Astronomy & Celestial Mechanics
the sky, computed
Concept

Geometric albedo — where it appears

The brightness of a body at zero phase angle, relative to a flat, perfectly white diffusing disc of the same size at the same distance. It measures how reflective a planet is towards its star, not how much light it reflects overall.

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

Also named here as phase angle — the same set of essays touches all of them, so they are one junction rather than several.

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 The polarised light HD 189733 b can send. The polarised part of HD 189733 b's reflected light, as a fraction of its star's light in parts per million, against orbital phase from one transit to the next, for an orbit of radius 0.0314 AU, a planet of 1.138 Jupiter radii and an inclination of 85.7°. A planet's reflected light is its geometric albedo times (R/a)² — here 3·10⁻⁴ — times a phase function, and the polarised part is that times the degree of polarisation, which for a scattering atmosphere peaks near quadrature where the planet is half lit. With an albedo of 0.4 and a peak polarisation of 0.5 the signal reaches 24 ppm (solid). The ceiling (dashed) gives the planet the largest albedo a scattering atmosphere can have, 0.75, and perfect polarisation: 91 ppm. An amplitude of 100 ppm (dotted), the order of the signal claimed in 2008, would need the planet to reflect and polarise more light than falls on it.

A polarisation capped by the light the planet catches

Light scattered by a planet's atmosphere is polarised, most strongly when the planet is half lit, and the polarisation angle turns as the planet orbits — so a planet's polarised light would weigh its atmosphere and draw its orbit on the sky without a transit. In 2008 it was reported for a hot Jupiter at a part in ten thousand of its star's light. That amplitude is more than the planet can send: no atmosphere reflects more than three-quarters of the light that falls on it or polarises more than all of it, and the light that falls on it is set by its size and its distance. The photons for a real detection are cheap; what limits it is a polarimeter stable to a part in a million.

sky · Apparent motion

Named alongside it

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

Phase anglePhase curveForward-scatteringLambert spherePhase functionAsymmetry parameterGreatest brilliancyGreatest elongationHot jupiterInferior conjunctionInstrumental polarisationLunar phase

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