Constructing the natural families of periodic orbits near irregular bodies

The periodic orbits in the potential of an irregular body have proved to be of various morphologies and distributed widely over a large scale. This study is motivated by finding a construction that organizes the periodic motion set in a natural way, which would contribute to our understanding of the orbital dynamics in the near-field regime of the minor planets in the Solar system. An analysis of the first-order local continuation condition presents the feasible solution set, which essentially depends on the eigen subspace of the monodromy affiliated with the trivial multiplier +1.We introduce a base category due to the algebraic multiplicity of multiplier +1 of any periodic orbit, ma, and then look at the details of the correlations between the uniqueness of the local continuation and the multiplicities of +1. A specific application of the periodic orbits around asteroid (243) Ida is shown as a demonstration. The data base of periodic orbits around Ida is established, based on which we performed numeric continuations and confirmed the existence of eight topological types of the double case (m_a = 2) as well as two of the quadruple case (m_a = 4). The local analysis outlines, in general, how a complete natural family is constructed in phase space. Sample natural families are presented, with special attention paid to the topological transitions. An analysis reveals several common features among these topological transitions, and we take these features as a sketch of an arbitrary natural family of periodic orbits, which serves as a bridge to understanding the general orbital behaviours near irregular bodies.