A temperature read against the light's long tail

A star’s effective temperature is defined by its luminosity and its radius: it is the temperature a blackbody of the same size would need to radiate the same power. Measured from the Earth, that is a statement about two observables — the bolometric flux arriving here, $F_{\rm bol}$, and the angular diameter of the star, $\theta$ — through

$$F_{\rm bol} = \frac{\theta^2}{4}\,\sigma T_{\rm eff}^4 .$$

The bolometric flux can be measured, by adding up the star’s light at every wavelength. The angular diameter, for all but a few hundred nearby stars, cannot: a star is a point to every telescope, and only interferometers with baselines of hundreds of metres resolve the brightest and largest. For everything else a temperature has to be read from the light alone, and a colour is the cheapest way — but a colour is a thermometer that needs calibrating, and the calibration’s zero point is the question.

In 1977 Donald Blackwell and Michael Shallis proposed a way to measure the temperature that needs neither the diameter nor a colour calibration. It rests on one property of the Planck curve.

The leverage in the tail #

Take the ratio of the bolometric flux to the flux in a single infrared band. Both are proportional to $\theta^2/4$ — the star’s apparent size dilutes every wavelength equally — so the ratio does not depend on the star’s size or its distance:

$$\frac{F_{\rm bol}}{F_\lambda} = \frac{\sigma T_{\rm eff}^4}{\Phi_\lambda(T_{\rm eff}, g, [{\rm Fe/H}])},$$

where $\Phi_\lambda$ is the flux the star’s surface emits at wavelength $\lambda$. The numerator rises as the fourth power of temperature. The denominator, at a wavelength well beyond the peak of the Planck curve, rises nearly as the first power, because on the long-wavelength side the curve flattens into the Rayleigh–Jeans tail where the emitted flux is simply proportional to the temperature. So the ratio rises as nearly the third power of temperature, and it can be inverted for the temperature with only a model of $\Phi_\lambda$.

The figure shows the exponent, $n$, for a blackbody, in four bands. In the K band at 2.2 microns it is 2.33 for a star of the Sun’s temperature and climbs toward three for hotter stars, as the band falls deeper into the tail; in J at 1.25 microns it is 1.70. In the visible V band it is negative near the Sun’s temperature: there the band sits on the steep short-wavelength side of the curve, its flux rises faster than the fourth power, and the ratio barely changes with temperature at all. The method works in the infrared and only there.

The exponent is also the method’s error divisor. The model enters only through $\Phi_\lambda$, and an error of one per cent in the modelled infrared surface flux shifts the inferred ratio by one per cent and the temperature by $1/n$ per cent — 0.43 per cent in K, or twenty-five kelvin at the Sun’s temperature. The method does not need the model to be right; it needs it to be nearly right in a part of the spectrum where nearly right is easiest, because the infrared continuum of a solar-type star is formed deep, is free of most absorption lines, and depends weakly on the details of the atmosphere.

Errors divided, and errors that remain #

The division is what the method offers, and the next figure shows how much it delivers.

The comparison with a colour is the instructive part. A V−K colour, from the visible to the infrared, spans the Planck peak, and for cool stars it changes quickly with temperature: a two-hundredths of a magnitude error in it is only twenty-seven kelvin at 5,000 K, less than the infrared flux method’s error from a two per cent model error. For hot stars the colour flattens and its error grows faster. On precision alone the colour wins for cool stars.

Halving both errors halves both curves and leaves their relation unchanged, which is the point: the comparison is between two kinds of error, not two precisions. A colour error of a hundredth of a magnitude is a statement about photometry, and photometry can be made that good. A model-flux error of one per cent is a statement about stellar atmospheres in the infrared, and whether they are that good is exactly what the comparison with interferometry tests.

But a colour is a relative measurement, and converting it to a temperature needs a calibration — a set of stars whose temperatures are known independently. The infrared flux method is one of the two ways those temperatures are obtained. Its errors are of the same order as a well-measured colour’s, and it needs no calibration of its own beyond the absolute flux scale. That is why it is used to calibrate the colour–temperature relations that most stellar temperatures are then read from: the colour is the thermometer, and the infrared flux method is one of the fixed points it is graduated against.

The method’s choice of band follows from the same figure. The K band gives the most leverage, but at 2.2 microns the light is faintest relative to the bolometric flux and the measurements are hardest; the J band is easier to measure and gives less leverage, and below about 4,500 K its leverage collapses as the band approaches the Planck peak of cool stars. In practice the method uses all three near-infrared bands and averages, weighting K where the photometry allows.

