Planet Occurrence Rates from Doppler and Transit Surveys Joshua N. Winn Abstract Prior to the 1990s, speculations about the occurrence of planets around other stars were informed only by planet formation theory, observations of circum- stellar disks, and the knowledge that at least one seemingly ordinary star — the Sun — had managed to make a variety of different planets. Since then, Doppler and transit surveys have exposed the population of planets around other Sun-like stars, especially those with orbital periods shorter than a few years. Over the last decade these surveys rose to a new level of perfection with Doppler spectrographs capable of 1 m s−1 precision, and space telescopes capable of detecting the transits of Earth- sized planets. This article is a brief introductory review of the knowledge of planet occurrence that has been gained from these surveys. Introduction If, in some cataclysm, all our knowledge of exoplanets were to be destroyed, and only one sentence passed on to the next generation of astronomers, what statement would contain the most helpful information? This challenge, a modified version of the one posed by Richard Feynman in his Lectures on Physics, is difficult to meet with words. Here is one possibility: Most Sun-like stars have planets, which display a wider range of properties — size, mass, orbital parameters — than the planets of the Solar System. The job would be easier if we could convey a mathematical function. The oc- currence rate density (or simply occurrence) is the expected number of planets per star that have certain properties. For example, transit observations reveal a planet’s radius R and orbital period P. We can summarize the findings of a transit survey with a function Princeton University, Department of Astrophysical Sciences, 4 Ivy Lane, Princeton, NJ 08544, USA, e-mail: [email protected] 1 2 Joshua N. Winn d2N G ≡ ; (1) R;P d logR d logP giving the mean number of planets per logarithmic interval of radius and period. A simple and accurate occurrence function would help our descendants design new instruments to detect planets, and inspire theories for planet formation. Occurrence depends on other planetary parameters, such as orbital eccentricity; and on the characteristics of the star, such as mass, metallicity, and age. Occurrence is also likely to depend on the properties of any other planets that exist in a particular system. No analytic function could possibly represent all these parameters and their correlations. Ideally we would transmit a computer program that produces random realizations of planetary systems that are statistically consistent with everything we have learned from planet surveys. What follows is an brief introductory review of the progress toward this goal that has come from Doppler and transit surveys. The cited works were chosen to be useful entry points to the literature, not to provide a comprehensive bibliography. The basics of the Doppler and transit methods themselves are left for other reviews, such as those by Lovis and Fischer (2010), Winn (2010), Wright (this volume) and Bakos (this volume). The next section describes methods for occurrence calcula- tions. Surveys have shown major differences in occurrence between giant planets and small planets, with a dividing line of about 6 R⊕ or 30 M⊕. Thus the results for giants and small planets are presented separately, in two sections. After that comes a review of what is known about other types of stars, followed by a discussion of future prospects. Methods Life would be simple if planets came in only one type, and we could detect them unerringly. We would search N stars, detect Ndet planets, and conclude that the in- tegrated occurrence is G ≈ Ndet=N. But detection is not assured, because small sig- nals can be lost in the noise. If the detection probability were pdet in all cases, then we would have effectively searched only pdetN stars, and the estimated occurrence would be Ndet=(pdetN). In reality, pdet depends strongly on the characteristics of the star and planet, as illustrated in Figure 1. Detection is easier for brighter stars, larger planets (relative to the star), and shorter orbital periods. For this reason we need to group the detected planets according to orbital period and other salient characteristics for detection: radius R, for transit surveys; and m ≡ M sinI for Doppler surveys, where M is the mass and I is the orbital inclination. Then our estimate becomes N −1 Gi ≈ Nd ∑ pdet;i j; (2) j=1 Planet Occurrence Rates 3 1000 100 ] ⊕ 3/2 Mmin~P mass [M 10 1/3 Survey Mmin~P duration 1 1 10 100 1000 10000 orbital period [days] Fig. 1 Idealized Doppler survey of 104 identical Sun-like stars. Each star has one planet on a randomly-oriented circular orbit, with a mass and period drawn from log-uniform distributions between the plotted limits. Each star is observed 50 times over one year with 1 m s−1 precision. The small dots are all the planets; the large blue dots are those detected with 5s confidence. For P shorter than the survey duration, the threshold mass is proportional to P1=3, corresponding to a constant Doppler semi-amplitude K ≈ 1 m s−1. For longer P, the threshold mass increases more rapidly, with an exponent depending on the desired confidence level (Cumming 2004). where the indices i and j specify the type of planet and star, respectively. Transit surveys have the additional problem that there is no signal at all unless cosI < R?