The cancellation that leaves with the hot stars

A star too far away for any interferometer to resolve still has an angular size, and its colour gives it. The method rests on one fact. How bright a unit of a star’s surface is depends almost entirely on the surface’s temperature, and the colour $V - K$, the ratio of visible to infrared light, measures that temperature. Calibrate the relation between colour and surface brightness on nearby stars whose diameters can be measured. Then any star’s angle follows from its magnitude and colour. Combined with the true radius of an eclipsing binary’s components, the angle gives a distance. Applied to late-type giants in the Large Magellanic Cloud, the method has reached about one per cent.

Late-type giants are faint at the distance of the next large galaxies. In Andromeda and Triangulum, at three-quarters of a megaparsec and beyond, the eclipsing binaries bright enough for spectroscopy of both components are hot, massive pairs: O and B stars at 20,000 to 40,000 kelvin. The method has to follow them there. This essay asks what the Planck function alone does to it on the way. The answer is that both properties that made it precise for cool giants weaken, and one of them reverses.

The relation turns towards the vertical #

The quantity calibrated is $F_V$, a combination of the V magnitude and the angular diameter that depends only on the surface:

$$F_V = 4.2207 - 0.1\,V_0 - 0.5\log\theta.$$

For late-type stars it is almost a straight line in colour, $F_V = 3.962 - 0.138\,(V-K)_0$, and that line was what gave giants their one-per-cent angles. The figure carries the relation for blackbodies, pinned to the Sun, from cool giants up to 40,000 kelvin.

Among cool stars the blackbody curve runs close to the empirical line. That is why the line exists: the colour and the surface brightness both follow temperature, and they do so at nearly proportional rates. Above about 10,000 kelvin the curve bends upward. Each doubling of temperature still raises $F_V$ by about the same amount, but buys less and less colour. From 5,000 to 10,000 kelvin $V-K$ changes by 1.75 magnitudes. From 10,000 to 40,000, a factor of four in temperature, it changes by 1.05. An infinitely hot blackbody would have $V - K = -1.03$, only 0.28 magnitudes bluer than one at 40,000 kelvin.

The reason is the part of Planck’s law that the classical one keeps. Far on the long-wavelength side of the spectrum’s peak, a blackbody’s brightness is $2ckT/\lambda^4$, with temperature as an overall factor. The ratio of two such brightnesses, which is what a colour is, contains no temperature at all. The K band at 2.2 microns is on that tail for every star the relation has ever been applied to. For a cool giant the V band at 0.55 microns is on the other side of the peak, and $V-K$ works as a thermometer because one band feels the exponential and the other does not. Once a star is hot enough to move its peak into the ultraviolet, the V band also slides onto the tail, and the colour stops changing.

The figure measures how much temperature information each band carries. It plots the logarithmic sensitivity of surface brightness to temperature, which approaches one on the Rayleigh–Jeans tail, where brightness is simply proportional to temperature. A colour measures temperature through the difference between its two bands’ sensitivities. For $V - K$ that difference is 3.47 at 5,000 kelvin and 0.39 at 30,000, a ninefold loss of leverage. The ultraviolet keeps a steep response, because a hot star’s peak lies there. A colour of the form UV minus V or UV minus K would measure a hot star’s temperature about as well as $V-K$ measures a cool giant’s. Such colours can only be observed from above the atmosphere.

A colour error costs more angle #

With the relation steepening, every error in the colour becomes a larger error in the angle. At fixed $V$, the change in $\log\theta$ for a magnitude of $V - K$ is twice the local slope of the relation. For the late-type empirical line that is 0.275. For blackbodies it is 0.30 at 5,000 kelvin, 0.39 at 10,000, 0.58 at 20,000 and 0.78 at 30,000.

That factor of 2.6 between a cool giant and a 30,000-kelvin star applies directly to every photometric error. The giants in the Magellanic Clouds were measured in the K band to about two hundredths of a magnitude. That precision costs them 1.3 per cent in angle and in distance, which is already the largest term in a single binary’s budget. A hot star measured equally well loses 3.6 per cent. The hot stars that matter are in galaxies fifteen times further away, so they are fainter in K by six magnitudes for the same luminosity. The K band is where a hot star is faintest relative to its visible light. Infrared photometry of individual B stars in Andromeda’s crowded discs does not reach two hundredths.

At five hundredths of a magnitude, the precision such a star can realistically be given, a 30,000-kelvin binary’s angle is uncertain by over nine per cent from colour alone. A single binary in Andromeda measured this way is a ten-per-cent distance before the relation’s own calibration enters. Averaging over many binaries reduces the photometric part as the square root of their number. It does nothing for the calibration, and the calibration is the hard part for hot stars.

