Lambert's Problem & Transfer Orbit Simulator

Q: What is Lambert's problem?

Lambert's problem asks: given two position vectors around a central body (departure point P1 and arrival point P2) and the time of flight between them, find the unique transfer orbit connecting the two points — its velocity vectors and orbital elements. Formulated by J.H. Lambert in 1761, it underpins all modern space-mission design: lunar and planetary launch-window computation, rendezvous, orbit transfers, and debris avoidance. This tool focuses on the minimum-energy solution (a_min = s/2) and on the 180-degree special case, the Hohmann transfer.

Q: What is the difference between the short way and the long way?

Lambert's problem normally has two solutions for a given transfer angle Δθ: the 'short way' (Δθ 180°). The short way arrives faster but tends to require more delta-v, while the long way takes longer and allows more freedom in departure direction — useful for phasing during rendezvous. At Δθ = 180° the two solutions coincide and reduce to the Hohmann transfer. The 'Transfer direction' selector in this tool switches between the two branches.

Visualise the transfer orbit that connects two points around a central body. Given r₁, r₂ and the transfer angle, the tool computes the minimum-energy ellipse, compares it with a Hohmann transfer, and reports the delta-v at each burn and the total — so you can build intuition for LEO-to-GEO, lunar and interplanetary transfer manoeuvres.

Parameters

Departure radius r₁

Distance from the central body to the departure point (default: LEO)

Arrival radius r₂

Distance to the destination (default: GEO geostationary radius)

Transfer angle Δθ

Central angle between P1 and P2. 180° = Hohmann transfer

Gravitational parameter μ

km³/s²

μ = GM. Earth: 398600, Moon: 4903, Sun: 1.327e11

Transfer direction

Two solutions exist between the same two points

Results

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Chord c (km)

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Min-energy semi-major a_min (km)

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Transfer time (h)

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Hohmann transfer time (h)

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First burn Δv (km/s)

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Total Δv (km/s)

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Transfer geometry — central body, initial orbit, target orbit, transfer ellipse

The central body sits at the centre. The blue circle is the initial orbit r₁, green is the target orbit r₂, and the orange ellipse is the transfer arc. Short way is drawn solid, long way is dashed. The yellow dot is the spacecraft sweeping along the transfer arc.

Transfer time vs transfer angle Δθ (30–330°)

Total Δv vs radius ratio r₂/r₁

Theory & Key Formulas

$$c = \sqrt{r_1^{2} + r_2^{2} - 2\,r_1 r_2 \cos\Delta\theta}, \qquad s = \frac{r_1 + r_2 + c}{2}, \qquad a_{\min} = \frac{s}{2}$$

Chord c, semi-perimeter s, and the semi-major axis a_min of the minimum-energy ellipse. a_min is the special solution of Lambert's equation for which the orbital energy (−μ/2a) is minimised among all ellipses connecting the two points.

$$t_{\min} = \frac{1}{\sqrt{\mu}}\,\frac{s^{3/2}}{\sqrt{2}}\,\left[\,\pi - \arcsin\sqrt{1-\frac{c}{s}} + \sqrt{\frac{c}{s}\!\left(1-\frac{c}{s}\right)}\,\right]$$

Flight time on the minimum-energy ellipse (Lagrange's theorem). μ = GM is the central-body gravitational parameter. Once c, s and μ are fixed, the flight time is uniquely determined.

$$\Delta v_{\text{total}} = \left|v_p - \sqrt{\mu/r_1}\right| + \left|\sqrt{\mu/r_2} - v_a\right|, \qquad v_{p,a} = \sqrt{\mu\!\left(\tfrac{2}{r_{p,a}} - \tfrac{1}{a_H}\right)}$$

Total delta-v for a Hohmann (180°) transfer. a_H = (r₁+r₂)/2, v_p is the perigee speed and v_a the apogee speed. The total is the sum of the speed differences from the two circular-orbit speeds.

Lambert's Problem and Transfer Orbits

🙋

"Lambert's problem" — I've never heard that name. What is it actually solving?

