The light that is missing from the edge

A photograph of the Sun is not uniformly bright. The centre of the disc is brightest and the edge is about forty per cent as bright, with the fall continuous and steepest right at the rim.

The obvious reading is that the limb is cooler than the centre. It is not, in the sense that matters: take a sample of gas from the middle of the disc and one from the edge, both at the same depth below the surface, and they are at the same temperature. What differs is not the gas. It is where along the sight line the photons that reach the observer came from.

Optical depth is measured along the ray #

A star has no surface. What it has is a region a few hundred kilometres thick where the gas goes from opaque to transparent — the boundary of the part of a star that boils, seen from outside, and “the surface” is wherever the optical depth along the observer’s line of sight reaches about one.

Optical depth is counted along the ray, not downward. A sight line to the centre of the disc goes straight down, so it reaches $\tau = 1$ at a vertical depth of one unit. A sight line to a point at angle $\theta$ from the centre enters at a slant, so it accumulates optical depth faster with vertical distance and reaches $\tau = 1$ at a vertical depth of only $\mu = \cos\theta$.

Higher up is cooler. That is the whole mechanism: the limb is not a cooler region, it is a shallower one.

The relation that makes this quantitative is Eddington–Barbier. Integrating the transfer equation with a source function $S(\tau)$ gives the emergent intensity

$$I(\mu) = \int_0^\infty S(\tau)\,e^{-\tau/\mu}\,\frac{d\tau}{\mu},$$

and if $S$ is linear in $\tau$ the integral evaluates exactly to $S(\tau = \mu)$. In a grey atmosphere in radiative equilibrium the source function is linear:

$$S(\tau) = \frac{3F}{4\pi}\left(\tau + \frac23\right),$$

so

$$\frac{I(\mu)}{I(1)} = \frac{\mu + 2/3}{5/3} = \frac{2 + 3\mu}{5}.$$

A linear law with coefficient exactly $u = 3/5$, and a limb exactly $2/5$ as bright as the centre. Both numbers come out of one differential equation with no free parameter in it.

Three measurements that share no assumption #

The Sun can be photographed and the darkening read directly. For every other star it has to be inferred, and there are three independent routes.

A transit light curve. The four contact points fix the geometry and the shape between them fixes the darkening. A planet crossing a uniform disc removes a constant fraction of the light and produces a flat-bottomed box; a planet crossing a darkened disc removes more light near the centre of the transit than near the limbs, and the bottom is curved. The curvature is a direct measurement of $I(\mu)$, sampled along whatever chord the planet happens to take — and it is also why a planet’s radius depends on the colour it is measured in for a second reason, quite apart from the planet’s own atmosphere. An eclipsing binary. The same geometry with a star doing the occulting instead of a planet. The occulted area is much larger, so the effect on the light curve is correspondingly larger, and the ingress and egress shapes carry the intensity profile of the eclipsed star.

An interferometric visibility. A stellar disc smaller than any telescope’s resolution is measured by the baseline at which its fringes vanish, and where the null falls depends on how the brightness is distributed across the disc. A uniform disc’s visibility is $|2J_1(x)/x|$ with its first null at $x = 3.8317$; a limb-darkened disc’s null sits further out, because the effective size of a centre-brightened source is smaller than its geometric size. Three methods. One occults with a small opaque circle, one with a large luminous one, one interferes light with itself. They share the star and nothing else, and they agree.

What was actually measured #

For the Sun, the observation is a photometric scan across the disc: intensity against position, at a stated wavelength, with the atmospheric transparency and the instrument’s scattering removed. The measurement is a ratio — the intensity at radius $r$ divided by the intensity at the centre — so no absolute calibration is needed, which is why good limb-darkening measurements predate good solar photometry by decades.

The scan is converted to $I(\mu)$ by the geometry $\mu = \sqrt{1 - (r/R)^2}$, and then fitted, usually with a polynomial in $\mu$ rather than a straight line, because the observed profile is not quite linear. The coefficients are tabulated by wavelength, and the wavelength dependence is strong: about 0.9 at 400 nm falling to about 0.35 at 1,600.

That dependence is the second thing the measurement contains, and it is the more informative one. The same temperature difference between two depths produces a larger intensity difference at short wavelengths, because the Planck function is steeper there — on the Wien side, a small change in $T$ is an exponential change in $B_\lambda$. So the run of $u$ with wavelength is a measurement of how steeply the Planck function varies, which is to say of the temperature itself, and it is a consistency check on the gradient rather than a new parameter.

