Astronomy & Celestial Mechanics
the sky, computed
Series

Extinction — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · starlight
  2. The Bouguer line, and the intercept nobody observed. Instrumental magnitude against airmass for one star of magnitude 10 outside the atmosphere, observed at 5 airmasses in 5 bands. Each slope is that band's extinction coefficient, computed from Rayleigh scattering, an aerosol term and ozone rather than assumed: U 0.493, B 0.246, V 0.126, R 0.061, I 0.027 magnitudes per airmass. Every line is fitted through its points and extended to X = 0, and the intercept there is the published magnitude — a measurement made at an airmass no observation is ever taken at, because the smallest airmass available is 1 and that is already a whole atmosphere. Two consequences follow and neither is a detail. The slope has to be re-measured every night, because the aerosol term changes with the weather and is not a property of the site. And the U-band line is 3.9 times steeper than the V-band one, so the extrapolation is 3.9 times longer in exactly the band where photons are scarcest — which is why ultraviolet photometry from the ground was always the least trustworthy part of a magnitude system.

    A magnitude measured where nothing was measured

    Every published brightness is an extrapolation off the end of a graph. A star is observed through one atmosphere at least, a line is fitted against airmass, and the number quoted is its intercept at zero — a place no observation is ever taken from.

    part 2 · starlight
  3. The light dust removes at 0.56 µm comes back at 124 µm. One energy budget drawn twice, on a wavelength axis spanning four and a half decades. The upper curve is starlight from a 6500 K photosphere; the shaded region under it is the part removed by a magnitude of visual extinction, computed from the same CCM law the other modes here draw, and weighted towards the ultraviolet exactly as that law says. The curve on the right is what the grains do with it: a modified blackbody at 20 K with an emissivity index of 1.8, normalised so that the energy under it equals the energy under the shaded region. Integrating the two curves actually drawn returns a ratio of 1.000. The starlight peaks at 0.56 microns and the re-emission at 124, a factor of 220, so nothing about the two is recognisable as the same photons and everything about them is the same joules. The practical consequence is a rule about arithmetic: a galaxy's ultraviolet luminosity and its far-infrared luminosity are not two independent measurements of how many young stars it has. One is the light that escaped and the other is the light that did not, and adding them without noticing counts part of the population twice.

    The dust is not lost light, it is moved light

    An extinction curve says how much starlight dust removes. It does not say the light is gone. Every photon a grain absorbs heats the grain, which radiates it back out at twenty kelvin — so an extinction measurement and a far-infrared spectrum are two halves of one energy budget, and the area under them is the same area.

    part 3 · starlight
  4. 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.

    part 4 · starlight
  5. How much light a fixed mass of dust scatters, as its largest grains grow. The scattering cross-section per unit mass of dust, on a logarithmic scale relative to the largest value drawn, against the size of the largest grains in a power-law distribution n(a) ∝ a^−3.5 from 0.005 µm, at wavelengths of 1.25, 2.2, 3.6 µm. With the largest grains at a quarter of a micron — the size the interstellar extinction curve implies — the dust scatters 3.6 µm light 33 times less efficiently than it does once they reach a micron: grains much smaller than the wavelength scatter as the fourth power of their size over the wavelength and are all but invisible. At 3.6 µm the scattering per mass peaks when the largest grains are 3.5 µm, and falls beyond, as the mass moves into grains too large to scatter efficiently per gram. A dense core that glows at 3.6 µm in scattered starlight is therefore a core whose grains have grown to a micron or more.

    A glow that weighs the grains reddening cannot see

    The interstellar extinction curve reads the sizes of dust grains from how much more they dim blue light than red, and it goes blind at the top end: grains larger than about a micron dim every visible colour alike. Those are the grains that hold most of the mass, and the ones from which planets begin. Dense cloud cores glowing faintly at 3.6 microns in scattered starlight are the first direct sign of them — because only grains grown to a micron or more scatter that light at all.

    part 5 · starlight

All series