Submitted for publication in the Astrophysical Journal A Preprint typeset using LTEX style emulateapj v. 5/2/11 CYCLIC TRANSIT PROBABILITIES OF LONG-PERIOD ECCENTRIC PLANETS DUE TO PERIASTRON PRECESSION Stephen R. Kane1, Jonathan Horner2, Kaspar von Braun1 Submitted for publication in the Astrophysical Journal 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. Subject headings: planetary systems – celestial mechanics – ephemerides – techniques: photometric 1. INTRODUCTION consequence of these precession effects is that a planet The realization that we have crossed a technol- that exhibits visible transits now may not do so at a ogy threshold that allows transiting planets to be de- different epoch and vice versa. tected sparked a flurry of activity in this direction af- Here we present a study of some precession effects on ter the historic detection of HD 209458 b’s transits known exoplanets. The aspect which sets this apart (Charbonneau et al. 2000; Henry et al. 2000). This has from previous studies is that we are primarily interested resulted in an enormous expansion of exoplanetary sci- in planets not currently known to transit, particularly ence such that we can now explore the mass-radius long-period eccentric planets which have enhanced tran- relationship (Burrows et al. 2007; Fortney et al. 2007; sit probabilities and larger precession effects. We inves- Seager et al. 2007) and atmospheres (Agol et al. 2010; tigate the subsequent rate of change of the transit prob- Deming et al. 2007a; Knutson et al. 2009a,b) of planets ability to show how they drift in and out of a transit- outside of our Solar System. Most of the known tran- ing orientation. We calculate the timescales and rates of siting planets were discovered using the transit method, change for the precession and subsequent transit prob- but some were later found to transit after first being abilities and discuss implications for the timescales on detected using the radial velocity technique. Two no- which radial velocity planets will enter into a transiting table examples are HD 17156 b (Barbieri et al. 2007) configuration, based upon assumptions regarding their and HD 80606 b (Laughlin et al. 2009), both of which orbital inclinations. We finally compare periastron argu- are in particularly eccentric orbits. Other radial velocity ment uncertainties to the expected precession timescales planets are being followed up at predicted transit times and suggest orbital refinement as a means to measure (Kane et al. 2009) by the Transit Ephemeris Refinement this effect. and Monitoring Survey (TERMS). Planets in eccentric orbits are particularly inter- 2. TRANSIT PROBABILITY arXiv:1208.4115v2 [astro-ph.EP] 7 Sep 2012 esting because of their enhanced transit probabilities Here we briefly describe the fundamentals of the geo- (Kane & von Braun 2008, 2009). This orbital eccentric- metric transit probability for both circular and eccentric ity also makes those planets prone to orbital precession. orbits. For a detailed description we refer the reader to In celestial mechanics, there are several kinds of pre- Kane & von Braun (2008). cession which can affect the orbital properties, spin In the case of a circular orbit, the geometric transit rotation, and equatorial plane of a planet. These have probability is defined as follows been studied in detail in reference to known transiting planets, particularly in the context of the precession Rp + R⋆ effects on transit times and duration (Carter & Winn Pt = (1) 2010; Damiani & Lanza 2011; Heyl & Gladman a 2007; Jord´an & Bakos 2008; Miralda-Escud´e 2002; P´al & Kocsis 2008; Ragozzine & Wolf 2009). One where a is the semi-major axis and Rp and R⋆ are the radii of the planet and host star respectively. More gen- erally, both the transit and eclipse probabilities are in- [email protected] 1 NASA Exoplanet Science Institute, Caltech, MS 100-22, 770 versely proportional to the star–planet separation where South Wilson Avenue, Pasadena, CA 91125 the planet passes the star-observer plane that is perpen- 2 Department of Astrophysics & Optics, School of Physics, dicular to the plane of the planetary orbit. The star– University of New South Wales, Sydney, 2052, Australia planet separation as a function of orbital eccentricity e 2 Stephen R. Kane et al. 3. AMPLITUDE OF PERIASTRON (APSIDAL) PRECESSION Periastron (or apsidal) precession is the gradual ro- tation of the major axis which joins the orbital ap- sides within the orbital