Lambert’s Theorem—A Complete Series Solution

Abstract

Lambert’s theorem states that for the two-body orbit determination problem, there is a particular value of the semimajor axis associated with a single conic transfer arc that will uniquely satisfy the initial conditions, which include the transfer time, the two ordered position vectors and the gravitational constant. The associated Lagrange equations for the orbital transfer time are expressed as series expansions for all cases. Many techniques exist to solve the Lagrange equations, but series solutions avoid the need for root finding and therefore they may allow for direct analytical optimization for mission design. Because of the case dependence, several different solution approaches are necessary among the three distinct forms of the series equations. Two new series solutions are given for each boundary of the long-way orbital case, which are also combined with an earlier result for hyperbolic and short-way transfers. Convergence properties of the new series solutions are investigated numerically for several examples.