Lambert's problem

In celestial mechanics, Lambert's problem is concerned with the determination of an orbit from two position vectors and the time of flight, posed in the 18th century by Johann Heinrich Lambert and formally solved with mathematical proof by Joseph-Louis Lagrange. It has important applications in the areas of rendezvous, targeting, guidance, and preliminary orbit determination.^[1]

Suppose a body under the influence of a central gravitational force is observed to travel from point P₁ on its conic trajectory, to a point P₂ in a time T. The time of flight is related to other variables by Lambert's theorem, which states:

The transfer time of a body moving between two points on a conic trajectory is a function only of the sum of the distances of the two points from the origin of the force, the linear distance between the points, and the semimajor axis of the conic.^[2]

Stated another way, Lambert's problem is the boundary value problem for the differential equation $${\displaystyle {\ddot {\mathbf {r} }}=-\mu {\frac {\hat {\mathbf {r} }}{r^{2}}}}$$ of the two-body problem when the mass of one body is infinitesimal; this subset of the two-body problem is known as the Kepler orbit.

The precise formulation of Lambert's problem is as follows:

Two different times ${\displaystyle t_{1},\,t_{2}}$ and two position vectors ${\displaystyle \mathbf {r} _{1}=r_{1}{\hat {\mathbf {r} }}_{1},\,\mathbf {r} _{2}=r_{2}{\hat {\mathbf {r} }}_{2}}$ are given.

Find the solution ${\displaystyle \mathbf {r} (t)}$ satisfying the differential equation above for which $${\displaystyle {\begin{aligned}\mathbf {r} (t_{1})=\mathbf {r} _{1}\\\mathbf {r} (t_{2})=\mathbf {r} _{2}\end{aligned}}}$$