A hundred kelvin under every radius

A star’s effective temperature is the temperature of a blackbody that would radiate the same flux from the same area, and it is the most basic number in stellar astrophysics after the mass. Catalogues quote it to tens of kelvin for millions of stars, and the precisions are real: the colours, spectra and line ratios from which temperatures are read are measured well enough to place one star relative to another to that accuracy. The accuracy is something else. Every relative method is calibrated against a small set of stars whose temperatures were measured absolutely, and the two absolute methods — reading the temperature against the infrared tail of a star’s light, and dividing its measured flux by an interferometrically measured angular size — disagree with each other, for cool stars, by about a hundred kelvin, or two per cent.

A hundred kelvin sounds like a detail beside six thousand. It is not, because temperature enters almost every other derived quantity with a power, and because a scale error is the same for every star measured on the scale. A random error of a hundred kelvin averages away over a sample of a hundred stars. A systematic one does not average at all.

The relative methods are the ones that reach millions of stars. A colour is a thermometer, and a calibrated colour–temperature relation turns a catalogue of colours into a catalogue of temperatures — after the shifts that a companion, a reddening and a metallicity impose on the colour have been removed, each of which can exceed a modern photometer’s precision. A spectrum gives a temperature from the shapes and strengths of its lines. Both are tied to the absolute scale by their calibrators, so both inherit its error whole: the relation is fitted to stars whose temperatures were measured absolutely, and if those are all a hundred kelvin too warm, so is every star the relation is applied to, however precise its colour.

A scale set by a handful of stars #

The absolute scale has always rested on few stars. Its first direct calibration came from an intensity interferometer in the Australian outback, which in the late 1960s and early 1970s measured the angular diameters of thirty-two bright hot stars; combined with their measured fluxes, they gave the first temperatures that depended on no model of a stellar atmosphere. In 1977 the infrared-flux method was introduced, which uses the fact that a star’s infrared flux depends on its temperature only weakly and its total flux strongly, so that their ratio is a thermometer nearly independent of the star’s size; it could be applied to any star with good photometry, and it became the workhorse for cool stars.

The two methods were then each refined for four decades, and in the 2010s, when long-baseline optical interferometers began measuring the diameters of cool dwarfs to a few per cent, the temperatures they implied came out cooler than the infrared-flux scale by tens of kelvin for Sun-like stars and by up to a hundred for the coolest dwarfs. Both methods have identified errors — the limb darkening that converts an interferometer’s fringes into a diameter, the absolute calibration of infrared photometry — and neither has been shown to be the one at fault. A hundred kelvin is the honest width of the disagreement.

The radius goes as the inverse square #

The cleanest propagation is into the radius. A star’s luminosity is its bolometric flux times the square of its distance, both of which are now measured well — the flux from calibrated photometry across the spectrum, the distance from a parallax. The luminosity is also $4\pi R^2\sigma T^4$. So the radius is

$$R = \sqrt{\frac{L}{4\pi\sigma}}\;T^{-2},$$

and an error of two per cent in the temperature is an error of four per cent in the radius, in the opposite direction. For a Sun-like star, a hundred kelvin too hot gives a radius three and a half per cent too small. The precision quoted for radii computed this way, from catalogues of parallaxes and fluxes, is a per cent or two. The scale error is larger than the precision.

This matters for more than the star. A radius computed from flux and temperature is the starting point for the stellar masses that come from evolutionary tracks, for the surface gravities that spectroscopic analyses fix, and for the comparison of radii between stars of different types — the measurement that shows red dwarfs to be larger than their models predict. If the scale is off by a hundred kelvin for the coolest dwarfs, part of that inflation is the scale’s, and the size of the discrepancy between models and stars depends on which of the two absolute methods the temperatures were tied to.

The same figure contains the cure. An interferometer measures a star’s angular diameter directly, and with a parallax that gives a radius that never touches the temperature. Comparing the two radii for the same star measures the temperature scale’s error, and that comparison is how the scale was calibrated: the angular diameter and the bolometric flux together give the temperature directly, $T^4 \propto F/\theta^2$, with no model. But it is possible only for stars large and near enough to resolve, a few hundred of them, and an angle of a few hundredths of an arcsecond is hard to measure to a per cent — the diameter itself depends on a model atmosphere through the limb darkening, which is where much of the disagreement between the two absolute methods comes from.

