Orbit Determination (Lambert's Problem)
Also known as: Lambert's problem, Lambert-Godstein trajectory problem
Lambert's problem is a classical astrodynamics boundary-value problem that determines an orbit connecting two points in space given a transfer time. Formulated by Johann Heinrich Lambert in the 18th century, it is fundamental to trajectory design for interplanetary missions and spacecraft maneuvers. The solution provides the orbital elements and velocities needed to transition between two positions.
Key highlights
- Provides exact solution under Keplerian assumptions
- Computationally efficient; fast evaluation suitable for trajectory optimization
- Reveals all possible transfers (multiple branches) for a given endpoint pair and time
- Well-established numerical algorithms (Battin, Vallado, Gooding)
Intuition
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How it works
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When to use it
Use Lambert's problem when planning spacecraft transfers between two points with a specified time window. It is essential for interplanetary missions, orbital rendezvous maneuvers, and quick trajectory feasibility checks. Assume Keplerian motion (no thrusting during transfer) and that only gravity acts. Use when the transfer time is significant but not too large to require perturbations.
Strengths & limitations
- Provides exact solution under Keplerian assumptions
- Computationally efficient; fast evaluation suitable for trajectory optimization
- Reveals all possible transfers (multiple branches) for a given endpoint pair and time
- Well-established numerical algorithms (Battin, Vallado, Gooding)
- Assumes gravitational field of a single central body; not valid near massive perturbations
- Does not account for thrusting during transfer
- Multiple solutions exist; selection requires additional criteria
- Singular when departure and arrival positions coincide
Common pitfalls
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Applications
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Frequently asked
How many solutions does Lambert's problem have?
For a given r₁, r₂, and Δt, there can be 0, 1, or 2 solutions. Short-way and long-way transfers correspond to different branches. If Δt is outside the feasible window, no real solution exists.
What is the time-of-flight equation?
It is a transcendental equation relating the orbital parameter (typically semi-major axis or eccentric anomaly difference) to the transfer time Δt. It must be solved numerically, usually with Newton–Raphson iteration.
How do I choose between multiple solutions?
Compare delta-v requirements, fuel consumption, and mission constraints. Short-way transfers are often preferred unless a specific trajectory shape is needed.
Sources
- 1.Lambert, J. H. (1761). Acta Helvetica. Physico-Mathematico-Anatomico-Botanico-Medica.
- 2.Vallado, D. A., Crawford, P., Hujsak, R., & Kelso, T. S. (2006). Revisiting Spacetrack Report #3. In AIAA/AAS Astrodynamics Specialist Conference.
- 3.Gooding, R. H. (1990). A procedure for the solution of Lambert's orbital boundary-value problem. Celestial Mechanics and Dynamical Astronomy, 48(2), 145-165.
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Cite this page
ScholarGate. (2026, June 3). Orbit Determination (Lambert's Problem). ScholarGate. https://scholargate.app/applied-physics/orbit-determination