The depth is not the area

A transit measures a planet by the light it removes, and the first thing anybody learns about it is that the depth is the ratio of the areas: a planet whose radius is a tenth of its star’s covers a hundredth of the disc and takes away a hundredth of the light.

That is not what happens. A star is not a uniformly bright disc. Its edge is dimmer than its centre, by a factor of two or more at visible wavelengths, because a sight line entering near the limb leaves the photosphere at a shallow angle and therefore from a cooler layer. So a planet crossing the middle of the disc blocks light that is brighter than average, and the transit is deeper than the area ratio.

What a limb-darkening law is #

The intensity emerging from a stellar atmosphere at an angle $\theta$ from the vertical is usually written as a polynomial in $\mu = \cos\theta$:

$$\frac{I(\mu)}{I(1)} = 1 - u_1(1-\mu) - u_2(1-\mu)^{2},$$

the quadratic law, with two coefficients. Other forms exist — linear, square-root, logarithmic, and a four-parameter version that fits model atmospheres best — and none of them is derived. They are fitting formulae for the output of a radiative-transfer calculation.

The underlying physics is straightforward and gives the linear coefficient exactly in one idealised case: a grey atmosphere in radiative equilibrium has $I(\mu)/I(1) = (2 + 3\mu)/5$, which is a linear law with $u_1 = 3/5$. Real atmospheres are not grey, so the coefficients depend on wavelength, on the star’s temperature, on its surface gravity and on its metallicity — and they are tabulated as functions of all four.

That is where the problem starts. The coefficients are not measurements. They are outputs of the same one-dimensional model atmospheres whose shortcomings the solar abundance revision exposed, and they are being used to interpret photometry precise to parts per million. There is a further wrinkle in the choice of law that is easy to miss and hard to escape. The coefficients of a two-parameter law are strongly correlated with each other: a larger $u_1$ with a smaller $u_2$ produces almost the same intensity profile over most of the disc. So quoting “the quadratic coefficients” for a star is quoting two numbers whose individual values are far less meaningful than a particular combination of them, and different tabulations that look very different in $u_1$ and $u_2$ separately can agree closely on the profile. Comparisons between analyses that use different laws are therefore harder than they look, and a paper that reports a radius without saying which law it used has withheld something material.

What the bias actually is #

The hero figure’s arithmetic is the point of this essay, so it is worth restating carefully.

A light curve was generated with a realistic quadratic law. It was then fitted for the radius ratio alone, holding the limb-darkening coefficients at four different values. Fitting with the coefficients the curve was made from returns the input to five figures — that is the control, and it says the bias is the mis-specification rather than the fitter. Fitting with no limb darkening at all returns a radius 3.6 per cent too large. Fitting with a linear law returns one within a hundredth of a per cent, by luck rather than by principle. Fitting with a plausible but wrong quadratic pair returns one half a per cent too large.

Three and a half per cent in a radius is a ten per cent error in a volume and therefore in a density, and a density is what a mass and a radius are combined to produce. For the best-measured planets — the ones with photometric precision at the parts-per-million level — this is the largest single term in the error budget, and it does not decrease with more data. That last clause is the important one. Almost every other term in a transit’s error budget — photon noise, scintillation, read noise — falls as more transits are collected. This one does not: observing a hundred transits with the wrong limb-darkening law gives a hundred times better precision on the wrong answer.

The direction of the bias is also fixed rather than random. Under-correcting for limb darkening — assuming a flatter star than the real one — always makes the recovered planet too large, because the model has to compensate for a depth it cannot otherwise explain. So a survey analysed with a single tabulated law does not merely scatter its radii; it shifts them all one way, and by an amount that varies systematically with stellar temperature, since the coefficients do.

Why the coefficients are not simply fitted #

The obvious response is to fit for the limb darkening as well as for the radius. It works, and only for the very best data.

The difficulty is degeneracy. The shape a limb-darkening coefficient imposes on a light curve is not very different from the shape a slightly different radius and impact parameter impose. With photometry good to a few hundred parts per million and a handful of transits, the posterior distribution is a long ridge: many combinations of radius, impact parameter and coefficients fit almost equally well, and the marginalised radius is broad.

