A Posteriori Transit Probabilities

Typically, when estimating the prior transit probability, one assumes a uniform distribution for the cosine of the inclination angle i of the companion's orbit, which yields the familiar estimate of ~R_*/a. 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 M_c. 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 with exponent alpha. For low transit probabilities, these probabilities reduce to a constant multiplicative factor of the corresponding prior transit probability. The prior and posterior probabilities are equal for alpha = -1, whereas the posterior transit probability is ~1.5 times larger and ~4/pi larger for for alpha = -3 and alpha = -2, but is less than the prior for alpha >= 0, and can be arbitrarily small for alpha > 1. We also calculate the posterior transit probability in different mass regimes for two physically-motivated mass distributions of companions around Sun-like stars, finding that the posterior is likely higher for Super-Earths, Neptunes and Super-Jupiters. 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.