🎓

Put simply: given a departure point P1, an arrival point P2 and a time of flight, "what orbit must my spacecraft fly to get there?" Lambert formalised it back in 1761 and it still sits at the heart of every space mission — lunar and Mars launch windows, ISS docking, debris avoidance, all of them. The inputs are just three numbers: r₁, r₂ and the transfer angle Δθ that separates them. From those, the ellipse — its semi-major axis a and eccentricity e — is fixed uniquely.

🙋

There must be infinitely many orbits connecting two points though, surely?

🎓

Sharp question. What Lambert actually proved is "once you fix the time of flight, the orbit is unique". So instead of asking you to enter time directly, this tool shows the easiest-to-compute case: the minimum-energy solution. That ellipse has a = s/2 with s = (r₁+r₂+c)/2, where c is the chord length. From there Gauss's formula gives the flight time. Try setting r₁=6678 (LEO), r₂=42164 (GEO) and Δθ=180°: you get a_min = 24421 km, t = 5.27 h, total Δv = 3.89 km/s — the textbook LEO-to-GEO Hohmann numbers.

🙋

What's the "short way" vs "long way" selector about?

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Between the same two points there are actually two ways to fly: the short arc on the Δθ < 180° side and the long arc going the other way around. Short way is fast but costly in Δv; long way takes more time but gives you more freedom for phasing — useful when you need to meet a moving target. The ISS "fast" 6-hour rendezvous is short-way; the older Soyuz 2-day rendezvous is more long-way. At exactly Δθ = 180° the two solutions merge and you get a Hohmann transfer.

🙋

So why exactly is the Hohmann transfer called "the most efficient"?

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Because for two coplanar circular orbits, it has been proven that two tangential burns at the ends — a Hohmann — minimise total Δv across almost every other impulsive transfer. The "Total Δv vs r₂/r₁" chart shows how Δv grows with the radius ratio. The catch is time: flight time scales with r₂^{3/2}, so for Mars (~258 days even with Hohmann) you may accept extra Δv for a faster transfer. That's why real mission designers build "pork-chop plots": thousands of Lambert solutions over (depart date × arrive date) to find the best practical trade-off.

🙋

If I change μ, can I also model transfers around the Moon or Mars with the same tool?

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Yes — μ tells the tool which central body you're orbiting. Earth: 398600, Moon: 4903, Mars: 42828, Sun: 1.327e11 km³/s². For example, around the Moon with r₁ = 1838 km (low lunar orbit), r₂ = 8000 km and Δθ = 180°, you get a Hohmann transfer of about 3.4 hours and total Δv around 0.6 km/s. Real Apollo-style Earth-to-Moon trajectories are bigger and use patched conics across the lunar sphere of influence, but they all start from this same two-body Lambert solution.

Frequently Asked Questions

Lambert's problem asks: given two position vectors around a central body (departure point P1 and arrival point P2) and the time of flight between them, find the unique transfer orbit connecting the two points — its velocity vectors and orbital elements. Formulated by J.H. Lambert in 1761, it underpins all modern space-mission design: lunar and planetary launch-window computation, rendezvous, orbit transfers, and debris avoidance. This tool focuses on the minimum-energy solution (a_min = s/2) and on the 180-degree special case, the Hohmann transfer.

Lambert's problem normally has two solutions for a given transfer angle Δθ: the "short way" (Δθ < 180°) and the "long way" (Δθ > 180°). The short way arrives faster but tends to require more delta-v, while the long way takes longer and allows more freedom in departure direction — useful for phasing during rendezvous. At Δθ = 180° the two solutions coincide and reduce to the Hohmann transfer. The "Transfer direction" selector in this tool switches between the two branches.

The Hohmann transfer is the minimum-delta-v 180-degree ellipse connecting two coplanar circular orbits of radii r1 and r2, with perigee at r1 and apogee at r2. It uses only two tangential burns and is provably the minimum over almost all impulsive transfer schemes. The canonical LEO-to-GEO case (r1=6678 km, r2=42164 km) gives total Δv ≈ 3.89 km/s and transfer time ≈ 5.27 hr — values this tool reproduces. The trade-off is the long flight time, which is why fast non-Hohmann transfers are sometimes preferred for crewed lunar and Mars missions.

