A wall across the porkchop

A launch window is a region on a plane: departure date across, flight time up, and at every point the energy of the one conic that connects the Earth at the first date to the target at the second. Drawn as contours, the energy makes the shape called a porkchop — two basins of cheap transfers separated by a narrow crease — and the whole art of choosing a mission’s dates is reading it. The windows come back every twenty-six months, and their costs do not: Mars’s eccentric orbit makes some windows much cheaper than others, in a cycle of fifteen years.

Those plots, like most first drawings of them, put Mars’s orbit in the plane of the Earth’s. The crease between the basins is then barely visible. Mars’s real orbit is tilted by 1.85 degrees to the ecliptic, and with the tilt put back, the crease becomes a wall.

The same plot, flat #

The comparison is worth making with the same calculation, so that the only difference between the two drawings is the tilt.

In the flat model there is nothing special about a transfer that goes exactly half-way round. The classic minimum-energy transfer between two circular orbits, the Hohmann transfer, is precisely such a transfer — it leaves from one side of the Sun and arrives at the other — and Lambert’s problem has a perfectly good solution there. The cheapest point in the flat porkchop sits close to the 180-degree line for exactly that reason. In the tilted model the same point is on top of the wall.

A plane through two nearly opposite points #

The reason is a statement about planes. Any Keplerian orbit about the Sun lies in a plane through the Sun. A transfer from the Earth to Mars must pass through the Earth at departure and through Mars at arrival, so its plane contains three points: the Sun, the departure point and the arrival point. Three points fix a plane — unless they are in a line.

When the transfer angle is 180 degrees, the departure and arrival points are on opposite sides of the Sun, and the three points are nearly collinear. If both points are in the ecliptic, any plane containing the line through them will do, and the ecliptic itself is the natural choice. If the arrival point is slightly above the ecliptic — at a heliocentric latitude $\beta$ of a fraction of a degree — the plane through the three points is forced, and it is steep.

The geometry is a single relation. For a plane through the Sun that crosses the ecliptic at the departure point, a point at longitude $\Delta\theta$ from departure lies at latitude $\beta$ if $\tan\beta = \tan i\,\sin\Delta\theta$, where $i$ is the plane’s inclination. At a quarter-turn, $\sin\Delta\theta = 1$ and the plane needs to be tilted only by the target’s latitude. As the transfer angle approaches half a turn, $\sin\Delta\theta$ goes to zero and the tilt needed goes to 90 degrees. For an arrival 1.85 degrees out of the ecliptic, a transfer of 170 degrees must be tilted by 10.5 degrees and one of 178 degrees by nearly 43.

A transfer orbit tilted by tens of degrees has to leave the Earth with a velocity pointing tens of degrees out of the Earth’s own orbital plane. The Earth moves at thirty kilometres a second, and turning a velocity that large costs twice the speed times the sine of half the angle — for ten degrees, more than five kilometres a second. The departure energy near the 180-degree line is therefore not the energy of a transfer but the energy of a transfer plus an enormous plane change, and it rises to hundreds of km²/s² on the line itself.

The wall is a plane change that geometry demands and nothing in the mission wants. It is not in the flat model because the flat model has no planes to change. It is in every real porkchop plot, because no target outside the Earth’s orbital plane can be reached by a transfer going exactly half-way round the Sun except at great cost.

Why a solver stumbles there #

The wall is visible in the mathematics before it is visible in the energy. Lambert’s problem asks for the conic through two points in a given time, and every formulation of its solution involves the angle between the two position vectors and the plane they define. When the angle is exactly 180 degrees and the points are exactly opposite, the plane is undefined: infinitely many orbits of the same shape pass through the two points, rotated about the line joining them, and the problem has no unique answer. Numerical Lambert solvers treat this as a singularity and have to be told what to do with it — in a flat model, choose the ecliptic; in a real one, there is no exactly-opposite case, because the target is never exactly in the plane, but there is a neighbourhood in which the solution is extremely sensitive to the target’s latitude.

