Concept

Porkchop plot — where it appears

A contour map of transfer cost against departure and arrival date, whose closed contours give the plot its name. Each pixel is one Lambert solve, and the basins are the launch windows, which repeat at the synodic period.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

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.

orbits · Lambert's problem

The cost of going to Mars, against when to leave and how long to take. Contours of departure energy C₃ over a grid of 64 × 56 solved Lambert problems: each point is a departure date, a flight time, and the unique single-revolution conic that connects the two planets between them. The cheapest transfer on this grid costs 5.1 km²/s², leaving in Mar 2001 with a flight time of 220 days, against 8.7 for the idealised Hohmann transfer between circular orbits of the same radii. A launch window is a region on this plane, not a moment, and its shape is what a launch period is negotiated against.

Two dates decide a mission

Given where a spacecraft leaves from, where it is going, and how long it may take, there is exactly one orbit joining the two. Solving that problem over every pair of departure and arrival dates produces a contour map, and the shape of the contours is what a launch window actually is.

spaceflight · Launch windows

Windows every 780 days, costing between 5 and 13. The cheapest departure energy available in each of 8 consecutive launch opportunities, each found by solving a grid of Lambert problems around the window and taking the minimum. Opportunities recur every 780 days — the synodic period of Earth and Mars, which is exact and which is why the interval between missions is always about twenty-six months. Their cost is not periodic on that interval. The cheapest here is 5.0 km²/s² and the dearest 12.9, a factor of 2.59, and the pattern repeats on a period of about fifteen years rather than on the synodic one. The cause is Mars's eccentricity of 0.093: a transfer that arrives near Mars's perihelion has less distance to cover and meets a faster-moving planet, and whether an opportunity does that depends on where Mars is in its own orbit — which drifts relative to the synodic cycle by a fixed amount each time and comes back into phase after seven windows. In launch mass the factor is larger than it looks: departure energy enters the rocket equation through an exponential, so a C₃ of 13 rather than 5 costs roughly 1.08 times the propellant at departure.

The window that comes back and the cost that does not

Launch opportunities to Mars recur every 780 days exactly, because that is the synodic period and a synodic period is arithmetic. What they cost does not repeat on that interval at all — the cheapest window is a factor of two and a half below the dearest, and the pattern comes back every fifteen years rather than every two.

spaceflight · Launch windows

Named alongside it

The objects these essays reach for when they reach for this one.

Characteristic energy Hohmann transfer Lambert's problem Launch window Synodic period Time of flight Transfer orbit Bifurcation Departure asymptote Departure energy Interplanetary transfer Lambert problem

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