The double lunar swing-by orbits are a special kind of orbits in the Earth-Moon system.These orbits repeatedly pass through the vicinity of the Moon and change their shapes due to the Moon’s gravity.In the synodic frame of the circular restricted three-body problem consisting of the Earth and the Moon,these orbits are periodic,with two close approaches to the Moon in every orbit period.In this paper,these orbits are revisited.It is found that these orbits belong to the symmetric horseshoe periodic families which bifurcate from the planar Lyapunov family around the collinear libration point L3.Usually,the double lunar swing-by orbits have k=i+j loops,where i is the number of the inner loops and j is the number of outer loops.The genealogy of these orbits with different i and j is studied in this paper.That is,how these double lunar swing-by orbits are organized in the symmetric horseshoe periodic families is explored.In addition,the 2n lunar swing-by orbits(n≥2)with 2n close approaches to the Moon in one orbit period are also studied.
Relative dynamics between the chief satellite and the deputy ones in for- marion flying is crucial to maintaining the formation. A good choice of the forma- tion usually requires a lower control frequency or less control energy. For formation flying missions in highly elliptic orbits, the well-known C-W equation is not accu- rate enough. Instead, Lawden＇s equation is often used. First, the solution to Lawden＇s equation with a very simple form is deduced. Then the J2 perturbation is added. It is found that Lawden＇s solution is not necessarily valid when the J2 perturbation is con- sidered. Completely discarding Lawden＇s solution and borrowing the idea of mean orbit elements, two rules to initialize the formation are proposed. The deviation speed is greatly reduced. Different from previous studies on the J2 perturbation, except for the relatively simple expression for the semi-major axis, the tedious formulae of the long period terms and the short period terms of other orbital elements are not used. In addition, the deviation speed is further reduced by compensation of the nonlinear effects. Finally, a loose control strategy of the formation is proposed. To test the ro- bustness of this strategy, a third body perturbation is added in numerical simulations.
Xi-Yun Hou, Yu-Hui Zhao and Lin Liu School of Astronomy and Space Science, Nanjing University, Nanjing 210093, China