What breaks the degeneracy is a high impact parameter or very high precision. A planet crossing near the limb samples the steep part of the intensity profile and constrains the coefficients directly; a planet crossing the centre samples only the flat middle and constrains them hardly at all. So the transits that are easiest to model precisely are the ones that carry least information about the thing that limits their precision.

In practice the field uses priors: the coefficients are taken from tables, with a Gaussian uncertainty of a few hundredths imposed, and the fit is allowed to move them within that. It is a reasonable compromise and it is also a way of making a model-dependent quantity look like a measurement with an error bar.

It is worth putting a number on what the degeneracy costs, because it is not a small effect on a marginal system. For a typical ground-based light curve of a hot Jupiter, fitting the two coefficients freely roughly doubles the uncertainty on the radius ratio compared with fixing them. Fixing them removes that uncertainty and replaces it with a systematic of unknown size. Neither choice is right, and the literature contains both, which is one reason published radii for the same planet from different groups sometimes differ by more than either quotes.

Colour, and the one honest check #

Limb darkening is strongly wavelength-dependent. In the blue, where the source function varies steeply with depth, the limb is much darker than in the infrared. A star’s disc at two microns is nearly uniform.

That gives a test. The true radius ratio does not depend on wavelength — a planet is the size it is — except through its atmosphere, which adds a wavelength-dependent annulus of a few scale heights. So a transit observed in several passbands should give radius ratios that agree once the limb-darkening treatment is correct — the same consistency argument that makes a magnitude have to say which light it means, and a systematic trend with wavelength that is not explicable by an atmosphere is a sign the treatment is wrong. This is more than a check. The wavelength dependence of the planet’s radius is transmission spectroscopy: the signal that says what the planet’s atmosphere is made of. It is a few hundred parts per million on top of a transit depth of a per cent, and it sits directly on top of the limb-darkening systematic. Several early claims of atmospheric detections were later attributed to limb-darkening treatment, and the standard practice now is to fit the two together and quote the covariance.

What was actually measured #

Nothing in the hero figure is an observation. What a transit survey records is a time series of relative flux, and everything else — the depth, the duration, the contact times, the radius ratio — is a fitted parameter.

The fit contains, at minimum: the radius ratio, the impact parameter, the scaled semi-major axis, the mid-transit time, the period, two limb-darkening coefficients, and a baseline flux with whatever instrumental trend the detector imposes. Some of those are well constrained by the data and some are not, and the ones that are not are constrained by priors from elsewhere.

The best individual measurements come from space photometry, where the absence of an atmosphere removes the dominant noise source. The precision achieved on a bright star is tens of parts per million per transit, which is enough that the limb-darkening coefficients can be fitted freely for a handful of systems — and where that has been done, the fitted coefficients disagree with the tabulated ones by more than the tables’ quoted uncertainties, systematically, in the sense that real stars are less limb-darkened than models predict.

The one case where the star cooperates #

There is a configuration in which limb darkening stops being a nuisance and becomes measurable from the transit itself, and it is worth knowing because it is the only clean one.

A planet on a highly inclined orbit, crossing the star at a large impact parameter, traces a chord that runs close to the limb along its whole length. The intensity varies steeply along that chord, so the light curve’s shape carries the profile directly, and the coefficients can be fitted without a prior. The cost is that a grazing transit is short, shallow, and strongly degenerate between radius and impact parameter — so what is gained on one parameter is lost on another.

The genuinely favourable case is a planet with a large radius ratio, so that the occulter is a substantial fraction of the disc and scans a wide swathe of it in one crossing. A giant planet transiting a small M dwarf covers ten per cent of the disc’s diameter, and its light curve does contain enough shape information to fit the limb darkening freely. Those systems are the ones used to test the tables, and the tests are how the disagreement between fitted and tabulated coefficients was found.

What the picture cannot show #

The model behind every curve here is a limb-darkened disc, which is an idealisation in three ways that matter at the current precision.

Stars have spots. A star’s magnetic cycle covers it in them, and a planet crossing a starspot occults a dark patch so the light curve shows a bump; a planet crossing a bright facula shows a dip. Unocculted spots change the baseline and bias the depth. The effect is large for active stars — parts in a thousand — and it is the reason radii for planets around young stars are quoted with wider error bars. It is also the reason an apparent planet is validated rather than confirmed more often around active stars: the astrophysics that mimics a transit and the astrophysics that distorts one are the same astrophysics.

