The Lick planet search: Detectability and mass thresholds

We present an analysis of 11 yr of precision radial velocity measurements o f 76 nearby solar-type stars from the Lick radial velocity survey. For each star, we report on variability, periodicity, and long-term velocity trends . Our sample of stars contains eight known companions with mass (M-p sin i) less than 8 Jupiter masses (M-J), six of which were discovered at Lick. Fo r the remaining stars, we place upper limits on the companion mass as a fun ction of orbital period. For most stars, we can exclude companions with vel ocity amplitude K greater than or similar to 20 m s(-1) at the 99% level, o r M-p sin i greater than or similar to 0.7M(J)(a/AU)(1/2) for orbital radii a less than or similar to 5 AU. We examine the implications of our results for the observed distribution of mass and orbital radius of companions. We show that the combination of intrinsic stellar variability and measurement errors most likely explains why all confirmed companions so far have K gre ater than or similar to 40 m s(-1). The finite duration of the observations limits detection of Jupiter-mass companions to a less than or similar to 3 AU. Thus it remains possible that the majority of solar-type stars harbor Jupiter-mass companions much like our own, and if so these companions shoul d be detectable in a few years. It is striking that more massive companions with M-p sin i > 3M(J) are rare at orbital radii 4-6 AU; we could have det ected such objects in similar to 90% of stars, yet found none. The observed companions show a "piling-up" toward small orbital radii, and there is a p aucity of confirmed and candidate companions with orbital radii between sim ilar to 0.2 and similar to 1 AU. The small number of confirmed companions m eans that we are not able to rule out selection effects as the cause of the se features. We show that the traditional method for detecting periodicitie s, the Lomb-Scargle periodogram, fails to account for statistical fluctuati ons in the mean of a sampled sinusoid, making it nonrobust when the number of observations is small, the sampling is uneven, or for periods comparable to or greater than the duration of the observations. We adopt a "floating- mean" periodogram, in which the zero point of the sinusoid is allowed to va ry during the fit. We discuss in detail the normalization of the periodogra m and the probability distribution of periodogram powers. We stress that th e three different prescriptions in the literature for normalizing the perio dogram are statistically equivalent and that it is not possible to write a simple analytic form for the false alarm probability, making Monte Carlo me thods essential.