## RV detections

Consider a star-planet system, where the planet's orbit is circular, for simplicity. By a simple application of Newton's laws, we can see that the star performs a reflex circular motion about the common centre of mass of the star and planet, with the same period (P) as the planet. The radius of the star's orbit is then given by:

where a is the radius of the planet orbit, and Mp and M+ are the planet mass and the star mass, respectively. The motion of the star results in the periodic perturbation of various observables that can be used to detect this motion. The RV technique focuses on the periodic perturbation of the line-of-sight component of the star's velocity.

Astronomers routinely measure RVs of objects ranging from Solar System minor planets to distant quasars. The basic tool to measure RV is the spectrograph, which disperses the light into its constituent wavelengths, yielding the stellar spectrum. Stars like our Sun, the so-called main-sequence stars, have well-known spectra. Small shifts in the wavelengths of the observed spectrum can tell us about the star's RV through the Doppler effect. Thus, a Doppler shift of AX in a feature of rest wavelength X in the stellar spectrum corresponds to an RV of:

where c is the speed of light.

The most obvious parameters which characterize the periodic modulation of the RV are the period, P, and the semi-amplitude, K (Figure 1.1(a)). These two parameters are related to the planet mass via the general formula (e.g., Cumming etal. (1999)):

In this formula G is the universal gravitational constant, and e is the orbital eccentricity. The inclination of the orbital axis relative to the line of sight is denoted

1.2 RV detections

1.2 RV detections

Time
Fig. 1.1. (a) A schematic illustration of a periodic RV curve of a planetary orbit, showing the two quantities P (period) and K (semi-amplitude). (b) Visualization of the inclination angle (i), the angle between the orbital axis and the line of sight.

by i (Figure 1.1(b)). In a circular orbit we can neglect e and, assuming that the planet mass is much smaller than the stellar mass, we can derive the empirical formula:

Mj denotes Jovian mass, and MQ stands for the Solar mass. (Extrasolar planets are typically the mass of Jupiter, hence we normalize our formulae using 'Jovian mass'.)

Close examination of Equation (1.4) reveals several important points. First, K has a weak inverse dependence on P, which means that the RV technique is biased towards detecting short-period planets. Second, the planet mass and the inclination appear only in the product Mp sin i, and therefore they cannot be derived separately using RV data alone. In principle, a planetary orbit observed edge-on (i close to 90°) will have exactly the same RV signature as a stellar orbit observed face-on (i close to 0). Statistics help to partly solve the conundrum, since values of sin i which are close to unity are much more probable than smaller values (e.g., Marcy and Butler (1998)). In fact, for a randomly oriented set of orbits, the mean value of sin i is easily shown to be 4/n ~ 0.785. Obviously, a better solution would be to seek independent information about the inclination.

Equation (1.4) shows the order of magnitude of the desired effect - tens or hundreds of metres per second. Detecting effects of this magnitude requires a precision of the order of metres per second. Such a precision was almost impossible to achieve before the 1990s. Before that time, the only claim of a very low-mass companion detected via RV was of the companion of the star HD 114762. The semi-amplitude of the RV variation was about 600 m s-1, and the companion mass was found to be around 10 Mj (Latham etal., 1989; Mazeh etal., 1996). Although the existence of this object is well established, the question of its planetary nature is still debated. Alternatively, it could be a brown dwarf - an intermediary object between a planet and a star. The detection of smaller planet candidates had to await the development of instruments that could measure precise RVs.

Campbell and Walker (1979) were the first to obtain RVs of the required precision. They introduced an absorption cell containing hydrogen fluoride gas in the optical path of the stellar light in order to overcome systematic errors in the RVs, using the known spectrum of the gas for calibration. They carried out a pioneering survey of 16 stars over a period of 6 years, which yielded no detections, probably because of the small sample size (Campbell etal., 1988).

Fig. 1.2. The phase-folded RV curve of 51 Peg, from Mayor and Queloz (1995).

Fig. 1.2. The phase-folded RV curve of 51 Peg, from Mayor and Queloz (1995).

1.3 Transit detections

The first planet candidate detected using precise RV measurements was 51 Peg b. Mayor and Queloz (1995) used the fibre-fed ELODIE spectrograph in the Haute-Provence Observatory (Baranne et al., 1996), and obtained an RV curve of 51 Peg corresponding to a planet with a mass of 0.44 MJ and an orbital period of 4.23 days (Figure 1.2). This short period means an orbital distance of 0.05 AU from the host star. The discovery was soon confirmed by Marcy et al. (1997), using the Hamilton echelle spectrograph at the Lick Observatory, with the iodine absorption cell technique (Butler et al., 1996). This proximity to the host star was a major surprise and it actually contradicted the previous theories about planetary system formation. This discovery, and those of many similar planets that followed (now nicknamed 'Hot Jupiters'), led to a revision of those theories, and to the development of the planetary migration paradigm (e.g., Lin et al. (1996)). The current state of the formation and evolution theories is reviewed in Chapter 3.

Since that first detection, several groups have routinely performed RV measurements. The most prominent groups are the Geneva group, using fibre-fed spectrographs (Baranne et al., 1996), and the Berkeley group, using the iodine-cell technique (Butler et al., 1996).