Lambertproblemet

Lambertproblemet, known in English as the Lambert problem, is a classical question in celestial mechanics and astrodynamics. It asks whether there exists a Keplerian orbit that connects two given position vectors r1 and r2 at two specified times t1 and t2, or equivalently, given the time of flight Δt = t2 − t1, determine the orbit that transfers a body from r1 to r2 under a central gravitational field with parameter μ. The problem is foundational for trajectory design and rendezvous planning in space missions.

Lambert’s theorem provides the core geometric relation: the time of flight between two points on a Keplerian orbit depends only on the chord length between the points, the distances from the central body, and the transfer angle between the position vectors. This leads to a family of possible solutions corresponding to different energies and numbers of revolutions, including short- and long-way paths as well as prograde or retrograde motions. Depending on Δt and the geometry, there can be up to multiple valid transfer orbits.

Solutions to the Lambert problem are typically numerical. Common approaches include the universal variable formulation using Stumpff functions, and various algorithms such as Gooding’s method, Izzo’s algorithm, and Battin’s method. These methods aim to find the orbital elements or the required velocity vectors at r1 (or r2) that produce a trajectory meeting the time constraint.

Applications are widespread in mission design, including orbital rendezvous, satellite reconfiguration, and interplanetary transfers, where planners must determine the velocity impulses needed to connect two positions in a given time. The problem remains a practical and active area of numerical astrodynamics research, with ongoing refinements to improve robustness and efficiency.