Detectability of extrasolar planets in radial velocity surveys

Radial velocity surveys are beginning to reach the time baselines required to detect Jupiter analogues, as well as sub-Saturn mass planets in close orbits. Therefore, it is important to understand the sensitivity of these surveys at long periods and low amplitudes. In this paper, I derive analytical expressions for the detectability of planets at both short and long periods, for circular and eccentric orbits. In the long-period regime, the scaling of the detection threshold with period depends on the desired detection efficiency. The 99 per cent velocity threshold scales as K proportional to P-2 proportional to a(3), whereas the 50 per cent velocity threshold scales as K proportional to P proportional to a(3/2). I suggest an extension of the Lomb-Scargle statistic to Keplerian orbits, and describe how to estimate the false alarm probability associated with a Keplerian fit. I use this Keplerian periodogram to investigate the effect of eccentricity on detectability. At short periods, detectability is reduced for eccentric orbits, mainly due to the sparse sampling of the periastron passage, whereas long-period orbits are easier to detect on average if they are eccentric because of the steep velocity gradients near periastron. Fitting Keplerian orbits allows the lost sensitivity at short orbital periods to be recovered for e less than or similar to 0.6. However, there remain significant selection effects against eccentric orbits for e less than or similar to 0.6, and the small number of highly eccentric planets discovered so far may reflect this. Finally, I present a Bayesian approach to the periodogram which gives a simple derivation of the probability distributions of noise powers, clarifies why the periodogram is an appropriate way to search for long-period signals, and emphasizes the equivalence of periodogram and least-squares techniques.