Lambert's Problem

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

Lambert's problem is one of the two canonical problems of orbital mechanics (the other being Kepler's problem): given an initial position vector $\vec r_1$ , a final position vector $\vec r_2$ , and a time of flight $\Delta t$ in a central gravity field, find the orbit connecting the endpoints together with the endpoint velocities $\vec v_1$ and $\vec v_2$ . It is at once a tool for initial orbit determination and a tool for transfer-orbit design (intercept, rendezvous, entry) (Vallado 2022; Battin 1999).

Calling it "Gauss's problem" is a misnomer. Euler first analysed the problem in 1744; Lambert extended it to elliptic and hyperbolic orbits during 1761–1771; Gauss in 1801 gave one solution method while recovering Ceres — he was solving Lambert's problem, not posing a new one (Vallado 2022, §7.6).

The two position vectors fix the orbital plane. Once a "short-way" or "long-way" choice is made ( $t_m=\pm1$ , corresponding to a transfer angle $\Delta\nu$ below or above $180^\circ$ ), the two-body Lambert problem has a unique solution; at $\Delta\nu=180^\circ$ the plane is undefined and the solution degenerates.

Lambert's Theorem (Time-of-Flight Theorem)

The central result of Lambert's geometric investigation, stated as a theorem by Battin (1999, p. 276):

The transfer time between two endpoints depends only on three quantities — the semi-major axis $a$ , the sum of the endpoint radii $r_1+r_2$ , and the chord length $c$ joining them — and not on the shape of the orbit (i.e., not on the eccentricity).

Equivalently, with $r_1+r_2$ , $c$ , and $\Delta t$ fixed, every admissible connecting orbit has the same semi-major axis; different eccentricities correspond to different members of the same ellipse family. This is the geometric reason Lambert's problem reduces to a single scalar equation in $a$ (or an equivalent variable). Lagrange gave the analytic form (Vallado 2022, Eq. 7-36):

\Delta t=\sqrt{\frac{a^3}{\mu}}\big[\,2N\pi+\alpha_e-\sin\alpha_e \mp (\beta_e-\sin\beta_e)\,\big]

where $N$ is the number of complete revolutions, $\sin(\alpha_e/2)=\sqrt{s/(2a)}$ , $\sin(\beta_e/2)=\sqrt{(s-c)/(2a)}$ , and the semi-perimeter is $s=(r_1+r_2+c)/2$ . The $\mp$ sign selects the short- or long-way transfer.

Minimum-Energy Solution and the Solvability Bound

Below a critical semi-major axis no connecting orbit exists. The critical value follows from the geometry of "two virtual-focus circles just touching" (Vallado 2022, Eq. 7-37):

a_{\min}=\frac{s}{2}=\frac{r_1+r_2+c}{4}

This is the minimum-energy orbit (specific mechanical energy $\varepsilon=-\mu/(2a)$ at its lowest), also called the fundamental ellipse. It gives a useful lower bound on the $\Delta v$ of a two-impulse transfer but is generally not the solution at a fixed time of flight.

Algorithm Families

Solving Lambert's problem is equivalent to solving a single transcendental equation in $a$ (or in the universal variable $z$ , or $x=\cos(\Delta E/2)$ ). Main algorithm families:

Gauss's original method (1809): based on the ratio $y$ of swept-sector area to triangle area; yields two independent equations in $y$ and $\Delta E$ and uses continued fractions to avoid trigonometric iteration. Originally elliptic only; extended to hyperbolic orbits by Bate, Mueller and White (Vallado 2022, §7.6.2).
Minimum-energy method: solve for the $a_{\min}$ orbit first, then use Eq. (7-36) to back out $\Delta t$ for calibration.
Universal-variable method (Battin-Vaughan 1984; Vallado Algorithm 58): a single variable $z=\Delta E^2$ (positive for ellipses, negative for hyperbolas, zero for parabolas) unifies all three conic classes, eliminating branch logic. The most common modern implementation.
Battin's method: solves the equation via hypergeometric continued fractions; elegant and robust across short/long-way and singular cases (Vallado 2022, §7.6.5).
Gooding's method (1988, 1990): the most robust implementation in practice, handles multi-revolution and near- $180^\circ$ cases; the default in commercial tools such as AGI ODTK.
Thorne's series solution (2004): gives $\Delta t(a)$ directly as a series, sidesteps iteration-initialisation problems, useful for visualising the solution space (the source of Vallado 2022, Fig. 7-9).

