Abstract. We study the orbits and manifolds near the equilibrium points of a rotating asteroid. The linearised equations of motion relative to the equilibrium points in the gravitational field of a rotating asteroid, the characteristic equation and the stable conditions of the equilibrium points are derived and discussed. First, a new metric is presented to link the orbit and the geodesic of the smooth manifold. Then, using the eigenvalues of the characteristic equation, the equilibrium points are classified into 8 cases. A theorem is presented and proved to describe the structure of the submanifold as well as the stable and unstable behaviours of a massless particle near the equilibrium points. The linearly stable, the non-resonant unstable, and the resonant equilibrium points are discussed. There are three families of periodic orbits and four families of quasi-periodic orbits near the linearly stable equilibrium point. For the non-resonant unstable equilibrium points, there are four cases; for the periodic orbit and the quasi-periodic orbit, the structures of the submanifold and the subspace near the equilibrium points are studied for each case. For the resonant equilibrium points, the dimension of the resonant manifold is greater than four, and we find at least 1 family of periodic orbits near the resonant equilibrium points. Besides, this theory is 2 applied to asteroids 216 Kleopatra,
The general three‐dimensional periodic orbits around asteroids were investigated to study orbital behaviours in the vicinity of irregular gravitational bodies. The orbital patterns around periodic orbits were determined through decomposition into corresponding local manifolds. In this paper, a topological classification of periodic orbits is presented to discriminate the stability and dynamical behaviours of neighbours. A hierarchical grid searching method was developed for systematically searching three‐dimensional periodic orbits around irregular bodies. The method was used to generate 29 basic families around the asteroid 216 Kleopatra as an example. By calculating the characteristic multipliers of periodic orbits, numerical evidence was generated to describe the denseness of periodic orbits around asteroids and to demonstrate the remarkable asymmetry among these orbits. The dependence of the topology of the periodic orbits on the Jacobi integral was examined, revealing the evolutionary features of the stability and orbital patterns nearby. The results can be used to assess the environment around 216 Kleopatra, and this method can be useful in the studies of other asteroids.
Halo orbits for solar sails at artificial Sun-Earth L 1 points are investigated by a third order approximate solution. Two families of halo orbits are explored as defined by the sail attitude. Case I: the sail normal is directed along the Sun-sail line. Case II: the sail normal is directed along the Sun-Earth line. In both cases the minimum amplitude of a halo orbit increases as the lightness number of the solar sail increases. The effect of the z-direction amplitude on x-or y-direction amplitude is also investigated and the results show that the effect is relatively small. In case I, the orbit period increases as the sail lightness number increases, while in case II, as the lightness number increases, the orbit period increases first and then decreases after the lightness number exceeds ∼ 0.01.
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