The wall sells inclination at the dearest price

A chart of the cost of going to Mars against the dates of departure and arrival has a wall across it: a ridge of prohibitive cost along the combinations of dates for which the transfer carries the spacecraft almost exactly half-way round the Sun. The cause is geometry. The transfer’s plane must contain the Sun, the Earth at departure and Mars at arrival, and when those three points are nearly in a line the plane through them is barely determined; Mars’s small inclination to the ecliptic then forces the plane to tilt steeply, towards the vertical, and launching into a steeply tilted orbit costs a great deal. Missions avoid the wall by choosing their dates on one side of it or the other, or by splitting the transfer with a plane change in the middle.

The wall is expensive because it produces something: a spacecraft on a transfer tilted by forty degrees reaches forty degrees of heliocentric latitude half-way to Mars, far out of the plane in which every planet and nearly every spacecraft travels. Some missions want exactly that. The Sun’s poles cannot be seen well from the ecliptic, where the Sun’s equator is only seven degrees from the line of sight; the fast solar wind streams from them, the magnetic field reverses there at the peak of each cycle, and what the poles are doing is one of the central uncertainties in predicting the next cycle. A spacecraft that reached even thirty degrees of latitude would see them. The question the wall raises is whether it is a way to get there — and the answer depends on what inclination costs, and where.

Why anyone wants to leave the plane #

The planets and nearly every spacecraft lie within a few degrees of one plane, and the Sun’s equator is tilted only seven and a quarter degrees to it. From anywhere near that plane, the Sun’s poles are seen at a grazing angle: the polar regions are foreshortened into a thin band at the limb, their magnetic field is measured along a line of sight nearly perpendicular to the field lines, and a fifth of the Sun’s surface is never seen face-on at all. The polar field is nevertheless the quantity that most needs measuring. It is built up after each maximum of the activity cycle from the remnants of decaying active regions carried poleward, and its strength at minimum sets the strength of the next cycle — the memory the dynamo keeps from one cycle to the next is stored there.

The solar wind is the other reason. Near solar minimum the Sun’s corona is dominated by large coronal holes over both poles, from which the fast wind streams outward at seven or eight hundred kilometres a second, while the slow, variable wind comes from the belt around the equator where closed magnetic loops and open field meet. A spacecraft in the ecliptic samples mainly the slow wind and the boundary; one at high latitude sits in the fast wind itself, where it can be measured undisturbed. Ulysses’s first polar passes, near solar minimum, found the fast wind filling the whole sky above about twenty degrees of latitude, a result no spacecraft in the ecliptic could have given.

The list price of a tilt #

The simplest way to leave the ecliptic is to launch into an inclined orbit directly. The Earth carries a spacecraft round the Sun at 29.8 kilometres a second, and an orbit of the same size inclined by $i$ requires that velocity to be rotated by $i$ — a change of $2v_\oplus\sin(i/2)$, supplied as the hyperbolic excess speed of the launch.

The direct price is brutal. Eight kilometres a second of launch excess — about as much as a large launcher with an upper stage can give a spacecraft of useful mass — buys fifteen degrees of inclination; twelve kilometres a second buys twenty-three. Thirty degrees would need sixteen. That is the cost of rotating a thirty-kilometre-per-second velocity, and the cheapest place to rotate a velocity is where it is smallest; the Earth’s orbit is not that place.

The wall charges the same list price. A transfer to Mars whose plane is tilted by $i$ leaves the Earth with the transfer’s perihelion speed rotated by $i$, and the departure excess needed is the vector difference between that and the Earth’s velocity. The curve for the wall sits just below the direct one, offset only by the fact that a transfer to Mars must first gain the few kilometres per second that reaching Mars at all requires; from there, each extra degree of latitude costs what it would cost in a direct launch. The wall does not sell inclination at a discount. It sells it at the Earth’s price, bundled with a trip to Mars.

How steeply the wall tilts #

The tilt grows rapidly only in the last few degrees before a half-turn. A transfer of 170 degrees is tilted by ten degrees when Mars is at its greatest distance from the ecliptic, and a transfer of 178 degrees by more than forty. Along the wall the inclination and its cost rise together, each extra degree of tilt charged at the direct rate, and the extra launch speed reaches twenty kilometres a second at 178 degrees — far beyond anything launched. The wall is steep in the chart because the geometry puts the whole of the rotation into a narrow range of dates, not because the rotation is expensive in some special way.

