The pole-on stars a brightness limit prefers

For most of the twentieth century the star Vega defined what a magnitude of zero meant. Its brightness through each filter anchored the scale against which other stars were measured, and its spectrum was the template for what a normal A-type star looks like. Its lines were narrow, which was taken to mean it rotated slowly.

Around the turn of the millennium, detailed line shapes and then interferometry showed otherwise. Vega is turning at roughly nine-tenths of the rate at which its equator would come loose, and it looks slow only because it is seen almost exactly along its rotation axis, within about five degrees of pole-on. Its pole is near 10,000 kelvin and its equator a couple of thousand degrees cooler. The standard star had been standard partly because of where the Earth happens to sit relative to its axis.

A single rapidly rotating star has a luminosity and temperature that depend on the observer. The question here is what that does to a population. Rapid rotation is common among stars hotter than about 7,000 kelvin, which lack the magnetised wind that spins cooler stars down, so any survey of such stars is full of them. If the bright, pole-on view of each star were systematically favoured, the errors would not scatter around zero; they would add up.

One star from every direction #

The swing has two causes that reinforce each other. From the pole the observer sees the hot polar cap face-on, and it dominates the disc. From the equator the observer sees the cool equatorial belt face-on and the hot poles foreshortened at the limb, and the star also shows a larger projected area, but a cooler one. An observer who assumes the star shines equally in every direction and converts its flux into a luminosity will overestimate it from the pole and underestimate it from the equator.

The inclination of five degrees makes almost no difference against pole-on, because the polar cap looks nearly the same from anywhere within a few degrees of the axis. Most of the change happens between thirty and ninety degrees, which is where most random axes point.

An average that is exactly right #

Before asking what a survey does, it is worth asking what random orientation does on its own, and the answer is a theorem rather than a calculation.

A star’s true luminosity is the total energy it emits per second. Every photon it emits leaves in some direction. An observer at a given inclination intercepts the photons going in their direction and infers a luminosity by multiplying by the full sphere. Averaged over all directions, weighted by how much of the sky each inclination occupies, those inferences add up the photons going in every direction — which is the total. The orientation average of the apparent luminosity is exactly the true luminosity, whatever the star’s shape and however its surface temperature varies.

The figures in this essay are computed by integrating the flux from a gravity-darkened Roche surface in each direction, and the integration is checked against the theorem before any population is built from it: over random orientations the apparent luminosity comes out equal to the true one to better than half a per cent. The residual is the resolution of the surface grid.

So a sample of rapid rotators with random axes and no selection has no luminosity bias at all. Each star is wrong by up to about ten per cent in either direction, and the errors cancel in the mean. The same is true of anything built linearly on the luminosity, which is the sense in which a population average is far more robust than any single star in it.

What a brightness limit does #

Real surveys are not volume-limited. They catalogue every star brighter than some limit, and a star that looks brighter is catalogued out to a greater distance. In a uniform, transparent volume the number of stars above a limit grows as the three-halves power of the luminosity they appear to have, so a star seen pole-on is counted over a larger volume than the identical star seen from its equator — the selection effect that also biases a survey of galaxies towards the intrinsically bright ones.

The shift is visible and small. At nine-tenths of critical, a star looks only 9 per cent brighter pole-on than equator-on at the measured exponent, and the three-halves power turns that into pole-on stars about 9 per cent over-represented. The share of the sample within thirty degrees of pole-on rises by one percentage point.

The exponent matters, since it sets the temperature contrast. Von Zeipel’s theoretical value of 0.25 is an upper bound on the contrast for a star in strict radiative equilibrium, and it is the natural pessimistic case.

Even at the edge of break-up and with the strongest darkening, a sample limited by brightness is still mostly made of stars seen from well away from their poles, simply because there is far more sky at high inclination. The favouring of pole-on stars is real, but it is a gentle reweighting of a distribution that random orientation fixes, not a replacement of it.