For a star with a transiting planet, the measurement is a light curve and the difficulty is that the darkening is degenerate with everything else. The transit depth, the impact parameter, the ratio of radii and the limb-darkening coefficients all shape the same curve, and fitting all of them at once produces correlated uncertainties. Standard practice is to fix the coefficients at values computed from a model atmosphere of the appropriate temperature and gravity — which means most published limb-darkening coefficients are theoretical inputs rather than measurements, and the handful of high-precision light curves where they can be fitted freely disagree with the models by a few per cent.

Where the model stops #

The atmosphere is not grey. The coefficient $3/5$ assumes the opacity is the same at every wavelength, which it emphatically is not: it is dominated by H⁻ in the Sun and varies by orders of magnitude across the spectrum. A real calculation solves the transfer equation wavelength by wavelength through a model atmosphere, and the answer differs from $3/5$ by tens of per cent depending on where in the spectrum one looks.

A linear law is not enough. The observed profile has curvature, particularly near the limb where $\mu \to 0$ and the sight line grazes. Quadratic, square-root, logarithmic and four-parameter laws are all in use, and the choice matters at the per cent level for transit fitting — which is above the precision of the best light curves, so it is a live systematic rather than a refinement.

The limb is not the edge of anything. At $\mu = 0$ the Eddington–Barbier relation says the emergent intensity is $S(0)$, which is finite, so the disc has a sharp edge in this treatment. It does not: the density falls off over a scale height, the transition is gradual, and the very limb is where the plane-parallel approximation underlying everything above fails. For the Sun the atmosphere is $10^{-4}$ of the radius and the failure is confined to the outermost fraction of a per cent of the disc. For a red supergiant with an extended atmosphere it is not confined at all, and the “diameter” of Betelgeuse depends on the wavelength by more than the measurement error.

Both extensions also make the same practical point: what is drawn on a stellar disc is never a picture of the surface, it is a map of where along each sight line the optical depth reached one.

The practical consequence is that the effective temperature a spectrum returns is a weighted average over a range of depths, weighted differently at every wavelength, and the single number quoted for a star is a property of that weighting as much as of the star.

The correction that changes a star’s size #

Limb darkening is a small effect with a disproportionate consequence, and the reason is that it sits between an observable and a quantity nobody can otherwise reach.

A star’s angular diameter is not a radius. Turning one into the other needs a distance, and the distance comes from a parallax; the two uncertainties compound. But before any of that, the angular diameter itself is model-dependent: a fringe null or a transit depth gives a number only after a brightness distribution has been assumed, and the difference between “uniform” and “limb-darkened” is seventeen per cent for a supergiant in the visible. The parallax that turns the angle into a length contributes its own error on top, and for Betelgeuse it is the larger of the two.

Seventeen per cent in radius is fifty-nine per cent in volume and, at fixed effective temperature, thirty-seven per cent in luminosity. A model of the outermost hundred kilometres of an atmosphere propagates into the position of a star on the Hertzsprung–Russell diagram, which is the diagram every statement about stellar evolution is made against. Between them the two sections make the general statement the essay has been circling. The emergent intensity at a given angle is the source function at the optical depth that angle reaches, so a disc is darkened where the source function falls outward, brightened where it rises, and uniform where it does neither — and every case in this collection is one of those three read off the same relation.

What the picture cannot show #

The third dimension. Every figure here treats the atmosphere as plane-parallel slabs, and the disc drawing shades a flat circle. The real object is a sphere with a stratified shell, and the reason $\mu$ appears at all is that the sight line’s angle to the local vertical changes across the disc — which is a three-dimensional fact drawn in two.

The granulation. The Sun’s surface is covered in convection cells a thousand kilometres across, each brighter in the middle and darker at its edges, with lifetimes of minutes. A limb-darkening curve is an average over that structure, and the structure itself changes the average, because the cells are seen from a different angle near the limb than at the centre. Three-dimensional hydrodynamic models of exactly this give limb-darkening coefficients differing from one-dimensional ones by several per cent.

Which star. The coefficients drawn are the Sun’s. A red giant’s atmosphere is far more extended and darkens far more; a hot star’s is more nearly grey and darkens less. A figure of one star’s disc is not a figure of stars.

