answer.

The core gives energy away #

The totals hide what happens inside, and the inside is where the textbook picture is not merely inaccurate but wrong in sign.

Divide the cluster into shells by mass, from the innermost tenth to the outermost, and measure each shell’s energy gain as a fraction of the instantaneous share it would have received. For short shocks every shell takes its share. As the shock lengthens the inner shells’ share falls first and fastest, and then does something no suppression factor can: it passes through zero and becomes negative. The innermost tenth of the mass reaches a net loss of nearly nine-tenths of its instantaneous share when the shock lasts about half the half-mass orbital time over two pi, and settles at a loss of over half for longer shocks. Every shell inside the 90 per cent radius ends, for the slowest shocks, within a few hundredths of zero.

A suppression factor is a positive number multiplying a positive energy, and it can make the heating small but never negative. The integration says the inner cluster’s stars, taken together, end a slow shock more tightly bound than they began. The energy did not vanish: the cluster’s total change is positive, so the outer stars gained more than the instantaneous estimate of their share would suggest, by exactly what the core lost.

Every star’s share, positive and negative #

The shell averages conceal a distribution, and the distribution shows how the core loses energy.

In the instantaneous limit every star’s second-order energy change is positive — half the square of its kick — and scattered about its expected value by the spread of heights above the mid-plane at its radius. In a shock lasting about one half-mass orbital time over two pi, forty per cent of the stars come out with less energy than they started with. Nearly all of them are inside the half-mass radius, where orbits are fast enough to follow the shock in part, and their losses, though individually small compared with what the outer stars gain, are enough to make the inner shells’ average negative.

Positive and negative second-order energy changes are familiar from linear response theory, where a slowly applied perturbation transfers energy between orbits that resonate with it in different ways — the same accounting that, in a plasma, produces Landau damping, and in a galaxy produces the drag a massive body feels from the orbits it perturbs. An orbit whose frequency sits on the falling side of the shock’s spectrum can give energy to it as readily as one on the rising side takes it, and the ensemble average depends on how many orbits sit on each side. In a cluster’s core, where orbits are fast and the shock is slow, enough sit on the losing side to make the average negative.

Two stars through the same shock #

What each kind of star does is clearest followed in time.

The halo star, at nearly eight scale radii, has an orbital period far longer than the shock. It barely moves while the force acts; it simply receives the impulse, and its energy steps up by the full instantaneous amount and stays there. The figure shows the second-order part only, so the dip and rise during the shock is the transient tidal energy of a star being squeezed and released; the permanent change after the shock is what matters.

The core star, half a scale radius from the centre, completes several orbits during the shock. Its energy rises and falls as the force builds and decays, following it adiabatically, and when the force has gone the star ends with almost the energy it began with — but not quite, and in this case slightly less. The figure’s two stars were chosen as the most extreme of their kinds, so the core star’s loss is the largest in the sample. Most core stars lose or gain only a little. Their average is what the shell figure measures.

Why the oscillator was the wrong model to start from #

The exponential suppression comes from a specific property of the harmonic oscillator: it has one frequency. A smooth pulse has a spectrum that falls off as a Gaussian at high frequency, and an oscillator responds only to the part of the spectrum at its own frequency, so a slow pulse, whose spectrum has almost nothing at that frequency, does almost nothing to it. The suppression is exponential because the pulse’s spectrum is.

A star in a cluster is not an oscillator except very near the centre, where a Plummer sphere’s density is nearly uniform and its potential nearly harmonic. Everywhere else the potential is anharmonic, and a star’s orbit has a spectrum of its own: a fundamental radial frequency and an azimuthal one, and harmonics and combinations of both. A slow pulse that misses a star’s fundamental can still couple through a low-frequency combination, and the response of an anharmonic orbit to a slow perturbation therefore falls off as a power of the duration rather than exponentially. That was the argument made in 1994 that the adiabatic protection of stellar systems had been greatly overstated, and it is why the star-by-star power law in the opening figure sits closer to the integration than the oscillator does at long durations.

The same multiplicity of frequencies is what allows energy to go the wrong way. A harmonic oscillator, averaged over its phase, always gains energy from a pulse, because there is only one way for it to respond. An orbit with several frequencies, some above the pulse’s characteristic frequency and some below, can be pushed along some and held back along others, and averaged over an ensemble whose population of orbits falls with energy, the net transfer at second order can take either sign. Inside a cluster, where the population is steep and the orbits fast, it comes out negative.

None of this required anything beyond Newton’s laws and a careful integration. What it required was not assuming that a cluster’s stars were simpler than they are — the habit that replaces an orbit with the straight line it would have followed in the impulsive limit, and that serves well there, carried into a regime where the orbit’s own shape is the whole answer.

Where real clusters sit #

All of this would be academic if real clusters sat at one end or the other. They sit in the middle.

