Kozai resonance in extrasolar systems

¹ Department of Mathematics FUNDP, 8 Rempart de la Vierge, 5000 Namur, Belgium e-mail: anne-sophie.libert@fundp.ac.be; tsiganis@astro.auth.gr
² Section of Astrophysics Astronomy and Mechanics, Department of Physics, University of Thessaloniki, 54 124, Thessaloniki, Greece

Received: 21 August 2008
Accepted: 13 October 2008

Abstract

Aims. We study the possibility that extrasolar two-planet systems, similar to the ones that are observed, can be in a stable Kozai-resonant state, assuming a mutual inclination of the orbital planes of order .

Methods. Five known multi-planet systems that are not in mean motion resonance were selected, according to defined criteria, as “possible prototypes” (υ Andromedae, HD 12661, HD 169830, HD 74156, HD 155358). We performed a parametric study, integrating several sets of orbits of the two planets, obtained by varying the (unknown) inclination of their orbital planes and their nodal longitudes, thus changing the values of their masses and mutual inclination. We also take into account the reported observational errors on the orbital elements. These numerical results are characterized using analytical secular theory and frequency analysis. Surface of section techniques are also used to distinguish between stable and chaotic motions.

Results. Frequency analysis offers a reliable way of identifying the Kozai resonance in a general reference frame, where the argument of the pericenter of the inner planet does not necessarily librate around ± as in the frame of the Laplace plane, through the non-coupling of the eccentricities of the two planets. We find that four of the five selected systems (υ Andromedae, HD 12661, HD 169830 and HD 74156) could in principle be in Kozai resonance, as their eccentricities and apsidal orientations are such that the system enters in the stability region of the Kozai resonance in 20-70% of the cases, provided that their mutual inclination is at least . Thus, a large fraction of the observed multi-planet systems has observed orbital characteristics that are consistent with stable, Kozai-type, motion in 3D. Unstable sets of orbits are also found, due to the chaos that develops around the stability islands of the Kozai resonance. A variety of physical mechanisms that could generate the necessary large mutual inclinations are discussed, including (a) planet formation; (b) type II migration and resonant interactions during the gas-dominated phase; (c) planetesimal-driven migration and resonance crossing during the gas-free era; (d) multi-planet scattering, caused by the presence of an additional planet.