Astronomy & Celestial Mechanics
the sky, computed
Concept

Colour index — where it appears

The difference between a source's magnitudes in two passbands, which for a star is a temperature. It is a difference of two magnitudes and therefore needs no distance, which is why reddening can be corrected without knowing how far away anything is.

Named by 11 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

starlight · Stellar colour The distance modulus. The difference between apparent and absolute magnitude against distance, on a logarithmic distance axis. It is a straight line of slope five per decade, passing through zero at ten parsecs — the definition of the absolute magnitude. Reading a distance off it requires the absolute magnitude, which is never measured and always inferred.

A brightness is a distance only if something is known

The inverse square law turns a brightness into a distance in one line. The line contains a quantity that has never been measured for any object outside the solar system.

starlight · Magnitudes 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 The colour of the zenith at twilight, with and without ozone. The colour of the zenith sky relative to sunlight — the 450 nm brightness against the 650 nm brightness, in magnitudes, bluer upward — against the Sun's depression, computed by single scattering with a 300 Dobson-unit ozone layer and again with none. Scattering alone favours blue by λ⁻⁴, which would make the sky 1.60 magnitudes bluer than sunlight if nothing were removed on the way; that is the dotted line. But after sunset every ray has travelled a long grazing path, and Rayleigh scattering removes blue from that path faster than red, so the two effects fight. Without ozone they very nearly cancel: the zenith is 0.01 magnitudes from sunlight's own colour at sunset and 0.06 at 6° — a pale, colourless sky — and only turns bluer, −0.13 at 10°, once the lit layer has climbed above most of the air the grazing ray used to cross. With ozone the zenith is −0.34 at sunset, −0.67 at 6° and −0.76 at 10°: bluer than without by 0.72 magnitudes at 6°, because the grazing ray also crosses the ozone layer near its tangent point, and ozone's Chappuis band absorbs orange and red rather than blue. The ozone cross-sections are approximate, and the conclusion does not depend on them to better than a factor of two.

Ozone keeps the twilight zenith blue

After sunset the sky overhead turns a deep blue, and scattering alone cannot explain it. The light that reaches the zenith has first grazed hundreds of kilometres of air, which strips blue out of the sunlight as fast as scattering puts it back, and the two very nearly cancel. What tips the balance is a gas that makes up a few parts in ten million of the atmosphere and absorbs the orange and red end of the spectrum.

sky · Twilight 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.

starlight · Stellar colour 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.

starlight · Stellar colour 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 A colour that predicts an angle. The surface-brightness parameter Fᵥ — a combination of the V magnitude and the angular diameter that depends only on how bright each unit of the star's surface is — against the colour V − K, the ratio of the star's light at 0.55 and 2.2 microns. The line is the relation log θ = 0.5175 + 0.2753 (V − K) − 0.2 V, fitted to interferometric diameters of dwarfs and subgiants: Fᵥ = 3.962 − 0.1376 (V − K). The dots are 40 calibrating stars drawn about it with a 1 per cent scatter in angular diameter, the precision of a good interferometric measurement; refitting them returns a slope of −0.1374 and a scatter of 0.0023 in Fᵥ, which is 1.1 per cent in angle. The dashed curve is what blackbodies of 3,500 to 9,000 K would give, pinned to the Sun: it departs from the stars' line by up to 0.035 over the calibrated range, because real atmospheres have molecular bands and limb darkening a blackbody lacks — which is why the relation is calibrated on measured stars rather than derived. Given a star's V and V − K, the line returns its angular diameter without resolving it.

An angle too small to measure, read from a colour

A giant star in the Large Magellanic Cloud subtends two microarcseconds, a hundred times below anything an interferometer resolves. Its angle is read instead from its colour, through a relation calibrated on nearby stars that can be resolved — and because dust dims a star and reddens it by amounts that nearly cancel in that relation, the distance it gives hardly depends on the dust at all.

starlight · Angular diameter A white dwarf cooling track in J−K and absolute V. The track a 0.6 solar-mass hydrogen-atmosphere white dwarf follows as it cools, absolute V magnitude against J−K colour, fainter downward. Dashed, a blackbody of the same radius, which slides steadily redward and fainter. Solid, with collision-induced absorption: the track reddens to J−K = 0.40 at 4250 K and absolute magnitude 16.1, then hooks back to the blue, reaching −0.86 at 3000 K and magnitude 18.1. Temperatures are marked along it. The oldest white dwarfs therefore sit in the same colour range as much younger, hotter ones and are told apart only by being fainter — which is why they were missed by surveys that selected by colour.

A colour that runs backwards at the end of cooling

A white dwarf reddens as it cools, like every other hot body — until its surface falls below about four thousand kelvin. Then pairs of hydrogen molecules, colliding in an atmosphere dense enough to crush them together, begin to absorb its infrared light, and the star turns blue again while it goes on getting fainter. The oldest white dwarfs in the Galaxy have the colours of young ones, and for years they were looked for in the wrong place.

stars · White dwarf cooling 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

Named alongside it

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

BlackbodyEffective temperatureReddeningPhotometric systemPhotometryAngular diameterApparent magnitudeBolometric correctionColour excessDistance modulusEclipsing binaryExtinction

All concepts