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.

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:

1. 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.

Stated another way, Lambert's problem is the boundary value problem for the differential equation 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:

are given.

Initial geometrical analysis

The three points

• , the centre of attraction,

• ,

• ,

. The geometrical problem to solve is to find all ellipses

define a hyperbola. This hyperbola is illustrated in figure 2.

Relative the usual canonical coordinate system defined by the major and minor axis of the hyperbola its equation is with

is

on the other branch of the hyperbola corresponding relation is i.e.

and the semi-major axis

is displayed in figure 3.

Solution for an assumed elliptic transfer orbit

First one separates the cases of having the orbital pole every orbital revolution.

.

are as in figure 1 with and the semi-major axis (with sign!) of the hyperbola discussed above is

The eccentricity (with sign!) for the hyperbola is and the semi-minor axis is ) where

)

is

The distance between the foci is

and the eccentricity is consequently

is positive or negative. In both cases one has that where

is the unit vector expressed in the canonical coordinates.

is positive then

is negative then With

• semi-major axis

• eccentricity

• initial true anomaly

is in the range that can be obtained with an elliptic Kepler orbit corresponding y value can then be found using an iterative algorithm.

with the equation

Equations () and () are then replaced with

() is replaced by and () is replaced by

Numerical example

Assume the following values for an Earth centered Kepler orbit

• r1 = 10000km

• r2 = 16000km

• α = 100°

These are the numerical values that correspond to figures 1, 2, and 3.

Selecting the parameter y as 30000 = 398603km3/s2. Corresponding orbital elements are

• semi-major axis = 23001km

• eccentricity = 0.566613

• true anomaly at time t1 = 7.577°

• true anomaly at time t2 = 92.423°

This y-value corresponds to Figure 3.

With

• r1 = 10000km

• r2 = 16000km

• α = 260°

one gets the same ellipse with the opposite direction of motion, i.e.

• true anomaly at time t1 = 7.577°

• true anomaly at time t2 = 267.577° = 360° − 92.423°

and a transfer time of 31645 seconds.

The radial and tangential velocity components can then be computed with the formulas (see the Kepler orbit

The transfer times from P1 to P2 for other values of y are displayed in Figure 4.

Practical applications

The most typical use of this algorithm to solve Lambert's problem is certainly for the design of interplanetary missions. A spacecraft traveling from the Earth to for example Mars can in first approximation be considered to follow a heliocentric elliptic Kepler orbit from the position of the Earth at the time of launch to the position of Mars at the time of arrival. By comparing the initial and the final velocity vector of this heliocentric Kepler orbit with corresponding velocity vectors for the Earth and Mars a quite good estimate of the required launch energy and of the maneuvers needed for the capture at Mars can be obtained. This approach is often used in conjunction with the patched conic approximation.

This is also a method for orbit determination. If two positions of a spacecraft at different times are known with good precision (for example by GPS fix) the complete orbit can be derived with this algorithm, i.e. an interpolation and an extrapolation of these two position fixes is obtained.

Parametrization of the transfer trajectories

.

exists

Open source code

• From MATLAB central

• PyKEP a Python library for space flight mechanics and astrodynamics (contains a Lambert's solver, implemented in C++ and exposed to python via boost python)

• A simple two-dimensional Lambert solver/orbit plotter in Microsoft Excel

References

External links

• Lambert's theorem through an affine lens. Paper by Alain Albouy containing a modern discussion of Lambert's problem and a historical timeline.

• Lambert's problem in orbital dynamics: a self-contained introduction. Paper by Lenox Baloglou, Parneet Gill and Tonatiuh Sánchez-Vizuet that presents a thorough description of the problem leading to Lagrange's solution algorithm. This expository text assumes a minimal background in physics and mathematics. (Journal version)

• Connect the dots...finding all possible orbits between two points. Eur. J. Phys. 46, 045004 (2025). Paper by P. R. Blanco that simplifies orbit fitting in the plane and suggests a numerical search on semi-latus rectum to solve the Lambert problem for direct, prograde transfers. With supplementary materials including a 2D orbit solver/plotter in MS Excel.

• Revisiting Lambert's Problem. Paper by Dario Izzo containing an algorithm for providing an accurate guess for the householder iterative method that is as accurate as Gooding's Procedure while computationally more efficient.

• Lambert's Theorem - A Complete Series Solution. Paper by James D. Thorne with a direct algebraic solution based on hypergeometric series reversion of all hyperbolic and elliptic cases of the Lambert Problem.