A profile that averages a decade of depth

What leaves a star at an angle is the source function evaluated where the slant optical depth is one. At the centre of the disc a sight line goes straight down and reaches optical depth one deep in the atmosphere, where it is hot; near the limb it goes in obliquely and reaches optical depth one high up, where it is cool. The limb is darker because it shows a cooler layer, and when the source function falls inward instead of rising, the limb is brighter.

That rule, the Eddington–Barbier approximation, turns a limb-darkening profile into an apparent depth sounding. Each position on the disc, labelled by $\mu$, the cosine of the angle between the sight line and the vertical, reads off the source function at optical depth $\mu$. Measure the intensity across the disc and the source function follows, depth by depth.

The approximation is exact only when the source function is a straight line in optical depth, and that exactness is the whole difficulty. A straight line is exactly what an average returns unchanged.

Each angle is an average #

The intensity leaving the surface at angle $\mu$ is not the source function at one depth but its average along the sight line, each layer weighted by how much of its light escapes:

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

The weight, $e^{-\tau/\mu}/\mu$, falls exponentially with depth below the layer at optical depth $\mu$ and is cut off above it by the scarcity of material. Plotted per unit of logarithmic depth, as in the first figure, it is a broad hump with its peak at $\tau = \mu$ and a width of a full decade at half height — a factor of ten in optical depth, which in the Sun’s photosphere is about a hundred and fifty kilometres. The width is the same for every angle; changing $\mu$ slides the hump along the depth axis without making it narrower.

If $S$ is linear in $\tau$, the average of $S$ over the hump is exactly $S(\mu)$, because the hump is centred, in the appropriate sense, on $\tau = \mu$. That is why the Eddington–Barbier relation is exact for a linear source function.

A grey atmosphere is the case in which the profile can be read as a sounding without error, and it is the case in which there is nothing to find: a straight line has no features. The inversion problem begins exactly where the atmosphere becomes interesting. Anything that is not linear — a bend, a plateau, a temperature minimum, a narrow layer of heating — is averaged with its neighbours over a decade of depth and comes out smoothed.

A feature the limb hides #

The figure shows what the averaging does to a feature. A bump in the source function a quarter of a decade wide — a narrow layer, a few tens of kilometres thick in the Sun, where something heats the gas above its surroundings — changes the emerging intensity at every angle, most near the angle whose kernel is centred on it, and nowhere by as much as the bump itself. The change has no edges, because the kernels have none. A bump twice as wide and half as tall, centred at the same depth, changes the profile almost identically; so does a smaller bump a little deeper with a broader shoulder. The profile has been made by a smoothing operation, and many different inputs have nearly the same smoothed output.

Reading the profile back #

Recovering $S(\tau)$ from $I(\mu)$ means undoing the smoothing, and the integral above has a name: it is a Laplace transform. Substituting $s = 1/\mu$ turns it into $\int S(\tau)\,e^{-s\tau}\,d\tau$ multiplied by $s$, and the limb profile is that transform sampled at values of $s$ from 1 at the disc centre to about 20 near the limb. Inverting a Laplace transform from samples is the textbook example of an ill-posed problem: the transform suppresses fine detail so strongly that any noise in the samples, inverted naively, explodes into enormous oscillations in the recovered function.

The cure is regularisation — adding to the inversion an assumption about what the answer should look like, so that among all the source functions consistent with the data within their errors, the smoothest is chosen. The assumption used in the figures is minimal: that the source function’s departure from that of a simple grey atmosphere is smooth, with a curvature of a given typical size. Everything sharper than that is not forbidden, but the data have to insist on it.

The recovered curves are the average of what many noisy measurements would give — the resolution of the method, separated from the scatter of any one measurement. At one per cent precision the bump is recovered as a broad swelling at roughly the right depth with about half its height. At a thousandth of a per cent — a precision no stellar limb profile has reached, and only the Sun’s approaches — it is recovered with four-fifths of its height and still too wide. The inversion converges on the truth, but slowly.

The recovered curves also go wrong above an optical depth of about a hundredth, and that is not a failure of the method but a statement about the data. The outermost angle measured, $\mu = 0.05$, has its kernel centred at optical depth 0.05; layers well above that contribute almost nothing to any measured intensity, and the inversion is free to put anything there that the smoothness assumption allows. The depth range a limb profile can sound is set by how close to the limb it is measured.

What each factor of ten buys #

This is the essay’s central result. The resolution of a limb-profile inversion improves with the precision of the data, but logarithmically: a factor of ten in precision narrows the resolved depth range by a fixed step, about a seventh of a decade here, rather than by a factor. Three and a half orders of magnitude in precision — from three per cent to a thousandth of a per cent — halve the resolved width.

