Astronomy & Celestial Mechanics
the sky, computed
Concept

Kelvin helmholtz timescale — where it appears

The time a star would take to radiate its own gravitational binding energy at its present luminosity, GM²/RL — thirty million years for the Sun. It is the timescale on which a star restores thermal equilibrium after being disturbed.

Named by 3 essays across 3 fields — 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 The same equation, run away at q = 0.8 and settled at q = 0.5. The donor's overfill of its own Roche lobe against time, integrated for 3 mass ratios at a donor adiabatic response of −0.33 — a star with a deep convective envelope, which expands as it loses mass. The overfill sets the transfer rate and the transfer rate changes the overfill, and the sign of that feedback is the stability criterion: where the lobe shrinks faster than the star does, ζ_L > ζ_ad, the overfill grows and the growth is exponential. At q = 0.8 the lobe responds at 0.03 and the overfill runs away; At q = 1.4 the lobe responds at 1.32 and the overfill runs away. At q = 0.5 it responds at -0.62 and the transfer throttles itself back. The critical ratio for this donor is 0.63 — below equal masses, which is the result the whole subject turns on: a giant transferring to a lighter companion is already unstable before the mass ratio has reversed, and what follows is not accretion but a common envelope. Nothing here is a fitted rate. The vertical scale is logarithmic and the runaway is a straight line on it, which is what an exponential is.

Three clocks and a runaway

Whether mass transfer between two stars is stable is a comparison of two logarithmic derivatives. How fast it runs is a separate question with three possible answers fourteen orders of magnitude apart — and the answer decides whether the companion accretes, is buried, or is swallowed.

stars · Mass transfer How fast the circulation a rotating star drives would stir it, against its lifetime. The classical Eddington–Sweet circulation time — the star's Kelvin–Helmholtz time divided by the ratio of centrifugal to gravitational acceleration at its surface — as a multiple of its main-sequence lifetime, against rotation rate as a fraction of critical, for stars of 2, 5, 15 solar masses, on logarithmic axes. Below the line at one, the circulation would turn the star over within its life. For 2 solar masses that happens above 0.101 of critical; for 5 solar masses that happens above 0.123 of critical; for 15 solar masses that happens above 0.115 of critical. The dot is the Sun, turning at 0.0084 of its critical rate, whose circulation would take 150 times its main-sequence life. By this estimate almost every star turning at more than a tenth of its critical rate should be stirred from core to surface; the measured surface compositions of such stars show that it is not, because the composition gradient left by core burning resists the circulation, and how strongly it resists is what the nitrogen at their surfaces measures.

The circulation that should have stirred every fast rotator

A rotating star in radiative equilibrium cannot be balanced in pressure and in heat at the same time, and the mismatch drives a slow circulation from pole to equator. The classical estimate of its speed says that any star turning at more than a tenth of its break-up rate should be stirred from core to surface within its life, bringing the nitrogen of hydrogen burning up with it. Some fast rotators show that nitrogen and some do not, and some slow rotators show it when they should not — including more than the angle of their axes can explain.

starlight · Gravity darkening

Named alongside it

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

AccretionAdiabatic responseBinary heatingCNO cycleCommon envelopeConvective envelopeCore-collapseCritical rotationDynamical timescaleGravity darkeningGravothermal catastropheMass ratio

All concepts