The Astrophysical Journal, 757:105 (6pp), 2012 September 20 doi:10.1088/0004-637X/757/1/105 C 2012. The American Astronomical Society. All rights reserved. Printed in the U.S.A. CYCLIC TRANSIT PROBABILITIES OF LONG-PERIOD ECCENTRIC PLANETS DUE TO PERIASTRON PRECESSION Stephen R. Kane1, Jonathan Horner2, and Kaspar von Braun1 1 NASA Exoplanet Science Institute, Caltech, MS 100-22, 770 South Wilson Avenue, Pasadena, CA 91125, USA; [email protected] 2 Department of Astrophysics & Optics, School of Physics, University of New South Wales, Sydney 2052, Australia Received 2012 July 19; accepted 2012 August 17; published 2012 September 6 ABSTRACT The observed properties of transiting exoplanets are an exceptionally rich source of information that allows us to understand and characterize their physical properties. Unfortunately, only a relatively small fraction of the known exoplanets discovered using the radial velocity technique are known to transit their host due to the stringent orbital geometry requirements. For each target, the transit probability and predicted transit time can be calculated to great accuracy with refinement of the orbital parameters. However, the transit probability of short period and eccentric orbits can have a reasonable time dependence due to the effects of apsidal and nodal precession, thus altering their transit potential and predicted transit time. Here we investigate the magnitude of these precession effects on transit probabilities and apply this to the known radial velocity exoplanets. We assess the refinement of orbital parameters as a path to measuring these precessions and cyclic transit probabilities. Key words: celestial mechanics – ephemerides – planetary systems – techniques: photometric 1. INTRODUCTION effects. We investigate the subsequent rate of change of the transit probability to show how they drift in and out of a The realization that we have crossed a technology threshold transiting orientation. We calculate the timescales and rates of that allows transiting planets to be detected sparked a flurry change for the precession and subsequent transit probabilities of activity in this direction after the historic detection of and discuss implications for the timescales on which radial HD 209458 b’s transits (Charbonneau et al. 2000; Henry velocity planets will enter into a transiting configuration, based et al. 2000). This has resulted in an enormous expansion upon assumptions regarding their orbital inclinations. We finally of exoplanetary science such that we can now explore the compare periastron argument uncertainties to the expected mass–radius relationship (Burrows et al. 2007; Fortney et al. precession timescales and suggest orbital refinement as a means 2007; Seager et al. 2007) and atmospheres (Agol et al. 2010; to measure this effect. Deming et al. 2007; Knutson et al. 2009a, 2009b) of planets outside of our solar system. Most of the known transiting planets 2. TRANSIT PROBABILITY were discovered using the transit method, but some were later found to transit after first being detected using the radial velocity Here we briefly describe the fundamentals of the geometric technique. Two notable examples are HD 17156 b (Barbieri et al. transit probability for both circular and eccentric orbits. For a 2007) and HD 80606 b (Laughlin et al. 2009), both of which are detailed description we refer the reader to Kane & von Braun in particularly eccentric orbits. Other radial velocity planets are (2008). being followed up at predicted transit times (Kane et al. 2009) In the case of a circular orbit, the geometric transit probability by the Transit Ephemeris Refinement and Monitoring Survey is defined as follows: (TERMS). R + R Planets in eccentric orbits are particularly interesting because P = p , (1) of their enhanced transit probabilities (Kane & von Braun 2008, t a 2009). This orbital eccentricity also makes those planets prone to orbital precession. In celestial mechanics, there are several where a is the semi-major axis and Rp and R are the radii of kinds of precession which can affect the orbital properties, the planet and host star, respectively. More generally, both the spin rotation, and equatorial plane of a planet. These have transit and eclipse probabilities are inversely proportional to the been studied in detail in reference to known transiting planets, star–planet separation where the planet passes the star-observer particularly in the context of the precession effects on transit plane that is perpendicular to the plane of the planetary orbit. times and duration (Carter & Winn 2010; Damiani & Lanza The star–planet separation as a function of orbital eccentricity 2011; Heyl & Gladman 2007;Jordan´ & Bakos 2008; Miralda- e is given by Escude´ 2002;Pal´ & Kocsis 2008; Ragozzine & Wolf 2009). a(1 − e2) r = , (2) One consequence of these precession effects is that a planet that 1+e cos f exhibits visible transits now may not do so at a different epoch and vice versa. where f is the