Astronomy & Celestial Mechanics
the sky, computed
Concept

The Rossiter–McLaughlin effect — where it appears

The apparent radial-velocity shift of a star as a transiting body covers part of its rotating disc. The shift is the first moment of a distorted line profile rather than a motion, and its asymmetry measures the angle between the orbit and the equator.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

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 A bump that crosses the line from -41 to 25 km/s. The residual of a rotationally broadened line profile during a transit, drawn at five epochs and offset vertically. The planet covers a strip of the stellar disc whose radial velocity is the projected rotation at that position, so it removes light from one velocity and leaves a bump in the residual there. As the planet crosses, the bump travels across the profile — and where it starts and ends is set by the geometry of the chord. An orbit aligned with the star's equator gives a track symmetric about the line centre; this one, tilted by 30 degrees, runs from -41 to 25 kilometres a second and is not. The measurement is of a path rather than of a centroid, which is why it works on rapidly rotating stars where the velocity anomaly is swamped by the line's own width.

A shadow crossing a rotating line

A transiting planet hides a strip of a rotating star, and that strip has a definite velocity. So the planet removes light from one place in the line profile and leaves a bump there — a bump that travels across the line as the transit proceeds, tracing the path the planet took across the disc.

exoplanets · Spin–orbit alignment The true obliquity a projected angle allows, against the tilt of the star. The true angle ψ between a transiting planet's orbit and its star's spin axis, against the inclination of that spin axis to the line of sight, for projected angles λ of 0°, 30°, 60°, 90°, 130° — from cos ψ = sin i★ cos λ for an orbit seen edge-on. At i★ = 90°, a star seen equator-on, the projection loses nothing and ψ = λ. As the star tips towards pole-on the true angle moves towards 90° from whichever side it started, so every curve ends at 90° when i★ = 0. Two things follow exactly. The projected angle is a lower bound for an aligned-looking system and an upper bound for a retrograde-looking one — ψ always lies between λ and 90°. And λ = 90° needs no inclination at all: a planet whose projected orbit crosses the stellar equator at right angles is on a polar orbit whatever the tilt, which is why the pile-up of polar orbits could be seen in projected angles alone.

An angle hidden by the tilt of the star

The line profile of a transited star gives the angle between orbit and spin as projected on the sky. The true angle needs the tilt of the star's spin axis, and every way of measuring that tilt is blind in the same place — a star seen equator-on, which is exactly where an aligned system sits.

exoplanets · Spin–orbit alignment A nearly aligned population, seen in projection and in three dimensions. 60,000 transiting systems whose stars are tilted from their planets' orbits by a small random amount: each of the tilt's two components is Gaussian with a width of 5°, and its direction around the orbit normal is random. Orange: the true obliquity ψ, which is never negative and is almost never near zero — its median is 5.9°, because a tilt with two components is small only when both of them are small at once. Blue: the projected angle λ the line profile measures, which is symmetric about zero and peaked there. The projected angles look better aligned than the true ones, and they are not hiding anything: for small tilts λ is one component of a two-component tilt, so the rms true angle, 7.04°, is √2 times the rms projected angle, 5.01°, with no knowledge of any star's inclination.

A spread the projection does not hide

A single projected spin–orbit angle says little about the true one, because the star's tilt is unknown. A population of nearly aligned systems is different: for small tilts the projected angle is one component of a two-component vector, so the rms true obliquity is exactly √2 times the rms projected angle — no star's inclination needed.

exoplanets · Spin–orbit alignment What DI Herculis's orbit should do, and what it does. The rate at which the periastron of the eclipsing binary DI Herculis advances, in degrees per century. Top: the prediction for stars whose spins are aligned with the orbit — 2.34 from general relativity, computed from the masses (5.17 and 4.52 solar masses), the 10.550164-day period and the eccentricity of 0.489; and 1.93 from the stars' tidal and rotational distortion, as published stellar models give it, of which the rotational flattening is 1.50 and the tides 0.43. Total 4.27. Bottom: the observed rate, between about 0.6 and 1.4 degrees per century in the published timing analyses — a quarter to a seventh of the prediction. For a quarter of a century after the discrepancy was established it was cited as possible evidence against general relativity, which contributes more than half the predicted total by itself.

A precession slowed by stars lying on their sides

The eclipsing binary DI Herculis turns the long axis of its orbit at a quarter to a seventh of the rate relativity and the stars' own distortion predict, and for a quarter of a century that gap was offered as evidence against general relativity. The answer was in the stars: both spin nearly on their sides, and a flattened star tipped past 54.7 degrees drags the orbit's periastron backwards instead of forwards. Explained, the slow precession becomes a measurement of something no spectrum shows — how far the spins lean towards the Earth.

stars · Binary stars

Named alongside it

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

Projected obliquityStellar inclinationStellar rotationLine-broadeningSpin–orbit alignmentStellar obliquityApsidal constantApsidal motionAsteroseismologyDegeneracyDoppler effectDoppler tomography

All concepts