Two extensions of the same relation are worth following, because between them they cover every way a stellar disc can fail to be uniform — and only one of them is about the atmosphere’s stratification.

The other variation across a disc #

Limb darkening is not the only reason a stellar disc is non-uniform, and the second reason has a completely different cause and a useful consequence.

A rotating star is oblate: the centrifugal effect reduces the effective gravity at the equator, so the star bulges there and its equatorial surface gravity is lower than its polar. The emergent flux from a stellar surface in radiative equilibrium is proportional to the local effective gravity — von Zeipel’s result — so the poles are brighter and hotter than the equator.

For a slow rotator the effect is negligible. For a star rotating near break-up it is enormous: the equatorial radius can exceed the polar by a third, and the pole can be two thousand kelvin hotter than the equator.

That is gravity darkening, and it differs from limb darkening in every respect that matters. Limb darkening is symmetric about the disc centre and depends on the viewing angle; gravity darkening is symmetric about the rotation axis and depends on where the axis points. A star seen pole-on shows gravity darkening as a centre-to-limb variation indistinguishable in form from limb darkening; the same star seen equator-on shows a bright band across the middle.

Two kinds of measurement have resolved it. Optical interferometry has imaged the discs of several nearby rapid rotators directly, showing oblate figures with bright poles, and the images match the predicted flux distribution.

And a transit does it indirectly. A planet crossing a gravity-darkened star traverses regions of different brightness in a pattern set by the angle between the orbit and the stellar equator, so the transit light curve becomes asymmetric — and fitting the asymmetry returns the spin–orbit alignment, which is otherwise measurable only spectroscopically.

A distortion of the disc that has nothing to do with the atmosphere’s stratification turns out to measure the orientation of a planetary orbit, which is a longer chain than most.

The first of the two has a different cause and the same appearance; the second has the same cause and the opposite appearance, which is the more instructive of the pair.

The limb that brightens #

The whole argument rests on the source function decreasing outward, and there are circumstances where it does not — and there the limb is brighter than the centre.

The condition is worth stating generally. The emergent intensity at angle $\mu$ is the source function at optical depth $\mu$ along the ray, so the disc is darkened at the edge if the source function falls outward and brightened if it rises. Limb darkening is therefore a statement about a temperature gradient’s sign rather than a universal property of spheres.

The clearest case of the other sign is an optically thin shell. If a star is surrounded by a tenuous emitting layer — a chromosphere, a corona, an extended envelope — the sight line at the limb passes through a much longer path within that layer than the sight line to the centre does, so the layer contributes more at the edge. The disc is limb-brightened in whatever wavelength the layer emits.

That is why the Sun’s corona is seen at the limb and not in front of the disc, and why an image of the Sun in an ultraviolet line formed in the transition region shows a bright ring rather than a dark one.

The same inversion happens in an emission line even without a separate layer. In the core of a strong line the opacity is enormous, so the emergent intensity comes from very high in the atmosphere — above the temperature minimum, where the temperature rises outward again — and the line core is limb-brightened while the continuum beside it is limb-darkened.

For a planet the sign is different again. A transiting planet’s atmosphere is seen only at the limb, in transmission, so what is measured is an annulus rather than a disc — and the annulus is bright in exactly the wavelengths the atmosphere absorbs, which is the whole of transmission spectroscopy.

One relation between a source function and an emergent intensity produces a darkened disc, a brightened ring and an absorbing annulus, and which appears depends only on which way the temperature runs.

Where the ladder goes next #

Later rungs on this anchor: the Eddington approximation derived, and where the $2/3$ in the source function comes from. Non-linear limb-darkening laws, and the fitting degeneracies that make the choice between them matter. Gravity darkening, in which a rapidly rotating star is oblate and its poles are hotter and brighter than its equator — a different intensity variation across a disc with a different cause. Interferometric imaging of a resolved stellar disc, which no longer needs the model at all. The Rossiter–McLaughlin effect, where limb darkening weights the velocity of the light a transiting planet hides. And the same physics on a planetary atmosphere, where limb darkening becomes limb brightening in an emission line.

The measurement is very old. Limb darkening is visible in a projected image of the Sun and was described by Christoph Scheiner in the 1620s, who used it — correctly — as evidence that sunspots were on the Sun rather than being small bodies in front of it. It took three hundred years for it to become a statement about a temperature gradient.