Stability Analysis of Single-Planet Systems and Their Habitable Zones

Abstract

We study the dynamical stability of planetary systems consisting of one hypothetical terrestrial mass planet (1 or 10 M ⊕ ) and one massive planet (10 M ⊕ − 10 M jup ). We consider masses and orbits that cover the range of observed planetary system architectures (including non-zero initial eccentricities), determine the stability limit through N-body simulations, and compare it to the analytic Hill stability boundary. We show that for given masses and orbits of a two planet system, a single parameter, which can be calculated analytically, describes the Lagrange stability boundary (no ejections or exchanges) but which diverges significantly from the Hill stability boundary. However, we do find that the actual boundary is fractal, and therefore we also identify a second parameter which demarcates the transition from stable to unstable evolution. We show the portions of the habitable zones of ρ CrB, HD 164922, GJ 674, and HD 7924 which can support a terrestrial planet. These analyses clarify the stability boundaries in exoplanetary systems and demonstrate that, for most exoplanetary systems, numerical simulations of the stability of potentially habitable planets are only necessary over a narrow region of parameter space. Finally we also identify and provide a catalog of known systems which can host terrestrial planets in their habitable zones.

The Stability and Fates of Hierarchical Two-Planet Systems

“…Even though the Hill stability might not determine the long-term stability of a two-planet system (see §2.1), we will use it as a benchmark (Barnes & Greenberg 2006, 2007. Kopparapu & Barnes (2010) numerically studied the relation between Hill and Lagrange stability and provided fitting expressions to determine the relation between these boundaries. Their results should be applicable to planetary systems consisting of one terrestrial-mass planet and one much more massive planet with initial eccentricities less than 0.6.…”

Three-body resonance overlap in closely spaced multiple-planet systems

“…All extrasolar planetary systems may experience epochs of dynamical instability (e.g. Ford, Havlickova & Rasio 2001; Barnes & Greenberg 2006; Chatterjee et al 2008; Gozdziewski, Breiter & Borczyk 2008; Thommes et al 2008; Gozdziewski & Migaszewski 2009; Raymond, Armitage & Gorelick 2009b; Kopparapu & Barnes 2010; Fabrycky & Murray‐Clay 2010).…”

The three-body problem

“…Prospects for the existence of a terrestrial planet in stellar HZ with known giant exoplanets were discussed by Jones et al [2005]. In more recent work, Kopparapu and Barnes [2010] identified 4 extra-solar planetary systems that can support a terrestrial planet in their HZ's.…”

“…Another set of systems has been proposed for the Jupiter-like planets discovered within a host star's HZ [Kasting et al, 1993, Kopparapu andBarnes, 2010]. Giant exoplanets within this region present a major challenge to the search for terrestrial exoplanets, which could support life.…”