Astronomy & Celestial Mechanics
the sky, computed
Concept

Self-gravitating system — where it appears

A system bound by its own gravity rather than by an external force, such as a star, a cluster or a galaxy. Such systems have negative heat capacity: losing energy makes them contract and grow hotter.

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

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.

gravitation · Virial theorem A cold sphere collapsing onto the virial theorem. The virial ratio 2K/|W| — twice the kinetic energy over the magnitude of the potential energy — counted over the bound shells, of 400 concentric shells released almost at rest (2K/|W| = 0.1) from a sphere whose density falls as one over the radius, against time in units of the sphere's free-fall time. It starts near zero; overshoots to 1.7 as the collapse reaches the centre, the inner shells first; rebounds and oscillates; and settles to 1.03, the virial value, staying within ten per cent of it from about 4.9 free-fall times on. The shells never collide with anything; each feels only the mass inside it. What damps the oscillation is phase mixing — shells of different energy oscillate with different periods and drift out of step — and what redistributes the energy is the potential changing while they move through it. A system of stars reaches equilibrium this way in a few dynamical times, where encounters between the stars would take far longer than the age of the universe.

An equilibrium reached without a single encounter

Every dynamical mass of a galaxy assumes its stars obey the virial theorem, and no two of those stars have ever exchanged energy. A collapse computed shell by shell shows how a system gets there anyway — in a few free-fall times, at half the size it fell from, by a potential that changes while its orbits are still moving through it.

gravitation · Virial theorem

Named alongside it

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

RelaxationVirial theoremBinary heatingCore-collapseDark matter haloFree-fall timeGravothermal catastropheKelvin helmholtz timescaleKinetic energyNegative heat capacityPhase mixingPotential energy

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