The Astrophysical Journal, 821:47 (15pp), 2016 April 10 doi:10.3847/0004-637X/821/1/47 © 2016. The American Astronomical Society. All rights reserved. EFFICIENT GEOMETRIC PROBABILITIES OF MULTI-TRANSITING EXOPLANETARY SYSTEMS FROM CORBITS Joshua Brakensiek1,2 and Darin Ragozzine3,4 1 Homeschool, Phoenix, AZ, USA; [email protected] 2 Carnegie Mellon University, Department of Mathematical Sciences, Wean Hall 6113, Pittsburgh, PA 15213, USA 3 University of Florida, Department of Astronomy, 211 Bryant Space Science Center, Gainesville, FL 32611, USA; dragozzine@fit.edu 4 Florida Institute of Technology, Department of Physics and Space Sciences, 150 West University Blvd., Melbourne, FL 32940, USA Received 2015 July 23; accepted 2016 February 22; published 2016 April 7 ABSTRACT NASA’s Kepler Space Telescope has successfully discovered thousands of exoplanet candidates using the transit method, including hundreds of stars with multiple transiting planets. In order to estimate the frequency of these valuable systems, it is essential to account for the unique geometric probabilities of detecting multiple transiting extrasolar planets around the same parent star. In order to improve on previous studies that used numerical methods, we have constructed an efficient, semi-analytical algorithm called the Computed Occurrence of Revolving Bodies for the Investigation of Transiting Systems (CORBITS), which, given a collection of conjectured exoplanets orbiting a star, computes the probability that any particular group of exoplanets can be observed to transit. The algorithm applies theorems of elementary differential geometry to compute the areas bounded by circular curves on the surface of a sphere. The implemented algorithm is more accurate and orders of magnitude faster than previous algorithms, based on comparisons with Monte Carlo simulations. We use CORBITS to show that the present solar system would only show a maximum of three transiting planets, but that this varies over time due to dynamical evolution. We also used CORBITS to geometrically debias the period ratio and mutual Hill sphere distributions of Keplerʼs multi-transiting planet candidates, which results in shifting these distributions toward slightly larger values. In an Appendix, we present additional semi-analytical methods for determining the frequency of exoplanet mutual events, i.e., the geometric probability that two planets will transit each other (planet–planet occultation, relevant to transiting circumbinary planets) and the probability that this transit occurs simultaneously as they transit their star. The CORBITS algorithms and several worked examples are publicly available. Key words: celestial mechanics – methods: analytical – methods: numerical – occultations – planetary systems – planets and satellites: detection Supporting material: machine-readable table 1. INTRODUCTION impossible to infer. Detecting all of the planets in multi-planet systems then becomes very unlikely due to natural deviations In the last couple of decades, astronomers have discovered from coplanarity, even when mutual inclinations are 1°. For thousands of exoplanets outside of our solar system through a variety of techniques. The transit method discovers exoplanets example, we show below that if an exact replica of the present by detecting periodic dimming in the amount of light coming solar system were being observed from any direction, at most three of the eight planets would be simultaneously aligned from a star caused by the planet blocking a portion of the star. fi ( ) NASA’s Kepler Space Telescope (hereafter “Kepler”) has used suf ciently well to transit Section 4.1 . Thus, multi-planet this method to detect thousands of planet candidates—a huge systems extend and complicate the known geometric bias step forward—due to its revolutionary data set that is far more inherent to the transit method. fi extensive in duration (∼4 years), duty cycle (∼90%), and We have developed a highly ef cient semi-analytical precision than any previous search (e.g., Burke et al. 2015). algorithm for the calculation of multi-transiting geometry Due to its ability to detect small planets, Kepler has uncovered which we call the Computed Occurrence of Revolving Bodies ( ) a rich vein of systems with tightly spaced inner planets (STIPs), for the Investigation of Transiting Systems CORBITS . which often exhibit multiple transiting planets. Observations of CORBITS is a practical and critical tool for calculating the systems with multiple transiting planets are extremely valuable geometric bias of multi-transiting systems which we make 5 for understanding the formation and evolution of this new freely available to the community. After motivating the need population of STIPs as well as other planetary