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CALTECH GE 133 - Effects of Secular Interactions in Extrasolar Planetary Systems

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arXiv:astro-ph/0606346 v1 14 Jun 2006Effects of Secular Interactions in Extrasolar Planetary SystemsFred C. Adams1,2and Gregory Laughlin31Michigan Center for Theoretical PhysicsPhysics Department, University of Michigan, Ann Arbor, MI 481092Astronomy Department, University of Michigan, Ann Arbor, MI 481093Lick Observatory, University of California, Santa Cruz, CA 95064ABSTRACTThis paper studies the effects of dynamical interactions among the planets in ob-served extrasolar planetary systems, including hypothetical additional bodies, with afocus on secular perturbations. These interactions cause the eccentricities of the planetsto explore a distribution of values over time scales that are long compared to observa-tional time baselines, but short compared to the age of the systems. The same formalismdetermines the eccentricity forcing of hypothetical test bodies (terrestrial planets) inthese systems and we find which systems allow for potentially habitable planets. Suchplanets would be driven to nonzero orbital eccentricity and we derive the distributionof stellar flux experienced by the planets over the course of their orbits. The generalrelativistic corrections to secular interaction theory are included in the analysis and sucheffects are important in systems with close planets (∼4 day orbits). Some extrasolarplanetary systems (e.g., Upsilon Andromedae) can be used as a test of general relativ-ity, whereas in other systems, general relativity can be used to constrain the sys temparameters (e.g., sin i>∼0.93 for HD160691). For the case of hot Jupiters, we discusshow the absen ce of observed eccentricity implies the absence of companion planets.Subject headings: Stars: Planetary systems1. IntroductionThe number of discovered planets in other systems is rapidly growing and the observationalsample show s an astonishing diversity of architectures (beginning with Mayor & Qu eloz 1995;Marcy & Butler 1996). One way to characterize these planetary systems is in terms of th eirorbital elements, where the semi-major axis a and orb ital eccentricity e are m ost often used. Thesevariables are equivalent to specifying the energy and angular momentum of the orbit. Previousexpectations, informed by the structure of our Solar S ystem, predicted planetary orbits with largervalues of a and smaller values of e than those found in the current observational sample. A majortheoretical effort is n ow being put forth to provide an explanation of the observed distributions of– 2 –orbital elements, e.g., the a −e plane. In multiple planet s ystems, however, it can be important tokeep in mind that the orbital eccentricities can vary dramatically through secular interactions ontime s cales that are long compared to observational baselines b ut short compared to the systemages. Instead of being described by a sin gle value of eccentricity, the orbits of planets in multipleplanet systems should generally be characterized by a complete distribution of eccentricity values(see also Takeda & Rasio 2005).Secular interactions, and the apparent lack of evidence for such interactions, can be used toplace constraints on observed planetary systems. For the known multiple planet systems, secularinteractions place strong constraints on the possib ility of additional as-yet undetected terrestrialplanets. For putative sin gle planet systems w ith zero or low eccentricity, the in ferred absenceof secular interactions (wh ich tend to prevent small eccentricity values e ∼ 0) implies that anyadditional planets in the system must have small masses, small eccentricities, and/or large semi-major axes; we quantify these constraints for observed systems. For mu ltiple planet systems withhot Jupiters (close planetary companions, e.g., in 4 day orbits), secular perturbations tend toincrease the eccentricity of the inner planet, wh ich also experiences tidal stressing forces from thestar. This interplay leads to continual energy dissipation in the planet.This paper follows a large body of previous work. Secular interactions have been studied forcenturies, primarily in the context of our Solar System (Brower & van Woerkom 1950; Laskar 1988;and many others). With the discovery of extrasolar multiple planet systems (starting with UpsilonAndromedae; Butler et al. 1999), secular perturbation theory can be applied to an ever growingcollection of planetary systems (e.g., Wu & Goldreich 2002). General schemes for studying secularinteractions have been developed (Mardling & Lin 2002) and used to study close (hypothetical)terrestrial planets (Mardling & Lin 2004). For hierarchical planetary systems, a higher order(octop ole) approximation scheme has been developed and applied to HD168443 and HD12661 (Lee& Peale 2003) and to triple star systems (Ford et al. 2000ab); an alternate approximation schemehas been developed and applied to the Upsilon Andromedae planetary system (Michtch en ko &Malhotra 2004). The effects of disk potentials, and the disappearance of the disk, h ave beenstudied (Nagasawa et al. 2003) with the goal of explaining observed orbital eccentricities (seealso Lubow & Ogilvie 2001). The effects of the Kozai mechanism have been explored for systemscontaining a binary companion (Takeda & Rasio 2005).This paper builds upon the stud ies outlined above with the modest goal of using secular in-teraction theory to extract additional information from extant multiple planet systems. The basictheory is outlined in §2, which includes the leading order corrections for general relativity (GR).The results and applications are then pr esented in §3. In §3.1, we determine the distributions ofeccentricity sampled by extrasolar planets over the course of their secular cycles, determine the sec-ular time scales, and study the signature of secular eccentricity variations in the a −e diagram. Weplace constraints on the possibility of additional terrestrial planets residing in currently detectedextrasolar planetary systems (§3.2) and place constraints on possible additional giant planets inobserved hot Jupiter systems (§3.4). We explore the effects of general relativity in secular interac-– 3 –tions and show that some systems can be u s ed to test relativity, while in other s y s tems relativitycan be used to p lace constraints on system parameters (§3.3). We conclude, in §4, with a discus sionand summary of results.2. Basic Theory of Secular InteractionsIn this section we outline the basic theory of secular interactions as


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CALTECH GE 133 - Effects of Secular Interactions in Extrasolar Planetary Systems

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