How much bias survives #

The quantities that matter to the rest of stellar astronomy are the mean luminosity, temperature and mass that such a sample yields, and they can be computed directly.

These are the numbers to set against the worry that gravity darkening biases a population because pole-on stars are over-represented. The sign of the worry is right, and the size is a fraction of a per cent in luminosity for stars below 0.9 of critical, rising to about one per cent only for a population entirely at break-up and with the strongest darkening theory allows. For comparison, the luminosity of a single such star is uncertain by ten to fifteen per cent from its unknown orientation alone, and the luminosities in most catalogues carry distance and extinction errors of several per cent.

The light a survey counts in #

The figures use bolometric light, the total over all wavelengths, and a real survey counts stars in a band. That changes the size of the effect, and in a direction that depends on the band.

A hotter surface is brighter at every wavelength, but not by the same factor. On the short-wavelength side of a star’s spectral peak the brightness rises very steeply with temperature, because the emission there comes from the exponential tail of the thermal distribution. On the long-wavelength side it rises only in proportion to the temperature. The ratio of two bands is a thermometer for exactly that reason. For a star near 9,000 kelvin, visual light sits close to the peak and grows roughly as the third power of the temperature, a little more gently than the fourth power of the total. Ultraviolet light, well to the short side of the peak, grows as the seventh or eighth power.

So the contrast between a hot pole and a cool equator is weaker in visual light than in total light, and much stronger in the ultraviolet. A survey selected in visual light favours pole-on rapid rotators even less than the figures show. A survey selected in the ultraviolet favours them considerably more, and its pole-on excess and luminosity bias could be several times larger. An infrared survey barely distinguishes pole from equator at all. The bias is not a property of the stars alone but of the stars and the filter together, which is one more way in which a magnitude has to say which light it means.

A survey with no brightness limit #

The cleanest remedy is not a correction but a different kind of survey. A brightness limit introduces the selection; a distance limit does not. Before parallaxes were available for large numbers of stars, a sample complete to a distance could be built only for the nearest few dozen, and every statistically useful sample of A and B stars was limited by brightness.

Parallaxes measured from space have changed that. Samples of hot stars complete out to a few hundred parsecs can now be drawn by distance, and the averaging theorem applies to them without qualification: the mean apparent luminosity of the rapid rotators in such a sample is the mean true luminosity, whatever their orientations. What remains is the scatter, which no choice of sample can remove, and the interstellar dust, which dims stars in the Galactic plane and reintroduces a weak brightness selection at the faint edge of any sample. The population-level bias from gravity darkening, already small, is one of the few systematic effects in stellar astronomy that better data can make disappear outright.

A spread that is a measurement #

That does not make rapid rotation harmless for populations. It moves the problem. The bias that survives averaging is small; the scatter that does not average away, star by star, is large, and it is scatter in exactly the quantities — luminosity and temperature — that place a star on the diagram that sorts stars by mass and age.

The scatter can be turned to use. In a young star cluster all the stars have the same age and composition, so stars of the same mass would sit at one point on the diagram if they did not rotate. Rapid rotators seen at random angles spread along short lines instead, and the width of the spread, for a known distribution of rotation rates, measures how strongly their surfaces are darkened. Clusters a few hundred million years old in the Magellanic Clouds show turnoffs from the main sequence that are much broader than their photometric errors, and in some a main sequence split in two. Rotation — through the darkening of each star’s surface and through its effect on how long each star lives — is the leading explanation for both, and the fraction of stars in each branch has been compared with the fraction of rapid rotators measured spectroscopically. The comparison works only because the orientations are random, so that the spread from darkening is a known function of the rotation rates rather than a free parameter.

The speed a spectrum hides #

Rotation speed is where orientation does the most damage, and the damage is not a selection effect at all. A spectral line is broadened by the component of the equatorial speed along the line of sight, $v\sin i$, and for random axes the average of $\sin i$ is π/4, so a population’s mean projected speed is 79 per cent of its mean true speed before any brightness limit is applied.

