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 , \theta _ { 1 } = \lambda _ { 1 } -2 \lambda _ { 2 } + \varpi _ { 1 } , \theta _ { 2 } = \lambda _ { 1 } -2 \lambda _ { 2 } + \varpi _ { 2 } , and \theta _ { 3 } = \varpi _ { 1 } - \varpi _ { 2 } , about 0 ^ { \circ } ( where \lambda _ { 1 , 2 } are the mean longitudes of the inner and outer planet and \varpi _ { 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 180 ^ { \circ } .
By considering the condition that { \dot { \varpi } } _ { 1 } = { \dot { \varpi } } _ { 2 } 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 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 45 ^ { \circ } 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 } \lesssim 0.06 and e _ { 2 } \lesssim 0.03 and the migration rate | \dot { a } _ { 2 } / a _ { 2 } | \lesssim 3 \times 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 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 time scale , whose inverse is about three orders of magnitude less than the above upper bound on | { \dot { a } } _ { 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 semi-major axes a _ { i } of only 7 % .
With eccentricity damping { \dot { e } } _ { i } / e _ { i } = - K| { \dot { a } } _ { 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 K \approx 100 if there is migration and damping of the outer planet only , but for K \approx 10 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 { \dot { a } } _ { i } as long as { \dot { e } } _ { i } / e _ { i } = - K| { \dot { a } } _ { 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 .