Astronomy & Celestial Mechanics
the sky, computed
Concept

Local thermodynamic equilibrium — where it appears

The condition in which a gas's level populations are set by collisions rather than by the radiation field, so the source function equals the Planck function. It holds well in a photosphere's continuum and badly in a line formed high in an atmosphere.

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

A better measurement that made the model worse: 0.9 per cent in the sound speed. The fractional difference between the Sun's sound speed as its own oscillations measure it and as a structural model predicts it, against fractional radius. Zero would be agreement. The lower curve is the model built on the solar abundances used until the mid-2000s, and it hugs the axis: a part in a thousand across most of the interior, which was for a long time the best-tested piece of stellar physics anybody had. The upper curve is the same model with the abundances re-measured using three-dimensional atmospheres and without assuming local thermodynamic equilibrium — better measurements by every methodological standard, which lowered carbon, nitrogen and oxygen by around thirty per cent. The disagreement grows to 0.9 per cent, and it is not spread through the star: it peaks at 0.683 of the radius, just beneath the base of the convection zone at 0.713. The same substitution moves the model's own convection-zone base from 0.715 to 0.729, against a seismic value known to about a thousandth. What is being tested here is not really the abundances but what converts a composition into a structure, which is the opacity: the metals whose abundances fell are exactly the ones whose bound–free absorption dominates at those temperatures, and an opacity larger by some fifteen per cent near that boundary would restore the agreement. Laboratory measurements of iron at those conditions have since come in high by about that much, which is a satisfying result to have arrived at by way of a discrepancy in the sound speed of the Sun. The curves are published inversions and model differences rather than anything computed here; what the figure adds is where they peak and by how much.

A better measurement that made the model worse

The Sun's composition was re-measured with better atmospheres and better physics, and the carbon, nitrogen and oxygen abundances fell by about thirty per cent. The improved model then disagreed with the Sun's own oscillations by ten times as much as the model it replaced, and it still does.

starlight · Opacity A sight line stops where its own optical depth is one. The source function of a grey atmosphere in radiative equilibrium — a straight line, S = (3/4)(τ + 2/3) in units of the flux — with the depth each sight line reads off it. The emergent intensity at an angle is the source function integrated along the ray with everything in front of it attenuating, and for a source function linear in optical depth that integral is exactly the source function evaluated at τ = μ. Not approximately: the Eddington–Barbier relation is an identity for a linear source function, and the numerical integral agrees with it here to a part in ten thousand, which is the quadrature's error and not the relation's. A vertical ray reads the source function at τ = 1 and a ray at 78 degrees reads it at τ = 0.2, which is higher in the atmosphere and therefore cooler — so the limb is dimmer than the centre by the amount the source function has risen between those two depths. Everything about limb darkening is the slope of this one line, and the famous 2/3 is the intercept: the average ray, over the whole disc, reads the source function at two thirds of a unit of optical depth, which is the depth a stellar "surface" actually means.

A sight line stops at two thirds

The whole of limb darkening is one sentence: what leaves a star at an angle is the source function evaluated where the slant optical depth is one. For a source function linear in depth that statement is exact rather than approximate, and it produces the grey atmosphere's (2+3μ)/5 with no fitting anywhere in it.

starlight · Limb darkening

Named alongside it

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

AbundanceBound-free absorptionConvectionEddington barbierEffective temperatureEquivalent widthGrey atmosphereHelioseismologyLimb darkeningMetallicityOpacityOptical depth

All concepts