systems like our for understanding multi-transiting geometry to determine the own (Ragozzine & Holman 2010, hereafter RH10). One frequency of planetary systems (Section 2), we describe the example of the value of these “multi-transiting systems” is that details of the CORBITS algorithm (Section 3). We demonstrate they are far more robust to false positives, which has allowed the value of CORBITS by applying it to example questions for a joint validation of 851 planets in 340 systems, nearly related to the multi-transiting probability of the solar system doubling the number of known exoplanets (Lissauer et al. and known Kepler multi-transiting systems (Section 4). After 2014; Rowe et al. 2014). some concluding remarks (Section 5), we present in an Despite its success, one clear drawback of the transit method Appendix some information on the geometry of exoplanet is that the planetary orbit must line up with the star from our perspective, otherwise the existence of the planet is nearly 5 See https://github.com/jbrakensiek/CORBITS 1 The Astrophysical Journal, 821:47 (15pp), 2016 April 10 Brakensiek & Ragozzine mutual events, some of which is relevant to transiting Johnson 2014). CORBITS allows for accurate calculations of circumbinary planets (TCBPs)(see the Appendix). the multi-transiting geometry, thereby enabling methods that can solve for the average multiplicity, the inclination distribu- tion, NPPS, FSWP, and other interesting and important 2. MOTIVATION information pertinent to the architectures of planetary systems. When planets are considered independently, the geometric detection bias can be computed on a planet-by-planet basis. However, for any exoplanetary properties that depend on 3. CORBITS ALGORITHM FOR MULTI-TRANSITING correlations between planets, the full multi-transiting geometry GEOMETRY must be understood. Extracting information-rich insights from Our goal is to understand how the geometric probability of fi multi-transiting systems therefore requires an ef cient method transiting in a system with multiple transiting planets affects for determining the geometric bias of multi-planet sys- our interpretation of multi-transiting systems. Although there ( ) tems RH10 . are many aspects to this interpretation, our focus here is on the To motivate the importance of understanding multi-transit- purely geometric component of calculating the bias toward ing geometry, we emphasize that a careful and critical detecting or not detecting planets in multiple systems. With this distinction must be made in distinguishing the average number in mind, we convert an astronomical question into a geometry ( ) of planets per star NPPS and the fraction of stars with planets problem. ( ) ( ) FSWP , following the notation of Youdin 2011 . Formally, the problem at hand is this: given a (conjectured) / Most frequency occurrence studies are actually calculating exoplanetary system with all the orbital properties of N planets the average NPPS. When occurrence is calculated planet by known, what is the probability that a random observer would planet without accounting for the complex correlations in be able to observe the transit of a particular m-planet subset? detecting planets in multiple systems, the result is the average This problem can be divided into two parts. NPPS. Understanding NPPS gives us huge insight into planet formation (e.g., period/radius distribution) and other important 1. For a system of planets on fixed orbits that is observed questions (e.g., the frequency of potentially habitable planets). perfectly for an infinite time, what is the probability that a Determining the FSWP requires a detailed analysis that self- specific subset of m out of N planets transit and the other consistently includes deriving the multiplicity and inclination N − m planets do not transit? distributions as discussed below. In this case, the frequency/ 2. How is this idealized case affected under actual occurrence is calculated system by system. FSWP is critical for observational limitations and for realistic planetary scientific questions (e.g., What is the efficiency of planet systems? ) ( formation? and observational questions e.g., How should I Our algorithm addresses the first question, an idealized design my exoplanet survey mission?). ( geometric problem that is very relevant to Kepler multi- The true multiplicity distribution i.e., the distribution of the transiting systems. This calculation is critical, but the ideal case number of planets that orbit each star) connects NPPS and / may not constitute a complete description of the probability of FSWP: the average true multiplicity is the ratio NPPS FSWP. observing real planet transits, primarily for three reasons: duty Even so, NPPS can be accurately calculated when multiple