Abstract: For the problem of planetary trajectory in a centered force field, it is solved by establishing equations with complex vectors as variables. In the process of solving, not only the general law of planetary motion is obtained, but also the speed formula of the planet and Hamilton's theorem are derived, and the second focus position of the planetary orbit and the trajectory based on the Laplace-Runge-Lenz vector are found, and thus the trajectory envelopes of satellites launched in different directions are obtained.

Key words:  centered force field, complex vector, Laplace-Runge-Lenz vector, focal point, envelope