Astronomy & Celestial Mechanics
the sky, computed
Starlight

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.

Assumes Limb darkening, Line formation and Opacity.

14 min read 5 figures Light is the only messengerScales that defeat intuition

A sight line stops at two thirds reduced limb darkening to one sentence: the emergent intensity at an angle is the source function at the depth where the slant optical depth is one, so a ray at the edge reads it higher up.

Nothing in that sentence says which way the source function runs. It runs one way in a photosphere, because temperature rises inward, and the entire literature of limb darkening is written for that case. Reverse the gradient and the same relation, unchanged, predicts a limb brighter than the centre.

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.
Fig. 1 The emergent profile across a disc for four gradients of the source function, every curve checked against a numerical integration of the transfer equation. A source function rising inward darkens the limb, a flat one gives a uniform disc, and one falling inward brightens the limb. The sign of the contrast is the sign of the gradient and nothing else — no new physics is added between the three cases, and no parameter is changed but one.

Where a stellar temperature stops falling

The prediction is not a curiosity, because the gradient does reverse, in every star with a photosphere.

Going outward from a stellar interior the temperature falls, as it must while energy is flowing outward by radiation. It keeps falling through the photosphere and reaches a minimum a few hundred kilometres above it — for the Sun, about 4,100 kelvin against an effective temperature of 5,772. Above that minimum it rises: slowly through the chromosphere to ten thousand kelvin over two thousand kilometres, then abruptly through a transition region to a million in the corona.

Why it rises is the outstanding unsolved problem of solar physics and does not matter here. What matters is that above the temperature minimum the source function increases outward, so any observation whose optical depth reaches one in that region sees a limb brighter than the disc’s centre.

Which observations do is decided by the opacity, and there are two families.

Long wavelengths. The free–free opacity of an ionised gas rises steeply with wavelength — as roughly its square — so at a millimetre the Sun becomes opaque far higher in the atmosphere than in the visible. At 1.3 millimetres the layer seen is in the chromosphere, and the Sun’s disc at that wavelength is brighter at the edge than in the middle, by ten to twenty per cent.

Strong lines. A line’s opacity is enormous at its centre, so the core of a strong line reaches optical depth one in the chromosphere while the continuum beside it forms in the photosphere. The cores of the calcium H and K lines and of hydrogen alpha therefore show a brightened limb where the continuum shows a darkened one — in the same image, a few tenths of a nanometre apart.

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.
Fig. 2 The same construction for a source function that falls inward as steeply as a chromospheric one rises outward. The disc’s profile inverts: the centre, which reads the deepest and therefore coolest layer of the inverted region, is now the faintest part. The contrast is smaller than the photospheric darkening because the chromospheric temperature gradient is gentler in optical depth, which is why solar limb brightening is a twenty per cent effect where limb darkening is a sixty per cent one.

The other mechanism, which is not a gradient at all

There is a second way to brighten a limb and it shares no physics with the first. Where a medium is optically thin, nothing is absorbed, the source function does not enter, and the emergent intensity is simply the emissivity integrated along the path.

A limb brightened by geometry alone. The path length a sight line traverses through a transparent shell, against where on the disc it strikes, for four inner radii. Where the material is optically thin the emergent intensity is simply the emissivity times the path, with no attenuation and no source function anywhere in it — so the picture is pure geometry. A filled sphere gives the longest chord through its centre and is brightest in the middle. A shell gives its longest chord tangent to its own inner surface, so it shows a ring, and the thinner the shell the sharper and brighter the ring: hollowing to 0.95 of the radius puts 6.2 times as much material along the tangent ray as along the central one. This is the second mechanism that brightens a limb and it has nothing to do with a temperature gradient — which matters, because a resolved image of a shell and a resolved image of an atmosphere with an inverted gradient look alike, and telling them apart needs the optical depth rather than the picture.
Fig. 3 Path length through a transparent shell, against where on the disc the sight line strikes. A filled sphere gives its longest chord through the centre and is brightest there. A shell gives its longest chord tangent to its own inner surface, so it shows a ring — and the thinner the shell, the sharper and brighter the ring. Hollowing to 0.95 of the radius puts six times as much material along the tangent ray as along the central one. Nothing about a temperature is involved.

The two mechanisms produce superficially similar pictures and are distinguished by three things.

Optical depth. The gradient mechanism requires optical depth of order one; the geometric one requires it to be small. A measurement of the optical depth — from the ratio of two lines of the same species, or from the spectrum’s shape — settles it directly.

