The Lick planet search: Detectability and mass thresholds

We present an analysis of 11 yr of precision radial velocity measurements of 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. For the remaining stars, we place upper limits on the companion mass as a function of orbital period. For most stars, we can exclude companions with velocity amplitude K greater than or similar to 20 m s(-1) at the 99% level, or 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 greater 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 should 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 detected 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 paucity of confirmed and candidate companions with orbital radii between similar to 0.2 and similar to 1 AU. The small number of confirmed companions means that we are not able to rule out selection effects as the cause of these features. We show that the traditional method for detecting periodicities, the Lomb-Scargle periodogram, fails to account for statistical fluctuations 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 vary during the fit. We discuss in detail the normalization of the periodogram and the probability distribution of periodogram powers. We stress that the three different prescriptions in the literature for normalizing the periodogram are statistically equivalent and that it is not possible to write a simple analytic form for the false alarm probability, making Monte Carlo methods essential.