## The Rossiter-McLaughlin Effect

A retrograde hot Jupiter (Credit: ESO)

It is a convenient fact that stars spin in space – a result of angular momentum imparted upon them from the collapse of the progenitor clouds from which they form. This fact, while easily taken for granted, can permit a great deal of information to be gained about surrounding planets whose orbits are fortuitously oriented to cause them to transit.

For the case of a rotating star, one hemisphere of the star will be approaching the observer, and the opposite hemisphere will be receding. The approaching hemisphere is therefore blueshifted, and the receding hemisphere is redshifted. Because the stellar spectrum will accordingly be blueshifted and redshifted, the spectral lines of a rotating star appear widened or broadened. This spectral line broadening is directly proportional to the star’s rotational velocity, and the angle between the line of sight and the stellar rotation axis. Much like with Doppler spectroscopic detection of planets, the constraints you can set to the rotational velocity of a star are only a lower limit because the rotation axis is unknown. Indeed, a rapidly rotating star viewed pole on (i = 0o), there will be no observed spectral line broadening because all of the surface rotation is perpendicular to the line of sight. Because of this inclination degeneracy, the rotational velocity of the star is represented in terms of v sin i.

The redshifting of the receding hemisphere is typically balanced by the blueshifting of the approaching hemisphere, and so there is no net change in the colour of the star. However the transit of a planet will cause an imbalance in this hemispheric Doppler shifting. For a planet in an equatorial, prograde transiting orbit the planet will first cover part of the approaching hemisphere. The Doppler shifting of the rotating stellar surface is no longer balanced and the stellar spectrum has a net redshift. This anomaly disappears as the planet moves to the centre of the stellar disc, as the redshifted and blueshifted hemispheres are both equally apparent again. Then as the planet moves over the receding hemisphere, there is a net blueshift in the spectrum as some of the redshifted light is occulted.

This radial velocity anomaly is known as the Rossiter-McLaughlin effect. It can be used to measure the projected stellar spin-orbit alignment angle (the angle between the spin axis of the star, and the orbital axis of the star+planet orbit). It can be thought of as the projected obliquity of the star relative to the planet orbit, or the inclination of the planetary orbit relative to the stellar equator. For an equatorial orbit (spin-orbit alignment), λ = 0°, and for a polar orbit (spin-orbit misalignment), λ = 90°.

The Rossiter-McLaughlin Effect

As we can see in the diagram above, the RM effect will have a different RV shape depending on the orientation of the planet’s orbit axis relative to the spin axis of the star. If the orbit is nearly polar and the planet transits across only one of the approaching or receding hemispheres, then the RM effect will be entirely blueshifted (if the planet crosses the receding hemisphere as is shown in the third example above) or redshifted (if the planet crosses the approaching hemisphere).

The amplitude of the RM effect, KR, relative to the amplitude of the star’s reflex velocity due to the influence of the planet over the course of the orbit, KO, can be estimated by
$\displaystyle \frac{K_R}{K_O} \approx 0.3\left(\frac{M}{M_J}\right)^{-1/3}\left(\frac{P}{3\text{ days}}\right)^{1/3}\left(\frac{v \sin i}{5\text{ km s}^{-1}}\right)$

Very few stars have their spin axes perfectly perpendicular to our line of sight. If the star is viewed pole-on, the RM effect amplitude will be zero. It’s worth remembering that it isn’t the absolute stellar rotation velocity that determines the Doppler shifting (and thus the RM effect amplitude), but the apparent rotational velocity, v sin i.

Another degeneracy affecting the RM effect is that impact parameters close to zero have the planet occupying both hemispheres more equally in time. This causes the RM effect to be much more symmetric than an off-centre transit with a high impact parameter. For the case of b = 0, the RM effect will be symmetric regardless of the value for λ, and only the RM effect amplitude will have changed. Since the RM effect amplitude is also a function of v sin i, some ambiguity is present in low- impact parameter systems over the value of λ.

