Extrasolar planetary systems near a secular separatrix

Extrasolar planetary systems display a range of behavior that can be understood in terms of the secular theory of classical celestial mechanics, including the motions of the major axes. Four planet pairs in the seventeen known extrasolar planetary systems with multiple planets (v And, 47 UMa, 55 Cnc, and HD 128311), have trajectories in orbital element space that lie close to the separatrix between libration and circulation. Here we examine the dynamics of the first two, which are not in mean motion resonance. The basics of secular theory are reviewed in order to develop insight into this behavior. The definition of a secular resonance is discussed, correcting misconceptions in the literature; it is not synonymous with libration and is not a commensurability of eigenfrequencies. The behavior of these two near-separatrix systems is evaluated with updated orbital elements by comparing both analytical and numerical results. We find that the apsidal motion from secular theory does not match the predictions from N-body simulations and conclude that first-order secular theory should be used with caution on extrasolar planetary systems. While the existence of one near-separatrix system could be explained simply by chance initial conditions, the fact that there are several is improbable unless some physical process tends to set up systems near the separatrix. Explanations based on an impulsive increase in the eccentricity of one planet are promising, but key issues remain open.