Sun-Earth triangular lagrange point orbit insertion and satellite station keeping

Minimizing the amount of fuel needed to place a satellite into Earth orbit and then inserting it into its operational altitude is an important driver of space mission analysis and design. This paper analyzes the velocity changes required to transfer satellites initially in Earth orbit to the Sun-Earth triangular Lagrange points and estimates the minimum velocity changes needed for station keeping at those points after the satellite's two-body motion around the Sun has been gravitationally perturbed. Times of flight required by each insertion and station keeping scenario investigated are also analyzed. Analytical, co-orbital rendezvous equations for an elliptical Earth orbit are derived as a function of Earth true anomaly and solved numerically due to their transcendental nature. These equations predict that total, minimum velocity changes are comparable to the solution of Lambert's problem for various times of flight. When these velocity change results are compared to the velocity changes required to transfer a satellite from Earth orbit to the planets, it can be concluded that a transfer from Earth orbit to the Sun-Earth L-4 and L-5 Lagrange points is comparable to a transfer from the Earth to Venus or the Earth to Mars. Station keeping velocity changes and times of flight are investigated by solving Lambert's problem for various initial conditions. The investigation shows that the minimum velocity changes needed to station keep a satellite positioned at L-4 or L-5 are in general inversely proportional to the corresponding times of flight needed to implement the station keeping maneuvers and directly proportional to the perturbation time elapsed when the maneuvers are commenced.