ar X iv : a st ro - p h / 01 08 10 4 v 2 2 D ec 2 00 1 Dynamics and Origin of the 2 : 1 Orbital Resonances of the GJ 876 Planets

The discovery by Marcy et al. (2001) of two planets in 2:1 orbital resonance about the star GJ 876 has been supplemented by a dynamical fit to the data by Laughlin & Chambers (2001) which places the planets in coplanar orbits deep in three resonances at the 2:1 mean-motion commensurability. The selection of this almost singular state by the dynamical fit means that the resonances are almost certainly real, and with the small amplitudes of libration of the resonance variables, indefinitely stable. Several unusual properties of the 2:1 resonances are revealed by the GJ 876 system. The libration of both lowest order mean-motion resonance variables and the secular resonance variable, θ 1 = λ 1 − 2λ 2 + ̟ 1 , θ 2 = λ 1 − 2λ 2 + ̟ 2 , and θ 3 = ̟ 1 − ̟ 2 , about 0 • (where λ 1,2 are the mean longitudes of the inner and outer planet and ̟ 1,2 are the longitudes of periapse) differs from the familiar geometry of the Io-Europa pair, where θ 2 and θ 3 librate about 180 •. By considering the condition that ˙ ̟ 1 = ˙ ̟ 2 for stable simultaneous librations of θ 1 and θ 2 , we show that the GJ 876 geometry results because of the large orbital eccentricities e i , whereas the very small eccentricities in the Io-Europa system lead to the latter's geometry. Surprisingly, the GJ 876 configuration, with θ 1 , θ 2 , and θ 3 all librating, remains stable for e 1 up to 0.86 and for amplitude of libration of θ 1 approaching 45 • with the current eccentricities — further supporting the indefinite stability of the existing system. Any process that drives originally widely separated orbits toward each other could result in capture into the observed resonances at the 2:1 commensurability. We find that forced inward migration of the outer planet of the GJ 876 system results in certain capture into the observed resonances if initially e 1 0.06 and e 2 0.03 and the migration rate | ˙ a 2 /a 2 | 3 × 10 −2 (a 2 / AU) −3/2 yr −1. Larger eccentricities lead to likely capture into higher order resonances before the 2:1 commensurability is reached. The planets are sufficiently massive to open gaps in the nebular disk surrounding the young GJ …