Radial Velocity Christophe Lovis Universite´ de Geneve` Debra A. Fischer Yale University The radial velocity technique was utilized to make the first exoplanet discoveries and continues to play a major role in the discovery and characterization of exoplanetary systems. In this chapter we describe how the technique works, and the current precision and limitations. We then review its major successes in the field of exoplanets. With more than 250 planet detections, it is the most prolific technique to date and has led to many milestone discoveries, such as hot Jupiters, multi-planet systems, transiting planets around bright stars, the planet-metallicity correlation, planets around M dwarfs and intermediate-mass stars, and recently, the emergence of a population of low-mass planets: ice giants and super-Earths. In the near future radial velocities are expected to systematically explore the domain of telluric and icy planets down to a few Earth masses close to the habitable zone of their parent star. They will also be used to provide the necessary follow-up observations of transiting candidates detected by space missions. Finally, we also note alternative radial velocity techniques that may play an important role in the future. 1. INTRODUCTION shifts with resepct to telluric lines. Assuming that telluric lines are at rest relative to the spectrometer, these absorp- Since the end of the XIXth century, radial velocities have tion lines would trace the stellar light path and illuminate been at the heart of many developments and advances in as- the optics in the same way and at the same time as the star. trophysics. In 1888, Vogel at Potsdam used photography to Although Griffin & Griffin did not obtain this high preci- demonstrate Christian Dopplers theory (in 1842) that stars sion, they had highlighted some of the key challenges that in motion along our line of site would exhibit a change in current techniques have overcome. color. This color change, or wavelength shift, is commonly By 1979, Gordon Walker and Bruce Campbell had a ver- known as a Doppler shift and it has been a powerful tool sion of telluric lines in a bottle: a glass cell containing hy- over the past century, used to measure stellar kinematics, to drogen fluoride that was inserted in the light path at the determine orbital parameters for stellar binary systems, and CFHT (Campbell & Walker 1979). Like telluric lines, the to identify stellar pulsations. By 1953, radial velocities had HF absorption lines were imprinted in the stellar spectrum been compiled for more than 15,000 stars in the General and provided a precise wavelength solution spanning about Catalogue of Stellar Radial Velocities (Wilson 1953) with 50 A.˚ The spectrum was recorded with a photon-counting a typical precision of 750 m s−1, not the precision that is Reticon photodiode array. Working from 1980 to 1992, they typically associated with planet-hunting. However, at that monitored 17 main sequence stars and 4 subgiant stars and time, Otto Struve proposed that high precision stellar radial achieved the unprecedented precision of 15 m s−1. Unfor- velocity work could be used to search for planets orbiting tunately, because of the small sample size, no planets were nearby stars. He made the remarkable assertion that Jupiter- found. However, upper limits were set on M sin i for orbital like planets could reside as close as 0.02 AU from their host periods out to 15 years for the 21 stars that they observed stars. Furthermore, he noted that if such close-in planets (Walker et al. 1995). were ten times the mass of our Jupiter, the reflex stellar ve- Cross-correlation speedometers were also used to mea- locity for an edge-on orbit would be about 2 km s−1 and sure radial velocities relative to a physical template. In detectable with 1950’s Doppler precision (Struve 1952). 