Astronomy & Celestial Mechanics
the sky, computed
Series

Stellar colour — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Blackbody curves at 3000, 5800, 10000 K. Thermal emission against wavelength, each curve scaled to its own peak so the shift can be seen on one plot. The peak moves to shorter wavelengths as the temperature rises, which is why colour is a thermometer.

    Colour is a thermometer, and it reads across the galaxy

    A star's colour gives its surface temperature, from two brightness measurements and no other information. It is the cheapest useful measurement in astronomy.

    part 1 · starlight
  2. The classical law gives every star the same colour. Planck's law and the Rayleigh–Jeans law at 3,000 K, 5,772 K, 10,000 K, both normalised to the 5,772 K Planck peak, on logarithmic axes. The classical law comes from counting standing waves in a cavity — 8πλ⁻⁴ of them per unit volume per unit wavelength — and giving each the kT that equipartition allows. It agrees with Planck's where the modes are crowded and each holds much less than kT, and it runs away where they are not: at 80 nm it exceeds the real spectrum by a factor of 1.1·10¹² while agreeing to within 55.9 per cent at 3000 nm, and the integral under it does not converge at all. That is the ultraviolet catastrophe, and it is the half everybody knows. The quieter half is that in 2ckT/λ⁴ the temperature is an overall factor, so the ratio of the law at two wavelengths is independent of it: the B − V index of a classical star comes out identical at 3,000 K, 5,772 K, 10,000 K — the same -0.968 magnitudes, to the last digit the quadrature carries — while Planck's law spreads the same three stars over 1.47 magnitudes. A classical universe has stars of every brightness and one colour. Colour is a thermometer only because the exponential in the denominator does not cancel, and the quantum of energy that put it there was fitted to this shape before anybody knew what it meant.

    The classical law gives every star one colour

    The ultraviolet catastrophe is the famous half. The quieter half is that in 2ckT/λ⁴ the temperature is an overall factor, so the ratio of the law at two wavelengths has no temperature in it — and a classical universe has stars of every brightness and one colour.

    part 2 · starlight
  3. Two laws that are the peak and the area of one curve. Planck curves at 3,000 K, 5,772 K, 9,600 K on logarithmic axes, with each peak marked. Normalised by its own peak, Planck's law is a universal function of x = hc/λkT, and three exponents follow from that alone and are fitted here off the drawn curves rather than quoted: the peak wavelength goes as T^-1.000, which is Wien's displacement law with a constant of 2.897772 mm K obtained by solving 5(1 − e^(−x)) = x for x = 4.965114; the peak HEIGHT goes as T^5.000; and the area goes as T^4.000. The third is the first two multiplied. A peak five powers high on a curve one power narrow encloses four powers of area, so Stefan–Boltzmann is not an independent fact about radiation — it is Wien's law and the height of the peak, taken together. That is also why the two are worth having at once. A colour gives the temperature and a flux gives the luminosity, and L = 4πR²σT⁴ then gives a radius: for the Sun at 5,772 K receiving 1361 W/m² at 1.000 AU, the arithmetic returns 6.957·10⁸ m against a measured 6.957·10⁸. A thermometer alone cannot do that, because a colour is a ratio and a ratio has no size in it; the radius comes from the one law that is an absolute quantity rather than a shape.

    Two laws that are one curve read twice

    Wien's displacement and Stefan–Boltzmann are the peak and the integral of the same function. The peak is five powers high and one power narrow, so the area is four — and the fourth power that is taught as a separate law is the first two multiplied.

    part 3 · starlight
  4. 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.

    part 4 · starlight
  5. How steeply the ratio of all the light to one band rises with temperature. The exponent n in (bolometric flux ÷ band flux) ∝ Tⁿ, for a blackbody, against temperature, for bands at V (0.55 µm), J (1.25 µm), H (1.65 µm), K (2.2 µm). The bolometric flux rises as T⁴; the flux in a band rises as T on the long-wavelength side of the Planck peak and much faster on the short side, so n is the difference. For the Sun's temperature it is 2.33 in K, 1.70 in J and −0.56 in V. An error of one per cent in the modelled band flux becomes an error of 1/n per cent in the temperature: 0.43 per cent, 25 K, in K. In the visible the ratio barely depends on temperature and carries no information about it; in the infrared it does, and the temperature is read with its model error divided.

    A temperature read against the light's long tail

    A star's total light depends on its temperature to the fourth power; its light in the infrared depends on it barely more than linearly, because the infrared lies on the long, flat tail of the Planck curve. The ratio of the two cancels the star's size and distance and leaves a steep function of temperature — and the only model it needs is the infrared surface flux, whose errors are divided by two and a half on the way to the answer. That is the infrared flux method, one of the two foundations of the stellar temperature scale, and the other foundation disagrees with it by up to a hundred and fifty kelvin.

    part 5 · starlight
  6. What a hundred kelvin moves, for a star like the Sun. The change in eight quantities derived for a Sun-like star and its planet when every temperature is raised by 100 K — 1.7% — as the uncertainty in the absolute temperature scale allows, against the precision each is quoted to (grey bars). The star's radius from its flux and parallax falls by 3.4%, because radius goes as the inverse square of temperature; the planet's radius follows, and its density rises by +12.8%. The oscillation-based mass rises by +2.6%, the main-sequence mass by +3.5% and the age falls by 9.8%. The iron abundance read from neutral iron lines rises by 0.08 dex, +20.7%. In every case the shift is comparable to or larger than the precision quoted, and in every case it moves all stars of a sample together.

    A hundred kelvin under every radius

    The temperatures of cool stars are quoted to tens of kelvin, and the scale they are all measured on is uncertain by about a hundred. That hundred is under almost everything else: a star's radius moves by twice its fractional error, a transiting planet's density by six times, a main-sequence age by ten per cent and an iron abundance by a fifth. And because a scale error moves every star of a sample the same way, no amount of averaging removes it.

    part 7 · starlight

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