Astronomy & Celestial Mechanics
the sky, computed
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Spin–orbit alignment — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · exoplanets
  2. The same orbit, realigned by one star and not by the other. The time an equilibrium tide takes to bring a planet's orbit into the plane of its star's equator, against orbital separation in stellar radii, for a planet of 1e-3 stellar masses. Both axes are logarithmic. The two curves differ only in how efficiently the star dissipates the tide, by a factor of 10⁴ — the contrast between a star with a convective envelope, where turbulence turns the tidal flow into heat, and one hotter than about 6,250 K, which has almost none. The lower curve is calibrated so that a Jupiter at 8 stellar radii realigns a cool star's orbit in 1 billion years, which is what the aligned systems require; the tidal quality factor of a star is not known from first principles and this is the honest way to say so. Everything else follows from the sixth power of the separation, which is measured off the drawn curve as 6.000. The two curves cross a Hubble time at 12.4 and 2.7 stellar radii, and their ratio is the sixth root of the dissipation contrast. Hot Jupiters sit between those two numbers. So the same arrival distribution of orbital tilts is erased around cool stars and preserved around hot ones, and a survey that finds cool hosts aligned and hot hosts scattered has measured the filter rather than the arrivals.

    A misalignment only cool stars forget

    A third of hot Jupiters orbit at a large angle to their star's equator, and some go round backwards. Sort the same planets by the temperature of their host and the picture changes — below about 6,250 kelvin almost all are aligned, and above it almost none are. The boundary is not about the planets.

    part 2 · exoplanets
  3. 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.

    part 3 · exoplanets
  4. 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.

    part 4 · exoplanets
  5. 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.

    part 5 · exoplanets
  6. A star brighter at its poles, and three paths a planet might take across it. A star rotating at 0.6 of its critical angular velocity, seen with its spin axis tipped 30° towards the observer from the plane of the sky, drawn with that axis vertical. Its outline is the Roche surface, 1.05 times as wide as it is tall in projection. The contours are the surface brightness, from von Zeipel's law with exponent 0.2 and a linear limb darkening: gravity at the equator is weaker, so the equator is cooler and its intensity there is 81 per cent of the pole's before limb darkening. Three transit chords at the same impact parameter, 0.3 polar radii, cross at projected spin–orbit angles of 0°, 45°, 90°. Only the first is symmetric about the star's axis; the others run from bright regions to dark ones or the reverse, and the light curve records which.

    A transit that runs differently backwards

    A star spinning near break-up is brighter at its poles than its equator, so a planet crossing it at an angle covers bright surface on the way in and dim surface on the way out. The light curve stops being symmetric in time, and the part that differs forwards and backwards measures the true spin–orbit angle from photometry alone — on exactly the stars where the line-profile method struggles.

    part 6 · exoplanets

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