Two places and a clock decide the path

An orbit has six numbers in it. Ask for the one that passes through a given point at a given time and arrives at a second given point at a second given time, and the question looks like a search in six dimensions with two three-dimensional constraints — which is to say a search with a great deal of freedom left in it and no obvious way to organise the freedom.

Lambert’s theorem says the search is one-dimensional, and it says so by asserting something that has no business being true.

The quantity that does not appear #

Take two positions $\mathbf r_1$ and $\mathbf r_2$ with a primary at the origin, and build three lengths from them: the two radii $r_1$ and $r_2$, and the straight-line chord $c$ between the points. From those, the semiperimeter of the triangle they make,

$$s = \frac{r_1 + r_2 + c}{2}.$$

Lambert’s theorem is that the time of flight along any conic through both points is a function of $a$, $s$ and $c$ alone. Two orbits with the same semi-major axis, connecting the same pair of points, take the same time — however differently shaped they are.

That is genuinely strange. Two ellipses can share a semi-major axis and differ in eccentricity by a factor of three, so that one hugs the primary and the other swings far out; they can be oriented so that one passes periapsis between the two points and the other does not go near it. Everything about how the body moves along the two arcs is different. The elapsed time is the same.

The explicit form, due to Lagrange, is worth writing because the case structure in it is the whole of what a Lambert solver has to decide:

$$\sqrt{\frac{\mu}{a^3}}\;t = (\alpha - \sin\alpha) - (\beta - \sin\beta),$$

$$\sin\frac{\alpha}{2} = \sqrt{\frac{s}{2a}}, \qquad \sin\frac{\beta}{2} = \sqrt{\frac{s-c}{2a}}.$$

Two angles, each a function of one length ratio. $\beta$ takes a negative sign when the transfer sweeps more than half a turn. $\alpha$ is replaced by $2\pi - \alpha$ on the branch whose arc passes through apoapsis. Nothing else varies.

Why the smallest orbit is the slowest #

The branch structure in the first figure contains a fact that reverses an intuition, and it is worth being explicit about it because the intuition is a good one everywhere else.

For a fixed pair of points, the semi-major axis cannot be arbitrarily small. There is a floor at $a = s/2$: below it, $\sin(\alpha/2) = \sqrt{s/2a}$ exceeds 1 and there is no ellipse through the two points at all. That floor is the minimum-energy transfer, since energy and semi-major axis are one quantity, and the least energy an orbit can have while still touching both points is the least $a$ it can have.

Now the reversal. Making $a$ larger — spending more energy — makes the flight faster, not slower, on the lower branch. The time falls monotonically as $a$ rises, towards a floor of its own at the parabola, which takes 103.2 days for the geometry drawn above. So the cheapest transfer between two fixed points is the slowest one available, and every faster transfer costs more energy. There is no trade to be made in the other direction: no ellipse is both smaller and quicker. The energy is the semi-major axis written another way, and the two branches meeting at $s/2$ are the two ways an orbit of the least possible size can join the two points.

For a Hohmann transfer that reads as familiar — the cheap transfer is the slow one — but the mechanism is different. A Hohmann transfer is cheapest because both burns are tangential and none of the velocity change is spent turning. This is a statement about a fixed pair of points, in which the direction of the velocity at each end is whatever the orbit delivers and is not free to be optimised. The two coincide only when the two points happen to be at opposite ends of a diameter.

One time, one orbit #

The monotonicity in the first figure is a claim with a consequence, and the figure checks it point by point rather than asserting it: on each branch the flight time is a strictly monotone function of $a$. A horizontal line at a requested time therefore cuts each branch at most once.

So for a given pair of points, a given sweep angle, and a given time of flight, there is exactly one ellipse. Not a family, not a two-parameter set: one. The problem that looked six-dimensional has one solution, and finding it is a root-find on a monotone function of a single variable.

This is the point at which Lambert’s problem stops being a curiosity and becomes the standard tool. Every interplanetary trajectory begins as a Lambert solve: pick a departure date, which fixes $\mathbf r_1$ from the Earth’s ephemeris; pick an arrival date, which fixes $\mathbf r_2$ from the target’s; solve for the unique orbit; difference its departure velocity against the Earth’s own to get the departure energy. One number out, two dates in.