A disc crossing at 200 kilometres a second through a layer 250 parsecs thick lasts about 1.2 million years. A typical globular cluster, a hundred thousand solar masses with a half-mass radius of three parsecs, has an orbital time at its half-mass radius of about two million years, and the product of the crossing time and its half-mass orbital frequency is about three and a half — well beyond the point at which the single-factor shortcut has fallen to a few per cent, and in the range where the integration shows the total heating still at a third of the instantaneous value. Only loose, low-mass clusters are shocked impulsively; dense, massive ones are deep in the slow regime at their half-mass radii and impulsive only in their outskirts.

That matters for the population. The globular clusters that survive today are the ones that withstood ten billion years of disc crossings and passages near the bulge, and models of how many there once were, and how many stars they shed into the Galaxy’s halo, depend on how much energy each crossing delivered. A prescription that underestimates the heating by an order of magnitude would conclude that shocks barely matter for any cluster but the loosest; one applied star by star recovers most of the heating, delivered to the outskirts, where it unbinds stars directly.

What was actually measured #

The heating of globular clusters by the disc and bulge is not measured directly. It is inferred from three kinds of evidence, each of which the calculation bears on.

Tidal tails. Several globular clusters trail streams of stars along their orbits, the material they have lost. The density of the tails varies along their length, with bunching that corresponds to the passages through the disc, and the rate of mass loss inferred from them is a test of how much energy each passage delivered. The best-studied, from the cluster Palomar 5, shows clumps along its tails that N-body models of the cluster on its orbit reproduce as the stars released at successive disc crossings.

Mass functions. A cluster that has lost a large fraction of its stars through the Galaxy’s tide has preferentially lost its low-mass stars, which had migrated outward through relaxation and were therefore in the outskirts where the shocks act. Clusters with orbits that cross the disc often or pass close to the bulge show depleted low-mass populations.

The survivors’ structure. The radii and concentrations of surviving clusters depend on their orbits, and the dependence is the signature of the heating the integrations compute. Numerical simulations of whole clusters on realistic Galactic orbits, with every star followed, now reproduce these trends, and they agree with prescriptions applied star by star rather than cluster-wide.

Where the experiment stops #

A fixed potential. The stars move in the cluster’s unchanging Plummer potential during the shock. In a real cluster the potential responds to the stars’ changing orbits, which the calculation leaves out; the response is second order and modifies the energy transfer by a correction of the same order as the effect itself in the inner cluster.

A one-dimensional squeeze. The shock compresses along one axis with a Gaussian time profile. A real disc crossing has a time profile set by the disc’s vertical density, which is flatter-topped, and a passage near the bulge squeezes in more than one direction. The scalings carry over; the exact suppression does not.

Twelve hundred stars. The shells and the scatter come from one sample, and the innermost tenth contains a hundred and twenty stars, so its average is uncertain by perhaps a tenth of its value. The sign of its slow-shock energy change is not in doubt, since it is negative at every duration beyond the impulsive regime and at both shock strengths tried.

No escape, no relaxation. Stars are not removed when they become unbound, and two-body relaxation, which moves energy between core and halo continually, is not included. After a shock, relaxation would begin to redistribute the energy the shock delivered unevenly.

An average dominated by the wrong place #

The general lesson is about where a correction is applied. A correction factor computed for a typical member of a population and applied to the population’s total is right only if the total is dominated by typical members. A tidal shock’s energy goes as the square of the distance from the mid-plane, so the total is dominated by the outermost tenth of the mass, which is not typical in any respect — not in its orbital frequency, its density, or its response to the shock.

A correction must be applied where the quantity it corrects is concentrated, not where the system is typical, and a quantity weighted by a high power of a coordinate is concentrated at the extremes. The same failure appears whenever a heavy-tailed weight meets a characteristic scale: the heating that survives the uniform part of a kick is concentrated far from the centre for the same reason.

The second lesson is that a factor between zero and one is a statement about sign as well as size. Suppression factors were derived for oscillators, for which second-order heating is always positive, and carried over to systems for which it is not. The integration shows a regime in which the inner parts of a cluster are cooled by an event that heats the whole. That redistribution — energy taken from the bound core and given to the loosely bound halo — is the direction in which a self-gravitating system with negative heat capacity is driven anyway, and a slow shock pushes it that way directly.

Still open: the core’s response over many crossings #

A single slow shock takes a little energy from the core and gives more to the halo. A cluster on a Galactic orbit is shocked every few tens of millions of years, hundreds of times over its life, and the core’s response accumulates. Whether repeated slow shocks accelerate the core’s contraction — taking energy from it each time, as relaxation already does — and so bring forward the gravothermal collapse that binary stars eventually halt, is a question for integrations of whole clusters through many crossings, with their potentials allowed to respond. If they do, the Galaxy’s disc is not only a stripping agent for the outskirts of its globular clusters but a driver of the evolution of their cores.