The coolest stars lose the leverage #

The leverage falls as the star cools, because a cooler star’s Planck peak moves into the infrared and the band climbs off the flat tail onto the steep side. At 4,000 kelvin the K-band exponent is 1.97 and the J-band 0.95; at 3,500 kelvin they are 1.79 and 0.58. For an M dwarf the J band has almost no leverage left, and the K band, the best available, divides a model error by less than two.

That is where the models are also worst. The infrared spectra of M dwarfs are cut by bands of water and carbon monoxide whose strengths depend on temperature, gravity and composition, and the “continuum” the method relies on is barely a continuum at all. Two effects compound: the model error grows and the divisor shrinks. For the red dwarfs that make up most of the Galaxy’s stars and host most of its small planets — stars whose models already disagree with their measured radii — the infrared flux method gives temperatures uncertain by a hundred kelvin or more, and interferometric diameters, available for a few dozen of the nearest, are the better foundation.

A zero point every star shares #

The infrared flux in the ratio is a measured quantity in physical units, and converting a star’s infrared magnitude into a flux requires the flux of a star of magnitude zero in that band — the absolute calibration of the photometric system. That calibration rests, ultimately, on a few standard stars and on models of Vega and Sirius in the infrared, and it is known to one or two per cent.

A random error in one star’s model flux averages away across a sample. An error in the zero point does not: every star measured through the same calibration moves the same way. A one per cent calibration error is a systematic shift of twenty to thirty kelvin in the whole temperature scale built from the method — not large by itself, but of the same order as the precision now claimed for the best samples, and in the same units as the disagreement between the two foundations of the scale.

The calibration is also where the definition of a magnitude band re-enters. The flux in “the K band” depends on the band’s exact transmission curve, which differs between photometric systems, and the same star measured through two telescopes gives two K magnitudes. The infrared flux method has to convert each survey’s magnitudes into fluxes with that survey’s own calibration, and a mismatch between survey and calibration is a zero-point error by another name.

A diameter that comes with the answer #

Once the temperature is known, the angular diameter follows from the defining equation: $\theta = 2\sqrt{F_{\rm bol}/\sigma T_{\rm eff}^4}$. The method therefore predicts a diameter for every star it measures — and for the few hundred stars that interferometers can resolve, the prediction can be checked.

A temperature error enters the diameter doubled, because the flux goes as the fourth power of temperature and the diameter as the square root of the flux. Interferometry measures the diameters of nearby stars to one or two per cent, which corresponds to a temperature precision of twenty-five to fifty kelvin. A disagreement of fifty to a hundred and fifty kelvin between the two scales is therefore a disagreement of two to five per cent in diameter — several times the interferometric error, and for nearby bright stars, several times the bolometric flux error too.

That is what the second foundation of the scale looks like. Interferometric temperatures use the same defining equation the other way round: measure $\theta$ directly, measure $F_{\rm bol}$, solve for $T$. They depend on no model of the infrared flux at all — but they depend on a model of the star’s limb darkening, because an interferometer measures the fringe visibility of a disc whose brightness falls toward its edge, and converting that into a diameter means assuming how it falls. For cool giants the correction is several per cent. The two foundations of the temperature scale each rest on a model, of different parts of the star.

The cool giants, where interferometry is easiest, are where the doubling bites hardest in absolute terms.

A cool giant is the interferometrist’s favourite target: large, bright and nearby enough that the longest baselines resolve it well beyond the first null of its visibility curve. It is also where a fixed disagreement in kelvin is the largest fraction of the temperature, and so the largest fraction of the diameter. A hundred-kelvin disagreement at 4,000 kelvin is a five per cent disagreement in diameter, which the interferometers would see clearly — if the limb-darkening correction, which for a cool giant’s extended atmosphere is itself several per cent and depends on a model, did not have to be subtracted first.

Two foundations that disagree #

The comparison has been made repeatedly, for the stars that both methods can measure, and the result has been stubborn. For cool dwarfs and giants the infrared flux method’s temperatures and the interferometric ones differ by fifty to a hundred and fifty kelvin, in a direction that has itself varied between studies as calibrations, model atmospheres and samples changed. Each successive improvement has moved the disagreement around rather than removing it.

