Stable Configurations of the υ Andromedae Planetary System

The υ Andromedae system is the first exoplanetary system to have the relative inclinations of two planets' orbital planes directly measured (McArthur et al. 2010), and therefore offers our first window into the 3-dimensional configurations of planetary systems. We present a full 3-dimensional, dynamically stable configuration for the 3 planets of the system, following up on McArthur et al (2010), which revealed that the orbits of the outer 2 planets are inclined by 30 deg. We used N-body simulations to search for stable 3-planet configurations that are consistent with the combined radial velocity and astrometric solution. The inner-most planet, b, could have only been detected by HST astrometry if it was at extremely low inclination. Because of this, its true mass and orbital plane are unconstrained by the observations, but our stability analysis limits their ranges significantly.The system appears to be close to the stability boundary, as we find that only 10 trials out of 1000 are robustly stable on 100 Myr timescales, or ~8 billion orbits of planet b. We find planet b's orbit must lie close to the fundamental plane of planets c and d, but can be either prograde or retrograde. These solutions predict b's mass is in the range 2 - 9 M_Jup and has an inclination angle from the sky plane of less than 45 deg. Crossfield et al. (2010) detected the planet via brightness variations in the combined light curve, and argued that such a configuration would require b's radius to be ~1.8 R_Jup, relatively large for a planet of its age. However, the eccentricity of b in several of our stable solutions reaches > 0.1, inducing upwards of 10¹⁹ watts in the interior of the planet via tidal dissipation. Ibgui et al. (2009) find that this energy source could inflate the radius to the amount required for Crossfield et al., and thus we have solutions that are consistent with all observational constraints.The Gaia telescope (Casertano et al. 2008) will refine the orbits of planets c and d and perhaps provide further constraints on b, such as a minimum inclination and maximum mass. Gaia will astrometrically determine the orbits of many other exoplanets, providing clues to the formation mechanisms and uniqueness of this system.