Astronomy & Celestial Mechanics
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The lambert-arc generator

One pair of points, 260 days, and the conics that fit
One pair of points, 260 days, and the conics that fit. Two positions 1 and 1.524 AU from the Sun, separated by 135°, and the transfer orbits that get from one to the other in 260 days. The short way has semi-major axis 1.2189 AU and eccentricity 0.3244; the long way, sweeping 225° between the same two points in the same 260 days, needs 1.2184 AU and e = 0.2680. Lighter is the minimum-energy transfer, a = s/2 = 1.2161 AU, which takes 244.2 days and is the slowest ellipse rather than the fastest: below it there is no ellipse through these points at all. Its vacant focus is constructed as the intersection of the circles of radius 2a − r about each endpoint, and it lands on the chord, which is what minimum energy means as geometry. Every arc here was solved from Lambert's formula and then checked by integrating Kepler's equation between its own two true anomalies — 260.0000 days against the 260 asked for. The chord is 2.3405 AU and the semiperimeter 2.4322; those two numbers and the semi-major axis fix the time, and nothing else in this drawing does.

Two positions 1 and 1.524 AU from the Sun, separated by 135°, and the transfer orbits that get from one to the other in 260 days. The short way has semi-major axis 1.2189 AU and eccentricity 0.3244; the long way, sweeping 225° between the same two points in the same 260 days, needs 1.2184 AU and e = 0.2680. Lighter is the minimum-energy transfer, a = s/2 = 1.2161 AU, which takes 244.2 days and is the slowest ellipse rather than the fastest: below it there is no ellipse through these points at all. Its vacant focus is constructed as the intersection of the circles of radius 2a − r about each endpoint, and it lands *on the chord*, which is what minimum energy means as geometry. Every arc here was solved from Lambert's formula and then checked by integrating Kepler's equation between its own two true anomalies — 260.0000 days against the 260 asked for. The chord is 2.3405 AU and the semiperimeter 2.4322; those two numbers and the semi-major axis fix the time, and nothing else in this drawing does.

3 essays call lambert-arc. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

Flight time against semi-major axis, for a fixed 135° sweep. Lambert's theorem drawn: the time to fly between two points 1 and 1.524 AU out and 135° apart, against the semi-major axis of the orbit that does it. Nothing else about the orbit enters — not its eccentricity, not where its periapsis is, not how it is oriented — which is the content of the theorem and the reason a two-point transfer is a one-dimensional search rather than a six-dimensional one. Two branches: the lower one is the ellipse whose arc stays short of apoapsis, falling towards the parabolic floor at 103.2 days as a grows without limit; the upper one is the ellipse of the same size whose arc runs through apoapsis, rising without limit. They meet at a = s/2 = 1.2161 AU, 244.2 days, which is the minimum-energy transfer and the slowest ellipse available — every faster one is bigger. Each branch is monotone, checked point by point across the drawn range, so a horizontal line cuts each at most once: for a given pair of points and a given time there is exactly one ellipse, and at 260 days it is a = 1.2189 AU on the upper branch. The freedom a mission designer has is not in this picture: it is the choice of the two points, which is what a porkchop plot sweeps. Orbits

Two places and a clock decide the path

The time to fly between two points depends on the semi-major axis, the chord between them, and the sum of their distances — and on nothing else about the orbit. Not the eccentricity, not where periapsis is, not the orientation. Lambert's theorem is why an interplanetary launch date is the root of one equation.

 5 transfers through the same two points in the same 1400 days. Time of flight against semi-major axis for every transfer through two points 135° apart at 1 and 1.524 AU, with the revolution count running from 0 to 2. Each count contributes two branches, and for N ≥ 1 the pair folds: the time has a minimum at a = 1.2426 AU for one revolution — only 2.2 per cent above the minimum-energy value of 1.2161, which is why the horizontal axis is the excess over that value and logarithmic — so a flight time above it is met twice and below it not at all. Reading the crossings of the 1400-day line off the drawn curves gives 5 of them — 0 revs high, 1 rev low, 1 rev high, 2 revs low, 2 revs high — which is 2N + 1 with N = 2, and the count is a property of the time rather than of the geometry. That is the practical content: a root-finder started from a single guess returns one of these 5 and gives no sign that the other 4 exist, and the cheapest of them is often not the one nearest the guess. Orbits

One time of flight and five ways round

Lambert's theorem says two positions and an interval fix the transfer. Allow the transfer to complete whole revolutions and that stops being true: the flight time folds, one number admits five arcs, and the cheapest of them is usually not the one a solver started nearest to.

 A Hohmann transfer, 2.6 to 1 in radius. Two circular orbits and the ellipse that touches both. The first burn raises the far point to the outer orbit; the second, half an orbit later, circularises. The costs are computed from the vis-viva relation. Spaceflight

The cheapest way between two orbits, and why it is so slow

Two burns and a long coast is the least fuel that will move a spacecraft between two circular orbits. It is also, for anything beyond the Moon, an unreasonably long wait.

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