Trajectory design using approximate analytic solutions of the N-body problem

The N-body problem as formulated by Sir Isaac Newton in the seventeenth century has been a rich source of mathematical and scientific discovery. Continuous attempts invested into the solution of this problem over the years have resulted in a host of remarkable theories that have changed the way the world is viewed and analyzed. A final solution in terms of an infinite time-dependent power series was finally discovered in the latter part of the twentieth century. However, the slow convergence of this result makes its implementation impractical in every day spacecraft trajectory design and optimization. The only feasible way to solve the N-body problem reliably is to numerically integrate the equations of motion.;This dissertation derives two new variable time step algorithms using time dependent power series solutions developed for the two-body problem. These power series solutions allow the space-dependent N-body problem to be transformed into a time-dependent system of equations that can be solved analytically. The analytic results do not yield global solutions, but rather approximate outcomes whose order of accuracy can be controlled by the user.;The two algorithms are used to investigate scenarios corresponding to a highly elliptical orbit in the two-body problem; periodic, central configuration scenarios in the three-body problem; and a non-periodic scenario in the restricted three-body problem. The results obtained are compared to the outcomes returned by a variable time step fourth-order, fifth-order Runge-Kutta numerical integration algorithm. The outcomes derived for each situation demonstrate that the two new variable time step algorithms are both more accurate and much more efficient than their Runge-Kutta counterpart.