A Posteriori Transit Probabilities

Abstract

ABSTRACTGiven the radial velocity (RV) detection of an unseen companion, it is often of interest to estimate the probability that the companion also transits the primary star. Typically, one assumes a uniform distribution for the cosine of the inclination angle i of the companion’s orbit. This yields the familiar estimate for the prior transit probability of ∼R∗/a, given the primary radius R∗ and orbital semimajor axis a, and assuming small companions and a circular orbit. However, the posterior transit probability depends not only on the prior probability distribution of i but also on the prior probability distribution of the companion mass Mc, given a measurement of the product of the two (the minimum mass Mc sin i) from an RV signal. In general, the posterior can be larger or smaller than the prior transit probability. We derive analytic expressions for the posterior transit probability assuming a power-law form for the distribution of true masses, , for integer values -3 ≤ α ≤ 3. We show that for low transit probabilities, these probabilities reduce to a constant multiplicative factor fα of the corresponding prior transit probability, where fα in general depends on α and an assumed upper limit on the true mass. The prior and posterior probabilities are equal for α = -1. The posterior transit probability is ∼1.5 times larger than the prior for α = -3 and is ∼4/π times larger for α = -2, but is less than the prior for α≥0, and can be arbitrarily small for α > 1. We also calculate the posterior transit probability in different mass regimes for two physically-motivated mass distributions of companions around Sun-like stars. We find that for Jupiter-mass planets, the posterior transit probability is roughly equal to the prior probability, whereas the posterior is likely higher for Super-Earths and Neptunes (10 M⊕ - 30 M⊕) and Super-Jupiters (3 MJup - 10 MJup), owing to the predicted steep rise in the mass function toward smaller masses in these regimes. We therefore suggest that companions with minimum masses in these regimes might be better-than-expected targets for transit follow-up, and we identify promising targets from RV-detected planets in the literature. Finally, we consider the uncertainty in the transit probability arising from uncertainties in the input parameters, and the effect of ignoring the dependence of the transit probability on the true semimajor axis on i.