Astronomy & Celestial Mechanics
the sky, computed
Series

Virial theorem — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · gravitation
  2. 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.

    part 2 · gravitation

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