The cancellation that leaves #

The single most useful property of the method for cool stars was its indifference to dust. An extinction $A_V$ left uncorrected moves the inferred angle in two directions at once. It makes the star fainter, so at fixed colour the inferred angle shrinks: $\log\theta$ falls by 0.2 per magnitude. It makes the star redder, since dust dims the K band only about a ninth as much as V, so $V - K$ grows by 0.886 magnitudes per magnitude of $A_V$. A redder star is assigned a cooler, dimmer surface and so a larger disc. For the late-type relation the second effect is $0.275 \times 0.886 = 0.244$ in $\log\theta$, and the two nearly cancel. A tenth of a magnitude of unknown dust costs one per cent. A standard candle measured in V would lose 4.7 per cent.

The reddening term is the colour lever times 0.886, and the lever is what grows for hot stars. The dimming term stays at 0.2. So the balance that made the method dust-proof shifts steadily as stars get hotter, and nothing restores it.

Through the blackbody relation’s local slope, the net effect of a tenth of a magnitude of dust is close to a per cent for cool giants, as the empirical relation gives. It grows smoothly with temperature, equals the error of a V-band standard candle near 13,500 kelvin, and at 30,000 kelvin is +11.9 per cent, two and a half times worse than the candle. For the hottest stars, the surface-brightness method is more sensitive to dust than the cruder method it was meant to replace. The error also has the opposite sign: unaccounted dust makes a standard candle look too far away and makes the angle method’s star look too close.

The geometry of the failure is visible in the first figure. The reddening vector in the plane of $F_V$ against $V - K$ has a fixed direction: 0.886 magnitudes to the right for each magnitude of $A_V$, and a fixed amount down in $F_V$ through the dimming. For cool stars the relation runs almost parallel to it, so a reddened star slides along the relation instead of leaving it. For hot stars the relation has turned steeply upward, the reddening vector cuts across it, and a reddened star lands on a different, cooler part of the curve. The dust that was nearly invisible to the cool-star method is fully visible to the hot one.

This matters more because hot stars sit where the dust is. The massive binaries in a spiral galaxy are young, close to the clouds they formed in and to the dust lanes of the arms. Their extinction is larger and patchier than that of the old giants in the Magellanic Clouds, and it has to be measured star by star. The usual way is from the colours, comparing observed colours with those expected for the star’s temperature. That temperature comes from the spectrum, so the extinction correction no longer cancels: it inherits the spectroscopic temperature’s errors directly.

How much dust a hot binary is behind #

The figure is per tenth of a magnitude left uncorrected, and the real question is how many tenths are typical. The old giants of the Large Magellanic Cloud that set the one-per-cent standard lie behind a few tenths of a magnitude of extinction, and most of it is foreground dust in the Milky Way that varies slowly across the sky. A massive binary in the disc of Andromeda is behind that foreground plus the dust of its own spiral arm, typically half a magnitude to a magnitude or more in V. It also varies from star to star on scales smaller than the separation between neighbouring binaries.

The correction is made from the star’s own colours. The observed colour minus the colour expected for the spectroscopic temperature gives the reddening, and the reddening times the ratio of total to selective extinction, $R_V$, gives $A_V$. Each step carries an error. A reddening known to three hundredths of a magnitude contributes a tenth of a magnitude of $A_V$. And $R_V$ is not a constant: it is set by the sizes of the grains, about 3.1 on average in the Milky Way, and observed to range from about 2.5 to 5 along different sightlines. For a star behind a magnitude of extinction, an error of 0.3 in $R_V$ is another tenth of a magnitude in $A_V$.

So a hot binary carries two or three tenths of a magnitude of residual extinction uncertainty before anything else is considered, and at 30,000 kelvin each tenth costs about twelve per cent through the colour relation. The figures do not make the method impossible for hot stars. They make it worse than the alternatives at the one task it was chosen for. Its value for cool stars was a distance that dust barely touched, and for hot stars dust is the largest term in the error budget.

Writing the relation in temperature instead #

There is an exact way round the colour, and the model-atmosphere distances take it. The surface-brightness parameter is not fundamentally a function of colour; it is

$$F_V = \log T_{\rm eff} + 0.1\,BC_V,$$

the logarithm of the effective temperature plus a tenth of the bolometric correction in V. A colour enters only because it is the cheap way to estimate the temperature. If the temperature is taken from the spectrum, from the ratios of lines of different ionisation stages of helium, silicon or nitrogen, the colour and its weak lever drop out. Dust then enters only through the V magnitude, with the plain 0.2 coefficient of a standard candle.

Here the Rayleigh–Jeans tail works for the method instead of against it. The property that ruins $V - K$ as a thermometer, that the visible brightness of a hot star is nearly proportional to its temperature, also makes the surface brightness forgiving of an error in that temperature. Along the blackbody’s V-band curve at 30,000 kelvin, a one-per-cent error in temperature moves $F_V$ by 0.0016 and the angle by 0.75 per cent; a three-per-cent spectroscopic temperature costs about two per cent. For a cool giant the same temperature error would cost several times as much, because there the V band sits on the steep Wien side. The weak lever that spoils the colour spoils the temperature error too, and in that direction it helps.

