Two dates decide a mission

The cheapest transfer between two circular orbits is a clean result about an idealised system: two coplanar circles, a body that may leave whenever it likes, and a transfer time that comes out of the geometry rather than being chosen. Real planets are on eccentric, inclined, moving orbits, and a launch date is a thing that has to be negotiated with a launch vehicle, a range and a budget.

What replaces the clean result is a two-dimensional map.

Lambert’s problem, and why it has one answer #

The underlying question is old and has a name. Given two position vectors and a time of flight between them, find the orbit.

It is not obvious that the answer is unique. Two points and a focus do not determine a conic — there is a one-parameter family of ellipses through any two points with the Sun at a focus, ranging from a thin cigar that barely reaches to a huge orbit that goes most of the way round. What picks one out is the time: a larger ellipse takes longer, monotonically, so specifying the flight time selects exactly one member of the family.

That monotonicity is Lambert’s theorem, stated by Johann Heinrich Lambert in 1761: the transfer time between two points depends only on the semi-major axis, the straight-line distance between them and the sum of their distances from the focus. It does not depend on the eccentricity, or the orientation, or anything else about the shape.

The practical consequence is that the problem can be solved by bisection. Parameterise the family by a single variable, compute the flight time it implies, and squeeze until the time matches. That is what the figure does — sixty-four by fifty-six times, once per grid cell — and the monotonicity is what guarantees it converges.

What the contours are measuring #

The quantity plotted is $C_3$, the characteristic energy: twice the specific orbital energy of the departure hyperbola, equal to the square of the speed the spacecraft has left over after escaping the Earth. It is quoted in kilometres squared per second squared, which is unfriendly until the reason is clear — $C_3$ is what a launch vehicle’s performance is specified in, because a rocket’s capability is naturally a curve of mass against energy rather than against speed.

Reading the hero figure: the cheapest transfer on the grid costs 5.1 km²/s², departing in March 2001 with a flight time of 220 days. That is close to a real mission. Mars Odyssey launched on 7 April 2001 and arrived on 24 October, a flight time of 200 days, at a $C_3$ of about 10.

The factor of two between the figure and the flight is not an error in either, and it is the most useful thing in this essay. The figure omits the inclination.

The ridge down the middle, and why this figure has not got one #

Mars’s orbit is inclined by 1.85° to the ecliptic. That sounds negligible and is not, because of where the transfer has to cross.

A transfer orbit lies in the plane containing the Sun, the departure point and the arrival point. If the two planets are in different planes, that transfer plane is inclined to both, and the spacecraft has to be launched into it — which costs energy. How much depends on the angle the transfer sweeps through: when the transfer angle passes 180°, the departure and arrival points and the Sun become nearly collinear, the plane is poorly determined, and the required inclination change goes to a maximum.

The result on a real porkchop plot is a ridge of high $C_3$ running diagonally across the middle, separating the “type I” transfers below 180° from the “type II” transfers above it. Every real Mars mission chooses one side of that ridge, and the two sides have quite different flight times — which is why some missions take six months and others eleven.

This figure is coplanar and therefore has no ridge. The omission is the reason its minimum is 5.1 where the flown value is 10, and it is stated rather than corrected because adding the third dimension would obscure what the contours are for.

The windows repeat, and the interval is not a year #

The Earth and Mars return to the same relative configuration once per synodic period,

$$\frac{1}{P_{\rm syn}} = \frac{1}{P_\oplus} - \frac{1}{P_{\rm Mars}},$$

which is 780 days, or about 25.6 months. So there is a launch window to Mars roughly every twenty-six months, and there is no possibility of arranging otherwise. Mission schedules are built around it: a delay of two weeks may be absorbed, a delay of two months costs two years.

That is a harder constraint than it looks, and it has visible consequences. The 2020 window carried Perseverance, Tianwen-1 and the Hope orbiter, launched within eleven days of each other by three different agencies — not because of any coordination but because the Sun and the planets did not offer an alternative. The ExoMars rover missed the same window by a matter of weeks and was delayed to the next one, then delayed again.

The synodic period also explains why the windows are not equally good. The window recurs every 25.6 months while Mars’s own year is 22.6 months long, so successive windows find Mars at different points of its eccentric orbit, and the cost cycles with a period of about seven synodic intervals — fifteen years. The 2003 window, with Mars near perihelion at opposition, was the cheapest in decades and carried four spacecraft.

Reading the shape rather than the minimum #

The single cheapest point is rarely what a mission flies, and the contours are drawn because the region matters more than the point.

A launch period, not a launch date. A vehicle cannot be guaranteed to launch on one particular day — weather, range conflicts and technical holds all intervene — so a mission is designed with a launch period of two or three weeks, and the vehicle must be capable of the worst $C_3$ within that period. The relevant number is therefore not the minimum on the plot but the maximum along a horizontal strip of it, and a broad shallow window is worth more than a narrow deep one.