The brightness limit again shifts the distribution only slightly towards small projected speeds. What makes $v\sin i$ hard is the random orientation itself, which smears a single true speed into a broad distribution with a long tail to zero. A statistical inversion recovers the distribution of true speeds from the distribution of projected ones, provided the axes are random, and the model says the brightness limit does not upset that proviso by more than a few per cent.

Gravity darkening does damage the measured speed in a second, more serious way that these figures do not include. The fastest-moving parts of a rapid rotator’s surface are at its equator, and the equator is the dimmest part of the star, so it contributes least to the wings of a rotationally broadened line that the speed is read from. Near critical rotation the measured $v\sin i$ therefore underestimates the true projected equatorial speed, and stars that are very close to break-up can be mistaken for stars with a comfortable margin. That is a bias in every star’s measurement, of the same sign, and unlike the selection effect it does not shrink when the sample grows.

One star, a line on the diagram #

The spread along each line is the individual-star error the population average removes. Its direction matters as much as its size. From pole to equator the star moves cooler and fainter together, but along a line much shallower than the main sequence, which at this mass climbs steeply in luminosity for a small change in temperature. Read by its luminosity alone, the star’s orientation looks like a three per cent error in the mass that decides everything else about a star; read by its temperature alone, the same orientation looks like an error of about ten per cent. A three per cent difference in mass is a difference of about seven per cent in main-sequence lifetime, and ten per cent in mass is a quarter of it, so an age inferred from where the star sits depends on which of its two coordinates the inference leans on. A cluster of rapid rotators seen at random angles shows the consequence as a smear across the diagram, one of the reasons the main sequence of a real cluster is a band with width rather than a line.

Vega, and an orientation that was luck #

With these numbers the question about Vega can be answered. The fraction of randomly oriented stars whose axes lie within five degrees of the line of sight is 0.38 per cent, about one in 260. A brightness limit increases the chance of catching such a star by the pole-on excess shown above, about nine per cent for a star like Vega at the measured darkening exponent and a third at the very most. Vega’s orientation is still a one-in-two-hundred accident.

Its prominence has a simpler explanation. It is bright because it is close, 7.7 parsecs away, and intrinsically luminous, and it would have been among the brightest stars in the northern sky from any orientation. Seen equator-on it would have been perhaps fifteen per cent fainter, a sixth of a magnitude, and it would have shown broad, shallow lines that would never have been chosen as the template for a normal star. The selection that made Vega the standard was a selection on narrow lines, and that is a selection on orientation far stronger than any brightness limit — which is why the star at the zero point of the magnitude scale turned out to be one of the least typical A stars in the sky.

The general lesson runs in both directions. Surveys of rapid rotators need not fear a large population bias from orientation, because random axes average the apparent luminosity to the truth and brightness selection barely disturbs that. But any selection that depends on how a star looks from one direction — narrow lines, low measured rotation, a spectrum that resembles a template — can pick out the pole-on stars with an efficiency no brightness limit approaches, and a sample built that way inherits their orientation wholesale.

Still open: do stellar axes point at random? #

Every result above rests on one assumption: that the rotation axes of stars point in random directions. For field stars that is expected, since they formed in many different clouds and have been scattered across the Galaxy. For stars born together in a cluster it is not obvious. The gas they formed from was turbulent but also rotating, and a strong enough coherent rotation could leave a common alignment in their spins. Measurements of stellar inclinations from the splitting of oscillation frequencies — the same stars used to calibrate asteroseismic scaling relations — were reported to show strong spin alignment among the red giants of two old open clusters, and a reanalysis of the same data found the evidence for alignment much weaker once the inclination measurements were treated with more care. Spectroscopic and photometric inclinations in other clusters have given mixed answers. If cluster stars do share their axes, a cluster’s rapid rotators are not a random sample of orientations at all, the exact average in this essay does not apply to it, and the whole cluster’s main sequence could be displaced in one direction by the single inclination of its common axis.