A sight line stops at two thirds

The light that is missing from the edge established what limb darkening measures — a temperature gradient — and the depth is not the area established what it costs a transit measurement. Both take the shape of the profile from a fitting formula and neither derives it.

It is derivable, and the derivation is one line long. It also produces, for the one case that can be solved exactly, a limb-darkening coefficient with no free parameter in it.

The transfer equation, integrated once #

Along a ray making an angle θ with the vertical, the intensity obeys

$$\mu\frac{\mathrm{d}I}{\mathrm{d}\tau} = I - S,$$

where $\tau$ is the optical depth measured vertically and $S$ is the source function — the ratio of emission to absorption, which in thermodynamic equilibrium is the Planck function at the local temperature.

Integrating outward from depth with the boundary condition that nothing comes in from above gives

$$I(0,\mu) = \int_0^\infty S(\tau)\,e^{-\tau/\mu}\,\frac{\mathrm{d}\tau}{\mu}.$$

The emergent intensity is the source function averaged over depth with an exponential weight whose scale is $\mu$. Deep layers are attenuated by everything in front of them; shallow ones contribute little because there is little of them.

Now suppose the source function is linear, $S = a + b\tau$. The integral is elementary:

$$I(0,\mu) = a + b\mu = S(\tau = \mu).$$

The emergent intensity is the source function at the depth where the slant optical depth is one. That is the Eddington–Barbier relation, and for a linear source function it is not an approximation at all — it is an identity, and the weighting function’s asymmetry cancels exactly against the linearity.

Why the grey atmosphere’s coefficient is 3/5 #

The grey case is the one that can be solved. Assume the opacity is independent of wavelength, require radiative equilibrium — the flux is the same at every depth — and the Eddington approximation gives

$$S(\tau) = \frac{3F}{4\pi}\left(\tau + \frac{2}{3}\right),$$

a straight line of slope $3F/4\pi$ and intercept two thirds of that.

Feed it through the relation. The intensity at angle μ is proportional to $\mu + 2/3$, so

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

which at the limb gives exactly 2/5. Writing that as a linear limb-darkening law, $I(\mu)/I(1) = 1 - u(1-\mu)$, gives $u = 3/5$ exactly.

That is worth pausing on. The coefficient tabulated for real stars is a fitted number that depends on wavelength, temperature, gravity and composition, and for the Sun in the visible it is around 0.6 to 0.9. The grey atmosphere’s is 0.6, from an argument with nothing adjustable in it.

What two thirds means #

The intercept in the grey source function is where the phrase “the surface of a star” comes from, and it is worth extracting.

A star has no surface. It has a region where the opacity falls rapidly enough that photons begin to escape, and the depth at which that happens depends on the wavelength and on the direction. The convention is to define the photosphere as the layer where the vertical optical depth is 2/3, and the reason is the relation above: averaged over the disc, weighting by area, the emergent flux comes from the source function at that depth.

The consequence is a definition rather than a discovery. The effective temperature of a star is the temperature at $\tau = 2/3$, and every radius quoted for a star is the radius of that layer. For the Sun the layer is a few hundred kilometres thick — the distance over which the optical depth goes from 0.1 to 10 — out of seven hundred thousand, which is why the disc has a sharp edge to look at and no edge at all in the physics.

It also explains why the number 2/3 appears in contexts that seem unrelated. A quantity defined at $\tau = 2/3$ will keep turning up wherever an atmosphere is being converted into a single temperature, and the 2/3 is not a rule of thumb but the intercept of a straight line that was computed once.

A disc drawn from the relation, and a transit measured with it #

The construction is worth carrying through to the two places those earlier essays ended up.

The first thing that falls out is a number for the total light. A transit measures a depth, and the depth is the fraction of the disc’s flux the planet covers, which needs the profile integrated over the area. An annulus at cosine μ has area proportional to $2\mu\,\mathrm{d}\mu$, so the disc-averaged intensity for the grey law is

$$\int_0^1 \frac{2+3\mu}{5}\,2\mu\,\mathrm{d}\mu \;=\; \frac{2}{5} + \frac{3}{5}\cdot\frac{2}{3} \;=\; \frac{4}{5}$$

of the central value. So a planet crossing the centre of a grey-atmosphere star blocks light 1.25 times brighter than the disc’s average, and the transit is 25 per cent deeper than the area ratio.

That is the same statement the transit essay makes with realistic coefficients and a fitted curve, arriving here from a source function and a boundary condition. It also shows where the 20 per cent quoted there comes from: the real coefficients are a little different from the grey ones, and the geometry of a real crossing is not exactly central.

