Astronomy & Celestial Mechanics
the sky, computed
Concept

Electron scattering — where it appears

The scattering of a photon by a free electron, whose cross-section is a constant independent of frequency, temperature and density. That constancy is why it gives a mass–luminosity exponent of exactly three and why the Eddington limit is proportional to mass.

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

Also named here as kramers opacity — the same set of essays touches all of them, so they are one junction rather than several.

Opacity against temperature, and the three things that supply it. The Rosseland mean opacity of a gas of composition X = 0.7, Y = 0.28, Z = 0.02, on logarithmic axes, at 10⁻⁷ g/cm³ and 10⁻⁶ g/cm³. The three faint curves are the separate processes at the first density — electron scattering, the Kramers bound-free and free-free term, and the negative hydrogen ion — and the solid curve is their sum. Every one of them is multiplied by the fraction of hydrogen the Saha equation says is ionised at that temperature and density, or by one minus it for H⁻, which is the only reason the low-temperature end is a picture of a star rather than of a formula outside its range: ungated, Kramers alone gives 10,343 cm²/g at 5,800 K, against the 0.40 drawn here. The peak sits at 15,400 K, where hydrogen is 78% ionised — that bump is not a detail, it is the engine of a Cepheid — and the flat floor at high temperature is electron scattering, which is the one term with no temperature in it at all.

The surface that is a depth

A star has no surface. What looks like one is the level at which the optical depth reaches about two-thirds — and the edge is sharp only because the opacity climbs so steeply that the transition takes a ten-thousandth of the radius.

starlight · Opacity An exponent between 2.3 and 4, not a number. The local logarithmic slope of the mass–luminosity relation against mass — the exponent that "L ∝ M^3.5" stands in for — read off the same fit to eclipsing binaries that the relation itself is drawn from. It is a step function with two corners, and the steps are 2.3 below 0.43 M☉, 4 0.43 to 2 M☉, 3.5 2 to 55 M☉. The horizontal lines are what homology predicts: a star in radiative equilibrium has L ∝ μ⁴M³/κ̄, which gives an exponent of 3 when the opacity is electron scattering and a constant, 5.5 when it is Kramers and depends on density and temperature, and 1 where radiation pressure holds the star up and the luminosity is pinned to the Eddington value. The measured steps sit between those predictions rather than on them, which is what a real star does — no star is homologous, the low-mass ones are convective throughout and the heavy ones have convective cores, and the fitted exponents are what is left after all of that. The useful statement is not that the exponent is 3.5 but that it is between 2 and 5 everywhere and 4 in the middle, and that the reason it moves is a change of which opacity is carrying the energy out.

An exponent that is a slope, not a law

The phrase “L goes as M to the three and a half” stands in for a curve with three straight pieces and two corners. The exponent is 2.3 below half a solar mass, 4 in the middle and falls towards 1 at the top — and each change of exponent is a change in which opacity is carrying the energy out.

stars · The mass–luminosity relation

Named alongside it

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

Kramers opacityOpacityConvectionEddington limitHomologyMain sequenceMass luminosity relationMean free pathMean molecular weightNegative hydrogen ionOptical depthPhotosphere

All concepts