arXiv:1107.2414v2 [astro-ph.EP] 19 Feb 2013 Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 20 February 2013 (MN L A T E X style file v2.2) Secular Dynamics in Hierarchical Three-Body Systems Smadar Naoz 1,2,† , Will M. Farr 1 , Yoram Lithwick 1,3 , Frederic A. Rasio 1,3 , Jean Teyssandier 1,4 1 CIERA, Northwestern University, Evanston, IL 60208, USA 2 Institute for Theory and Computation, Harvard-Smithsonian Center for Astrophysics, 60 Garden St.; Cambridge, MA, USA 02138 3 Department of Physics and Astronomy, Northwestern University 4 Institut d’Astrophysique de Paris, UMR 7095, CNRS, UPMC, 98 bis bd Arago, F-75014 Paris † Einstein Fellow 20 February 2013 ABSTRACT The secular approximation for the evolution of hierarchical triple configurations has proven to be very useful in many astrophysical contexts, from planetary to triple-star systems. In this approximation the orbits may change shape and orientation, on time scales longer than the orbital time scales, but the semimajor axes are constant. For example, for highly inclined triple systems, the Kozai-Lidov mechanism can produce large-amplitude oscillations of the eccentric- ities and inclinations. Here we revisit the secular dynamics of hierarchical triple systems. We derive the secular evolution equations to octupole order in Hamiltonian perturbation theory. Our derivation corrects an error in some previous treatments of the problem that implicitly assumed a conservation of the z-component of the angular momentum of the inner orbit (i.e., parallel to the total angular momentum of the system). Already to quadrupole order, our re- sults show new behaviors including the possibility for a system to oscillate from prograde to retrograde orbits. At the octupole order, for an eccentric outer orbit, the inner orbit can reach extremely high eccentricities and undergo chaotic flips in its orientation. We discuss applica- tions to a variety of astrophysical systems, from stellar triples to merging compact binaries and planetary systems. Our results agree with those of previous studies done to quadrupole order only in the limit in which one of the inner two bodies is a massless test particle and the outer orbit is circular; our results agree with previous studies at octupole order for the eccentricity evolution, but not for the inclination evolution. 1 INTRODUCTION Triple star systems are believed to be very common (e.g., Tokovinin 1997; Eggleton et al. 2007). From dynamical stabil- ity arguments these must be hierarchical triples, in which the (inner) binary is orbited by a third body on a much wider or- bit. Probably more than 50% of bright stars are at least double (Tokovinin 1997; Eggleton et al. 2007). Given the selection ef- fects against finding faint and distant companions we can be rea- sonably confident that the proportion is actually substantially greater. Tokovinin (1997) showed that 40% of binary stars with period < 10 d in which the primary is a dwarf (0.5 − 1.5 M⊙) have at least one additional companion. He found that the frac- tion of triples and higher multiples among binaries with period (10 − 100 d) is ∼ 10%. Moreover, Pribulla and Rucinski (2006) have surveyed a sample of contact binaries, and noted that among 151 contact binaries brighter than 10 mag., 42±5% are at least triple. Many close stellar binaries with two compact objects are likely produced through triple evolution. Secular effects (i.e., coherent interactions on timescales long compared to the or- bital period), and specifically Kozai-Lidov cycling (Kozai 1962; Lidov 1962, see below), have been proposed as an impor- tant element in the evolution of triple stars (e.g. Harrington 1969; Mazeh and Shaham 1979; S¨ oderhjelm 1982; Kiseleva et al. 1998; Fabrycky and Tremaine 2007; Perets and Fabrycky 2009; Thompson 2011; Shappee and Thompson 2012). In addition, Kozai-Lidov cycling has been suggested to play an important role in both the growth of black holes at the centers of dense star clusters and the formation of short-period binary black holes (Wen 2003; Miller and Hamilton 2002; Blaes et al. 2002). Re- cently, Ivanova et al. (2010) showed that the most important formation mechanism for black hole XRBs in globular clusters may be triple-induced mass transfer in a black hole-white dwarf binary. Secular perturbations in triple systems also play an impor- tant role in planetary system dynamics. Kozai (1962) studied the effects of Jupiter’s gravitational perturbation on an inclined asteroid in our own solar system. In the assumed hierarchical con- c  0000 RAS