Lambert's Problem: Orbital Rendezvous Transfer Solver

Departure & Target Orbits

Departure radius r₁ 7000 km

Target radius r₂ 15000 km

Transfer angle Δν 150°

Time of flight (TOF) 2.00 h

Solver Output

solving…

Δv depart

— km/s

Δv arrive

— km/s

Total Δv

— km/s

Universal anomaly z

—

Newton iterations

—

Propagation miss

— km

How it works

Lambert's problem: given two position vectors r₁, r₂ and a transfer time Δt, find the single two-body orbit that connects them. Unlike a Hohmann transfer (which only works for two aligned circular orbits), Lambert's solution works for any two points and any flight time — it is what mission planners actually solve to schedule a rendezvous or an interplanetary transfer window.

This solver uses the universal-variable formulation (works for ellipses, parabolas and hyperbolas alike) via the Stumpff functions C(z), S(z):

A = sin(Δν) · sqrt(r1·r2 / (1 − cos Δν))
y(z) = r1 + r2 + A·(z·S(z) − 1) / sqrt(C(z))
F(z) = (y(z)/C(z))^1.5 · S(z) + A·sqrt(y(z)) − sqrt(μ)·Δt

Solve F(z) = 0 for z by Newton–Raphson, then:
  f = 1 − y/r1        g = A·sqrt(y/μ)        ġ = 1 − y/r2
  v1 = (r2 − f·r1) / g          v2 = (ġ·r2 − r1) / g

r1 / r2 sliders — set the departure and target orbital radii (Earth's μ = 398,600 km³/s²).
Δν — the transfer angle swept between departure and arrival; at exactly 180° the geometry is singular (A → 0, no unique transfer plane), which is why the slider avoids it.
TOF — the time budget for the transfer. Every (r1, r2, Δν, TOF) combination has its own unique Δv cost — that's the whole point of the solver.
The yellow arc is not drawn from the algebra directly — it is the trajectory obtained by numerically integrating Newton's law of gravitation (d²r/dt² = −μr/|r|³) forward from the solved departure velocity. The "propagation miss" readout is the residual distance between that independently-integrated endpoint and the true target — a live accuracy check on the solver.

Real station-keeping relevance: this is the exact class of problem solved before a Soyuz or Cygnus rendezvous burn, or before choosing an interplanetary launch window — a fixed arrival date (TOF) plus known departure/target ephemerides (r1, r2, Δν) determines one required burn budget.

Departure orbit (r1) Target orbit (r2) Transfer trajectory

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