Part of the problem is that the two methods are compared on different stars. Interferometry resolves bright, nearby, mostly large stars — giants and the nearest dwarfs — and those are the hardest for the infrared flux method’s photometry, because the infrared detectors of the large surveys saturate on them. The overlap is a few dozen stars measured carefully by both, and the systematic errors of each method, including the zero point and the limb-darkening correction, are of the size of the disagreement.

What would settle it is a sample of fainter stars resolved by interferometry and measured photometrically without saturation, and the longest-baseline interferometers are slowly reaching such stars. Until then, the temperature scale has an uncertainty of order a hundred kelvin at its base, and every quantity derived from a temperature — a radius, a luminosity, an age from a turn-off, an abundance from line strengths — inherits it.

A method that predates the instruments it needed #

When Blackwell and Shallis proposed the method, the infrared photometry it needed existed for a few hundred bright stars, measured one at a time with single-element detectors, and the bolometric fluxes had to be assembled from heterogeneous spectrophotometry. The temperatures it produced were good to about two per cent and were used chiefly to calibrate model atmospheres for the Sun-like stars that dominated abundance work.

The all-sky infrared surveys of the 1990s and 2000s changed its scale. Near-infrared magnitudes for hundreds of millions of stars, measured through one well-characterised set of filters with one calibration, made it possible to apply the method to any star with good visible photometry, and the large calibrations of colour–temperature relations for dwarfs and giants that most surveys now use rest on it. The space astrometry catalogues added the last ingredient, distances, which turn the bolometric flux into a luminosity and the implied angular diameter into a physical radius that can be checked against stellar models.

What did not change is the foundation the method shares with every other: the absolute flux of a star of magnitude zero, set by a handful of standards. The method’s precision has improved by a factor of several in forty years; its accuracy is still limited by that one calibration and by the comparison with interferometry it cannot do without.

Why the tail is the right place to look #

The method’s insight is general and worth stating apart from its application. A measurement is most robust when the quantity sought enters it strongly and the quantity modelled enters it weakly. The Planck function, read across its peak, is a strong function of temperature on its short-wavelength side and a weak one on its long-wavelength side; its peak and its integral are one curve read twice, and the tail is a third reading. The infrared flux method puts the model where the curve is weak and the measurement where it is strong — all the light, which goes as the fourth power — and divides one by the other.

The same structure appears wherever a quantity has to be extracted from a ratio. A distance from surface brightness works because a cool star’s surface brightness, in a red band, depends weakly on its colour in the right combination. A redshift from photometry fails where the colours are flat in redshift. In each case the method succeeds or fails according to whether the dependence it relies on is steep where the measurement is and flat where the model is.

What the figures leave out #

The surface flux is a blackbody. Real stars have absorption by molecules in the infrared — carbon monoxide in K, water in cool dwarfs — and an infrared continuum that departs from a blackbody by several per cent. The method’s leverage is close to the blackbody value for solar-type stars and lower for the coolest, where the infrared is rich in molecular bands.

The bolometric flux is exact. Measuring it means integrating the star’s spectrum from the ultraviolet to the far infrared, and the ultraviolet part, where the flux is small for cool stars and large for hot ones, has to be modelled where it is not observed. An error of a per cent or two in the bolometric flux is common and enters the temperature at a quarter of its size.

Extinction is ignored. Dust between the star and the observer dims the visible far more than the infrared, so it lowers the bolometric flux more than the infrared flux and lowers the inferred temperature. For nearby stars the correction is small; for distant ones it is the largest single uncertainty.

The comparison of scales is drawn, not measured. The shaded range in the diameter figure is the span of reported disagreements, not a single measurement.

Still open: a hundred kelvin at the base of everything #

The effective temperature is the most basic quantity in stellar astrophysics after the mass, and its absolute scale for cool stars is uncertain by about a hundred kelvin, or two per cent. That uncertainty is invisible in most catalogues, which quote precisions of tens of kelvin, because those are the relative precisions of a method calibrated against one or the other foundation. The two foundations, the infrared flux method and interferometry, are the only ways to fix the absolute scale without assuming a model atmosphere is right, and they disagree.

The resolution will probably come from a third route: spectrophotometry calibrated directly against laboratory standards, which would give the infrared flux scale without relying on Vega, combined with interferometric diameters of fainter dwarfs, which would give the comparison stars the infrared surveys can measure. Until both are in hand, a star’s temperature is good to a hundred kelvin in absolute terms however precisely it is quoted, and the error sits beneath every result that uses one.