A planet inherits it, cubed #

A transiting planet’s radius is the star’s radius times the square root of the transit’s depth, so a planet radius is a stellar radius and inherits its error one for one. The planet’s mean density is its mass divided by the cube of its radius, and the cube triples the error.

The planet’s mass enters too, through the star’s mass, which sets how large a planet produces a given radial-velocity signal; with the star’s mass taken from its oscillations, a hotter star is slightly more massive and the planet slightly heavier, and that partly offsets the radius. The net effect is still large: a hundred kelvin moves the planet’s density by twelve or thirteen per cent. The difference between a super-Earth made of rock and iron and one with a thick layer of water is about thirty per cent in density at the same mass. A hundred kelvin of scale error moves every planet a third of the way across that difference, and it moves all of them in the same direction. A population of super-Earths around stars measured on a scale that is a hundred kelvin too cool would look systematically less dense — more water-rich — than it is. The question of how many small planets carry substantial water, which is one of the central questions about planet formation inside and outside the snow line, is posed in exactly the density range where this shift matters, and the answer for the whole population moves with the scale.

The oscillations pass it on unevenly #

Asteroseismology was meant to free stellar masses and radii from the temperature, and it nearly does — unevenly.

The two oscillation frequencies a star’s spectrum gives — the spacing of its overtones, which measures its mean density, and the frequency of its strongest oscillations, which measures its surface gravity — combine through two scaling relations into a mass and a radius. The temperature enters only to convert the gravity, as its square root, and so it reaches the radius weakly, as the square root of the temperature, and the mass more strongly, as the three-halves power. A hundred kelvin moves the seismic radius by under a per cent and the seismic mass by over two and a half.

The asymmetry has a use. A seismic radius and a flux radius respond to the temperature in opposite directions — one up by a per cent, the other down by three and a half — so their ratio for the same star moves by four and a half per cent for a hundred kelvin, more than either. Comparing them across the thousands of stars that space photometry has measured oscillations for is a second way of measuring the temperature scale, and it reaches stars far too small and distant for an interferometer. The comparison has its own systematic errors — the scaling relations are themselves calibrated on the Sun and corrected empirically — but they are unrelated to the interferometric ones, and the two calibrations of the scale, if they agree, constrain it better than either.

An age read from the wrong place on the diagram #

A main-sequence star’s age is estimated from where it sits in the plane of temperature and luminosity, against model tracks. Shift its temperature and it sits on a different track.

On the main sequence a hotter star is a more massive one, and a more massive star burns its hydrogen faster, so a temperature that is too high places the star on a track for a heavier star with a shorter life. In the crude homology of the figure — temperature rising as the square root of the mass, luminosity as its fourth power — a hundred kelvin is a tenth of the age. Real isochrones are flatter in some places and steeper in others, and near the main-sequence turnoff, where ages are most precise, the dependence is stronger still.

The figure also shows why the cool stars suffer most. A fixed error in kelvin is a larger fraction of a cool star’s temperature, and the cool stars’ ages are the ones used to study the slow processes — the spin-down of stars, the evolution of their activity, the ages of the planets around them. A hundred kelvin applied to the whole scale ages the cool stars by more than the hot ones, and a trend with temperature in a sample’s ages could be the scale’s error rather than the stars’ history.

Two kinds of iron disagree about the temperature #

The most sensitive of all the propagations runs into chemistry, and it is also the one that gives spectroscopists their own thermometer.

In a Sun-like photosphere iron is almost entirely ionised. The neutral atoms that make the Fe I lines are a minority, and the Saha equation makes their share fall steeply as the temperature rises: a hotter model atmosphere holds fewer neutral iron atoms at the same total abundance, so to reproduce the same observed line it needs more iron. The inferred abundance rises by about eight hundredths of a dex for a hundred kelvin, a fifth more iron. Ionised iron, the majority, barely changes in number; its lines respond only through the population of their excited levels, which moves the other way.

