Astronomy & Celestial Mechanics
the sky, computed
Series

Limb darkening — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The limb is 40% as bright as the centre, and that is a temperature gradient. Left: a stellar disc shaded by the grey-atmosphere law I(μ)/I(1) = (2 + 3μ)/5, in 26 steps, with μ = cos θ read from the centre outwards. Right: that law against μ, with linear laws at the measured solar coefficients from 400 to 1600 nm. The grey law's coefficient is exactly 3/5 and its very limb is exactly 2/5 of the central brightness — both read off the drawn curve rather than quoted — because the Eddington–Barbier relation makes the emergent intensity at angle μ the source function at optical depth τ = μ, and in radiative equilibrium that source function is linear in τ. The limb is not a cooler part of the star. A sight line entering at the edge reaches unit optical depth higher up, where the gas is cooler, so what the darkening measures is the run of temperature with depth; a star with an isothermal atmosphere would show a uniform disc, and one with a steeper gradient a darker limb. The measured coefficients fall from 0.9 at 400 nm to 0.35 at 1600 — the same gradient seen through a less steep Planck function — which is why a radius measured from a transit or a fringe null has to say which colour it was measured in.

    The light that is missing from the edge

    The Sun's limb is forty per cent as bright as its centre, and the reason is not that the edge is cooler. A sight line entering at the edge stops higher up, so what the darkening measures is the temperature gradient — and it is worth seventeen per cent on Betelgeuse's radius.

    part 1 · starlight
  2. A transit depth of 1.200 per cent for a planet of area 1.000 per cent. Three transits of the same planet across the same star, differing only in how the star's brightness falls toward its edge. A planet of radius ratio 0.1 covers 1.000 per cent of the stellar disc's area, and if the disc were uniformly bright that would be the depth. It is not uniformly bright: a sight line near the limb leaves the photosphere at a shallow angle and therefore from a cooler layer, so the edge is dimmer than the centre, and a planet crossing near the middle blocks light that is brighter than average. The transit drawn with realistic coefficients is 1.200 per cent deep — 20 per cent deeper than the area — and it is also rounder, because the covered brightness changes through the crossing instead of staying flat. The consequence is stated in the numbers beside the curves. Each is a least-squares fit of the radius ratio to the realistic curve, performed with a different assumed limb-darkening law, and the recovered radius moves by up to 3.6 per cent depending on which law is assumed. Fitting with the law the curve was made from returns the input to five figures, which is the control: the bias is the mis-specification and not the fitter. Since the coefficients come from a model atmosphere rather than from the light curve, every published planetary radius carries a systematic from stellar physics that no amount of photometric precision removes — and it is the dominant one for the best-measured planets. The picture holds the impact parameter fixed; a grazing transit is worse, because it samples only the limb, where the disagreement between laws is largest.

    The depth is not the area

    A planet covering one per cent of its star's disc does not make a transit one per cent deep. The star is brighter in the middle, so a planet crossing the middle blocks more than its share — and the correction depends on coefficients that come from a stellar atmosphere model rather than from the light curve.

    part 2 · starlight

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