Wavelength dependence. The gradient mechanism’s contrast changes with wavelength because the depth probed does. The geometric one’s does not, because it is geometry.

And the shape of the profile. A gradient brightens the limb smoothly, in the way the second figure shows. A shell produces a ring with a maximum at a definite radius and a fall beyond it, which is a qualitatively different curve.

That third test is the one that identified the structure around several evolved stars as detached shells rather than as extended atmospheres, and it works on an unresolved source too: the ring shows up as a characteristic signature in the visibility curve an interferometer measures.

Reading a height off a wavelength

The reason millimetre observations of the Sun are worth making at all is that the wavelength chooses the height, and the choice is continuous.

The free–free opacity of an ionised gas goes as roughly λ2\lambda^2 at these wavelengths, so doubling the wavelength quadruples the opacity and moves the layer where τ=1\tau = 1 upward by whatever distance quadruples the column. In a chromosphere whose density falls with a scale height of a few hundred kilometres, that is a move of several hundred kilometres per octave in wavelength.

So an instrument that observes at several millimetre bands is sounding the chromosphere directly. At 0.35 millimetres it sees the upper photosphere and the disc is flat or slightly darkened; at 1.3 it sees the low chromosphere and the disc is brightened; at 3 it sees higher still and the brightening is larger. Reading the brightness temperature at each wavelength gives the temperature at each height, with no model of the atmosphere in the chain at all.

That is a nearly model-free thermometer for a region where every other method is model-laden, and it is why the millimetre arrays built for extragalactic work have solar programmes on them. The chromospheric temperature structure inferred from ultraviolet lines requires a treatment of level populations that are demonstrably out of equilibrium; the millimetre continuum requires the free–free opacity, which is textbook, and the assumption that the gas is thermal, which at these densities it is.

The catch is calibration, and it is the one named in the section below: a brightness temperature is an absolute flux, and an absolute flux at a millimetre needs a standard. The profile across the disc needs none, which is why the shape of the solar millimetre disc was settled before its temperature was.

The same sign flip elsewhere

Once the rule is stated as the sign of the contrast is the sign of the gradient, its other appearances are easy to find and each is a measurement.

A planet’s limb in an emission line. A transiting planet’s atmosphere is optically thin in most of the infrared and optically thick in the cores of its strongest bands. Where it is thick and the temperature rises outward — as it does in a hot Jupiter’s upper atmosphere, heated by the star — the emission spectrum shows the bands in emission rather than in absorption. Detecting a molecular band in emission is therefore a detection of a thermal inversion, and it was the first evidence that some hot Jupiters have stratospheres — a claim that sits directly on top of the limb-darkening systematic that biases every transit radius.

A sunspot’s umbra. Within a spot the temperature structure is different and so is the limb darkening, and the contrast between spot and photosphere changes across the disc in a way that measures the spot’s own depth.

And the solar corona. At radio wavelengths above a few centimetres the corona is optically thick and its temperature rises outward, so the radio Sun is larger than the optical one, brighter at its edge, and has a size that depends on the observing frequency — which makes what a stellar radius means a question with a different answer in every band.

A limb brightened by geometry alone. The path length a sight line traverses through a transparent shell, against where on the disc it strikes, for four inner radii. Where the material is optically thin the emergent intensity is simply the emissivity times the path, with no attenuation and no source function anywhere in it — so the picture is pure geometry. A filled sphere gives the longest chord through its centre and is brightest in the middle. A shell gives its longest chord tangent to its own inner surface, so it shows a ring, and the thinner the shell the sharper and brighter the ring: hollowing to 0.99 of the radius puts 14.1 times as much material along the tangent ray as along the central one. This is the second mechanism that brightens a limb and it has nothing to do with a temperature gradient — which matters, because a resolved image of a shell and a resolved image of an atmosphere with an inverted gradient look alike, and telling them apart needs the optical depth rather than the picture.
Fig. 4 The same construction for a shell hollowed almost to its outer radius, which is what a thin detached shell around an evolved star looks like. The ring narrows and sharpens and the centre of the disc goes nearly dark, so the object presents as an annulus rather than as a disc. A brightness distribution shaped like a ring is a strong statement about geometry and says nothing about a temperature, and confusing it with an inverted gradient is the error this figure exists to make hard.

What was actually measured

The solar limb brightening at millimetre wavelengths is measured by scanning a resolved disc, in the same way the visible-light darkening is, and the two have been done with the same technique at the same site — the difference being that at a millimetre the atmosphere above the telescope is itself emitting, so the sky is a background that has to be subtracted rather than ignored.

