Patched Three-Body Approximation: Application to Interplanetary Trajectory Planning

In this chapter, in the first instance the circular, planar, restricted three-body problem is first revisited. Following the enunciation of the key features of the three-body problem, including the transformation of the reference coordinate frame to a rotating frame, the calculation of the equilibrium points, the definition of the Hill sphere and its comparison with sphere of influence introduced earlier as well as other features, the transformation from one synodic reference frame to another is considered in some detail with examples. The methods of genera rating trajectories by solving the circular, restricted three-body problem are then discussed and compared with representative solutions to the multi-body problem. Based on the solutions to the circular, restricted three-body problem, the patched three-body approximation method is introduced. In this method the trajectory is first constructed by solving circular, restricted three-body problem using an appropriate set of three bodies including the departure planet. Then, when the contribution of the second primary or the departure planet is insignificant, it is substituted by employing a transformation, to include the arrival planet as the second primary. A trajectory correction manoeuvre is applied at the patch point to switch from one trajectory to the other. The application of this method including the influence of impulsive and continuous control forces is then considered. Optimal control theory is used to construct segments of the trajectories which are patched together to construct a suitable interplanetary trajectory.