What the temperature route cannot escape is the zero point. Writing $F_V$ in temperature requires knowing how bright, in absolute terms, a 30,000-kelvin surface is in the V band. That is the bolometric correction, three magnitudes and more for such stars, because most of their light is in the ultraviolet where V does not look. It comes from model atmospheres. The colour relation was valued because an interferometer measured its zero point directly. The temperature relation’s zero point is computed, and it is the same for every star, so no number of binaries averages it away. Dust, meanwhile, enters only through V, with a standard candle’s coefficient of 0.2. That is better than the colour relation’s at 30,000 kelvin but none of the cancellation the cool giants enjoyed.

What the hot-star distances have done instead #

The eclipsing-binary distances published for Andromeda and Triangulum in the first decade of this century did not use a colour relation at all. They took each component’s surface brightness from a model atmosphere at the temperature and gravity given by its spectrum, and fitted the extinction together with the light curves. Their quoted uncertainties are five or six per cent per galaxy. Both the model atmosphere and the extinction law sit inside that number. Neither is reduced by observing more binaries, since the same models and the same law serve every star.

That is a reasonable choice given the figures here. A model atmosphere computes the surface brightness in every band from the physics. It does not need the colour to measure temperature, because the spectrum’s lines already do. What it gives up is the one advantage the empirical relation had: independence from the model atmospheres whose limb darkening and line blanketing it was calibrated to sidestep. For hot stars there is a second cost. Their atmospheres are less settled than the Sun’s: winds, departures from local thermodynamic equilibrium, and for the fastest rotators a surface whose temperature varies from pole to equator. A one-per-cent hot-star distance would need a model atmosphere correct to that level in absolute surface brightness, and nothing yet shows that one is.

The calibration nobody has finished #

The empirical route would need interferometric diameters of nearby hot stars, and those are hard to get for a reason the figures do not show. B stars are rare, so the nearest are hundreds of parsecs away. They are also small, a few solar radii on the main sequence, so their discs are a fraction of a milliarcsecond: near or below what the longest optical baselines resolve. The calibrations published so far for early-type stars rest on a few dozen objects, many of them rapid rotators whose shape and temperature vary across the disc. Their scatter is several times the one per cent achieved for giants.

The K band adds a problem of its own. Many B stars, the Be stars, carry discs of ionised gas that radiate free–free emission, strongest in the infrared. A Be star is too bright in K for its photosphere, too red in $V - K$, and is assigned too large an angle. Winds from O stars do the same thing more weakly. Every hot star whose K light includes some gas besides its photosphere moves the colour in the direction the steep relation magnifies. The K band chosen for cool stars because it sees through dust is, for hot stars, the band most contaminated by everything around them.

What the Planck function leaves out #

The figures use blackbodies through narrow bands, and real hot stars differ from both. The Balmer jump, the drop in brightness below 0.365 microns caused by hydrogen absorbing from its second level, makes the U band and the near ultraviolet behave very differently from a blackbody between about 8,000 and 20,000 kelvin. Line blanketing redistributes ultraviolet flux into the visible. Broad filters smear the sensitivities over their passbands. The empirical relation’s slope for late-type stars, 0.275, is a little shallower than the blackbody’s 0.30 at 5,000 kelvin for exactly these reasons.

None of this can restore the leverage at the hot end. Every atmosphere, however blanketed, has a Rayleigh–Jeans tail at long enough wavelength, and at 30,000 kelvin both V and K are on it. The atmospheres change the numbers on the curves by tens of per cent. They do not change which way the curves go.

Still open: whether the method reaches past the Local Group #

The surface-brightness method’s best property, a one-per-cent geometric distance almost immune to dust, belongs to cool stars and does not carry over to hot ones. Beyond the Magellanic Clouds the bright binaries are hot. For them $V - K$ is a weak thermometer, colour errors are magnified two to three times, and the dust cancellation has turned into a dust sensitivity worse than a standard candle’s. The distances to Andromeda and Triangulum that used such binaries replaced the colour relation with model atmospheres and stand at five or six per cent.

The distances already obtained can at least be checked against each other. Andromeda’s distance from its Cepheids, from the tip of its red giant branch and from its eclipsing binaries agree within their errors, as four routes to one galaxy should if none has a hidden systematic larger than its quoted one. Agreement at five per cent does not show that any of them could reach one.

Two routes could change that. One is ultraviolet photometry from space, which would give hot stars a colour with the leverage $V - K$ gives giants. It would bring its own extinction problem, since dust absorbs most strongly in the ultraviolet and with a law that varies from galaxy to galaxy. The other is the empirical calibration that has not yet been made: interferometric diameters of a few dozen non-rotating, non-emission B stars to one per cent. That would let the method use a measured relation instead of a computed one, and it needs baselines long enough to resolve fractions of a milliarcsecond. Whether either reaches the precision that would make a hot binary in Andromeda as good a yardstick as a cool giant in the Large Magellanic Cloud, and so tie the chain of distances to the next spiral galaxy with the same rigour, is still open.