Stars are not spherical. A rapidly rotating star is oblate and gravity-darkened: its poles are hotter and brighter than its equator. A planet on an inclined orbit then crosses a chord whose brightness profile is asymmetric, producing a distorted transit that has actually been used to measure the star’s obliquity.

The atmosphere is three-dimensional. The coefficients tabulated from one-dimensional models are known to be wrong; three-dimensional simulations give different ones, and the differences are of the same size as the disagreements described above. Tables from those simulations exist for a limited range of stars and are gradually replacing the older ones.

The star scanned by a caustic #

The tables are model outputs, and there are two configurations in which the intensity profile is measured directly rather than computed. One of them scans a stellar disc at a resolution nothing else approaches.

When a foreground mass passes in front of a distant star, the star’s light is magnified — and if the lens is a binary, the magnification pattern on the sky contains closed curves along which the magnification is formally infinite. As the source star drifts across one of those curves, its limb crosses first and its centre later, so the magnification rises in a way that depends on the brightness profile of the part of the disc currently on the curve.

The effective resolution is set by how sharply the magnification varies across the curve, and it is far finer than the source’s angular diameter — which for a star in the Galactic bulge is a microarcsecond. So the light curve of a caustic crossing is a scan of a stellar disc that no telescope could resolve, and fitting it returns the limb-darkening coefficients directly.

Several dozen such events have been analysed, mostly for giants in the bulge, and the coefficients come out systematically different from the tabulated ones — in the same direction as the transit fits find, which is that real stars are less limb-darkened than the models say.

The measurement’s limitation is the sample. A caustic crossing is a rare event, it cannot be scheduled, it lasts hours, and the stars it happens to are the bright giants that are large enough to resolve — which are not the stars that host transiting planets.

The profile is measurable and it is measurable for the wrong stars, which is why the transit analyses still use tables.

The bias depends on the geometry as much as on the star, so it is worth reading the same fit at a large impact parameter and the same family at a longer period.

What the bias does to a histogram #

The essay has priced the bias for one planet. Applied to a survey it does something worse than adding scatter, and the case that shows it is the sharpest feature in the exoplanet census.

The distribution of planetary radii has a deficit at about 1.8 Earth radii — a gap between the rocky planets and those with retained hydrogen envelopes. Its position is quoted to a few per cent, and its position is what the theories of atmospheric loss are compared against.

A few per cent is the size of the limb-darkening bias. So a systematic error in the treatment does not merely blur the gap; it moves it, and it moves it by an amount comparable with the precision the measurement is quoted at.

Worse, the bias is not uniform across the sample. The coefficients depend on stellar temperature, so a survey containing stars from late K to early F has a bias that varies systematically along the sample — which means a gap that is genuinely at one radius for all stars can appear tilted, and a gap that is genuinely tilted can appear flat.

The response has been to re-derive the radii with consistent stellar parameters and a single treatment, which removes the differential part of the problem and leaves the overall shift. The gap survived that exercise and got sharper, which is the strongest evidence that it is a feature of the planets rather than of the analysis.

A systematic that is common to a whole sample is invisible in the sample’s internal consistency, and the only defence is to vary the treatment deliberately and see how far the answer moves.

Two more settings bracket where the bias is worst, since it depends on both how large the planet is and how much of the disc it crosses.

Where the ladder goes #

The first rung of this anchor established what limb darkening is and what it measures about a stellar atmosphere: the light missing from the edge is a temperature gradient. This one is about what it costs somewhere else.

The pattern is worth naming because it recurs throughout this collection. A quantity that is the signal in one measurement is the systematic in another, and the second measurement inherits every uncertainty of the first. A planetary radius carries a stellar atmosphere’s error bar; a planetary density carries a stellar radius’s; and a claim about a planet’s composition therefore rests, several links back, on how well a one-dimensional model reproduces the run of temperature in the outer few hundred kilometres of a star.

The next rungs of this anchor go toward the cases where the darkening is measured rather than assumed: eclipsing binaries, where a stellar companion scans the disc with a much larger occulter; interferometry, which resolves the disc directly on the nearest giants; and microlensing, where a caustic sweeps across a source star and traces its brightness profile point by point. All three give limb-darkening coefficients that a transit fit can then use, and all three are available for a small number of stars.