The second thing is the shape of the ingress. A planet entering at the limb covers material at two fifths of the central intensity, so the light curve’s corners are rounded by an amount the relation predicts rather than describes — and the rounding is what the four contact points have to be extracted from.

Where the relation is only approximate #

The identity holds for a linear source function. Real source functions are not linear, and the departures are where the useful physics is.

Near a spectral line, the source function drops. A line’s opacity is large, so the optical depth reaches one much higher in the atmosphere, where it is cooler — which is the whole reason absorption lines are dark. But close to the line’s core the source function also departs from the Planck function, because the transition is no longer in equilibrium with the gas, and the Eddington–Barbier reading of “the line core shows the temperature at its own $\tau = 1$” becomes an approximation with a computable error.

In a convective atmosphere, the gradient is not the radiative one. The linear source function assumed radiative equilibrium. Where convection carries the flux — which for the Sun is everywhere below the photosphere and partly within it — the temperature gradient is shallower than the radiative calculation gives, so the source function is flatter and the limb darkening weaker.

And the atmosphere is not one-dimensional. A real photosphere is a field of granules: hot rising columns and cooler sinking lanes, with a horizontal temperature contrast of hundreds of kelvin at the same depth. The one-dimensional source function is an average over that structure, and averaging a non-linear function of temperature is not the function of the average — which is one of the reasons three-dimensional models give different limb-darkening coefficients from one-dimensional ones, in a direction that matters at the precision transit photometry now reaches.

The same relation on something that is not a star #

Nothing in the derivation used a star. It used a semi-infinite absorbing medium with a source function that varies with depth, and that description fits several other objects this collection cares about.

A planet’s atmosphere in thermal emission. The infrared brightness temperature of a giant planet is the temperature at the depth where its own optical depth reaches one, and that depth moves with wavelength through the methane and ammonia opacities. A brightness-temperature spectrum is therefore a sounding: each wavelength reports the temperature at its own $\tau = 1$, and the run between them is the atmosphere’s thermal profile.

An accretion disc. A disc’s spectrum is a stack of blackbodies because each annulus radiates from its own photosphere at its own temperature, and each annulus’s emergent intensity is its own source function at its own $\tau = 2/3$. The complication there is that the disc’s vertical structure is not in radiative equilibrium — energy is being dissipated within it — so the source function’s gradient is steeper than the stellar one, and disc atmospheres are more limb-darkened than stars.

And a molecular cloud. A line from a cloud reports the excitation temperature at the depth where that line becomes optically thick, which is why a survey in an optically thick line measures a temperature and one in an optically thin line measures a column density. The two are different observables from the same cloud, and which is which is decided by the same relation.

The recurring statement is that an observation reports conditions at the depth its own opacity chooses, and changing the observation changes the depth. That is what makes an atmosphere soundable rather than merely a surface, and it is also the trap: a set of measurements at different wavelengths is a set of measurements at different places, and treating them as one is the most common way to build a wrong temperature.

What was actually measured #

The Sun’s limb darkening is measured directly, by pointing a photometer at a resolved disc and scanning, and it has been done in dozens of passbands over a century. That is the one star for which the profile is an observation rather than an inference.

The quantity behind it — the source function’s run with depth — is not measured at all. It is computed, from a model atmosphere that solves the transfer equation and the equations of hydrostatic and radiative equilibrium simultaneously, with an opacity taken from atomic physics.

Three ingredients of that computation are worth naming.

The opacity. In a solar-type photosphere the dominant continuum opacity in the visible is the negative hydrogen ion, which is a bound state of an extra electron on a hydrogen atom held by polarisation — a genuinely awkward quantum-mechanical calculation, and one whose value has been revised. The opacity sets where $\tau = 1$ is at each wavelength, so it sets the whole of the wavelength dependence.

The assumption of local equilibrium. The source function is set equal to the Planck function, which requires the gas’s level populations to be collisionally controlled. That holds well in the deep photosphere and less well higher up, where the radiation field decouples.

And the composition. The opacity depends on the abundances, and the solar abundance revision changed them by enough to break the agreement with helioseismology. Limb darkening is one of the observables that revision has to remain consistent with, and it is a weak constraint compared with the seismology.

Where the model stops #

The Eddington approximation is an approximation. The grey solution above uses a closure relation between the mean intensity and the radiation pressure that is exact only for an isotropic radiation field, and the field near the top of an atmosphere is not isotropic — it is hemispherical, with nothing coming down. The exact grey solution, obtained by a much harder method, differs from the Eddington one by about one per cent in the temperature at the surface, and the exact limb-darkening law is not quite linear.

