Astronomy & Celestial Mechanics
the sky, computed
Concept

Rayleigh scattering — where it appears

Scattering by particles far smaller than the wavelength, whose cross-section goes as the inverse fourth power and which is what the air itself does. It polarises the scattered light perpendicular to the scattering plane, maximally at a right angle, which is what makes the daytime sky's polarisation a map of the Sun's position.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

starlight · Extinction A radius that depends on the colour it is measured in. Transit depth against wavelength for a planet of 1.38 Jupiter radii at 1400 K, whose atmosphere has a scale height of 538 km — computed from H = kT/µg, not assumed. One scale height of extra opacity adds 153 parts per million to a transit of 1.40 per cent, so the whole spectral signal is 862 ppm at its strongest: one part in 16 of the transit that carries it. The features are at real band centres — sodium at 0.589 µm, water at 1.4 µm, carbon dioxide at 4.3 µm — with the rise at the blue end the Rayleigh slope of scattering off the smallest particles.

A radius that depends on the colour it is measured in

Measure a transit in one colour and then another, and the planet is a different size. The difference is a few atmospheric scale heights, which is a few hundred parts per million of an already tiny signal.

exoplanets · Exoplanet atmospheres Three planets that are the same spectrum. A model transmission spectrum, in scale heights of apparent radius, drawn three times: once as it is, once with the reference radius raised by 0.45 scale heights and the abundance reduced to compensate, and once with a cloud deck truncating the features. The three differ by 0.21 scale heights root-mean-square against features of 2.1, which is well inside the error bars of any real observation. The reason is structural rather than observational: a transmission spectrum measures a difference in apparent radius with wavelength and never an absolute radius, so the level is a free parameter, and shifting the level trades against the abundance almost exactly. Adding a cloud deck adds a third parameter that flattens features and trades against both. Three unknowns and one curve is why the quoted abundance uncertainties from transmission spectroscopy are so much larger than the photometric precision suggests.

A spectrum flattened by cloud, or by nothing

A transmission spectrum measures how a planet's apparent radius changes with wavelength, and never the radius itself. That missing level is a free parameter, it trades almost exactly against the abundance of whatever is absorbing, and a cloud deck adds a third unknown to a curve that constrains two.

exoplanets · Exoplanet atmospheres The zenith sky after sunset, as single scattering predicts it. The brightness of the zenith sky at 550 nm, in magnitudes per square arcsecond with brighter upward, against the Sun's depression below the horizon, computed by single scattering in a spherical atmosphere: US Standard Atmosphere densities, Rayleigh scattering, a 300 Dobson-unit ozone layer, and every photon scattered exactly once. The Earth's shadow climbs the zenith as R(sec d − 1) — 8.7 km at 3°, 35.1 at 6°, 142 at 12° — through air that thins by a factor of e every eight kilometres or so, so the brightness does not fall steadily: it falls 1.4 magnitudes a degree between 3° and 6°, and 2.7 a degree between 7° and 10°. The model gives 14.3 at the end of civil twilight and 27.0 at the end of nautical. The dashed line is the natural night sky, 21.9 magnitudes per square arcsecond. Single scattering reaches it at 9.25° of depression — 8.8 degrees before the 18° at which astronomical twilight is observed to end. The difference is not an error in the arithmetic; it is the light this model leaves out, scattered more than once.

The shadow that climbs the zenith

After sunset the sky overhead is lit only above the Earth's own shadow, and the shadow climbs as the square of the Sun's depression. Scattering computed once from that geometry predicts a sky that dims faster and faster and is as dark as night by nine degrees. The real sky takes eighteen, and the difference is light that has been scattered more than once.

sky · Twilight 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

Named alongside it

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

OzoneScale heightMean molecular weightSingle-scatteringTwilightAbundance degeneracyAerosolAirmassAtmospheric compositionAtmospheric extinctionBayesian inferenceBouguer line

All concepts