Astronomy & Celestial Mechanics
the sky, computed
Concept

Reddening — where it appears

The preferential removal of blue light by dust, detectable without knowing any distance, which is the only reason extinction can be corrected. It is why a distance derived from an uncorrected magnitude is always too large, and the correction is made in a colour rather than in a brightness because a colour needs no distance.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

The extinction law, for three kinds of dust. How much of a star's light dust removes, against inverse wavelength, normalised to one at the V band. Blue light is extinguished more than red, which is why reddening and extinction are the same measurement — and the hump at 4.6 inverse microns is a feature of the grains themselves, present on almost every sight line and still without an agreed carrier. Larger grains give a flatter law and a larger R_V.

Dust makes everything look further away

A star behind dust is fainter, so a distance taken from its brightness comes out too large. The same dust also makes it redder, and the reddening is measurable where the dimming is not — which is the only reason the correction can be applied at all.

starlight · Extinction B − V against temperature, computed and measured. B − V against effective temperature. The curve is the colour index of a blackbody, obtained by integrating Planck's law against B and V response functions and subtracting a constant so that the index is exactly zero at 9600 K — the convention that an A0V star has every colour index zero, which is a choice and not a measurement. The 15 points are the main sequence as it is actually measured, and they do not lie on the curve: at the Sun's temperature the blackbody gives 0.446 where the sky gives 0.653, and at M0V 0.995 against 1.40. The model is too blue almost everywhere, and least wrong near 9600 K — which is why the zero point is put where it is. A colour index is a difference of two magnitudes, so it is a difference of two integrals, and changing either filter changes the number.

A magnitude has to say which light

The same star is a different magnitude in every filter, and a colour index is a difference of two conventions rather than a property of the star. Both are integrals of a spectrum against a piece of glass, and the zero point is a choice somebody made in 1953.

starlight · Photometric systems Three shifts larger than the error bar, and two that cancel. The blackbody B − V index against temperature, with three systematic shifts marked at 5,772 K. All three are computed from the same Planck integrals the relation itself is: a 3,800 K companion contributing 25 per cent of the V light reddens the index by 0.093 magnitudes, because the companion is relatively brighter in the redder band; 0.35 magnitudes of visual extinction at a total-to-selective ratio of 3.1 adds 0.113 directly, since the colour excess is the extinction divided by that ratio; and a metallicity of -1 dex subtracts 0.200, because the metal lines that eat the B band are the ones a metal-poor star is short of. Read as temperatures, the same star comes back at 5331 K, 5243 K and 7021 K against a true 5,772. Two of the three have opposite signs, and that is the difficulty rather than the relief. All three together shift the index by 0.005 magnitudes against 0.405 of combined magnitude — very nearly nothing, because the opposing pair removes almost all of it — and return 5744 K, which is 28 K from the truth by cancellation and not by accuracy. A reddened metal-poor star and an unreddened solar-metallicity one are the same point on this curve, and no amount of photometry in two bands separates them. What does separate them is a third band, or a spectrum — which is where the cheapest measurement in astronomy stops being cheap.

Three shifts larger than the error bar, and two that cancel

An unresolved companion, a reddening and a metallicity each move a colour index by more than any modern photometer's precision. Two of them move it in opposite directions, so the three together can return the right temperature by cancellation rather than by accuracy.

starlight · Stellar colour The extinction curve of a mixture, and of single sizes. Extinction relative to the V band against inverse wavelength, from the infrared at 0.3 inverse microns to the near ultraviolet at 3.3. The thick grey curve is the observed Galactic average, the Cardelli–Clayton–Mathis law at Rᵥ = 3.1. The coloured curves are computed: MRN, 0.005–0.25 µm, Rᵥ = 2.4; all grains 0.05 µm, Rᵥ = 1.8; all grains 0.3 µm, Rᵥ = 18.4. Grains of any single size make a curve with a knee: steep where the wavelength is longer than the grain, flat where it is shorter. Only a spread of sizes, with many more small grains than large ones, makes the nearly straight rise with inverse wavelength that is observed, and the first model follows the observed curve to an r.m.s. of 0.19 in these units. The ultraviolet bump beyond this range needs a separate population of small carbonaceous grains and is not modelled.

A size read from the slope of a colour

Interstellar dust dims blue light more than red, and the way the dimming rises toward the blue is nearly a straight line in inverse wavelength. No grain of any single size makes that line. A grain only removes light selectively when it is comparable in size to the wavelength, so the extinction curve read across wavelength is the population of grains read across size — and it says there are vastly more small grains than large ones, in the same proportion that grinding rocks together produces.

starlight · Extinction The cancellation that protects the method, and where it fails. The error in distance from 0.1 magnitudes of extinction left uncorrected, against temperature, for the angle read from V and V − K through the blackbody relation's local slope, and for a standard candle in V (dashed, −4.7 per cent). For late-type stars the dimming and the reddening nearly cancel — the empirical relation gives +1.0 per cent. For hotter stars the reddening term grows and nothing offsets it: the error matches the standard candle's near 13,500 K and reaches +7.5 per cent at 20,000 K and +11.9 at 30,000 K, worse than the candle the method was meant to improve on.

The cancellation that leaves with the hot stars

The angular size of a cool giant can be read from its colour, and dust barely spoils the answer: the dimming and the reddening dust causes push in opposite directions and nearly cancel. Carried to hot stars the method gets weaker on both counts. Above about 10,000 K the colour V − K stops tracking temperature, a colour error costs more angle, and the reddening outweighs the dimming. Near 13,500 K leaving the dust uncorrected costs as much as it would a standard candle.

starlight · Angular diameter One population, two period–luminosity relations. The period–luminosity relation of 400 synthetic Cepheids spread across the instability strip, each behind a random amount of dust up to E(B − V) = 0.6: absolute magnitude in V (upper scatter) and in the Wesenheit index W = V − 2.44(V − I) (lower). In V the scatter about the fitted line is 0.57 magnitudes — 0.20 from the strip's width alone, the rest from the dust. In W it is 0.07: the dust is gone by construction, and most of the strip's width has gone with it, because a star that is hotter at a given period is bluer by about as much as dust would make it redder. That leftover 0.07 magnitudes is 3 per cent in distance per star.

A scatter that points the way dust does

A Cepheid's period fixes its luminosity only to within the width of the instability strip, and at a given period the hotter stars are the brighter ones. Dust moves stars in the colour–magnitude diagram along a line too, and the two lines point nearly the same way. A magnitude built to cancel the dust — the Wesenheit index — therefore cancels most of the strip's width as well, which is why the tightest period–luminosity relations are the ones that ignore the extinction law's own advice.

stars · Variable stars

Named alongside it

The objects these essays reach for when they reach for this one.

Colour indexBlackbodyInterstellar extinctionColour excessEffective temperaturePhotometric systemPhotometryAngular diameterApparent magnitudeBinary starBolometric correctionCepheid variable

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