A diameter that depends on a model atmosphere

The first rung of this anchor obtained an angular diameter from the baseline at which a star’s fringes vanish, and noted in passing that what comes back is a uniform-disc diameter. This rung is about that qualification, because it is worth several per cent and several per cent is what the measurement is for.

The size of the qualification is easy to state and easy to underestimate. A uniform-disc diameter is two to six per cent smaller than the star’s true limb-darkened diameter, depending on the wavelength and on how cool the star is. Two to six per cent is nothing at all if the diameter is wanted to establish that a star is resolved, and it is the dominant systematic if the diameter is wanted for a temperature, a radius or a distance — which is what it is always wanted for.

A star is not a uniform disc. Looking towards its centre one sees deep and hot; looking towards the limb the same optical depth is reached higher up and cooler, so the edge is fainter. That is limb darkening, and it means the star’s brightness distribution is not the top-hat whose Fourier transform is the Airy pattern the first rung fitted.

The fit in that figure is least squares on the drawn curves rather than a published conversion table, and the zero-coefficient case returns the uniform disc to within 0.02 per cent — which is the procedure checking itself.

Why limb darkening exists at all #

The mechanism is worth a paragraph, because it makes the size of the coefficient predictable rather than fitted.

A line of sight towards the centre of a stellar disc goes straight down into the atmosphere; a line of sight towards the limb enters at a grazing angle. Both stop being transparent at the same optical depth — about two thirds — but the grazing line reaches that depth after a shorter vertical distance, so it sees a shallower and cooler layer. The temperature falls outward, so the limb is cooler, so it is fainter.

How much fainter depends on how steeply the Planck function responds to temperature at the observing wavelength, and that is where the wavelength dependence comes from. In the visible, on the Wien side of a cool star’s peak, the intensity goes as a high power of the temperature and a small temperature drop is a large brightness drop. In the infrared, on the Rayleigh–Jeans side, the intensity goes as the first power and the same temperature drop barely shows.

So limb darkening is strong in the blue and weak in the infrared, strong for cool stars and weak for hot ones, and the coefficient is computed rather than adjusted. The uncertainty in it is the uncertainty in the temperature gradient near the surface, which is where convection and non-local thermodynamic equilibrium both matter and where a one-dimensional model is at its weakest.

Why the first lobe cannot tell #

The reason the two models agree so well where the signal is strong is not a coincidence. It is the same reason a low-resolution measurement of anything returns its second moment and nothing else.

The visibility is the Fourier transform of the brightness distribution. At small baselines only the low spatial frequencies are sampled, and every compact distribution of the same total flux and the same second moment has the same low-frequency behaviour — the transform’s expansion begins $1 - \langle r^2\rangle (\pi B/\lambda)^2/4 + \ldots$, with the shape entering only at fourth order. So the first lobe measures a size, in the sense of a second moment, and it does not care how the light is arranged inside it.

Distinguishing the arrangements needs the higher frequencies, which is the second lobe, which is where the visibility is a few per cent of unity and where an interferometer’s calibration errors live.

It is worth putting the sizes of the two regimes side by side. The first lobe runs from zero baseline out to the first null and holds visibilities from one down to zero; a real measurement takes a dozen points across it, each good to one or two per cent. The second lobe runs from the first null to the second and holds visibilities from zero up to about four per cent and back; a measurement there takes points whose absolute errors are the same one or two per cent of unity, which is a relative error of fifty per cent.

So the second lobe is not merely harder. It is harder in the specific way that makes a model comparison useless — the discrepancy between the models is 4 per cent of unity and the error bar on a point is 1 to 2 per cent of unity, so distinguishing them needs many points and an instrument whose systematics are understood at the per-cent level across a null. That is a small number of nights on a small number of instruments.

Where the correction comes from #

Since the observation cannot supply it, the correction is taken from a model.

A one-dimensional model atmosphere — a run of temperature with optical depth, in hydrostatic and radiative equilibrium, at an assumed effective temperature, surface gravity and composition — predicts the specific intensity as a function of the angle at which the surface is viewed. Integrating that over the disc gives a brightness profile, and transforming it gives the visibility curve that should be fitted instead of a uniform disc.