The overwhelming majority of planets which have had RM effects measured are hot Jupiters, whose short orbital periods and high mass cause high orbital RV amplitudes, leaving the RM effect amplitude to be small in comparison. Interestingly, however, the magnitude of the RM effect amplitude is not dependent on the planet’s mass or semi-major axis, but on only the stellar v sin i and Rp. A planet of a given radius will have the same RM effect amplitude whether it is close to the star, producing a high orbital RV amplitude, or far from the star, with a much smaller orbital RV amplitude.

Accordingly, transiting planets in long-period orbits may have RM effect amplitudes several times their orbital RV amplitude. The advantages of this become quite apparent when one takes into consideration the timescales involved. Without the Rossiter-McLaughlin effect, a Doppler spectroscopic detection of an Earth-analogue around a solar-type star (with v sin i = 5 km s-1) would require at least a year to secure a full orbit, and furthermore, would require a Doppler precision on the order of 10 cm s-1. With noise issues in the RV data, it may require several orbits to build up enough data to confirm the planet. However, the Rossiter-McLaughlin effect for the same planet would be on the order of ~30 cm s-1 and would last less than a day. The far shorter time needed to observe the RM effect and the far greater amplitude make this a promising tool to validate extrasolar planet candidates whose existence are known from transits alone, but whose mass may take a considerable amount of effort to determine. Instead of wasting months trying to validate a planet candidate and run the risk that the candidate is a false-positive, a single night can conclusively demonstrate that the object is an orbiting planet.

Rossiter-McLaughlin Effect for a transiting Earth analogue

Because the rotational velocity of a star is correlated with its age and mass, which itself is correlated to the spectral type of the star, the true rotational velocity of a star is loosely correlated to the spectral type. Comparing a measured v sin i to the expected rotational velocity can give you a loose estimate for the inclination of the stellar rotation axis to the line of sight. This, combined with λ, can permit you to estimate the true spin-orbit alignment angle, typically denoted with φ (though variations on this definitely exist).

Constraining the spin-orbit alignment angle of extrasolar planetary systems provides clues to their dynamical histories. It is still not clear how hot Jupiter systems form. Giant planets may tidally interact with the planet-forming disk to migrate inward toward the star, or they may be gravitationally scattered inward by close encounters with other planets. Either scenario has testable predictions in the context of the observed spin-orbit alignment of a planetary system.

Peaceful migration through disk-interaction is expected to result in well-aligned planetary systems. Like in our Solar System, planets that form in a protoplanetary disk coplanar with the stellar equator will maintain that orientation, and you’ll end up with planets orbiting close to the stellar equatorial plane. However if hot Jupiters are the result of the scattering of planets through close encounters with each other, then the migration behaviour of hot Jupiters will have been much more chaotic in the early history of the planetary system and the spin-orbit alignment angle may have a wide range of values.

The first few hot Jupiters to have their RM effects measured were all fairly well aligned, consistent with calm migration as was generally believed to be the most likely explanation of their origin. Later work with planets from the SuperWASP survey uncovered several misaligned planets, including even retrograde hot Jupiters. Further work since has found planets across a wide range of values for λ.

Hot Jupiter Alignments

As spin-orbit alignment angles came in over the years it became apparent that misaligned planets are preferentially around hotter stars. Specifically above stellar effective temperatures of Teff > 6250 K, the distribution of λ seems to be much more random. This is clearly a clue to understanding the dynamical histories of hot Jupiters, but it’s not currently clear what it means.

Not only will measuring λ for extrasolar planet systems help us understand their formation histories, but it can also prove to be a powerful tool for confirming small, long-period planets with relatively little effort. It can require fairly high quality RV data, but it is well worth the effort to obtain these values, especially for smaller planets to see how the dynamical histories of low-mass, short-period planets compare to those of hot Jupiters. Do these multi-planet systems of low-mass planets form in a similar way to hot Jupiters? So far the evidence appears to point to “no,” but more data is needed to understand this question.