Astronomy & Celestial Mechanics
the sky, computed
Concept

Disc shocking — where it appears

The heating of a star cluster as it crosses the Galactic disc, whose extra pull compresses the cluster for a few million years. Each crossing gives stars kicks proportional to their height, removing loosely bound stars and leaving a smaller, denser cluster.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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.

gravitation · Impulse approximation 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.

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.

gravitation · Impulse approximation

Named alongside it

The objects these essays reach for when they reach for this one.

Globular clusterImpulse approximationPlummer sphereTidal shockAdiabatic invariantCoulomb logarithmCrossing timeEnergy diffusionEscape energyTidal forceTwo-body relaxationVirial theorem

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