Astronomy & Celestial Mechanics
the sky, computed
Concept

Energy diffusion — where it appears

The slow increase in the binding energy spread of a stellar system as many weak encounters each add a small random kick. Repeated tidal shocks heat a cluster or a satellite galaxy this way, and the outer stars, kicked hardest, are lost first.

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

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 A slow shock still heats the cluster. The energy a Plummer star cluster gains from a compressive tidal pulse, as a fraction of what the same impulse delivered instantaneously would give, against the pulse's duration times the orbital frequency at the half-mass radius. The points come from integrating 1200 stars through the pulse one by one. The lines are prescriptions applied to the same stars: an oscillator's exponential suppression and a power law, each applied star by star at the star's own orbital frequency, and the shortcut of one power-law factor for the whole cluster, evaluated at the half-mass radius. At the longest pulse drawn the cluster keeps 0.28 of the impulsive energy; the star-by-star rules give 0.24 and 0.36, bracketing it, and the half-mass shortcut gives 0.017, 17 times too little. The total is set by the outermost stars, which carry most of the impulsive energy and are always slower than the shock.

A slow shock heats the halo and robs the core

The textbook treatment of a slow tidal shock multiplies the instantaneous answer by a suppression factor that falls steeply as the shock lengthens. Integrated star by star through shocks of intermediate length, a cluster behaves differently. Its total heating falls slowly, because the outer stars are always slower than any shock and carry most of the energy, and its core does not merely escape the heating: on balance its stars lose energy, which no factor that multiplies a positive number can describe.

gravitation · Impulse approximation

Named alongside it

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

Globular clusterImpulse approximationPlummer sphereTidal shockAdiabatic invariantDisc shockingEscape energyGalactic discLinear responseOrbital frequencySecond order perturbationVirial theorem

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