Inputs
Visualization
Departure (r₁) and arrival (r₂) radii dashed; the solved transfer arc in blue; burn points marked. The solver finds the conic joining the two points in exactly the given time.
v₁—v₂—Δv total—
Given two positions and a time of flight, solve the transfer orbit that connects them — the workhorse behind interplanetary targeting. First-cut: coplanar, single revolution.
Departure (r₁) and arrival (r₂) radii dashed; the solved transfer arc in blue; burn points marked. The solver finds the conic joining the two points in exactly the given time.
Lambert’s problem asks: given two position vectors and the time to fly between them, what orbit connects them? It is the boundary-value counterpart to ordinary propagation, which takes a position and velocity and marches forward. The result is the two velocity vectors — at departure and arrival — which immediately give the transfer’s Δv against any starting and ending orbit.
The problem is named for Johann Heinrich Lambert, who stated in 1761 the theorem that the transfer time depends only on the semi-major axis, the chord between the two points, and the sum of the two radii — not otherwise on the orbit’s shape. Gauss gave an early practical method; modern solvers use “universal variables” (via the Stumpff functions) so one formulation handles elliptic, parabolic, and hyperbolic transfers, iterating on a single variable until the computed time of flight matches the requested one. This tool uses that approach and is validated against a standard textbook case.
Lambert’s problem is the computational heart of interplanetary mission design: sweep departure and arrival dates over a grid, solve Lambert for each pair, and the total-Δv contours form the “porkchop plot” that reveals launch windows. It is equally central to targeting, orbit determination, and the terminal phase of rendezvous.