Astronomy & Celestial Mechanics
the sky, computed
Concept

Transits — where it appears

Passages of a planet across the face of its star, which remove a fraction of the light equal to the squared radius ratio. Four contact times fix the geometry, and their spacing gives a stellar density with no assumption about the star.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The limb is 40% as bright as the centre, and that is a temperature gradient. Left: a stellar disc shaded by the grey-atmosphere law I(μ)/I(1) = (2 + 3μ)/5, in 26 steps, with μ = cos θ read from the centre outwards. Right: that law against μ, with linear laws at the measured solar coefficients from 400 to 1600 nm. The grey law's coefficient is exactly 3/5 and its very limb is exactly 2/5 of the central brightness — both read off the drawn curve rather than quoted — because the Eddington–Barbier relation makes the emergent intensity at angle μ the source function at optical depth τ = μ, and in radiative equilibrium that source function is linear in τ. The limb is not a cooler part of the star. A sight line entering at the edge reaches unit optical depth higher up, where the gas is cooler, so what the darkening measures is the run of temperature with depth; a star with an isothermal atmosphere would show a uniform disc, and one with a steeper gradient a darker limb. The measured coefficients fall from 0.9 at 400 nm to 0.35 at 1600 — the same gradient seen through a less steep Planck function — which is why a radius measured from a transit or a fringe null has to say which colour it was measured in.

The light that is missing from the edge

The Sun's limb is forty per cent as bright as its centre, and the reason is not that the edge is cooler. A sight line entering at the edge stops higher up, so what the darkening measures is the temperature gradient — and it is worth seventeen per cent on Betelgeuse's radius.

starlight · Limb darkening A 27.5 m/s velocity the star does not have. Left: a rotating stellar disc, approaching on one side and receding on the other, with the chord a planet of 0.1 stellar radii takes across it at impact parameter 0.5 and a sky-projected obliquity of 0°. Right: the apparent radial velocity that results, computed by covering the disc cell by cell — the flux hidden at each phase and its mean line-of-sight velocity — rather than from a fitted formula. The star's centre of mass does not move at any point in this: the anomaly is entirely a statement about which parts of the line profile are missing. Its amplitude is 27.5 m/s at v sin i = 4.5 km/s, and the two numbers are related by the depth of the transit, since blocking a fraction f of light of mean velocity v shifts a flux-weighted centroid by f·v. The curve is antisymmetric about mid-transit to 0.00% of its own amplitude, which is what an aligned transit gives: equal time on the blue half and the red. Limb darkening is included at u = 0.6, and it matters: it weights the hidden light towards the centre of the disc, where the rotation velocity is smallest.

A velocity measured from a shape

A transiting planet hides part of a rotating disc, so the star's line profile loses a slice at one velocity and its fitted centroid moves. The star has not moved at all — and the lopsidedness of that motion is the whole measurement of whether the orbit lies in the star's own equatorial plane.

exoplanets · Spin–orbit alignment

Named alongside it

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

Limb darkeningAngular diameterDoppler effectEclipsing binaryEnergy transportLine-broadeningObliquityOptical depthPlanet migrationRadial velocityRadiative transferRossiter mcLaughlin effect

All concepts