The models used are the same grids that underpin every spectroscopic analysis in the subject, and the limb-darkening coefficients are tabulated from them for every filter anybody uses. Different grids give coefficients differing by five to ten per cent of themselves, which on a correction of four per cent is a residual of a few tenths of a per cent — small, and it is the number that ought to be quoted as the modelling uncertainty and usually is not.

The circularity is visible and is real. The model needs an effective temperature; the effective temperature is what the diameter is being measured to obtain. In practice the loop is entered with a spectroscopic temperature, run once, and not iterated — the dependence is weak enough that it does not matter, which is a statement that has been checked and is true.

The quarter power is the one piece of good news. An eleven per cent spread in diameter becomes a five per cent spread in temperature, because $T \propto \theta^{-1/2}$ and the square root halves the fractional error. It is not enough good news: 170 kelvin on a star of 3,400 is comparable with the disagreements between spectroscopic temperature scales that the direct method exists to arbitrate.

The correction is not large and it is not negligible, and its size can be stated once and for all: for a linear coefficient between 0.2 and 0.7, the true diameter exceeds the uniform-disc diameter by between 1.6 and 6.5 per cent. Published measurements quote the corrected value with an uncertainty that usually includes something for the model, typically half a per cent — which is a judgement rather than a propagated error, because there is no ensemble of model atmospheres from which a variance could be computed.

There is one check available and it is a good one. A star observed at two wavelengths has two different limb-darkening coefficients and must have one diameter. Requiring the corrected diameters to agree tests the model’s wavelength dependence, which is the part of it doing the work, and where it has been done the agreement is at the one per cent level. That is not a test of the absolute correction — a model wrong by a constant factor at every wavelength would pass — but it is the only test there is.

What else is in the number #

The limb-darkening correction is the term this essay is about and it is not always the largest.

The bolometric flux. The direct temperature needs the star’s total flux, integrated over all wavelengths, corrected for interstellar extinction. For a bright nearby star that integration is done over real photometry with a modest extrapolation; for anything reddened it requires an extinction law and a colour excess, and a five per cent error in the flux is a 1.25 per cent error in the temperature.

Surface structure. A cool giant’s surface is not smooth. Convection produces a small number of enormous cells, so the disc is mottled at the ten per cent level and changes on a timescale of months, and the “diameter” measured on one night is not the diameter measured on the next. This is not a modelling deficiency; it is the star.

Non-sphericity. A rapid rotator is oblate, by up to twenty-five per cent for the fastest, and its poles are hotter than its equator — the gravity darkening that makes a fast rotator’s spectrum depend on where the observer stands. There is no diameter to measure: there is a projected shape and an orientation, and both have to come out of the fit. For the fastest rotators an interferometer resolves the shape directly and the “angular diameter” ceases to be a meaningful entry in a catalogue.

The distance to the second lobe. Reaching a visibility of a few per cent requires a baseline nearly twice the one that reaches the first null, and doubling a baseline is not a small change to an interferometer — the light from each telescope must be brought together with the path lengths matched to a fraction of a wavelength over that distance. So the observations that would settle the correction are not merely noisier; they need an instrument twice the size.

The wavelength dependence within a band. A measurement made through a filter integrates the visibility over a range of wavelengths, and the visibility depends on wavelength both through the baseline in units of λ and through the limb darkening. For a narrow band this is negligible; for the broad bands that give enough photons on a faint star it is a smearing of the fringe that must itself be modelled.

What is actually measured #

An interferometer does not measure a visibility. It measures a fringe contrast, on a baseline, at a wavelength, through an atmosphere that is destroying the fringe faster than it can be recorded.

The raw contrast is multiplied by an instrumental and atmospheric transfer function that must be calibrated by observing a star assumed to be unresolved — and “assumed to be unresolved” means a star whose diameter is estimated from its spectral type and its brightness, which is a diameter that was calibrated on interferometry. That loop is closed by using calibrators much smaller than the targets, so the residual error is second order, and it is the largest single term in most published diameters.