Arrival conditions. The contours show departure energy. A second plot of the same grid shows arrival speed, and the two minima are not in the same place: a faster transfer arrives faster, and arriving faster costs propellant for an orbiter or thermal protection for a lander. Mission design is a negotiation between two overlapping regions on the same plane.

And the flight time itself is a cost. Every extra month is a month of consumables, of radiation dose for a crew, and of opportunities for something to fail. The flat bottom of a porkchop region is often traded away for a shorter flight up its steep left-hand side.

Departing from a planet, not from a point #

The plot solves the heliocentric leg and reports a $C_3$, and there is a second geometry hidden in that number which decides when in the day a rocket may leave.

A departure hyperbola has a direction — the departure asymptote, the direction the vehicle is travelling once the Earth’s gravity has stopped mattering. To reach it, the parking orbit around the Earth must contain that direction, which constrains the orbit’s plane, which constrains the launch azimuth and the moment of launch. That is the daily launch window, and it is typically one to two hours long. Miss it and the next opportunity is the next day, at a slightly different $C_3$.

So a mission has two nested windows: an interplanetary one of a few weeks, set by the contours above, and a daily one of a couple of hours, set by the rotation of the Earth carrying the launch site into the right plane. Every trajectory is stitched from three two-body problems, and the daily window is a property of the first stitch rather than of the middle one.

There is a variant worth knowing because it removes the constraint. A vehicle can launch into any convenient parking orbit and then coast until it reaches the right point before performing the injection burn — which converts the daily window into a choice of coast duration and is why upper stages are built to restart. It costs propellant boil-off and battery life, and the trade is made mission by mission.

The synodic period, and the other bodies it applies to #

The 780-day interval is not special to Mars, and putting the numbers side by side shows what the arithmetic does.

Venus’s synodic period is 584 days, so its windows come nearly three months more often than Mars’s. Jupiter’s is 399 days — barely longer than a year — because Jupiter moves so slowly that the Earth almost catches it up in one of its own orbits. For the outer planets the synodic period tends toward one Earth year, so windows to Neptune recur annually and the reason a mission there is rare has nothing to do with waiting for one.

$$\frac{1}{P_{\rm syn}} = \frac{1}{P_\oplus} - \frac{1}{P_{\rm target}}$$

is the same beat frequency that makes an eclipse repeat after three periods nearly share a multiple and that produces the great inequality of Jupiter and Saturn. Two circulating angles, one difference, one long period — the third appearance of the same construction in this collection, and here it sets a launch manifest.

What was actually measured #

Nothing on this plot is measured; every point is computed. What is measured is the input, and the accuracy of the input is worth registering.

The planets’ positions come from a numerically integrated ephemeris, fitted to decades of radar ranging, spacecraft tracking and lunar laser ranging. The Earth’s position relative to Mars is known to better than a hundred metres at the relevant epochs. That is nine orders of magnitude smaller than the distance, and it means the transfer geometry is not a source of error in any modern mission.

What the trajectory design is limited by is the execution: the launch vehicle’s injection accuracy, which is of order a metre a second in $\Delta v$ and translates into thousands of kilometres of arrival error at Mars, and the small forces acting during cruise — solar radiation pressure, propellant venting, the thermal recoil of the spacecraft’s own radiators. Every interplanetary cruise carries several trajectory-correction manoeuvres for exactly this reason, and their total budget is tens of metres a second against a departure of thousands.

Why the shape is called what it is #

The name is not a joke somebody added afterwards; it is a description, and it says something about the mathematics.

The contours close into a lobed region because $C_3$ has a minimum in the interior of the plane and rises in every direction from it — steeply toward short flight times, where the transfer has to be forced, and steeply toward departures early or late in the window, where the geometry is wrong. Between those two directions the rise is gentle, so the level sets are elongated and lopsided rather than circular. Add the inclination ridge and the region is cut nearly in half, leaving two lobes joined at a waist.

That is the shape of a cut of meat, and the name stuck in the 1960s. It is worth keeping because it names the useful property: a porkchop region is not a peak with a summit worth standing on, it is a broad low area with an awkward boundary, and a design lives in the middle of the area rather than at the lowest point.

Buying the ridge back #

The plot’s central feature is a ridge of high energy where the transfer angle passes through a straight angle, and there is a manoeuvre that makes it much cheaper — which is why real missions fly through the region the plot says to avoid.

The difficulty at that angle is geometric. Departure and arrival are nearly opposite, so the plane containing them is poorly defined, and any inclination difference between the two orbits has to be removed by a transfer that is steeply inclined to both. The velocity cost of that inclination is what the ridge is made of.