Two things about that measurement are worth stating because they took decades to settle.

The contrast was disputed for thirty years. Different instruments gave different answers — some finding brightening, some finding a flat disc, a few finding darkening — and the reason is that a single-dish measurement of a limb is a convolution of the true profile with the telescope’s beam, and a beam comparable with the width of the brightening smears it away entirely. The modern measurements are interferometric, which resolves the profile directly, and they agree.

And the absolute brightness temperature is harder than the profile. The profile is a ratio and needs no flux calibration; the temperature at the centre of the disc needs one, and a calibration at millimetre wavelengths against a planet whose own brightness temperature is a model is a chain with a model at the end of it.

For the emission-line case the measurement is a spectrum rather than an image, and the detection of a band in emission is a statement about a line-to-continuum ratio at a level of a few hundred parts per million on a source that is not resolved at all. Several early claims of stratospheres in hot Jupiters were retracted when the instrument’s systematics were better understood, and the current set rests on data from a telescope built after the first claims were made.

What a ring does to an unresolved source

Most of the objects this applies to are not resolved, and a brightness distribution that cannot be imaged still leaves a signature.

An interferometer measures the Fourier transform of the sky brightness, and the transform of a uniform disc is one function while the transform of a ring is another. A uniform disc’s visibility falls smoothly to a first null and stays small; a ring’s oscillates, with a first null at a different baseline and a substantial rebound afterwards. The presence and depth of that rebound is a detection of a ring in a source that is never imaged.

The same statement holds for an occultation. When a body passes in front of a source, the light curve is the source’s brightness profile convolved with the occulting edge — so a limb-brightened source produces a light curve with shoulders that a uniformly bright one does not, and the shoulders are the measurement. That technique has resolved stellar diameters for a century using the Moon as the occulter, and it resolves a brightness profile as well as a diameter whenever the signal-to-noise allows.

And the same again for a microlensing caustic crossing, which scans a source’s disc point by point at a resolution no telescope approaches. A limb-darkened source produces one shape of crossing and a limb-brightened one produces another, and the handful of events analysed this way are the only measurements of a stellar brightness profile for a star other than the Sun that owe nothing to a model atmosphere.

Three techniques that never form an image all measure a brightness profile, and each does it by exploiting something that moves across the source — a baseline, an edge, a caustic. The image is a convenience; the profile is what the data contain.

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.
Fig. 5 And the construction the whole essay turns on, drawn for the ordinary case so the reversal has something to be a reversal of. A ray at the limb reads the source function high up; a ray at the centre reads it deep. Which of the two is brighter is settled entirely by whether the line slopes up or down, and this figure draws the case where it slopes up. Tilt the line the other way and every label stays where it is while the disc turns inside out.

Where the model stops

A chromosphere is not one-dimensional and not static. The solar chromosphere is a violently dynamic structure of spicules, fibrils and shocks, with temperature varying by thousands of kelvin over hundreds of kilometres horizontally and on timescales of minutes. The smooth inverted source function in these figures is an average over that, and averaging a steeply non-linear function of temperature is not the function of the average — so the mean chromospheric temperature inferred from a limb profile is not the mean chromospheric temperature.

The source function is not the Planck function there. In a photosphere the collision rate is high enough to keep the level populations thermal. In a chromosphere it is not, so the source function decouples from the local temperature and becomes a function of the radiation field arriving from elsewhere. Everything in this essay treats the source function as a proxy for temperature, and in the one region the essay is about, that proxy is at its weakest.

The geometry is a sphere. The shell figure assumes a spherically symmetric shell, and real detached shells are clumpy and often bipolar. A ring in an image is evidence of a shell; the shell’s thickness read off the ring’s width is a fit to a model with a symmetry in it.

And nothing here is resolved except the Sun. Limb brightening is measured directly on one object. Everywhere else it is inferred from a spectrum, from a visibility curve, or from the shape of a light curve — and each of those is a one-dimensional projection of a two-dimensional brightness distribution, which is the same underdetermination an interferometric image has with fewer data still — and a resolved stellar disc is a short list of nearby giants even before any of that.

The disc that is neither

Between a photosphere and a transparent shell there is a case that shows both mechanisms at once, and it is the commonest configuration in the sky.