The fringe itself is not seen either, in the sense of being held still. The atmosphere moves the optical path by many wavelengths on a timescale of milliseconds, so the fringe sweeps across the detector faster than it can be integrated, and what is recorded is a squared visibility averaged over that motion — which is why the phase is lost and why a phase that survives what corrupts it had to be invented. Recovering an amplitude from a squared quantity averaged over an unknown jitter is the calibration problem restated, and it is why the transfer function has to be measured every few minutes on a star assumed to be a point.

What the diameters are for #

It is worth being explicit about why several per cent on a few dozen stars is worth this much trouble.

The temperature scale. Every spectroscopic temperature is calibrated, directly or through several steps, against stars with direct temperatures. A systematic error in the direct scale propagates into every stellar temperature in every catalogue, and from there into every stellar mass and age derived from a colour–magnitude diagram.

Testing stellar models. A model predicts a radius for a star of a given mass, composition and age. Comparing that with a measured radius is one of very few tests of stellar structure that does not go through a luminosity, and the tests that exist — mostly on the only stars whose masses are known, the eclipsing binaries — find the models a few per cent too small for cool stars — a discrepancy the same size as the correction this essay is about, which is a reason to want the correction right.

Transiting planets. A planet radius is a stellar radius multiplied by a transit depth, so every planet radius inherits its host’s. Interferometry is the only route to a host star’s radius that does not pass through a stellar model, and it works for a handful of the nearest hosts — which are then used to calibrate the model-based radii of the rest.

The distance scale. An angular diameter and a measured radius together give a distance, and for the classical Cepheids this is the Baade–Wesselink method: the radius from integrating the pulsational velocity, the angle from interferometry, and a distance with no rung under it. The limb-darkening correction enters that distance directly, and the projection factor that converts a spectroscopic velocity into a pulsational one is its analogue on the other side.

The generalisation #

The structure worth extracting is that a low-order measurement returns a moment, and the shape lives in the orders that were not measured.

Every fit of a one-parameter model to data that only constrain a moment returns that moment wearing the model’s units. A uniform-disc diameter is not wrong; it is a correct statement about the second moment of the brightness distribution, mislabelled as a length. The correction is the difference between the moment and the length, and it can only come from knowing the shape.

The same shape recurs constantly. A temperature that depends on where the observer stands is the same star measured through a different projection of the same unresolved structure. A radius that depends on the colour it is measured in is the same problem for a planet’s atmosphere rather than a star’s, and the spin that is one length in disguise is a third instance: a single fitted length carrying the whole of what a spectrum was allowed to say.

There is a second reading about the value of an insensitive measurement. The first lobe’s indifference to the brightness profile is exactly what makes the diameter measurable at all: a quantity that depended on the profile would need the profile to be known, and it is not. The insensitivity is a feature until it is time to correct for it, and then it is the whole problem. A measurement’s robustness and its incompleteness are the same property seen from two ends.

The corollary is a rule for reading a measured size. Ask what model was fitted, and whether the data constrained anything the model did not have. A one-parameter fit to data that would support three parameters is reporting a number with two assumptions folded into it, and the assumptions do not appear in the error bar.

Where the ladder goes next #

The next rung goes to the imaging, where the model stops being fitted and starts being reconstructed. With enough baselines and enough closure phases an interferometer produces a picture rather than a diameter, and the pictures of the largest stars show what the fits could not: spots, asymmetries, and an oblateness that no single number describes.

Further rungs on this anchor: the diameters of stars with atmospheres so extended that there is no surface to measure, where the visibility curve has no first null; the use of lunar occultations, which give a one-dimensional profile at high angular resolution for any star the Moon happens to pass in front of; the interferometric radii of transiting-planet hosts, which is the one route to a planet’s radius that does not pass through a stellar model; and the diameters of stars measured during a total eclipse of the Sun by an asteroid, which is the same idea with a worse-known occulter.