Tool 7 · Maneuver & Mission Design

Lambert Solver

Solve the two-position, time-of-flight transfer problem. Enter departure and arrival position vectors, choose the transfer time, and compute the required departure and arrival velocity vectors with an estimated maneuver cost relative to circular reference orbits.

What this tool computes

Lambert's problem asks: if a spacecraft must travel from one position vector to another in a specified time, what transfer orbit connects those two points? This is one of the most important problems in mission design because it connects geometry, timing, and velocity requirements.

  • Departure velocity vector \(\mathbf v_1\)
  • Arrival velocity vector \(\mathbf v_2\)
  • Transfer trajectory estimate
  • Departure and arrival speed
  • Approximate \(\Delta v\) relative to circular reference velocities

Core mission-design idea

The inputs are the initial position \(\mathbf r_1\), the final position \(\mathbf r_2\), the time of flight \(\Delta t\), and the gravitational parameter \(\mu\).

\[ \mathbf r_1,\ \mathbf r_2,\ \Delta t,\ \mu \quad \Longrightarrow \quad \mathbf v_1,\ \mathbf v_2 \]

Unlike a Hohmann transfer, a Lambert transfer is not limited to two circular coplanar orbits. It is used for rendezvous, phasing, interplanetary transfers, and trajectory targeting.

Universal-variable Lambert formulation

This educational implementation uses a universal-variable style solution. The geometry is controlled by the transfer angle and the Lambert parameter \(A\):

\[ A = \sin(\Delta \theta) \sqrt{\frac{r_1 r_2}{1-\cos(\Delta \theta)}} \]

The solver searches for the universal variable that matches the requested time of flight, then computes the Lagrange coefficients and velocity vectors.

\[ \mathbf v_1 = \frac{\mathbf r_2 - f\mathbf r_1}{g}, \qquad \mathbf v_2 = \frac{\dot g\mathbf r_2 - \mathbf r_1}{g} \]

How to interpret the result

  • Short time of flight usually requires higher transfer speed.
  • Long time of flight may produce a lower-energy arc, but not always a practical mission.
  • Prograde and retrograde options can give very different velocity vectors.
  • The plotted curve is a numerical propagation from the computed departure state.

Important limitation

This is a zero-revolution educational Lambert solver. Multi-revolution Lambert solutions are not included yet.

Interactive Lambert solver

Departure position \(\mathbf r_1\) in km

Arrival position \(\mathbf r_2\) in km

Choose a preset, then adjust vectors and time manually.

Results

Departure velocity \(\mathbf v_1\)

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Arrival velocity \(\mathbf v_2\)

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Departure speed

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Arrival speed

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Approx. departure Δv

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Approx. arrival Δv

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Interpretation

Run the solver to compute the transfer velocity and plot the trajectory.

Lambert transfer trajectory

Assumptions and limitations

This tool assumes:

  • Two-body point-mass gravity
  • Impulsive departure and arrival velocity changes
  • Zero-revolution Lambert transfer
  • No drag, J2, thrust arcs, finite burns, or third-body effects
  • Approximate \(\Delta v\) uses local circular reference speed for interpretation only
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