THE LAMBERT PROBLEM OF ORBITAL DYNAMICS

When considering space travel between two celestial bodies under a gravitational potential, Lambert's problem asks if there exists a way to free fall between the two bodies. Keplerian dynamics ensures that this free fall trajectory describes a conical arc. The Lambert equation provides a solution to this problem that relates travel time and the energy necessary for this fall to occur. We will introduce the physical concepts of motion under a gravitational potential and a characterization of conic sections in terms of a parameter known as eccentricity. Using conservation properties, we will show that free fall under a gravitational potential must follow a conical trajectory. Therefore, Lambert's problem can be rephrased as determining the energy required to place the vessel on the desired conical orbit connecting the two bodies. We will then show Lagrange's solution algorithm which yields the required energy as a function of the departure and arrival dates. We conclude by showing how this is a relevant problem in orbital mission planning.