Convergence of a Periodic Orbit Family Close to Asteroids During a Continuation

In this work, we study the continuation of a periodic orbit on a relatively large scale and discover the existence of convergence under certain conditions, which has profound significance in research on asteroids and can provide a total geometric perspective to understanding the evolution of the dynamic characteristics from a global perspective. Based on the polyhedron model, convergence is derived via a series of theoretical analyses and derivations, which shows that a periodic orbit will evolve into a nearly circular orbit with a normal periodic ratio (e.g., 2:1, 3:2, and 4:3) and almost zero torsion under proper circumstances. As an application of the results developed here, three asteroids, (216) Kleopatra, (22) Kalliope and (433) Eros, are studied, and several representative periodic orbit families are detected, with convergence in three different cases: bidirectional, increasing-directional and decreasing-directional continuation. At the same time, four commonalities among these numerical examples are concluded. First, a (pseudo) tangent bifurcation arises at the cuspidal points during the variations in the periodic ratios in a single periodic orbit family. In addition, these cuspidal points in the periodic ratio coincide with the turning points during the variations in the average radius, the maximal torsion and the maximal radius of curvature. Furthermore, the periodic ratio increases (or decreases) with an increase (or decrease) in the Jacobian constant overall. We find the relationship between periodic ratio and Jacobian constant. The results implies that the periodic ratios for a fixed resonant status have an infimum and a supremum. Finally, if the periodic orbit converges to a point, it can be an unstable equilibrium point of the asteroid.