The logarithm is characteristic of every inversion of a Laplace transform, and it has a simple origin. The transform’s kernels are smooth exponentials, and the information about a structure of width $w$ in log depth is carried by differences between kernels that fall off exponentially as $w$ decreases. Detecting a structure twice as sharp requires measuring differences exponentially smaller, which requires exponentially better data. Precision buys resolution only through its logarithm.

That is the surprising connection. The limb profile of a star is mathematically the same problem as recovering a distribution of relaxation times from a decay curve, a distribution of particle sizes from scattered light, or the run of a planet’s interior from its gravity field — every problem in which the measurement is a smooth average over the quantity of interest. All of them share the logarithmic resolution, and all of them share the habit of reporting smooth answers, not because the truth is smooth but because smoothness is all the data can support.

Kernels with edges #

Not every inversion in astronomy is condemned to the logarithm, and the contrast shows what makes this one hard. The Sun’s interior is mapped by helioseismology, which inverts the frequencies of its acoustic oscillations for the sound speed and the rotation at each depth. Each oscillation mode samples the interior with its own kernel, as each angle on the disc samples the atmosphere with its own, and the inversion combines thousands of them.

Helioseismic inversions resolve the interior to a few per cent of the solar radius, far better than a limb profile resolves the atmosphere, and the reason is the shape of the kernels. An acoustic mode is trapped between the surface and a turning point, below which the sound waves are refracted back up; its kernel is large above the turning point and cut off sharply below it. Modes with slightly different turning points have kernels whose difference is concentrated in the thin layer between the turning points, and subtracting them isolates that layer. The kernels have edges, and edges localise.

The limb-darkening kernels have no edges. Each is a smooth exponential hump, and the difference between two of them is another smooth hump, almost as broad. No combination of smooth exponentials makes a sharp one without amplifying noise exponentially, and that is the whole of the logarithm. An inversion is only as sharp as the sharpest feature its kernels contain.

More wavelengths, more depths #

A single wavelength’s limb profile sounds a range of optical depth set by the angles measured. Measuring at several wavelengths extends it. The opacity of a stellar atmosphere varies with wavelength — the hydrogen anion that dominates the Sun’s opacity absorbs more strongly in the blue than near 1.6 microns, and much more strongly in the cores of spectral lines than in the continuum between them — so optical depth one is reached at different geometric depths in different colours. A limb profile in the near-infrared opacity minimum sounds the deepest layers; one in the core of a strong line sounds layers hundreds of kilometres higher.

Combining wavelengths therefore extends the inversion’s reach, from the deep photosphere up into the chromosphere. It does not sharpen it. Each wavelength’s kernels are still a decade wide in that wavelength’s optical depth, and the depth scales of different wavelengths are related through the opacity, which must itself be known — any error in the opacity’s wavelength dependence becomes an error in where each wavelength’s information belongs. Multi-wavelength limb scans of the Sun are how the temperature minimum and the rise into the chromosphere were first mapped, and the maps carry that resolution: smooth curves, with the structure of the temperature minimum resolved to a few hundred kilometres at best.

The same averaging governs the formation of spectral lines, where each point in a line profile is formed over a range of depths, and much of a line is not in its depth. Inversions of line profiles for the temperature structure meet the same logarithmic wall, and they add more data — many lines, many points in each — to push against it.

The Sun’s limb, measured #

The Sun is the one star whose limb profile is measured directly, point by point across a resolved disc, at hundreds of wavelengths. Its profile has been measured since the early twentieth century, and the measurements were among the first evidence that the Sun’s atmosphere is close to radiative equilibrium: the profile in the visible is close to the straight line of the grey model, the departures small and smooth.

Those departures are where the information is. Inverted across wavelengths from the near-infrared to the near-ultraviolet, the Sun’s limb profiles gave empirical temperature structures for the photosphere that agreed with radiative-equilibrium models in the deep layers and departed from them above — the first hint of the temperature minimum and of heating above it. The empirical models built from them have the resolution the fourth figure predicts: smooth curves, with the temperature minimum placed to a few hundred kilometres and its depth to a few hundred kelvin.

The precision of the solar data is extraordinary — profiles measured to a fraction of a per cent across the whole disc — and the resolution it buys is still about half a decade of depth. That is not a failure of the observations. It is the logarithm.

