The Rossiter-McLaughlin Effect

The intuition: hiding half a spinning star

Point a spectrograph at a star and you do not resolve its disk — you collect the summed light of millions of surface patches into a single spectrum. The star rotates, so one half of its visible disk is sweeping toward you and the other half is sweeping away. By the Doppler effect, the light from the approaching limb is shifted slightly blue and the light from the receding limb slightly red. Each narrow absorption line therefore gets smeared out into a broad, roughly symmetric trough whose width encodes the projected rotation speed, v sin i★. This is ordinary rotational broadening, and it is symmetric: the blue and red contributions balance, so the line centroid sits at the star's true velocity.

Now slide an opaque planet across the disk. When the planet sits on the blueshifted (approaching) limb, it is blocking blue-shifted starlight. That light is missing from the summed profile, so the surviving light is, on average, slightly redder — and the spectrograph reports the star as receding a touch faster than it really is. A few hours later the planet has crossed to the redshifted (receding) limb; now it hides red-shifted light, the surviving profile skews blue, and the apparent velocity flips to a blueshift. The star itself never changed its motion. What you are watching is the spectroscopic shadow of a planet selectively covering a rotating surface. That sign-flipping wobble, superimposed on the transit, is the Rossiter-McLaughlin effect.

The punchline is geometric. The exact timing and shape of the anomaly — which limb is covered first, how symmetric the blue-to-red swing is, where the zero-crossing falls relative to mid-transit — encode the angle between the planet's transit chord and the star's rotation axis. That angle is the system's spin-orbit alignment, and the Rossiter-McLaughlin effect is the single most productive way we have of measuring it.

The mechanism and the governing equation

Treat the stellar disk as a brightness map I(x, y) with a line-of-sight velocity field set by rotation. For solid-body rotation seen with the spin axis projected along the sky's y-direction, the line-of-sight velocity of a surface element at projected position x (in units of the stellar radius R★) is

v_los(x) = x · v sin i★

where v sin i★ is the equatorial rotation velocity projected onto the line of sight. A planet of radius R_p covering a small patch at position (x_p, y_p) removes a flux fraction roughly δ = (R_p/R★)² (modulated by limb darkening) that was carrying velocity v_los(x_p). The flux-weighted velocity centroid of the whole star then shifts by approximately

ΔV_RM(t) ≈ − δ(t) · x_p(t) · v sin i★ · f(limb darkening)

The minus sign is the heart of the effect: removing blue light (x_p < 0) raises the apparent velocity, removing red light (x_p > 0) lowers it. The track x_p(t) of the planet across the disk is a straight chord whose orientation is the projected obliquity λ. For an aligned orbit (λ = 0°) the chord runs perpendicular to the spin axis, x_p sweeps cleanly from −1 to +1, and ΔV_RM traces a clean antisymmetric blue-then-red curve. For a polar orbit (λ = 90°) the chord runs parallel to the spin axis at fixed x_p, so the anomaly has a single sign throughout the transit. The full velocity-anomaly curve is thus a direct readout of the transit geometry.

A subtlety that the simple formula above hides: a partially occulted rotating line profile is not merely shifted, it is deformed — the planet carves a localized bright "bump" out of the dark absorption trough at the velocity of the patch it covers. The classical centroid formula (the Ohta-Taruya-Suto 2005 and Hirano et al. 2011 analytic models) captures the leading behaviour, but precise modern work fits the full line-shape distortion, which is what Doppler tomography and the "reloaded" technique exploit.

The key numbers

The amplitude of the anomaly follows directly from the transit depth and the rotation speed. To order of magnitude,

ΔV_peak ≈ (R_p / R★)² × v sin i★ × C

where C ≈ 0.6–0.8 folds in limb darkening and the geometry of partial coverage. The relevant numbers across real systems:

• Projected rotation speed v sin i★. Sun-like dwarfs rotate at the equator at ~2 km/s (the Sun: 1.6–2.0 km/s); young or hot stars reach 10–150 km/s. The anomaly amplitude is directly proportional to this.
• Transit depth (R_p/R★)². A hot Jupiter (R_p ≈ 1 R_Jup) around a Sun-like star (R★ ≈ 1 R☉, with R_Jup/R☉ ≈ 0.10) gives R_p/R★ ≈ 0.10, depth ≈ 1%. An Earth-size planet gives depth ≈ 0.008% — too small for an RM detection with current instruments.
• Resulting anomaly. 10–100 m/s for typical hot Jupiters; well over 1 km/s for giants around fast-rotating A/F stars. Compare with the orbital reflex (Keplerian) RV signal of a hot Jupiter, ~50–200 m/s, on top of which the RM bump rides during the few-hour transit.
• Instrument precision. HARPS (ESO 3.6 m, La Silla) and ESPRESSO (VLT) reach ~1 m/s and sub-m/s respectively; HIRES (Keck) and SOPHIE (OHP) are at the few-m/s level. This is why even modest anomalies are routinely measurable.
• Timescale. The whole signature plays out over the transit duration — typically 2–4 hours for a hot Jupiter on a ~3-day orbit — which is why a single transit night can deliver an obliquity.

How it is observed and modelled

An RM campaign collects a dense time series of high-resolution spectra spanning a full transit plus baseline before and after. Each spectrum is reduced to a radial velocity (classically) or to a cross-correlation function / line profile (for the modern techniques). The observed RV time series is the sum of three things: the constant systemic velocity, the slowly varying Keplerian orbital reflex motion, and — only during the transit window — the RM anomaly. Subtract a fitted Keplerian and the RM bump stands out.

The data are then fit with a model parameterised by λ and v sin i★ (plus the well-known transit geometry from photometry: R_p/R★, impact parameter b, inclination). Three main methods coexist:

Method	What it fits	Best for	Key reference
Classical RM (velocity anomaly)	RV centroid vs. time	Slow rotators, legacy data	Ohta+ 2005; Hirano+ 2011
Doppler tomography	Planet "bump" tracking across the line profile	Fast rotators (broad lines)	Collier Cameron+ 2010
Reloaded / Revolutions RM	Local stellar surface spectrum behind the planet	Differential rotation, convection, μ-dependence	Cégla+ 2016; Bourrier+ 2021

The output is λ, the sky-projected angle. To turn that into the true three-dimensional obliquity ψ you also need the stellar inclination i★. Combining v sin i★ with an independent rotation period (from photometric spot modulation or asteroseismology) and the stellar radius gives i★, and then cos ψ = cos i★ cos i_orb + sin i★ sin i_orb cos λ.

Worked example: HD 209458 b

Take the first transiting exoplanet found, HD 209458 b, a 0.69 M_Jup hot Jupiter on a 3.52-day orbit around a G0 V star with R★ = 1.16 R☉ and v sin i★ ≈ 4.5 km/s. The transit depth is about 1.5%, so R_p/R★ ≈ 0.12.

Estimate the peak anomaly:

ΔV_peak ≈ (R_p/R★)² × v sin i★ × C
        ≈ (0.12)² × 4500 m/s × 0.7
        ≈ 0.0144 × 4500 × 0.7
        ≈ 45 m/s

The measured semi-amplitude of the anomaly is indeed a few tens of m/s. Winn et al. (2005), using Keck/HIRES, fit the antisymmetric blue-then-red shape and found λ = −4.4° ± 1.4° — the orbit lies essentially in the stellar equatorial plane, prograde and aligned. This was the template case: a textbook well-behaved system that confirmed planets can form and migrate while keeping the disk's original alignment.

Contrast that with HAT-P-7 b. Here the anomaly does not flip from blue to red — it stays one-signed and then reverses the expected order, the spectroscopic fingerprint of a planet that crosses the receding limb first. Narita et al. and Winn et al. (2009) independently derived λ ≈ 180° (within errors): the planet orbits retrograde, against the star's spin. No quiet disk migration can produce that; it demands a violent dynamical history.

Discovery and the rise of obliquity science

The effect was seen first not in planets but in eclipsing binary stars. In 1924 Richard A. Rossiter, in his University of Michigan doctoral thesis, reported the anomalous velocity excursions during the eclipses of Beta Lyrae; in the same year and at the same institution Dean B. McLaughlin reported them for Algol (β Persei). The physical interpretation — selective occultation of a rotating disk — was understood almost immediately, and the signature became a tool for studying stellar rotation in eclipsing binaries for decades.