An accretion disc seen face-on is optically thick and shows the ordinary darkening of any atmosphere with a temperature gradient. Seen edge-on it presents a long path through its own outer, cooler, thinner material and shows the geometry of a shell. And at intermediate inclinations it shows a combination whose sign varies across the image, because the optical depth along a sight line depends on where that sight line strikes.

The same is true of a stellar wind. Close to the star it is optically thick in the strong lines and darkens; far out it is thin and brightens; and the transition between the two happens at a radius that differs from line to line, so a single object shows both behaviours in one spectrum at different wavelengths.

Two mechanisms with no physics in common can operate in one object at one moment, distinguished only by which part of it a particular photon came from. That is why the discriminating measurement named above — the optical depth — has to be made as a function of position rather than once, and why a resolved image is worth so much more than an integrated measurement even when the integrated one is more precise.

The habit this produces is worth naming. When a picture has two possible explanations, the first question is not which is right but whether both are, in different parts of it — because a mixture is the normal case and a pure one is the special case somebody chose to draw.

The generalisation

The shape to carry is that a phenomenon named for one of its signs is usually a two-sided relation with a convention attached.

“Limb darkening” names the case that happens in the passband human eyes use on the one star that can be resolved by them. The underlying statement — the emergent intensity is the source function where the slant optical depth is one — contains no sign at all, and the darkening is a fact about photospheric temperature gradients rather than about limbs. A field that had first observed the Sun at a millimetre would have named the effect the other way and had to explain the visible case as an exception.

When a relation’s named effect has a sign, ask what fixes it, because the answer is almost always a property of the systems that happened to be studied first rather than of the relation. The same applies to a redshift, to a positive heat capacity, to a lag rather than a lead in a tidal response — each of which has the other sign somewhere and each of which is the same equation.

The second reading is about telling apart two causes of one appearance. A brightened limb has two explanations with no physics in common, and the observation that distinguishes them is neither a better image nor a longer integration: it is a measurement of the optical depth, which is a different quantity entirely. When two mechanisms predict the same picture, the discriminating observation is usually not a better picture — and looking harder at the thing that is ambiguous is the most reliable way to stay confused.

Still open: the profile as a sounding

What comes next turns the profile from a phenomenon into an instrument. Each position on a disc reports a different depth, and a profile measured at many wavelengths reports a two-dimensional grid of depths — so a resolved, spectrally dispersed limb scan is in principle an inversion for the whole run of temperature through a photosphere and a chromosphere.

In practice it recovers only a few numbers, because the weighting functions at neighbouring angles overlap almost completely, and the useful question is how many independent numbers a given set of observations contains. That is the same question an underdetermined image reconstruction asks, arriving from the other direction, and the answer here is smaller than it looks.

Where this goes next

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. 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. Dissipation against viscosity for Io: a peak at 10^13.5 Pa s, and two solutions at the observed rate. The imaginary part of the Love number — the part that turns tidal work into heat — against the viscosity of the body's interior, for a Maxwell rheology at Io's size, density and forcing period. Both axes are logarithmic and the curve is not monotonic, which is the whole content of the figure. At high viscosity the body is elastic: it stores the energy the tide puts in and gives it back, and dissipates nothing. At low viscosity it is fluid: it deforms all the way and does so in phase with the forcing, and dissipates nothing again. Everything happens in between, at viscosities for which the Maxwell time — viscosity divided by rigidity — is comparable to the orbital period, and the peak here is 0.735 at 10^13.5 pascal seconds. Two consequences follow, and they pull in opposite directions. The peak is an upper limit: a homogeneous body of this size cannot dissipate more than that however its viscosity is chosen, so a measured heat flow above it would refute the model rather than constrain it. And below the peak the observed value is met twice — at 10^11.5 and at 10^15.5 pascal seconds — so a heat flow alone does not say which side of the peak the interior is on. Breaking that degeneracy needs a second observable, and the usual one is the phase of the response rather than its size. The picture treats the body as one homogeneous Maxwell solid, which is certainly wrong for a moon with a molten layer; a partial melt concentrates the dissipation and shifts the peak. One heat flow, and two viscosities Io radiates a hundred thousand gigawatts of tidal heat, and that number is supposed to say something about the rock inside it. It does, and not what one would expect — because dissipation vanishes at both extremes of viscosity, the measured heat is produced by two different interiors and cannot choose between them.

The objects this essay names

Each one links to every other essay that touches it.

Brightness temperatureChromosphereEddington barbierEmission measureLimb brighteningOptical depthOptically thinRadiative transferSource functionTemperature inversion