The vacant focus, and what minimum energy looks like #

There is a construction that makes the theorem visible rather than merely true, and it is the oldest way into the subject.

An ellipse has two foci, and the primary occupies one of them. The other — the vacant focus — obeys the defining property: the sum of distances from the two foci to any point on the curve is $2a$. So if the transfer passes through $\mathbf r_1$, the vacant focus lies at distance $2a - r_1$ from that point; if it also passes through $\mathbf r_2$, it lies at distance $2a - r_2$ from that one. It is therefore an intersection of two circles, and given $a$ it is constructed rather than chosen.

That construction is where the theorem comes from. The flight time depends on the shape of the swept region, and the swept region is fixed once the two endpoints and the vacant focus are fixed — and the vacant focus is fixed by $a$ and the two radii, with the chord entering because the two circles’ intersection depends on how far apart their centres are. Three lengths, and the geometry is determined.

At $a = s/2$ the two circles are tangent and the vacant focus lands exactly on the chord, which is what the light curve in the second figure shows. Below $s/2$ they do not meet: that is the same statement as “no ellipse exists”, arrived at with a compass instead of an inequality.

What was actually measured #

Nothing on this page is an observation, and that is worth saying plainly, because Lambert’s problem is where celestial mechanics is at its most purely deductive: two positions in, one orbit out, with no data anywhere in the arithmetic.

The observations are one level down, and there are two kinds.

The first is the pair of positions themselves. $\mathbf r_1$ and $\mathbf r_2$ come from the planetary ephemeris, which is a fit to two centuries of meridian-circle observations, a few decades of radar ranging to Venus and Mars, and — for the inner planets since the 1970s — Doppler tracking of spacecraft in orbit around them. The Earth’s position at a given instant is known to a few hundred metres and Mars’s to a few tens of metres along track. A Lambert solution inherits those errors directly.

The second is $\mu$, and this is the sharper case. The transfer time depends on $\sqrt{a^3/\mu}$, so the answer is only as good as the Sun’s gravitational parameter — which is not the Sun’s mass and cannot be turned into one without $G$. What is measured, by ranging, is $GM_\odot$ as a single quantity, to about ten significant figures — the same distinction the gravitational constant essay is entirely about, arriving here as a limit on how precisely a date can be chosen. That is enough for the solve to be limited by the ephemeris rather than by the constant, which is a comfortable position to be in and was not always the case.

The measurement that actually closes a mission is neither of these. It is the post-injection tracking: after the burn, the spacecraft’s own Doppler and ranging give the orbit it is really on, which differs from the Lambert solution by the execution error of the engine. Every deep-space mission budgets a mid-course correction for exactly that difference, of a few metres per second against a departure of several kilometres per second. The Lambert solve says where to aim; it does not say where the vehicle went.

When there is more than one answer #

The uniqueness of the solution holds for a transfer completed in less than one revolution, which is the case a first transfer is always designed as. Allow longer and it stops holding.

If the flight time exceeds one orbital period, the spacecraft may reach its destination after completing one or more complete revolutions on the way — and each additional revolution allowed brings a pair of new solutions rather than one. For a flight time permitting $N$ complete revolutions there are $2N+1$ trajectories connecting the same two points in the same time.

The pairs come from a fold in the problem: for a given number of revolutions the flight time as a function of orbit size has a minimum, so times above that minimum are achieved by two different orbits, one on each side of it. At the minimum the two coincide, which is a genuine singularity of the solver and a place every implementation has to handle specially.

These are not curiosities. A rendezvous with a target in a similar orbit — resupplying a station, catching an asteroid on a nearby path — often has a flight time of several revolutions, and the multi-revolution solutions are cheaper than the direct one because they let the transfer orbit differ less from both endpoints.

So the essay’s central statement needs one qualification: two places and a clock decide the path uniquely, provided the clock has not run long enough for a second path to become possible. After that they decide a small, enumerable set, and choosing among them is the design problem.

The enumeration is also what makes an automated trajectory search possible: a solver that returns every solution for a given pair of dates can be run over a grid of dates without any risk of missing the cheap option, which a solver returning only the direct transfer cannot.

It is also why every published Lambert solver states its revolution limit in the interface: the number of answers is part of the specification rather than a detail of the algorithm.