Its exoplanet career began in 2000, when Queloz and collaborators reported a first tentative measurement for HD 209458 b and noted the technique could probe spin-orbit alignment; Gaudi and Winn (2007) later spelled out clearly how the anomaly's shape constrains λ. It began in earnest in 2005 with the precise HD 209458 b measurement. The field then exploded. The discovery of clearly misaligned (XO-3 b, 2008) and retrograde (HAT-P-7 b and WASP-17 b, 2009) hot Jupiters overturned the comfortable assumption of universal alignment. By around 2010–2012 Amaury Triaud, Simon Albrecht, and collaborators had assembled enough measurements to reveal a startling pattern: misalignment correlates with host-star temperature. Planets around hot stars (T_eff > ~6250 K, the Kraft break) are frequently misaligned; planets around cool stars are usually aligned. Today the catalog runs to hundreds of measured obliquities, gathered by HARPS, ESPRESSO, HIRES, SOPHIE, and SUBARU/HDS, and the technique is being pushed down toward smaller planets and even to mapping the stellar surface itself.

Variants and related phenomena

• Doppler tomography. Rather than collapsing each spectrum to one velocity, this tracks the dark planetary "bump" as it migrates across the broad, rotationally smeared line profile. Because broad lines (fast rotators) carry more spatial information, tomography is the method of choice for hot, rapidly rotating A/F stars — exactly the regime where the classical centroid method struggles.
• The "reloaded" Rossiter-McLaughlin (RM^reloaded) and RM Revolutions. Developed by Cégla, Bourrier, and colleagues, these isolate the spectrum of the precise surface element behind the planet at each moment, directly recovering the local rotation velocity. They can detect differential rotation (equator faster than poles), center-to-limb convective shifts, and resolve degeneracies between λ and v sin i★.
• Gravity darkening. A rapidly rotating star is oblate and has hotter, brighter poles (von Zeipel effect). The asymmetric brightness map distorts the transit light curve itself, providing an independent — and out-of-the-plane — handle on obliquity, as exploited for KOI-13 b by Barnes and others.
• Spot crossings and starspot tracking. Repeated transits over a spotted star can show the planet occulting the same spots, mapping the latitude band the chord crosses and constraining alignment photometrically (used extensively for Kepler systems).
• Stellar obliquities of binaries. The original 1924 application — measuring the mutual alignment of stellar spins and orbits in eclipsing binaries — remains active and informs theories of binary formation.

Common misconceptions and subtleties

• "It measures the true obliquity." No — it measures only the sky-projected angle λ. A system with λ ≈ 0° could still be substantially misaligned in 3-D if the star's pole points toward or away from us. The true obliquity ψ needs the stellar inclination as well.
• "The star's velocity actually changes during transit." It does not. The bulk radial velocity of the star is unchanged; the apparent shift is an artifact of removing part of a rotating, spatially unresolved surface from the line profile.
• "The line is just Doppler-shifted, so a centroid fit is exact." A partly occulted line is deformed, not cleanly shifted, so the classical centroid model is biased. The bias is small for slow rotators but grows for fast ones, which is why line-profile (tomographic) modelling is preferred there.
• "Retrograde means the planet physically reversed its motion." Retrograde here is relative to the star's spin: the orbit runs counter to the stellar rotation. The planet's own orbital motion is perfectly ordinary; it is the spin-orbit relationship that is backward, pointing to scattering or Kozai-Lidov migration rather than smooth disk migration.
• "Any transiting planet can be measured." The anomaly scales as (R_p/R★)² v sin i★. Small planets, or planets around very slow rotators, can push the signal below the m/s noise floor — so RM obliquities are still dominated by giant planets, though ESPRESSO is steadily reaching smaller worlds.
• "Limb darkening can be ignored." Limb darkening reshapes both the brightness map and the effective velocity weighting; ignoring it skews the inferred λ and v sin i★. It is a standard nuisance parameter in every modern fit.