1989, an object with M sin i of 11 M was discovered Two decades later, Griffin & Griffin (1973) identified a Jup in an 84-day orbit around HD 114762 (Latham et al. 1989). key weakness in radial velocity techniques of the day; the The velocity amplitude of the star was 600 m s−1 and al- stellar spectrum was measured with respect to an emission though the single measurement radial velocity precision spectrum. However, the calibrating lamps (typically tho- was only about 400 m s−1, hundreds of observations effec- rium argon) did not illuminate the slit and spectrometer tively beat down the noise to permit this first detection of a collimator in the same way as the star. Griffin & Griffin substellar object. outlined a strategy for improving Doppler precision to a re- In 1993, the ELODIE spectrometer was commissioned markable 10 m s−1 by differentially measuring stellar line on the 1.93-m telescope at Observatoire de Haute-Provence 1 (OHP, France). To bypass the problem of different light paths for stellar and reference lamp sources described by r1 cos f r1 = ; (3) Griffin & Griffin (1973), two side-by-side fibers were used r1 sin f at ELODIE. Starlight passed through one fiber and light r_ cos f − r f_ sin f _r = 1 1 : (4) from a thorium argon lamp illuminated the second fiber. 1 _ r_1 sin f + r1f cos f The calibration and stellar spectrum were offset in the _ cross-dispersion direction on the CCD detector and a ve- We now need to express r_1 and f as a function of f in locity precision of about 13 m s−1 enabled the detection of order to obtain the velocity as a function of f alone. In a a Jupiter-like planet orbiting 51 Pegasi (Mayor & Queloz first step, we differentiate Eq. 2 to obtain r_1: 1995). In a parallel effort to achieve high Doppler preci- 2 _ 2 _ sion, a glass cell containing iodine vapor was employed by a1e(1 − e )f sin f er1f sin f r_1 = 2 = 2 : (5) Marcy & Butler (1995) to confirm the detection of a plane- (1 + e cos f) a1(1 − e ) tary companion around 51 Peg b. Both Doppler techniques Replacing r_1 in Eq. 4, we obtain after some algebra: have continued to show remarkable improvements in preci- sion, and have ushered in an era of exoplanet discoveries. 2 _ r1f − sin f _r1 = 2 (6) 2. DESCRIPTION OF THE TECHNIQUE a1(1 − e ) cos f + e h1 − sin f 2.1. Radial Velocity Signature of Keplerian Motion = 2 ; (7) m1a1(1 − e ) cos f + e The aim of this section is to derive the radial velocity 2 _ equation, i.e. the relation between the position of a body where h1 = m1r1f is the angular momentum of the first on its orbit and its radial velocity, in the case of Keplerian body, which is a constant of the motion. It can be ex- motion. We present here a related approach to Chapter 2 pressed as a function of the ellipse parameters a and e as on Keplerian Dynamics. As shown there, the solutions of (see Chap. 2): the gravitational two-body problem describe elliptical orbits around the common center of mass if the system is bound, s m Gm2m4a(1 − e2) with the center of mass located at a focus of the ellipses. 2 1 2 h1 = h = 3 : (8) Energy and angular momentum are constants of the motion. m1 + m2 (m1 + m2) The semi-major axis of the first body orbit around the center Substituting h1 in Eq. 6, we finally obtain: of mass a1 is related to the semi-major axis of the relative orbit a through: s Gm2 1 − sin f m 2 2 _r1 = 2 : (9) a1 = a ; (1) m1 + m2 a(1 − e ) cos f + e m1 + m2 where m1 and m2 are the masses of the bodies. As a final step, the velocity vector has to be projected In polar coordinates, the equation of the ellipse described onto the line of sight of the observer. We define the inclina- by the first body around the center of mass reads: tion angle i of the system as the angle between the orbital plane and the plane of the sky (i.e. the perpendicular to the line of sight). We further define the argument of periastron 2 2 a1(1 − e ) m2 a(1 − e ) ! as the angle between the line of nodes and the periastron r1 = = · ; (2) 1 + e cos f m1 + m2 1 + e cos f direction (see Fig. 5 of Chapter 2 for an overview of the ge- ometry). In a Cartesian coordinate system with x- and y- where r is the distance of the first body from the center 1 axes in the orbital plane as before and the z-axis perpendic- of mass, e is the eccentricity and f is the true anomaly, i.e. ular to them, the unit vector of the line of sight k is given the angle between the periastron direction and the position by: on the orbit, as measured from the center of mass. The true anomaly as a function of time can be computed via Kepler’s 0 sin ! sin i 1 equation, which cannot be solved analytically, and therefore k = @ cos ! sin i A : (10) numerical methods have to be used (see Chapter 2). cos i In this section we want to obtain the relation between The radial velocity equation is obtained by projecting the position on the orbit, given by f, and orbital velocity. More velocity vector on k: specifically, since our observable will be the radial veloc- ity of the object, we have to project the orbital velocity onto the line of sight linking the observer to the system.