Stability of multiplanet systems in binaries

Dipartimento di Fisica, University of Padova, via Marzolo 8, 35131 Padova, Italy
e-mail: Francesco.Marzari@pd.infn.it

Received: 19 February 2016
Accepted: 19 July 2016

Abstract

Context. When exploring the stability of multiplanet systems in binaries, two parameters are normally exploited: the critical semimajor axis a_c computed by Holman & Wiegert (1999, AJ, 117, 621) within which planets are stable against the binary perturbations, and the Hill stability limit Δ determining the minimum separation beyond which two planets will avoid mutual close encounters. Both these parameters are derived in different contexts, i.e. Δ is usually adopted for computing the stability limit of two planets around a single star while a_c is computed for a single planet in a binary system.

Aims. Our aim is to test whether these two parameters can be safely applied in multiplanet systems in binaries or if their predictions fail for particular binary orbital configurations.

Methods. We have used the frequency map analysis (FMA) to measure the diffusion of orbits in the phase space as an indicator of chaotic behaviour.

Results. First we revisited the reliability of the empirical formula computing a_c in the case of single planets in binaries and we find that, in some cases, it underestimates by 10–20% the real outer limit of stability and it does not account for planets trapped in resonance with the companion star well beyond a_c. For two-planet systems, the value of Δ is close to that computed for planets around single stars, but the level of chaoticity close to it substantially increases for smaller semimajor axes and higher eccentricities of the binary orbit. In these configurations a_c also begins to be unreliable and non-linear secular resonances with the stellar companion lead to chaotic behaviour well within a_c, even for single planet systems. For two planet systems, the superposition of mean motion resonances, either mutual or with the binary companion, and non-linear secular resonances may lead to chaotic behaviour in all cases. We have developed a parametric semi-empirical formula determining the minimum value of the binary semimajor axis, for a given eccentricity of the binary orbit, below which stable two planet systems cannot exist.

Conclusions. The superposition of different resonances between two or more planets and the binary companion may prevent the existence of stable dynamical configurations in binaries. As a consequence, care must be devoted when applying the Holman and Wiegert criterion and the Hill stability against mutual close encounters for a multiplanet system in binaries.