Where the model stops #

The theorem is exact for a two-body problem and the solar system is not one. Three limits, in order of how soon they bite.

The sweep angle of 180°. For coplanar orbits and a transfer of exactly half a turn, the plane of the transfer is undefined — any plane containing the two points and the primary will do, and they are collinear. Lambert’s equation does not fail there so much as decline to answer, and the ridge across the middle of a porkchop plot is that indeterminacy showing up as a wall of expensive solutions on either side. Real transfers between real planets never sweep exactly 180° because the orbits are not quite coplanar, which converts a singularity into a merely expensive region.

Multiple revolutions. Everything above assumes the transfer completes less than one full turn. Allow $N$ complete revolutions before arrival and the flight-time function acquires $N$ further pairs of branches, each pair meeting at a minimum, so a given time of flight can be met by $2N + 1$ different orbits. Low-thrust and low-energy missions live in exactly that regime, and choosing among the solutions is a real design problem rather than a formality.

Perturbations. The elements do not stay constant, so an orbit that satisfies Lambert’s equation at the moment of the burn does not arrive where the equation says. In practice the Lambert solution is the first iterate of a full numerical trajectory optimisation, which integrates all the bodies and re-solves. What Lambert’s theorem provides is a starting guess good enough that the optimisation converges — and a starting guess is not nothing: without one, the optimiser is searching six dimensions with no idea where to begin.

The time-of-flight curve has the same shape for every geometry and the geometry decides where the interesting parts of it fall.

What the picture cannot show #

The plane. Every figure here is drawn face-on, and a real transfer between two planets is not coplanar with either. The inclination change is folded into the departure energy in a porkchop plot and appears nowhere in the geometry above.

The direction of travel. The arcs in the second figure would be identical if traversed the other way. Which end is the departure is asserted by a label, and the whole content of “short way” versus “long way” is about direction rather than about shape.

The cost. Flight time against semi-major axis is not flight time against Δv. The velocity change a transfer demands depends on the difference between its endpoint velocities and the planets’ own, which is a vector subtraction the first figure has no axis for. A shorter flight is a larger orbit and a larger orbit is usually dearer — but “usually” is doing work there, and the exceptions are what a porkchop plot is drawn to find.

The problem was posed about comets #

Johann Heinrich Lambert stated the theorem in 1761, and not about spaceflight, which did not exist, nor about transfers, which nobody wanted. He was working on comets.

The problem then was the reverse of the modern one. A comet had been seen at two epochs, its two heliocentric positions had been estimated by other means, and the question was what orbit it was on. Lambert’s insight was that the elapsed time between the two sightings, which was known exactly, was enough to close the system — because it depends on so little. Three lengths and a time give the semi-major axis, and the semi-major axis with the two positions gives the whole orbit.

That is one of two classical routes into an orbit from observations, and this site carries the other: the determination from five sight lines and no distance at all, which is Gauss’s method and needs no positions to begin with. The two are complementary in an exact sense. Gauss’s method gets distances out of angles and is badly conditioned when the observed arc is short; Lambert’s method needs distances to start and is then exact. In modern practice they run together: Gauss for the first orbit, Lambert to refine it, both inside a least-squares fit that uses every observation. Two more arcs, for the two geometries the essay has not drawn.

Where the ladder goes next #

Later rungs on this anchor: the Battin and Gooding formulations, which reparameterise the solve so that the iteration converges in three steps rather than thirty and which are what every flight-dynamics library actually contains. The multiple-revolution case in full, with its $2N+1$ solutions and the minimum-time branch point between each pair. The $B$-plane, which turns the arrival end of a Lambert solution into two targeting coordinates a navigator can steer. Lambert with a perturbing body, where the two-body arc becomes the first term of a series. And the inverse question — given a departure energy rather than a time of flight, which pairs of dates are reachable — which is the porkchop plot read the other way and is how a launch period is actually chosen.

Lambert’s theorem is a statement that a certain quantity does not depend on things it obviously should. That kind of statement is rare and it is always worth asking what symmetry is behind it. Here it is the same one that makes the third law depend only on the semi-major axis and no other element: the period of an inverse-square orbit is a function of its energy alone, and Lambert’s theorem is that fact applied to an arc instead of to a whole turn.