Astronomy & Celestial Mechanics
the sky, computed
Series

Limb darkening — the series

5 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
  3. 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.

    part 3 · starlight
  4. The sign of one gradient decides which way the disc is shaded. The emergent intensity across a stellar disc, against the fractional radius, for four gradients of the source function with depth. Every curve is the Eddington–Barbier relation evaluated at the appropriate μ and checked against a numerical integration of the transfer equation. The contrast between centre and limb is the gradient, and its sign is the gradient's sign. A source function rising inward — the ordinary case, since temperature rises inward and the source function follows it — darkens the limb: a sight line at the edge leaves from higher up, where the gas is cooler. A flat source function produces a uniformly bright disc whatever the geometry. And a source function that falls inward brightens the limb, because now the shallower ray is reading a hotter layer. That last case is not hypothetical: above a star's photosphere the temperature stops falling and begins to rise, so at wavelengths that see those layers — millimetre continuum, the cores of strong lines, the ultraviolet — the Sun's limb is brighter than its centre. Limb darkening is therefore not a property of stars but of a gradient, and the same atmosphere shows both signs at different wavelengths.

    A limb brighter than the middle

    Limb darkening is not a property of stars. It is a property of a gradient, and the same relation that produces it produces the opposite whenever the source function falls inward — which it does above every stellar photosphere. The Sun's limb is darker in the visible and brighter at a millimetre, and both are the same equation.

    part 4 · starlight
  5. A feature read back from the limb. A bump in the source function at optical depth 0.1, a quarter of a decade wide (black dashed), and what an inversion of the limb profile recovers on average when the profile is measured at 24 angles with errors of 1, 0.1, 0.01, 0.001 per cent, the inversion assuming only that the departure from a grey atmosphere is smooth. At one per cent the recovered bump has 56 per cent of its height and is spread over about twice its width; at a thousandth of a per cent, 82 per cent, and still wider than the truth. A thousandfold gain in precision buys a modest gain in resolution — each factor of ten a step, not a factor.

    A profile that averages a decade of depth

    Each point on a star's disc reports the temperature at a different depth, so a limb-darkening profile looks like a depth sounding of the atmosphere waiting to be read. It is one, but a blurred one. Every angle averages a whole decade of optical depth, the averages overlap, and recovering the temperature structure from them is the inversion of a Laplace transform — a problem whose resolution grows only as the logarithm of the data's precision.

    part 5 · starlight

All series