Dynamics and Origin of the 2:1 Orbital Resonances of the GJ 876 Planets

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Abstract
(Abridged) A dynamical fit has placed the two planets about the star GJ 876 in coplanar orbits deep in 3 resonances at the 2:1 mean-motion commensurability with small libration amplitudes. The libration of both lowest order mean-motion resonance variables, theta_1 and theta_2, and the secular resonance variable, theta_3, about 0 deg. differs from the familiar geometry of the Io-Europa pair, where theta_2 and theta_3 librate about 180 deg. By considering a condition for stable simultaneous librations of theta_1 and theta_2, we show that the GJ 876 geometry results because of the large orbital eccentricities e_i, whereas the very small e_i in the Io-Europa system lead to the latter's geometry. Surprisingly, the GJ 876 resonance configuration remains stable for e_1 up to 0.86 and for amplitude of libration of theta_1 approaching 45 deg. with the current e_i. We find that 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 |(da_2/dt)/a_2| < 0.03(a_2/AU)^{-3/2} yr^{-1}. The bound on the migration rate is easily satisfied by migration due to planet-nebula interaction. If there is no eccentricity damping, eccentricity growth is rapid with continued migration within the resonance, with e_i exceeding the observed values after a further reduction in the semi-major axes a_i of only 7%. With eccentricity damping (de_i/dt)/e_i = -K|(da_i/dt)/a_i|, the e_i reach equilibrium values that remain constant for arbitrarily long migration within the resonances. The equilibrium e_i are close to the observed e_i for K=100 (K=10) if there is migration and damping of the outer planet only (of both planets). It is as yet unclear that planet-nebula interaction can produce the large value of K required to obtain the observed eccentricities.

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