Greyness is a fiction. No real opacity is independent of wavelength. What the grey calculation gives is a useful fiction whose parameter is a suitably weighted mean opacity, and the choice of weighting is another approximation.

The figure shows a semi-infinite plane. A star is a sphere, and the curvature matters when the atmosphere’s thickness is a non-negligible fraction of the radius. For a main-sequence star it is not; for a red supergiant with an extended atmosphere it is, and the profile has to be computed spherically — which produces a limb that is not sharp and a radius that depends on what one means by it.

And nothing here is a spectrum. The relation gives the intensity at one wavelength from the source function at one wavelength. Everything a stellar spectrum contains is the variation of the opacity from wavelength to wavelength, which the grey assumption threw away on the first line.

How far a sight line actually sees #

The relation says which depth is reported and it is worth asking how sharply, because “a sight line sees to optical depth one” is a statement about a mean rather than about a boundary.

The weighting function is $e^{-\tau/\mu}/\mu$, whose mean is $\mu$ and whose width is also $\mu$. So a vertical sight line reports a weighted average over a full decade of optical depth, from about 0.1 to about 3 — which in a solar photosphere is a couple of hundred kilometres and a temperature range of a thousand kelvin.

That width is why the relation is only exact for a linear source function. A curved source function is being averaged over a range wide enough for the curvature to matter, and the emergent intensity is then the source function at $\tau = \mu$ plus a correction proportional to its second derivative. The correction is small in a photosphere and it is not small in a chromosphere, where the source function turns over.

It also decides what limb darkening can measure. A profile across a disc constrains the source function’s gradient over a range of depths comparable with the depths probed, which is a factor of twenty in $\tau$ between the centre and the extreme limb. That is a genuine sounding — it is why the profile carries information about the temperature structure rather than just about one temperature — and it is a coarse one, because the weighting functions at different μ overlap heavily.

An inversion from limb darkening alone recovers about two numbers, which is why the fitting laws have two coefficients and why the arguments about which law to use are arguments about a parameterisation rather than about physics.

What a temperature at a depth is worth #

The relation delivers a source function, and what is usually wanted is a temperature — so the last step is worth examining because it is where the assumptions concentrate.

Setting the source function equal to the Planck function at the local temperature requires local thermodynamic equilibrium, which requires the level populations of whatever is absorbing to be set by collisions rather than by the radiation field. In a photosphere the density is high enough that this holds for the continuum to a per cent or better.

It holds much less well for a line, and it fails badly for a line formed high in the atmosphere. There the radiation field arriving from below is hotter than the local gas, the populations are driven by it, and the source function is smaller than the Planck function — so a line’s core is darker than a naive reading of the relation would give, and inferring a temperature from its depth without correcting for that overstates how cold the layer is.

The correction requires solving the population equations simultaneously with the transfer, for every level of every species that matters, which is the difference between a one-line relation and a research code. The relation says which depth is reported; what is reported from it is a separate and much harder question, and conflating the two is the standard way to get a wrong temperature from a right formula.

The generalisation #

The structure worth carrying is that an exponentially weighted integral of a linear function is the function evaluated at the weight’s scale, and that this is why so much of stellar atmospheres reduces to seeing down to optical depth one.

The pattern recurs wherever a quantity is emitted throughout a medium that absorbs it. The observed value is the source evaluated where the medium becomes opaque, and the depth at which that happens is a function of whatever the observation is sensitive to — the wavelength, the angle, the line strength. Changing any of those changes the depth probed, which is what turns a single spectrum into a sounding of an atmosphere: a strong line sees high, a weak line sees deep, and the run of temperature between them is read off the relative depths.

The second reading is about exactness. The relation is usually presented as an approximation and stated with a caveat, and for a linear source function it is exact — the approximation is in the linearity, not in the relation. That distinction matters practically, because it says where to look when the answer is wrong: not at the weighting function, which is handled exactly, but at whether the source function is straight over the range the weight spans. Knowing which half of an approximation is doing the approximating is most of knowing when it fails.

Still open: the gradient with the other sign #

What comes next takes the same relation and reverses one thing. Every statement here assumed the source function rises inward, because temperature does — and above a star’s photosphere it stops rising inward and starts falling, so at any wavelength that sees those layers the relation predicts a limb that is brighter than the centre. That prediction is correct and it is observed, on the Sun, at millimetre wavelengths and in the cores of strong lines.

Beside it lies the case where the source function is not the issue at all: an optically thin shell, where the emergent intensity is the emissivity times the path length and a limb is brightened by geometry with no gradient anywhere in it.