That sensitivity is what the wall is. A solver asked for a transfer at 179.9 degrees to a target 1.85 degrees out of the ecliptic returns an orbit tilted by almost ninety degrees, correctly; a solver asked for the same transfer to a target in the ecliptic returns one tilted by nothing. The difference between the two answers is the difference between a real porkchop and a flat one, and it is carried entirely by the $\sin\Delta\theta$ in the relation for the plane.

Two basins, named by the wall #

The wall does something useful as well: it divides the transfer space into two kinds of transfer that behave differently. Transfers that go less than half-way round the Sun are called Type I, those that go more than half-way Type II, and the wall runs between them. On the porkchop the Type I basin lies below and to the left of the wall — shorter flights, departing later for a given arrival — and the Type II basin above and to the right.

The two types trade different things. Type I transfers are shorter, so they spend less time in interplanetary space and arrive sooner; Type II transfers are longer but sometimes cheaper, and their arrival geometry often suits a lander’s approach better. In any given window one basin usually holds the cheapest transfer, and which one depends on where Mars is along its eccentric orbit when the spacecraft arrives. Real missions to Mars have flown both; the choice between them is often made on the arrival end — the approach speed, the direction of the incoming hyperbola, the lighting at the landing site — rather than on the departure cost, once the wall has made the half-way transfer unavailable.

Lambert’s problem with more than one revolution adds Types III and IV and beyond, going round the Sun more than once, and each type boundary at a whole or half-number of turns has its own wall for the same reason. For Mars the multi-revolution types are rarely competitive; for missions that loop round the Sun to meet a planet again, they are the whole design.

How wide the wall is #

The wall is thinner than its height suggests, and how thin depends on the window.

A single row of the porkchop — one flight time, a range of departure dates — shows the wall in cross-section. Far from the half-turn the flat and tilted models agree: when the transfer plane can be chosen with a transfer angle of 120 degrees, reaching a target 1.85 degrees out of the ecliptic costs a tilt of about two degrees, which the departure absorbs for almost nothing. Near the half-turn the tilted curve rises steeply, and in this window, where Mars is met almost two degrees out of the ecliptic, it more than doubles the departure energy over a span of 82 days of departure dates.

That span is the practical cost. The flat model’s cheapest transfer on this row lies close to the wall, and in the tilted model it is gone: the cheapest departure is on the far side, about two and a half times more expensive, weeks away. A mission planned on the flat model would have chosen a date that the real Solar System does not allow. The wall does not only add a line of forbidden dates; it moves the optimum.

Turning half-way #

The wall can be crossed, and the way to cross it follows from the geometry that made it. The plane change is expensive near 180 degrees because it is being made at the start, where the whole tilt must be put in at once. It need not be. A spacecraft can be launched into a transfer in the ecliptic — the flat model’s transfer, with no tilt at all — and turned later, at the point where the turn is cheapest.

The broken-plane manoeuvre makes that turn about a quarter of an orbit before arrival. There, the plane needed to reach Mars differs from the ecliptic by exactly Mars’s latitude at arrival — the $\sin\Delta\theta = 1$ case of the relation above — so the turn is only a degree or two, and the spacecraft’s heliocentric speed there, around 23 kilometres a second, makes it cost about 0.86 kilometres a second. Added to the flat transfer’s departure, the total crosses the wall as if it were not there: 4.3 kilometres a second from low Earth orbit where the direct transfer costs more than thirty. Splitting a rotation between two burns is the general form of the idea, and here the split is between a departure that does none of the turning and a midcourse burn that does all of it where it is cheapest.

Away from the wall the broken-plane transfer is more expensive than the direct one, by most of its midcourse burn, because a direct transfer at 120 or 240 degrees absorbs its small tilt in the departure almost for nothing. So the broken plane is a tool for one region of the plot: departure dates near the half-turn, where it replaces an impossible transfer with an ordinary one plus one extra burn in deep space. Several missions have used it to keep a launch period open across dates that would otherwise have been lost to the wall.

The launch site’s share #

The wall is drawn here in departure energy, which is only part of what a launch vehicle cares about. The other part is the direction in which the departure hyperbola must leave the Earth — in particular its declination, the angle of the outgoing asymptote above or below the Earth’s equator. A launch site at latitude $\phi$ can reach an asymptote of declination up to about $\phi$ by launching due east, which costs nothing extra; steeper declinations need the rocket to launch into a more inclined parking orbit, and every degree beyond the site’s latitude costs payload — a turn is cheapest where the vehicle is slowest, and at launch it is never slow. From a site at 28.5 degrees north, departures with asymptote declinations beyond about thirty degrees are expensive.