true anomaly, which describes the location of the Here we present a study of some precession effects on known planet in its orbit, and so is a time-dependent variable as the exoplanets. The aspect which sets this apart from previous planet orbits the star. For a transit event to occur the condition studies is that we are primarily interested in planets not currently of ω + f = π/2 must be fulfilled (Kane 2007), where ω is the known to transit, particularly long-period eccentric planets argument of periastron, and so we evaluate the above equations which have enhanced transit probabilities and larger precession with this condition in place. The geometric transit probability 1 The Astrophysical Journal, 757:105 (6pp), 2012 September 20 Kane, Horner, & von Braun Figure 1. Transit probability for a sample of the known exoplanets as a Figure 2. Calculated GR periastron precession rates plotted as a function of function of orbital period. In cases where a change in ω from current to 90◦ eccentricity for the known exoplanets with Keplerian orbital solutions. The results in a transit probability improvement >1%, a vertical arrow indicates the radius of the points is logarithmically proportional to the orbital period of the improvement. planet. The symbol for Mercury is used to indicate its position on the plot. ◦ − may thus be re-expressed as (0.0119 century 1). By comparison, the precession due to per- turbations from the other solar system planets is 532 century−1 ◦ −1 (Rp + R)(1 + e cos(π/2 − ω)) (0.148 century ) while the oblateness of the Sun (quadrupole P = , (3) − t a(1 − e2) moment) causes a negligible contribution of 0.025 century 1 (0◦.000007 century−1; Clemence 1947;Iorio2005). which is valid for any orbital eccentricity. Note that these Here we adopt the formalism of Jordan´ & Bakos (2008)in equations are independent of the true inclination of the planet’s evaluating the amplitude of the periastron precession. We first orbital plane. define the orbital angular frequency as Given the sensitivity of transit probability to the argument of periastron, it is useful to assess how the probabilities for GM 2π n ≡ = , (4) the known exoplanets would alter if their orientation was that a3 P most favorable for transit detection: ω = 90◦. We extracted data 3 from the Exoplanet Data Explorer (Wright et al. 2011) which where G is the gravitational constant, M is the mass of the host include the orbital parameters and host star properties for 592 star, and P is the orbital period of the planet. The total periastron planets and are current as of 2012 June 30. For each planet, we precession is the sum of the individual effects as follows: calculate transit probabilities for two cases: (1) using the current ◦ ˙ =˙ ˙ ˙ ˙ value of ω, and (2) using ω = 90 . The transit probabilities for ωtotal ωGR + ωquad + ωtide + ωpert, (5) case (1) are shown in Figure 1. Those planets whose case (2) where the precession components consist of the precession probabilities are improved by >1% are indicated by a vertical due to GR, stellar quadrupole moment, tidal deformations, and arrow to the improved probability. There are several features planetary perturbations, respectively. Jordan´ & Bakos (2008) of note in this figure. The relatively high transit probabilities conveniently express these components in units of degrees between 100 and 1000 days are due to giant host stars whose −2 per century. The components of ω˙ quad and ω˙ tide have a and large radii dominates the probabilities (see Equation (3)). There −5 are several cases of substantially improved transit probability, a dependencies, respectively. Since we are mostly concerned most particularly HD 80606 b, which is labeled in the figure. The with long-period planets in single-planet systems, we consider following sections investigate the periastron precession required here only the precession due to GR since this is the dominant to produce such an increase in transit probability. component in such cases. This imposes a lower limit on the total precession of the system, particularly for multi-planet systems. 3. AMPLITUDE OF PERIASTRON This precession is given by the following equation: (APSIDAL) PRECESSION −1 −1 7.78 M a P ω˙ GR = , (6) Periastron (or apsidal) precession is the gradual rotation of (1 − e2) M 0.05/AU day the major axis which joins the orbital apsides within the or- bital plane. The result of this precession is that the argument with units in degrees per century. of periastron becomes a time-dependent quantity. There are a To examine this precession effect for the known exoplanets, variety of factors which can lead to periastron precession, such we use the data extracted from the Exoplanet Data Explorer, as general relativity (GR), stellar quadrupole moments, mu- described in Section 2. The GR precession rates for these planets tual star–planet tidal deformations, and perturbations from other are shown in Figure 2 as a function of eccentricity, where the planets (Jordan´ & Bakos 2008).