Rossiter–McLaughlin Effect

The Rossiter–McLaughlin (RM) effect is a spectroscopic phenomenon observed when an eclipsing companion star or a transiting exoplanet passes in front of a rotating host star, sequentially blocking portions of the star's approaching (blueshifted) and receding (redshifted) hemispheres and thereby producing a characteristic S-shaped distortion in the star's disk-integrated radial velocity (RV) curve during transit. Independently discovered and measured in 1924 by American astronomers Richard A. Rossiter (in the eclipsing binary Beta Lyrae) and Dean B. McLaughlin (in Algol, Beta Persei) at the University of Michigan Observatory, and first detected for an exoplanet (HD 209458 b) by Queloz et al. (2000), the Rossiter–McLaughlin effect is the primary astrophysical diagnostic for measuring stellar obliquity (spin–orbit alignment, λ or ψ)—revealing whether a planetary system formed quiescently inside a flat protoplanetary disk or suffered violent dynamical scattering into inclined, polar, or retrograde orbits (Winn, 2010; Albrecht et al., 2022).

A rotating star of equatorial radius R_⋆, rotational period P_rot, and inclination I_⋆ between its spin axis and the observer's line of sight has a projected equatorial rotation velocity of v sin I_⋆ = (2π R_⋆ / P_rot) sin I_⋆:

• One half of the visible stellar disk rotates toward the observer, emitting light that is Doppler blueshifted (v_los < 0).
• The opposite half of the stellar disk rotates away from the observer, emitting light that is Doppler redshifted (v_los > 0).
• Outside of transit, the blueshifted light from the approaching limb and the redshifted light from the receding limb exactly cancel, so the centroid of every photospheric absorption line sits at the star's center-of-mass orbital velocity.

When an opaque planet of radius R_p transits across the stellar disk, it acts as a moving dark mask of fractional area (R_p / R_⋆)² (minus limb darkening):

1. Prograde, Well-Aligned Orbit (Sky-Projected Obliquity λ ≈ 0°):
   • First Half of Transit (Ingress to Mid-Transit): The planet crosses the approaching (blueshifted) hemisphere first. By blocking a slice of blueshifted starlight, the remaining unblocked starlight has a net excess of redshifted photons, causing the measured disk-integrated radial velocity to shift anomalously positive (an apparent redshift, Δv_RM > 0).
   • Mid-Transit: As the planet crosses the projected stellar rotation axis, it blocks zero net rotational Doppler velocity (Δv_RM = 0).
   • Second Half of Transit (Mid-Transit to Egress): The planet crosses the receding (redshifted) hemisphere, blocking redshifted light and shifting the disk-integrated radial velocity anomalously negative (an apparent blueshift, Δv_RM < 0).
2. Retrograde Orbit (λ ≈ 180°): The planet enters across the receding (redshifted) limb first and exits across the approaching (blueshifted) limb, inverting the RM waveform (first blueshift Δv_RM < 0, then redshift Δv_RM > 0).
3. Polar Orbit (λ = ±90°) with Impact Parameter b ≠ 0: If the planet moves perpendicular to the stellar equator along a chord displaced from the disk center (b > 0), it stays entirely on either the blueshifted or redshifted hemisphere throughout the entire transit, producing a single-signed (monopolar) symmetric bump rather than an antisymmetric S-wave!

To first order (neglecting differential rotation and macroturbulence), the instantaneous radial-velocity anomaly Δv_RM(t) when the planet obscures a small fraction f(t) ≈ (R_p / R_⋆)² of the limb-darkened stellar flux at sub-planet rotational velocity v_p(t) is (Ohta, Taruya & Suto, 2005; Gaudi & Winn, 2007):

Δv_RM(t) ≈ − [ f(t) / (1 − f(t)) ] v_p(t)

yielding a maximum semi-amplitude K_RM of:

K_RM ≈ 0.65 (v sin I_⋆) (R_p / R_⋆)² √(1 − b²)

For a typical hot Jupiter (R_p / R_⋆ = 0.10, transit depth 1%) transiting a sun-like star (v sin I_⋆ = 3–10 km s⁻¹), K_RM ≈ 20 to 70 m s⁻¹—easily resolved at high signal-to-noise by extreme-precision echelle spectrographs such as HARPS, ESPRESSO, HIRES, NEID, and MAROON-X (which achieve σ_RV ≈ 0.1–1.0 m s⁻¹). For an Earth-sized planet (R_p / R_⋆ ≈ 0.01, transit depth 10⁻⁴), K_RM is only ~20–50 cm s⁻¹.

Whereas the RM effect alone measures the sky-projected spin–orbit angle λ (the angle on the plane of the sky between the stellar spin axis and the planet's orbital normal), combining λ with the stellar inclination I_⋆ (derived from rotational period P_rot, radius R_⋆, and v sin I_⋆, or asteroseismic rotational splitting) and the planet's orbital inclination i_orb (≈ 90° for a transiting planet) yields the true three-dimensional obliquity angle ψ via spherical trigonometry:

cos ψ = cos I_⋆ cos i_orb + sin I_⋆ sin i_orb cos λ ≈ sin I_⋆ cos λ

Measuring λ and ψ across more than 200 transiting exoplanet systems has overturned the classical expectation that all planets orbit in the equatorial plane of their host star (as the eight Solar System planets do to within ψ_☉ = 7.15°):

Because classical radial-velocity cross-correlation templates can suffer systematic biases when v sin I_⋆ exceeds the spectrograph resolution (the line distortion is no longer a simple shift of a Gaussian), Collier Cameron et al. (2010) and Cegla et al. (2016) introduced Doppler Tomography (and the "Reloaded Rossiter–McLaughlin" technique):

• By subtracting the in-transit cross-correlation function (CCF) from the out-of-transit stellar master CCF, astronomers directly isolate the "missing starlight" spectrum emitted by the tiny patch of the stellar photosphere hidden behind the planet at each 5-minute exposure.
• Tracking this dark planet-occulted Doppler shadow as a diagonal streak across a time-versus-velocity spectrogram not only yields λ free of rotational-broadening template bias, but also measures stellar differential rotation (latitudinal shear between the star's equator and poles) and center-to-limb convective blueshift variations across the host star's surface.

1. Rossiter, R. A. (1924). "On the detection of an effect of rotation during eclipse in the velocity of the brighter component of beta Lyrae, and on the constancy of velocity of this system." Astrophysical Journal, 60, 15–21. https://doi.org/10.1086/142825
2. McLaughlin, D. B. (1924). "Some results of a spectrographic study of the Algol system." Astrophysical Journal, 60, 22–31. https://doi.org/10.1086/142826
3. Queloz, D., Eggenberger, A., Mayor, M., Perrier, C., Beuzit, J. L., Naef, D., Sivan, J. P., & Udry, S. (2000). "Detection of a spectroscopic transit by the planet orbiting the star HD209458." Astronomy & Astrophysics, 359, L13–L17. https://arxiv.org/abs/astro-ph/0006213
4. Winn, J. N., Fabrycky, D., Albrecht, S., & Johnson, J. A. (2010). "Hot stars with hot Jupiters have high obliquities." Astrophysical Journal Letters, 718(2), L145–L149. https://doi.org/10.1088/2041-8205/718/2/L145
5. Albrecht, S. H., Dawson, R. I., & Winn, J. N. (2022). "Stellar obliquities in exoplanetary systems." Publications of the Astronomical Society of the Pacific, 134(1038), 082001. https://doi.org/10.1088/1538-3873/ac6c09