What the wall does offer is a combination. A mission that must go to Mars anyway, and that wants a modest excursion out of the ecliptic on the way — to look at the Sun’s poles from ten or fifteen degrees for a few months, or to sample the solar wind above the current sheet that lies near the Sun’s equator — can take a transfer on the flank of the wall and pay a few kilometres per second for latitude it would otherwise not get at all. For anything more than that, the price at the Earth’s orbit is simply too high.

The discount at Venus #

A flyby changes a spacecraft’s direction without propellant, and it can change it out of the plane as easily as in it. The flyby cannot change the speed relative to the planet; it rotates the relative velocity, and the heliocentric velocity, which is the planet’s velocity plus the relative one, is rotated with it. A flyby of Venus can therefore tilt a spacecraft’s orbit — but only by as much as rotating its Venus-relative velocity can tilt the sum.

The trade is the one every flyby makes. A slow encounter is turned through a large angle, but there is little relative velocity to turn; a fast one has plenty of velocity, but Venus hardly bends it. The best single flyby adds about ten degrees of inclination at an encounter speed near nine kilometres a second. More is available by repeating the flyby. If the spacecraft’s orbit after one encounter has a period that is a simple ratio of Venus’s — one to one, four to three, three to two — it meets Venus again at the same point, with the same relative speed, and can be turned further. Each encounter turns the relative velocity a little further out of the plane, and the inclination rises in steps.

The steps stop. The heliocentric velocity is Venus’s thirty-five kilometres a second plus the relative velocity, and once the whole relative velocity points perpendicular to the plane, no further turning can tilt the sum more: the ceiling is set by the ratio of the encounter speed to Venus’s orbital speed. At fifteen kilometres a second it is about twenty-three degrees. To go higher, the encounter speed has to be raised — which the flybys themselves can do, by trading some of their turning for a change in the orbit’s size — and a mission that climbs to thirty degrees and beyond does it by a long sequence of flybys that alternately pump the encounter speed and spend it on tilt. That is how Solar Orbiter is climbing: from the ecliptic to seventeen degrees of solar latitude after its first years of Venus flybys, and to more than thirty by the end of the decade, with each step separated from the next by one or more orbits of the Sun.

The whole turn at Jupiter #

Jupiter offers a much larger discount, for two reasons. It is massive enough to turn a relative velocity through a large angle even at high encounter speeds, and its own orbital speed is only thirteen kilometres a second — less than half the Earth’s. A spacecraft that arrives at Jupiter with a relative speed comparable to Jupiter’s own can be turned to leave with a heliocentric velocity pointing almost straight out of the plane.

The arithmetic has a surprising lever. A launch that only just reaches Jupiter’s orbit arrives there near its aphelion, moving slowly, and meets Jupiter at five or six kilometres a second — enough to leave on an orbit inclined by about twenty-five degrees. A slightly faster launch carries the aphelion beyond Jupiter, and the spacecraft crosses Jupiter’s orbit with a growing radial speed; the encounter speed rises several times faster than the launch speed. By 10.7 kilometres a second at launch, the encounter speed equals Jupiter’s orbital speed, and the flyby can turn the spacecraft onto a polar orbit of the Sun. The turn available at a flyby six Jupiter radii out is always more than enough; the limit is the encounter speed.

That is what Ulysses did, launched in 1990 at close to that speed, turned at Jupiter in early 1992, and sent onto an orbit inclined by 79 degrees to the ecliptic, which carried it over both of the Sun’s poles three times in the following eighteen years. At the Earth’s distance the same inclination would have required a change of velocity of nearly forty kilometres a second. At Jupiter’s it cost the launch to Jupiter, and the flyby did the rest.

What each route costs in time #

The three routes trade launch energy against time and against the inclination they can reach. The direct launch is immediate and shallow and needs a large launch. The Jupiter route needs a large launch too — a little more than the direct one’s for the same speed — but reaches Jupiter in a year or so and leaves on any inclination up to polar. The Venus route needs only the modest launch that reaching Venus requires, and reaches its first tens of degrees over several years, a flyby at a time; it cannot reach the poles without pumping its encounter speed up, which takes years more. The wall, in this comparison, is a variant of the direct route: an immediate inclination, bought at the Earth’s price.