Mission designers solve Lambert's problem for thousands of (departure date, arrival date) pairs and plot the resulting delta-v as contour lines on a "pork-chop plot". The chart shows departure C3 (escape energy from Earth) and arrival V-infinity at the target planet, letting engineers pick the launch date that satisfies rocket capability, spacecraft mass and science-window constraints. NASA missions from Mariner and Voyager through Mars Reconnaissance Orbiter to the upcoming Europa Clipper all rely on this method as standard practice.

Real-World Applications

Geostationary satellite injection (GTO → GEO): Commercial communication satellites are lifted by a launcher into a geostationary transfer orbit (GTO, perigee at LEO, apogee at GEO) and then circularise at apogee with their own apogee-kick burn. That is precisely the default case in this tool — r₁ = 6678, r₂ = 42164, Δθ = 180° — giving a LEO-to-GEO Hohmann with total Δv ≈ 3.89 km/s and a flight time of about 5.3 h. The launch-capacity curves for H-IIA, Falcon 9 and Ariane 5 are all sized against this Δv.

Interplanetary probes (Mars Reconnaissance Orbiter and friends): For Mars, designers connect Earth's heliocentric orbit (r₁ ≈ 1.496e8 km) and Mars's (r₂ ≈ 2.279e8 km) with a Sun-centred Hohmann transfer, then run Lambert solutions over many departure and arrival dates to pick the launch window. The Hohmann theoretical minimum is about 258 days and Δv ≈ 5.6 km/s (Sun, μ = 1.327e11 km³/s²). Real missions usually trade some Δv for non-Hohmann arrivals that give better landing-site visibility or shorter cruise.

ISS rendezvous and docking: Soyuz and Crew Dragon catch up with the International Space Station (altitude ≈ 400 km) using 6-hour or 2-day transfer profiles. Both are designed with Lambert solutions that compute the phasing and approach speed at each burn. Fast profiles re-solve Lambert in real time during the chase to correct for tracking errors and ISS ephemeris updates.

On-orbit servicing and debris removal: Missions for satellite refuelling, repair or debris capture often visit several targets in a tour. Each leg is its own Lambert problem, and the overall mission is a travelling-salesman-style optimisation over total Δv and time-of-flight. Genetic algorithms and AI-based optimisers are now an active area of research for designing such multi-target tours.

Common Misconceptions and Pitfalls

The most common misconception is that "Δθ = 180° is always the most efficient". That is true only for circle-to-circle coplanar transfers. As soon as the orbits are inclined or eccentric, the optimum transfer angle drifts away from 180°. Worse, real missions are dominated by phasing — your target has to be in the right place when you arrive. A Hohmann opportunity for Mars exists only every ~26 months (the synodic period), so missions routinely sacrifice some Δv to hit a usable launch window. Treat the numbers from this tool as an idealised lower bound on Δv, not a turnkey schedule.

A second pitfall is forgetting that Lambert's problem is a two-body model. The equations here ignore every gravitational influence except the central body. In practice, an Earth-to-Moon transfer switches to a Moon-centred two-body inside the lunar sphere of influence (a patched-conic approximation), and interplanetary missions chain three Lambert solutions across the departure planet, the Sun and the arrival planet. High-fidelity missions add J2 oblateness, solar radiation pressure and third-body perturbations from the Moon and Jupiter. Lambert is always the initial guess; numerical integration refines it.

Finally, do not assume that "a small Δv means low fuel mass" without doing the rocket equation. The Δv reported here is an instantaneous velocity increment; the fuel mass follows from Tsiolkovsky's equation, Δv = Iₛₚ·g·ln(m₀/m₁). A 3.89 km/s burn with a 320-s specific impulse (hydrazine bipropellant) consumes about 71 % of the initial mass as propellant. That is why more than half of a typical commercial-satellite launch mass is propellant for orbit changes — and why even a 10 % Δv reduction in mission design can double the payload science return.

Lambert's Problem & Transfer Orbit Simulator

Lambert's Problem and Transfer Orbits

Frequently Asked Questions

Real-World Applications

Common Misconceptions and Pitfalls

How to Use

Worked Example

Practical Notes