Rossiter-McLaughlin Effect: Exoplanet Spin-Orbit Alignment

Spin-Orbit Geometry

Sky-projected obliquity λ 30°

Impact parameter b 0.30

Planet / star radius R_p/R_s 0.10

Stellar v sin i_* 6.0 km/s

Transit phase (scrub) 0.00

Live Readout

RV anomaly

0 m/s

Relative flux

100.0%

On-disk position (x, y in R_s)

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Anomalous RV vs. transit phase

How it works

As an exoplanet transits, it blocks a small patch of the star's rotating disk. Because the star rotates, that blocked patch was contributing a slightly Doppler-shifted sliver of light to the star's integrated spectral lines. Removing it shifts the disk-averaged line centroid — the star briefly appears to have an anomalous radial velocity, even though it never actually moved. This is the Rossiter-McLaughlin (RM) effect.

Local line-of-sight velocity (rigid rotation,
spin axis in sky plane):
  v(x) = v sin i* · (x / R_s)

Occulted flux fraction (limb-darkened):
  f(x,y) = [A_overlap / (π R_s²)] · [1 − u(1 − μ)]
  μ = sqrt(1 − (x² + y²)/R_s²)

Anomalous RV:
  ΔRV(t) ≈ − v sin i* · (x(t)/R_s) · f(t)

The planet's true orbital path is a straight chord at fixed impact parameter b. The obliquity λ is the angle between that chord and the star's projected spin axis. Rotating the chord by λ mixes it into (x, y): a small λ (well-aligned system) blocks the approaching hemisphere first and the receding hemisphere second, giving the classic antisymmetric "S-curve" anomaly. Near λ = ±90° (a polar orbit) the planet stays on one Doppler hemisphere for the whole transit, so the anomaly no longer changes sign — this asymmetry is exactly how astronomers measure λ from real RM spectroscopy, most famously revealing that many hot Jupiters orbit misaligned with — or even opposite to — their star's spin.

λ slider — spin-orbit angle; try 0° (aligned), 90° (polar), 180° (retrograde).
b slider — how close the chord passes to disk centre; b → 0 gives the largest peak amplitude.
R_p/R_s — bigger planets occult more flux, scaling the anomaly by (R_p/R_s)².
v sin i* — faster stellar rotation directly scales the anomaly amplitude.

Approaching limb (blueshifted) Receding limb (redshifted) Transiting planet

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