The time matters for more than impatience. The Sun’s magnetic poles reverse at the maximum of each eleven-year cycle, and the polar field built up afterwards is the best single predictor of the strength of the next cycle, stored — as the solar-cycle argument puts it — as one step of memory kept at the poles. A mission that reaches high latitude at solar minimum sees the strong, orderly polar field and the fast wind from the polar coronal holes; one that arrives at maximum sees the reversal. Ulysses, by luck of launch and the length of its orbit, made polar passes at both; a mission climbing by repeated Venus flybys arrives at high latitude whenever its sequence delivers it, and its science depends on which phase of the cycle that turns out to be.

Avoiding the wall rather than using it #

For a mission that only wants to reach Mars, the wall is an obstacle, and its cost can be avoided in two ways that illuminate what the wall is. The first is to choose dates away from it: transfers of 150 or 200 degrees need little tilt, and the chart’s two basins of cheap dates lie on either side of the wall. The second is to take the transfer the dates suggest but break it: fly the first part in the ecliptic, and make the plane change later, when the spacecraft is far from the Sun and moving more slowly. Splitting a rotation between two burns always costs less than making it in one, and making it at a slower speed costs less still, which is why a broken-plane transfer through the wall can cost almost nothing extra.

Both remedies work by not buying inclination at the Earth’s orbit. The broken-plane transfer buys the plane change it needs at the aphelion side of the orbit, at Mars-like speeds rather than Earth-like ones; an extreme version of the same idea is to fly further out in order to turn where the turning is cheap. The out-of-ecliptic missions follow the same principle to its conclusion. They go somewhere slow — Jupiter’s orbit, where the speed is thirteen kilometres a second — or somewhere with a flyby that will do the turning, and they make the turn there.

What the arithmetic leaves out #

Each route is drawn from its simplest arithmetic. The direct and wall prices assume a tangential, circular-orbit geometry at 1 AU, and a real launch into an inclined orbit can do a little better by combining the tilt with a change in orbit size; the difference is a few per cent, not a factor. The Venus figure takes the whole turn out of the plane at each flyby, which is the most inclination any flyby can add, and ignores the need to keep meeting Venus: real flyby sequences spend some of each turn keeping the resonance, and gain inclination more slowly than the staircase drawn. The Jupiter calculation takes a transfer from a circular Earth orbit to Jupiter’s distance with no deep-space manoeuvre, and treats Jupiter’s orbit as circular; both are good to a few per cent in the launch speed.

It leaves out the radiation of Jupiter’s belts, which a close flyby crosses and which limits how close a spacecraft can go; the power available so far from the Sun, which is why Ulysses carried a radioisotope generator; and the thermal environment near the Sun for a mission climbing by repeated Venus flybys, whose perihelion sits inside Mercury’s orbit so that it can image the Sun from close by. Each of those is a cost of the route not measured in kilometres per second. And it leaves out the launch vehicles, which are the real constraint on the direct and Jupiter routes: a launch excess of ten or eleven kilometres a second for a spacecraft of several hundred kilograms is at the limit of what any launcher has provided, and was provided for Ulysses only by stacking two upper stages on a space shuttle’s payload.

The wall’s honest use #

The wall across the launch chart is where the geometry of a transfer forces a plane change that nobody asked for, and the price of that plane change is the price of any rotation at the Earth’s orbit. It can be used deliberately, but only as what it is — a way to combine a trip to Mars with a small excursion out of the plane, priced at the full rate. For reaching high latitudes, the lesson is the one that runs through every plane change: rotate a velocity where it is small, or where something else will rotate it for nothing. The Tisserand parameter of each flyby says what orbits are reachable from it for free; Venus reaches a few tens of degrees in years, Jupiter the poles in one.

Still open: whether a small, cheap spacecraft can see the poles #

The two missions that have left the ecliptic by large angles were large and expensive, and they went by the two routes the figures show to work. A small spacecraft on a modest launch has only the Venus route, and it is slow; the alternative that has been proposed is a solar sail, which can tilt its orbit continuously by leaning against sunlight, trading months for the rotation, with no propellant and no flybys. The figures here set the benchmark such a spacecraft has to beat: twenty-odd degrees in a few years by Venus, or polar in a year by Jupiter for the price of a heavy launch. Whether the sails now being developed reach the lightness at which they could climb to thirty degrees in comparable time — the materials problem that has kept the idea waiting for fifty years — is the open question, and the first missions to try will be judged against these curves.