Near the wall the transfer plane is steep, and so is the departure asymptote: the velocity the spacecraft needs at departure points well out of the ecliptic, and the ecliptic is itself tilted 23.4 degrees to the Earth’s equator. The declination constraint therefore closes in from the sides of the wall, and the dates a real mission can fly from a given site are bounded by both — the energy the rocket can supply and the declination the launch site can reach. On published mission porkchops the two appear as separate sets of contours, and the launch period is the region where both are satisfied. The wall is where both fail at once.

The way back #

The same wall appears in the other direction. A spacecraft leaving Mars for the Earth — a sample-return mission, or a crew going home — must meet the Earth, which is in the ecliptic, starting from Mars, which is not. Now it is the departure point that lies out of the ecliptic and the arrival point that lies in it, and the transfer plane through the Sun and two nearly opposite points is forced steep in the same way. Return porkchops have their own wall along their own 180-degree line, with a height set by Mars’s latitude at departure rather than at arrival. For a round trip, the outbound and return windows are linked by how long a crew stays at Mars, and both walls have to be avoided at once — a constraint that rules out a surprisingly large share of the conceivable round-trip dates in some cycles.

A wall whose height changes #

The wall’s height depends on how far out of the ecliptic Mars is when the spacecraft arrives, and that changes from window to window.

Mars’s orbit crosses the ecliptic at two points, its nodes, and is 1.85 degrees above or below it a quarter of an orbit from each. A transfer that arrives when Mars is near a node reaches a target almost in the ecliptic, and the wall is thin: the plane through three nearly collinear points can still be almost flat. A transfer arriving a quarter-orbit from a node meets Mars at its full latitude and the wall is as wide as it gets. The cheapest transfers of successive windows arrive at different points of Mars’s orbit — the same cycle that makes their costs vary — so each window’s porkchop has a wall of a different height. In the eight windows drawn, the arrival latitude ranges from 0.58 degrees to 1.85; the 2001 window drawn in the figures above is one of the worst.

The flat model cannot show this, and it is the one respect in which consecutive porkchop plots genuinely differ in shape rather than merely in cost. Mission planners do not use the flat model, of course; every real porkchop is computed with full three-dimensional ephemerides, and the wall is there from the start. What the flat model hides is the reason for it, and the comparison is the clearest way to see that a fraction of a degree of inclination is responsible for a feature hundreds of times higher than the valley floor.

What the drawing leaves out #

The planets here move on Keplerian orbits with fixed elements, and Mars’s orbit is tilted about a fixed line of nodes; a real ephemeris adds the slow precession of both and the small perturbations of each by the others. The Earth is placed exactly in the ecliptic, which is correct by definition. The departure energy is computed from the heliocentric transfer alone, patched to the planets’ sphere of influence in the patched-conic approximation, and it ignores the declination of the departure asymptote, which a launch site’s latitude constrains and which near the wall is steep. And the broken-plane comparison places the midcourse turn exactly a quarter-turn before arrival and prices it at the mean of the transfer’s speeds at its two ends, where an optimised manoeuvre would choose both the point and the split more carefully and do slightly better.

None of these touches the wall’s existence. The wall is a consequence of three points nearly in a line, and it would be there with any ephemeris and any launch site.

Still open: using the wall #

The wall is usually described as an obstacle, a line of forbidden dates. It has also been proposed as a tool. A transfer that must be steeply inclined anyway reaches high heliocentric latitudes on the way, and for a mission that wants to leave the ecliptic — to observe the Sun’s poles, or to sample the solar wind far from the plane — a departure near the half-turn gives a large out-of-plane component for the price the wall charges. The question of whether that price is ever worth paying, compared with the gravity assists that have usually been used to leave the ecliptic, turns on the same geometry: a plane through three nearly collinear points, and a fraction of a degree of latitude that decides how steep it has to be.