A knife edge sweeping across a star #

One method samples a distant star’s profile with much finer resolution in angle than any other. When a star is gravitationally lensed by a binary lens, the magnification is concentrated along caustic curves, and as a caustic sweeps across the star’s disc it acts as a knife edge, magnifying one strip of the disc at a time. The light curve during the crossing, lasting hours, traces the integral of the star’s brightness across the strip, and differentiating it recovers the profile across the disc in fine steps of $\mu$.

That resolves the profile in angle, not the atmosphere in depth. A profile measured at a hundred values of $\mu$ instead of twenty-four is still a sampled Laplace transform, and the inversion from it to the source function is still bound by the logarithm. Caustic crossings have measured the limb darkening of giant stars in the Galactic bulge with a precision that tests model atmospheres in detail, and the tests are forward comparisons: the model predicts the profile, the caustic measures it, and a mismatch says the model is wrong without saying where in depth.

Stars other than the Sun #

For the Sun the limb profile is measured directly, point by point across the resolved disc. For other stars it must be inferred, and the inference passes through instruments that measure the profile only indirectly.

Interferometry measures the star’s visibility at several baselines, and the drop of visibility beyond its first null depends on the limb darkening; the diameter it gives depends on a model atmosphere for the same reason. A handful of interferometric visibilities constrain one or two parameters of the profile, not its shape.

Transits of planets across their stars sample the profile along the planet’s path, and the depth of the transit depends on where on the disc the planet crosses. A well-measured transit light curve constrains two limb-darkening coefficients to a per cent or so.

Eclipsing binaries sample it as one star’s limb passes over the other’s. Microlensing samples it as the caustic sweeps across the star’s disc, sometimes with extraordinary resolution.

In every case the data deliver a few parameters of a smooth law, and the logarithmic resolution of the inversion means that no foreseeable improvement will turn them into a depth sounding. What they test is whether a model atmosphere’s predicted profile is consistent with the measured one — the forward problem — which is much easier than recovering the atmosphere from the profile.

Why forward models are trusted more than inversions #

That distinction is how stellar atmospheres are actually studied, and it follows from everything above. A model atmosphere computes the temperature structure from physics — radiative equilibrium, convection, the opacities — and predicts the limb profile. The prediction is compared with the measurement. If they agree, the model is consistent with the data; if they disagree, something in the physics is wrong.

The inversion runs the other way, and the figures show what it can deliver: a structure resolved to half a decade at the best precision imaginable. The forward model can be wrong in its physics, but it is not limited by resolution; the inversion is free of physical assumptions beyond smoothness, but it is limited to the resolution the data allow. How much of the picture is the prior is the question every inversion must answer, and for a limb profile the answer is: everything finer than half a decade of depth.

What the figures assume #

The atmosphere is plane-parallel. The kernel $e^{-\tau/\mu}$ assumes that the atmosphere’s thickness is negligible compared with the star’s radius. Near the extreme limb of a giant star, whose atmosphere is a few per cent of its radius, sight lines curve through layers of changing depth and the kernels change shape.

The source function is the same at every point on the disc. A real star’s surface has granulation, spots and faculae, and the limb profile averages over all of them; an inversion recovers the average atmosphere, which is not the atmosphere of any particular point.

The inversion’s smoothness assumption is a single number. The typical curvature allowed for the departure from a grey atmosphere sets how strongly the inversion prefers smooth answers, and the recovered heights and widths depend on it; the logarithmic scaling with precision does not.

The data have independent errors at 24 angles. Real limb profiles are measured on a fine grid near the limb and coarsely near the centre, with errors that are correlated through the instrument’s scattered light and the seeing, which blurs the extreme limb where the profile is steepest.

Still open: the chromosphere read from the limb #

The layers the figures show going wrong — above an optical depth of a hundredth — are where the Sun’s temperature reaches its minimum and turns upward into the chromosphere. That upturn is the signature of non-radiative heating, and its height and steepness are among the most constraining facts about how the Sun’s outer atmosphere is heated.

Reaching those layers needs wavelengths where the opacity is large enough for optical depth one to lie high up — the far ultraviolet, the cores of the strongest lines, and the millimetre continuum, where the free–free opacity of the chromosphere’s electrons makes the radiation a nearly linear thermometer. Millimetre interferometers have begun to measure the Sun’s limb profile at those wavelengths, where the source function is the local temperature and the inversion is as well-posed as limb inversions get. Whether the temperature rises in a smooth gradient or in thin sheets — whether the chromosphere is heated everywhere or in narrow layers — is a question about structure narrower than a decade of depth, and the logarithm in the fourth figure says how much precision it will take to answer.