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S. J. Peale, M. H. Lee (Dept. of Physics, UCSB)
Ease of capture into low-order mean-motion resonances from nebula induced differential migration means that such resonances are likely to be ubiquitous among extrasolar planetary systems. The libration of the lowest order 2:1 mean-motion resonance variables, \theta1 = \lambda1-2\lambda2+\varpi1 and \theta2 = \lambda1-2\lambda2+\varpi2, and the secular resonance variable, \theta3 = \varpi1-\varpi2 (\lambdai and \varpii are the mean longitudes and the longitudes of periapse), about 0\circ in the GJ 876 system differs from the geometry of the Io-Europa pair, where \theta2 and \theta3 librate about 180\circ. The large eccentricities, ei, of the GJ 876 system cause its libration geometry to differ from that of Io and Europa, and, remarkably, the GJ 876 geometry is stable for e1 up to 0.86 (Lee and Peale 2002). Following a differentially migrating, coplanar two-planet system through capture into the 2:1 resonances for a selection of parameter values reveals the conditions for a wide variety of unusual 2:1 resonance configurations.
For (m1+m2)/m0 \la 10-3 (m0, m1, and m2 are the masses of the primary, inner and outer planets respectively) and very small initial ei, the system is captured automatically into all resonances as the orbits are forced together, with \theta1 librating about 0\circ and \theta2 and \theta3 librating about 180\circ --- like the Io-Europa pair. Continued migration forces the ei to larger values until the system passes smoothly over to the GJ 876 configuration with all \thetai librating about 0\circ. For m1\la m2, the ei continue to grow with continued migration with libration amplitudes about 0\circ remaining very small until instability ensues at very large ei. The ei at which the system becomes unstable decrease with increasing planetary masses. For m1 \ga m2, the passage through the Io-Europa configuration is as above, but when the increasing e1 reaches ~0.1--0.2 (depending on m1/m2), the libration centers depart by tens of degrees from 0\circ leading to stable libration of the \thetai far from either 0\circ or 180\circ. As eccentricities continue to grow, the libration centers return to 0\circ and remain there until instability is induced. If (m1+m2)/m0 \ga 10-3, which forces large ei before resonance encounter, then the system passes directly into the GJ 876 configuration upon resonance capture. The libration of both \theta1 and \theta2 forces the \varpii to precess at the same rate normally in a retrograde sense, but this changes to prograde above some large value of e1 while librations of the \thetai are maintained. Within each type of evolution, the details of the path depend only on m1/m2 and not on the total planetary mass, provided mi are not too large. Support for this work is NASA PG&G Grant NAG5-3646.
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Bulletin of the American Astronomical Society, 34, #3
© 2002. The American Astronomical Society.