plane. The result of this pre- cession is that the argument of periastron becomes a time dependent quantity. There are a variety of factors which can lead to periastron precession, such as gen- eral relativity (GR), stellar quadrupole moments, mu- tual star–planet tidal deformations, and perturbations from other planets (Jord´an & Bakos 2008). For Mercury, the perihelion precession rate due to general relativistic effects is 43′′/century (0.0119◦/century). By compari- son, the precession due to perturbations from the other Solar System planets is 532′′/century (0.148◦/century) while the oblateness of the Sun (quadrupole mo- ment) causes a negligible contribution of 0.025′′/century Fig. 1.— Transit probability for a sample of the known exoplanets ◦ as a function of orbital period. In cases where a change in ω from (0.000007 /century) (Clemence 1947; Iorio 2005). current to 90◦ results in a transit probability improvement > 1%, Here we adopt the formalism of Jord´an & Bakos (2008) a vertical arrow indicates the improvement. in evaluating the amplitude of the periastron precession. We first define the orbital angular frequency as is given by GM 2π a(1 − e2) n ≡ ⋆ = (4) r = . (2) r a3 P 1+ e cos f where G is the gravitational constant, M⋆ is the mass of where f is the true anomaly, which describes the location the host star, and P is the orbital period of the planet. of the planet in its orbit, and so is a time dependent The total periastron precession is the sum of the individ- variable as the planet orbits the star. For a transit event ual effects as follows to occur the condition of ω + f = π/2 must be fulfilled (Kane 2007), where ω is the argument of periastron, and ω˙ total =ω ˙ GR +ω ˙ quad +ω ˙ tide +ω ˙ pert (5) so we evaluate the above equations with this condition where the precession components consist of the pre- in place. The geometric transit probability may thus be cession due to GR, stellar quadrupole moment, tidal re-expressed as deformations, and planetary perturbations respectively. Jord´an & Bakos (2008) conveniently express these com- (Rp + R⋆)(1 + e cos(π/2 − ω)) Pt = (3) ponents in units of degrees per century. The components a(1 − e2) −2 −5 ofω ˙ quad andω ˙ tide have a and a dependencies respec- which is valid for any orbital eccentricity. Note that these tively. Since we are mostly concerned with long-period equations are independent of the true inclination of the planets in single-planet systems, we consider here only planet’s orbital plane. the precession due to general relativity since this is the Given the sensitivity of transit probability to the argu- dominant component in such cases. This imposes a lower ment of periastron, it is useful to assess how the proba- limit on the total precession of the system, particularly bilities for the known exoplanets would alter if their ori- for multi-planet systems. This precession is given by the entation was that most favorable for transit detection: following equation ◦ − − ω = 90 . We extracted data from the Exoplanet Data 7.78 M a 1 P 1 Explorer3 (Wright et al. 2011) which include the orbital ω˙ = ⋆ (6) GR − 2 parameters and host star properties for 592 planets and (1 e ) M⊙ 0.05/AU day are current as of 30th June 2012. For each planet, we with units in degrees per century. calculate transit probabilities for two cases: (1) using To examine this precession effect for the known exo- ◦ the current value of ω, and (2) using ω = 90 . The planets, we use the data extracted from the Exoplanet transit probabilities for case (1) are shown in Figure 1. Data Explorer, described in Section 2. The GR preces- Those planets whose case (2) probabilities are improved sion rates for these planets are shown in Figure 2 as a by > 1% are indicated by a vertical arrow to the im- function of eccentricity, where the radius of the point proved probability. There are several features of note in for each planet is logarithmically scaled with the or- this figure. The relatively high transit probabilities be- bital period. As a Solar System example, the precession tween 100 and 1000 days are due to giant host stars whose rate for Mercury is shown using the appropriate symbol. large radii dominates the probabilities (see Equation 3)). There are two distinct populations apparent in Figure There are several cases of substantially improved tran- 2 for which the divide occurs at a periastron precession sit probability, most particularly HD 80606 b, which is of ∼ 0.1◦/century.