Astronomy & Celestial Mechanics
the sky, computed
Series

Spin–orbit alignment — the series

3 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

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