Dynamics and origin of the 2 : 1 orbital resonances of the GJ 876 planets

The discovery by Marcy and coworkers 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, 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, theta(1)=lambda(1)-2lambda(2)+pi(1), theta(2)=lambda(1)-2lambda(2)+pi(2) and theta(3)=pi(1)-pi(2), about 0degrees (where lambda(1,2) are the mean longitudes of the inner and outer planet and pi(1,2) are the longitudes of periapse) differs from the familiar geometry of the Io-Europa pair, where theta(2) and theta(3) librate about 180degrees. By considering the condition that (pi)over dot(1) = (pi)over dot (2) for stable simultaneous librations of theta(1) and theta(2), we show that the GJ 876 geometry results from 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 theta(1), theta(2), and theta(3) all librating, remains stable for e(1) up to 0.86 and for amplitude of libration of theta(1) approaching 45degrees 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) less than or similar to 0.006 and e(2) less than or similar to 0.03 and the migration rate \(a)over dot(2)/a(2)\ less than or similar to 3x10(-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 876 and to clear the disk material between them, and the resulting planet-nebular interaction typically forces the outer planet to migrate inward on the disk viscous timescale, whose inverse is about 3 orders of magnitude less than the above upper bound on \(a)over dot(2)/a(2)\ for certain capture. 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 semimajor axes a(i) only 7%. With eccentricity damping (e)over dot(i)/e(i) = -K\(a)over dot(i)/a(i)\, the eccentricities reach equilibrium values that remain constant for arbitrarily long migration within the resonances. The equilibrium eccentricities are close to the observed eccentricities for Kapproximate to100 if there is migration and damping of the outer planet only, but for Kapproximate to10 if there is also migration and damping of the inner planet. This result is independent of the magnitude or functional form of the migration rate (a)over dot(i) as long as (e)over dot(i)/e(i) = -K\(a)over dot(i)/a(i)\. Although existing analytic estimates of the effects of planet-nebula interaction are consistent with this form of eccentricity damping for certain disk parameter values, it is as yet unclear that such interaction can produce the large value of K required to obtain the observed eccentricities. The alternative eccentricity damping by tidal dissipation within the star or the planets is completely negligible, so the observed dynamical properties of the GJ 876 system may require an unlikely fine-tuning of the time of resonance capture to be near the end of the nebula lifetime.