So the two kinds of iron line disagree about the abundance unless the temperature is right, and spectroscopists use exactly that. A temperature is chosen so that lines of every excitation potential give the same abundance — the excitation balance — and so that neutral and ionised lines agree — the ionisation balance. Both are thermometers internal to the spectrum, needing no photometry, and they give a third determination of the temperature scale whose errors come from the model atmosphere rather than from fluxes or diameters. Their weakness is that neutral iron in a real photosphere is not in the local equilibrium the Saha equation assumes; it is over-ionised by ultraviolet radiation from deeper layers, by an amount that depends on the star and that the simplest models omit. That correction is itself a few tens of kelvin, and it is one of the places where an atom’s minority status makes it a sensitive and treacherous messenger.

The one star without the problem #

One star is exempt, and it is the reason relative methods work at all. The Sun’s effective temperature follows from its total irradiance at the Earth, measured by radiometers in orbit, and its radius, measured from its disc, and both are known so well that the Sun’s temperature is fixed to within a few kelvin: 5,772 by convention. A star analysed relative to the Sun — its spectrum compared line by line with the Sun’s, its colours with solar colours — inherits the Sun’s precision rather than the scale’s uncertainty, as long as it is similar enough to the Sun that the comparison is nearly exact. That is why the most precise abundances and ages in stellar astrophysics are for solar twins, stars within a hundred kelvin of the Sun, and why their precision falls steeply as stars become less Sun-like.

So the scale’s uncertainty is not uniform. It is a few kelvin at the Sun and grows with distance from it in temperature, gravity and composition — tens of kelvin for typical dwarfs and a hundred or more for the coolest dwarfs and the giants. The figures above, drawn for a Sun-like star, overstate the effect there and understate it for the stars that most planets orbit, which are cooler, smaller and redder than the Sun.

What the scale does not touch #

There is a quantity the scale leaves alone, and it is the one that matters most for whether a planet is habitable. A planet’s equilibrium temperature is set by the flux it receives, which is the star’s luminosity divided by the square of the planet’s distance. The luminosity comes from the bolometric flux and the parallax, the distance from the orbital period and the star’s mass, and the temperature enters neither, except through the star’s mass, whose influence on the planet’s distance is weakened by a cube root. A hundred kelvin of scale error moves a planet’s equilibrium temperature by a fraction of a kelvin.

The scale shrinks or swells planets without warming or cooling them. Its effect on the census of habitable planets is therefore through their sizes and compositions — whether a planet in the right place is a rocky one — and not through which planets are in the right place. That division is a reminder that a systematic error has a direction in the space of conclusions as well as a size, and that some of what a survey concludes is protected from it entirely.

What the propagation leaves out #

The propagations here are first-order and independent. In practice the derived quantities are computed together — a spectroscopic analysis fits temperature, gravity and abundance at once, and an error in one shifts the others in a correlated way — and some of the effects shown partly cancel. A spectroscopic temperature that is too high also gives a gravity and metallicity that are too high, and the isochrone age then responds to all three. The sizes in the figures are what each quantity does on its own; the combined response of a given analysis can be smaller or larger.

The scale’s error also need not be a constant offset. The disagreement between the infrared-flux and interferometric temperatures is a few tens of kelvin for Sun-like stars and grows toward the coolest dwarfs and the giants, so the error is a function of temperature, and its effect on any trend across a sample depends on its slope as well as its size.

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

The absolute temperature scale of cool stars rests on a few hundred stars measured by two methods that disagree by about a hundred kelvin, and the disagreement propagates into everything else: stellar radii, planet densities, ages, abundances. The precisions quoted in catalogues are smaller than the scale’s uncertainty, because they measure how well stars are placed relative to one another, which is a different and easier thing.

What would fix it is more stars measured absolutely, and better agreement between the methods. Interferometers with longer baselines and the next generation of space-based spectrophotometry — absolute fluxes across the whole spectrum, calibrated against laboratory standards — attack the two halves of the direct method. Asteroseismology supplies an independent route for stars too faint for interferometry, and the ionisation balance of iron, corrected for its departures from equilibrium, a third. Three routes with unrelated errors could pin the scale to a few tens of kelvin. Until they do, every planet’s density and every cool star’s age carries an uncertainty that is quoted nowhere in its catalogue entry, because it is the same for every entry — the one kind of error a catalogue’s own statistics can never reveal.