Astronomy & Celestial Mechanics
the sky, computed
Concept

Crossing time — where it appears

The time a body takes to traverse a system once, its size divided by a typical speed. It is the natural unit against which every other dynamical timescale is quoted, and the relaxation time is this multiplied by roughly a tenth of the number of members.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

An average that arrives, and a snapshot that never does. 2T/|U| against time for Burrau's problem — three masses released from rest, and never repeating, in two forms: the instantaneous ratio, and the ratio of the running time-averages. The instantaneous one runs between 0.03 and 1.97 — the system is a long way from equilibrium at almost every moment, because the bodies are alternately falling together and flying apart. The averaged one settles: after 50 time units it is 1.0142, against the exact 1 the virial theorem requires of any bound system. This system is not periodic and is not even permanently bound — Burrau's problem ejects its lightest body — so the average is drifting rather than converged, and that is the theorem's own condition made visible: it is a statement about bound systems and says nothing about anything that is leaving. What the figure cannot show is the error: a cluster observed once gives 2T/|U| with a scatter of this size, and what makes its mass believable is not the measurement but the relaxation.

An average that weighs what cannot be watched

A cluster's mass can be had from its speeds alone — no orbit followed, no period observed, no distance to any single star. The theorem that allows it is an average over time, which is exactly what a photograph is not.

gravitation · Virial theorem 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.

How long a system takes to forget

A star crossing a galaxy is deflected by every other star, and the deflections add as a random walk. The time for that walk to change a star's energy by its own amount is a hundred million billion years for a galaxy and a billion for a globular cluster, and everything about how the two are modelled follows from which side of a Hubble time that falls on.

gravitation · Two-body relaxation 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

Named alongside it

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

Coulomb logarithmGlobular clusterImpulse approximationTwo-body relaxationAdiabatic invariantBound systemThe collisionless Boltzmann equationCollisionless systemCore-collapseDark matterDisc shockingDynamical mass

All concepts