  Following the general numerical analysis of Melita and Woolfson (1996), I
showed in a recent paper that a restricted, planar, circular planetary system
consisting of Sun, Jupiter and Saturn would be captured in a near (2:1)
resonance when one would allow for frictional dissipation due to interplanetary
medium (Haghighipour, 1998). In order to analytically explain this resonance
phenomenon, the method of partial averaging near a resonance was utilized and
the dynamics of the first-order partially averaged system at resonance was
studied. Although in this manner, the finding that resonance lock occurs for
all initial relative positions of Jupiter and Saturn was confirmed, the
first-order partially averaged system at resonance did not provide a complete
picture of the evolutionary dynamics of the system and the similarity between
the dynamical behavior of the averaged system and the main planetary system
held only for short time intervals. To overcome these limitations, the method
of partial averaging near a resonance is extended to the second order of
perturbation in this paper and a complete picture of dynamical behavior of the
system at resonance is presented. I show in this study that the dynamics of the
second-order partially averaged system at resonance resembles the dynamical
evolution of the main system during the resonance lock in general, and I
present analytical explanations for the evolution of the orbital elements of
the main system while captured in resonance.
