Astronomy & Celestial Mechanics
the sky, computed
Concept

Phase function — where it appears

How a body's brightness varies with phase angle, normalised to one at full phase. It encodes how the surface or cloud distributes reflected light in angle, separately from how much it reflects in total.

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

Venus is brightest 51 days from inferior conjunction, 38 per cent lit. The brightness of Venus through one synodic period of 584 days, in magnitudes below its brightest, against days from superior conjunction — inferior conjunction at the two ends of the axis. The planet is treated as a matte, Lambert-scattering sphere on a circular orbit of 0.723 AU, so its flux is its phase function divided by the square of its distance from the Earth. The two factors fight: near inferior conjunction the planet is closest but shows only a thin crescent, and near superior conjunction it is fully lit but 1.72 AU away. For this orbit the contest has an interior winner. The brightest moment is 51 days either side of inferior conjunction, at an elongation of 44.6° from the Sun, with 38 per cent of the disc lit and the planet 0.539 AU away; greatest elongation, at 46.3°, comes 71 days from inferior conjunction, after the brightness peak on the way out from inferior conjunction. At superior conjunction the planet is 0.82 magnitudes fainter than its best. The marked point is the observed greatest brilliancy, about 36 days from inferior conjunction at an elongation near 39° — closer to conjunction and to a thinner crescent than any matte sphere on this orbit can be brightest at. A surface that sends more light forward, towards large phase angles, would move the peak exactly that way. Very near either conjunction the planet is lost in the Sun's glare, and the curve there describes light nobody sees.

Brightest as a crescent, and not as a disc

Venus is fully lit when it is furthest away and nearest when it is barely lit, and it is brightest in between, as a crescent weeks from inferior conjunction. A matte planet only has such a peak if its orbit is wider than about 0.46 of the Earth's — Mercury's is not, and Mercury is brightest full — and Venus's real peak sits closer to conjunction than any matte sphere allows, which is its clouds throwing light forward.

sky · Apparent motion Phase curves that agree where they can be measured. Modelled reflected light of a cloud-covered planet — geometric albedo times phase function, on a logarithmic axis — against phase angle, for clouds of asymmetry parameter 0.00 and 0.65, from scattering evenly to throwing most light forward. Each curve's single-scattering albedo — with the planet's radius known, as a transit measures it — has been chosen to match the first curve between 30° and 110° (shaded), the range the measurement is assumed to cover: the fitted albedos are 0.9900, 0.9998, and inside the window the matched curve differs from the first by at most 1%. Beyond it they part: at 165°, a thin crescent, the forward-scattering cloud is 6.0× as bright as the isotropic one. The model is an optically thick cloud with exact single scattering and a similarity-scaled multiple-scattering term.

Two clouds that shine alike until the crescent

A grey cloud that scatters light evenly and a whiter one that throws it forward return almost the same light at every phase angle from thirty degrees to a hundred and ten. They part only at the thin crescent, where a planet round another star is hardest to see — and the measurement that tells them apart in the middle of the orbit is the polarisation, a ratio with no albedo in it.

sky · Apparent motion A glow that switches off when the shadow climbs past the shell. The brightness that 100,000 objects in orbit add to the zenith sky, as a share of a natural moonless sky of 22 magnitudes per square arcsecond, against how far the Sun is below the horizon, for shells at 550 and 1200 km. Each object reflects sunlight with an effective area of 0.5 square metres; together they spread a glow whose brightness does not depend on the shell's distance, because more objects are in each patch of sky the further away the shell is. At the end of astronomical twilight the two shells add 3.0% and 2.5%. The lower shell's zenith goes dark when the Earth's shadow passes it, at 23 degrees of depression; the higher one at 33.

A glow counted in square metres of sunlit metal

Satellites too faint to see one at a time still reflect sunlight, and spread across the sky they add a glow that no mask removes. Its brightness does not depend on how far away the satellites are, only on how much sunlit area is in orbit and whether the Earth's shadow has reached it. A hundred thousand satellites as bright as today's add a few per cent to the darkest sky — and the estimate that the sky is already a tenth brighter rests on an area of debris nobody has measured.

sky · Twilight

Named alongside it

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

Forward-scatteringGeometric albedoLambert spherePhase anglePhase curveAstronomical twilightAsymmetry parameterEarths shadowGreatest brilliancyGreatest elongationInferior conjunctionLow earth orbit

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