The remedy is to abandon the requirement that the transfer be a single conic. A broken-plane manoeuvre flies the first part of the trajectory in one plane, performs a small burn somewhere near the midpoint, and completes the journey in another — so the plane change is made where it is cheapest rather than at departure where the speed is highest.

The saving is substantial precisely because it exploits the same rule that governs plane changes everywhere: the cost is proportional to the speed at which the turn is made, and the midpoint of a transfer to Mars is the slowest part of it.

In practice this converts the porkchop plot from a map with a forbidden ridge into one with a shallower ridge and an extra parameter. Every recent Mars mission has carried such a manoeuvre in its baseline, budgeted at a few tens of metres a second, and the launch periods that result are wider than a single-conic analysis would allow.

So the contours are the answer to a question slightly narrower than the one a designer asks. They price the two-burn transfer exactly; the flown trajectory is a three-burn one, and the third burn is placed where this plot has no axis for it.

The general lesson is one this collection keeps arriving at: a plot prices the option it was drawn for, and an option it does not contain can be cheaper than every point on it.

That is worth carrying beyond trajectory design, since the same shape of error is easy to make with any optimisation whose search space was fixed before the question was asked.

The practical form of the warning is to ask what the axes are, and then to ask what a designer would be allowed to vary that has no axis.

Applied here it identifies the mid-course burn immediately, and applied to a launch-window study it identifies the launch vehicle’s own performance curve, which is the other axis nobody draws.

Both of those are in every mission’s actual trade study and in none of its published contour plots.

What the picture cannot show #

The inclination, as above. It is the largest omission and it accounts for roughly a factor of two in the minimum $C_3$.

Multi-revolution solutions. Lambert’s problem has one answer for a transfer of less than one full revolution and a family of further answers if the spacecraft is allowed to go round more than once. Those appear on a real porkchop plot as additional islands at long flight times, and they are occasionally the cheapest option — several asteroid missions have flown them.

And the fact that the destination is not a point. The plot solves for arrival at Mars’s centre. A mission arriving at a planet has to hit a specific point of a specific approach hyperbola in order to enter the orbit it wants or land where it intends, and that requirement narrows the acceptable region considerably — sometimes to a strip much thinner than the contours suggest. And it assumes the burns are instantaneous. Every point on the plane is a two-impulse transfer: a departure burn, a coast on one conic, an arrival burn. A vehicle with electric propulsion has no such trajectory. Its thrust is a hundredth of a gravity and it accelerates continuously for months, so its path is not a conic and its cost is not a $C_3$ — it is the propellant integrated along a thrust profile that has to be optimised as a whole. The window structure softens with it: a low-thrust mission can leave over a span of months rather than weeks, at a cost that varies gently instead of steeply, and the arithmetic that produces this plot has no way to say so. That is the second sense in which the axes decide the answer, and it is the reason a solar-electric mission’s schedule is argued about in different terms from a chemical one’s.

One region of the plot has been cropped out of every figure so far, and it is where the other half of Lambert’s problem lives.

The existence of that second family is a consequence of the geometry rather than of the arithmetic. Two points and a time of flight admit a conic through both, and for long enough times they admit more than one — the spacecraft may cross the transfer angle the short way or the long way, and for still longer times it may complete a revolution before arriving. Each of those is a separate branch of the same Lambert solution, and each has its own cost surface.

Missions almost never take them, and the reason is not the Δv. A three-year cruise costs three years of operations, three years of radiation dose, three years of component failures and three years during which the launch that produced it cannot be repeated. The porkchop plot prices the propellant and prices nothing else, so a branch that wins on the contour map can lose comprehensively on the programme.

Where the long branches are used is in the outer solar system, where the direct route is not merely expensive but impossible with any available launcher, and a decade of cruise is the baseline rather than the penalty. There the plot is read differently: the question is not which departure is cheapest but which departures exist at all, and the answer is usually a narrow band running through a flyby of something else on the way. That turns the two-dimensional plot into a search over a sequence of dates rather than a pair of them, and the search is combinatorial: every candidate flyby body multiplies the space, and every additional leg adds two more axes. What makes it tractable is that each leg is still a Lambert problem with a closed-form answer, so the cost of evaluating one complete itinerary is measured in microseconds and millions of them can be enumerated.

Where the ladder goes next #

The rung above is the multi-body trajectory: a route that uses one or more gravity assists on the way, which turns a two-dimensional plot into a search over a sequence of encounter dates and is how every mission to the outer planets is designed. The rung beside it is low-thrust trajectory optimisation, where there is no departure impulse to plot and the entire trajectory is the answer to an optimal-control problem — and where the porkchop plot’s role is taken by a solver whose output nobody can draw on one page.