Multi-Revolution Solutions

When the specified $\Delta t$ allows the transfer orbit to complete $N\ge1$ full revolutions before arriving, each $N$ admits $2N+1$ candidate solutions ( $N$ -revolution short-way + $N$ -revolution long-way + 1 minimum-energy degenerate arc). Prussing (1992) and Shen-Tsiotras (2003) describe the structure of multi-revolution solutions and the minimum- $\Delta v$ selection: at fixed time, allowing more revolutions can reduce $\Delta v$ , but the optimum must be selected among $2N_{\max}+1$ candidates. Shen-Tsiotras further show that the fixed-endpoint transfer reduces to an auxiliary problem whose global optimum is read off a $\Delta v$ contour map over separation angle and time of flight (Shen & Tsiotras 2003).

Perturbed Lambert Problem

When the two-body assumption fails (Earth oblateness $J_2$ , third-body gravity, solar radiation pressure, etc.), Lambert's theorem no longer holds and the problem must be solved numerically. Standard procedure:

Use the two-body Lambert solution as the initial guess;
Numerically integrate under the full force model to obtain the terminal position error $\delta\vec r_2$ ;
Iterate on $\vec v_1$ via differential correction or a homotopy method until the terminal position satisfies the tolerance.

The resulting arc is called a perturbed Lambert arc. It is the basic tool for midcourse corrections in cislunar transfers, multi-arc patching, and intermediate-arc calculations in two-impulse cislunar transfers. The formal equivalent is the more general two-point boundary value problem under Cowell integration (see Cowell's perturbation method). Criscola et al. (2024) give a direct solution via the theory of functional connections that bypasses iteration.

Uncertain Lambert Problem and the Linear-Variational Form

In practice $\vec r_1$ and $\vec r_2$ (from radar or optical measurements) carry errors, so $\vec v_1$ and $\vec v_2$ are random variables. Schumacher et al. (2015) formalise this as the Uncertain Lambert Problem (ULP):

General nonlinear form: the map $\vec v_1=L(\vec r_1,\vec r_2)$ pushes forward the position density to a velocity density; its structure is governed by Liouville's equation and is usually resolved by Monte Carlo.
Linear-variational Lambert problem: linearise about a nominal solution; the initial and final velocity variations are then related to the position variations by partitions of the nominal state transition matrix $\Phi_{rr},\Phi_{rv},\Phi_{vr},\Phi_{vv}$ . Given a position covariance, the velocity covariance follows without iteration and is equivalent to a weighted batch least-squares differential correction (Schumacher et al. 2015).
Higher-order methods: Armellin et al. use differential algebra (COSY-Infinity) to expand the solution in a Taylor series; Hall-Singla (2020) build higher-order sensitivity matrices via the conjugate unscented transform — derivative-free approximations of the output distribution.
Multi-revolution case (UMRLP): in the multi-revolution case the distribution is further constrained by orbital-dynamical boundaries; the per-revolution solution fraction $p_N$ decreases with $N$ . Duan et al. (2025) handle this constrained distribution with an adaptive polynomial-chaos expansion.

Robust Lambert Solver (RLS): an engineering-grade implementation that converges without an initial guess for both single- and multi-revolution cases, typically combining the strengths of Izzo's vectorised algorithm and Blanchard's algorithm. Within this family the RLS, UMRLP, the complete uncertain-solution distribution, and the per-revolution fraction $p_N$ are all sub-constructs of the ULP.

Application Notes

Intercept and rendezvous: the standard computation module for two-impulse transfers and fixed-time intercepts.
Initial orbit determination: recover an orbit from two timed position vectors (e.g., from radar trilateration or two position vectors).
Midcourse correction: use a perturbed Lambert solution as the reference to determine midcourse impulse vectors (see multi-impulse manoeuvres and two-impulse orbit transfers).
Lambert guidance: embedding Lambert solutions in a real-time closed-loop guidance law — see Lambert Guidance Routine.

References

Vallado, 2022, Fundamentals of Astrodynamics and Applications, §7.6 (survey of Lambert's problem, theorem, and algorithm families).
Battin, 1999, An Introduction to the Mathematics and Methods of Astrodynamics (modern statement of Lambert's theorem; hypergeometric continued-fraction algorithm).
Shen & Tsiotras, 2003, Optimal Two-Impulse Rendezvous Using Multiple-Revolution Lambert Solutions (structure of multi-revolution solutions and optimal selection).
Schumacher et al., 2015, Uncertain Lambert Problem (formulation and linear-variational solution).
Hall & Singla, 2020, Higher-order Sensitivity Matrix Method for Probabilistic Solution to Uncertain Lambert Problem (higher-order sensitivity approach).
Duan et al., 2025, Adaptive Polynomial Chaos Expansion Method for Uncertain Multiple-Revolution Lambert Problem (multi-revolution uncertain form).
Criscola et al., 2024, Application of the Theory of Functional Connections to the Perturbed Lambert's Problem (direct solution of the perturbed case).
Izzo, 2006, Lambert's Problem for Exponential Sinusoids (a Lambert-type problem for exponential-sinusoid shapes).