Rossiter–McLaughlin Effect

The spin shadow, in one picture

Picture a Sun-like star spinning on its axis. One edge — one limb — rotates toward you, so its light is slightly blueshifted; the opposite limb rotates away and is redshifted. Add up the light from the whole disk and those shifts cancel: the line in the spectrum is symmetric, just broadened by rotation. This rotational broadening is parameterised by v·sin(i), the projected equatorial rotation speed.

Now slide an opaque planet across the disk during a transit. Wherever the planet sits, it subtracts light from that part of the line. Cover the blueshifted limb and you remove blue light, so the surviving averaged light looks too red — the apparent radial velocity jumps positive. Cover the redshifted limb and the disk looks too blue — the velocity dips negative. The planet is casting a "spin shadow," and the velocity anomaly it produces over the few hours of a transit is the Rossiter–McLaughlin effect.

The crucial insight is that the path the planet takes across the rotating disk determines the shape of that anomaly. If the orbit is aligned with the stellar equator, the planet crosses the blue limb and red limb symmetrically, giving a clean anti-symmetric blue-then-red (or red-then-blue) signal that crosses zero at mid-transit. If the orbit is tilted, the planet might spend most of the transit over one hemisphere, producing a lopsided, sign-skewed curve. A retrograde planet flips the whole pattern. Reading that shape is how astronomers measure spin-orbit alignment without ever resolving the star as anything more than a point of light.

What λ actually measures

The quantity extracted from the fit is λ, the angle on the plane of the sky between the projected orbital axis and the projected stellar spin axis. It is a projected obliquity, not the full three-dimensional tilt. Conventionally:

λ (projected angle)	Geometry	Interpretation
≈ 0°	Prograde, aligned	Planet orbits in the star's equatorial plane, same sense as spin
~20°–60°	Mild misalignment	Tilted orbit; common around hot stars
≈ 90°	Polar	Orbit nearly perpendicular to the stellar equator
≈ 180°	Retrograde	Planet orbits opposite the star's rotation

To convert the sky-projected λ into the true 3D obliquity ψ, you also need the inclination of the stellar spin axis, i_star. That extra angle is harder to get: it can be inferred by combining the measured v·sin(i) with an independently determined rotation period and stellar radius, or directly from asteroseismology, which senses the orientation of pulsation modes. Without it, a system reported at λ ≈ 0° could still be substantially tilted out of the plane of the sky — a caveat that matters when interpreting large alignment surveys.

How loud is the signal?

To first order the velocity anomaly scales as the fraction of starlight the planet blocks multiplied by the local rotational velocity it covers:

ΔV_RM ≈ (R_p / R_s)² · v·sin(i)

The squared radius ratio is just the transit depth. The table below shows why this is a giant-planet game with today's instruments — Earth-sized planets produce signals buried beneath sub-m/s noise floors.

Planet	Transit depth (Rp/Rs)²	Assumed v·sin(i)	Approx. RM amplitude
Hot Jupiter	~1.0%	5 km/s	~50 m/s
Hot Jupiter, slow rotator	~1.0%	1.5 km/s	~15 m/s
Hot Neptune	~0.1%	5 km/s	~5 m/s
Earth analog	~0.01%	2 km/s	~0.2 m/s

This is why the discovery target was a hot Jupiter and why the technique grew up alongside high-resolution spectrographs. HARPS reaches ~1 m/s and ESPRESSO on the VLT pushes toward 0.1 m/s, opening RM measurements to smaller, slower-spinning systems. A complementary approach, Doppler tomography, skips the disk-averaged velocity and instead watches the planet's shadow march line-by-line through the rotationally broadened stellar line profile — especially powerful for rapidly rotating stars where the classical RM curve becomes shallow and degenerate.

Why obliquities rewrote planet formation

Planets are born in a flat protoplanetary disk that should share the star's equatorial plane, so the naive expectation was near-zero obliquity everywhere. The first surprise came in 2009 when several hot Jupiters turned up badly misaligned — and a few, like WASP-17 b, were outright retrograde. That single observation killed the idea that all hot Jupiters arrived by gentle disk migration.

• Disk migration drags a planet inward through the gas disk and preserves alignment, predicting λ ≈ 0°.
• High-eccentricity migration — planet–planet scattering or the Kozai–Lidov mechanism from a distant companion — throws a planet onto a tilted, eccentric orbit that tides later shrink and circularise, naturally producing large λ.
• Tidal re-alignment can erase the evidence: cool stars (below the Kraft break, T_eff ≈ 6250 K) have thick convective envelopes that tidally damp misalignment, so they appear aligned even if they started tilted; hot stars retain their tilts.

The observed correlation — hot stars host misaligned hot Jupiters, cool stars host aligned ones — is one of the strongest pieces of evidence that violent migration is widespread and that tides reshape the record afterward. The Rossiter–McLaughlin effect, a method first used on eclipsing binaries a century ago, became the key witness.

Common misconceptions

• "It's a brightness dip like a transit." No — it's a velocity distortion measured spectroscopically; the photometric transit happens at the same time but is a separate signal.
• "λ is the planet's tilt." λ is the sky-projected angle; the true 3D obliquity needs the stellar spin inclination too.
• "Bigger planets always give cleaner signals." Amplitude depends on v·sin(i) as well — a giant planet around a slow rotator can be harder than a smaller planet around a fast spinner.
• "Aligned means the orbit is circular and tidy." Alignment around cool stars may be a tidally re-set state, not the original geometry.
• "It only works for exoplanets." The effect was first seen in eclipsing binary stars in 1924 and is still used to study stellar spin in binaries.
• "It needs a space telescope." It's a ground-based, spectroscopic measurement; the limiting resource is radial-velocity precision, not photometric stability.