=ac, where ac is the orbital distance at the time of inferior conjunction. Thus we must also divide by the probability for this condition to be met, which is equal to R?=a for randomly-oriented circular orbits. This conceptually simple method has been the basis of many investigations. The results of Doppler surveys are often presented as a matrix of G values for rectangular regions in the space of m ≡ M sinI and P; for transit surveys the regions are in the space of R and P. Ideally, each region should be large enough to contain many detected planets, and yet small enough that the detection probability does not vary too much from one side to the other. In practice these conditions are rarely achieved, and more complex methods are preferable. One approach is to posit a functional form for the occurrence rate density, such as a power law d2N G = = Cma Pb ; (3) m;P d lnm d lnP 4 Joshua N. Winn 9 8 7 2 x survey 6 duration 5 ] ⊕ 4 3 radius [R 2 1/6 Rmin ~ P 1 1 10 100 1000 10000 orbital period [days] Fig. 2 Idealized transit survey of 104 identical Sun-like stars. Each star has one planet on a randomly-oriented circular orbit, with a radius and period drawn from log-uniform distributions between the plotted limits. Each star is observed continuously for one year with a photon-limited photometric precision corresponding to 100 parts per million over 6 hours. The small dots are all the planets; the large blue dots are those detected with 10s confidence based on at least two transit detections. Compared to the Doppler survey, the transit survey finds fewer planets and is more strongly biased toward short periods. This is because the geometric transit probability is low and is proportional to P−2=3. The threshold radius varies as P1=6 out to twice the survey duration (Pepper et al. 2003). and use this function to construct a likelihood function for the outcome of a survey. The likelihood function must take into account the detection probability, the proper- ties of the detected systems, and the properties of the stars for which no planets were detected. Then the values of the adjustable parameters C, a, and b are determined by maximizing the likelihood. Details of this method are provided by Tabachnik and Tremaine (2002) or Cumming et al. (2008) for Doppler surveys, and Youdin (2011) for transit surveys. Foreman-Mackey et al. (2014) generalized this technique to cope with uncertainties in the planet properties such as R and P. They also cast the problem in the form of Bayesian hierarchical inference. Most studies report the probability for a star to have a planet with certain prop- erties, regardless of any other planets in the system. Accounting for multiple-planet systems is more difficult. For Doppler surveys, the main problem is that the star is pulled by all the planets simultaneously. As a result, the detectability of a given signal to depend on the properties of any other detectable planets — especially their periods — and on the timespan and spacings between the data points. This makes it difficult to calculate the detection probability. Planet Occurrence Rates 5 For transit surveys, the different planetary signals do not overlap very much. In- stead, the problem is the degeneracy between multiplicity and inclination dispersion (Tremaine and Dong 2012). A star with only one detected planet could lack addi- tional planets, or it could have a system of many planets only one of which has an inclination close enough to 90◦ to exhibit transits. In principle this degeneracy can be broken by combining the results of Doppler and transit surveys and solving for both the occurrence rate density as well as the planetary mass-radius relationship. Doppler surveys have uncovered a total of about 500 planets. The most infor- mative surveys for planet occurrence were based on observations with the High Resolution Echelle Spectrometer (HIRES) on the Keck I 10-meter telescope (Cum- ming et al. 2008; Howard et al. 2010) and the High Accuracy Radial-velocity Planet Searcher (HARPS) on the La Silla 3.6-meter telescope (Mayor et al.