A timetable that orbits the Sun

A launch to Mars has its date set by the planets, and the window opens every 780 days, for a few weeks. So every mission has been a one-off effort, in which everything it will need is accelerated out of the Earth’s gravity at great expense. For a crew that means propellant, supplies, shielding from radiation and a habitat large enough for months of travel, and a return ticket that has to be carried at every stage. The rocket equation charges the same price for each. It charges again for coming back.

In 1985 Buzz Aldrin, then long retired from NASA, published a suggestion based on an observation about orbital mechanics. Trains do not stop their engines and then carry everything from one city to another each time. A cruise ship stays on a repeating route and passengers board and leave at each port. If there were an orbit round the Sun that visited the Earth and Mars repeatedly, then a large habitat could stay on it for decades, with a few tonnes of crew and supplies coming up each time on small and fast vehicles to meet it.

The idea needs an orbit with the property that it comes back to the Earth, at the right moment, every 2.135 years. The question is whether such an orbit exists, and it has a neat answer: nearly.

What it means to meet the Earth again #

Take Earth and Mars on circular coplanar orbits. The synodic period is

$$S=\frac{1}{1-1/P_M}=2.135\ \text{years},$$

for a Mars period of $P_M=1.881$ years. In that time the Earth goes round twice and on by another 48.7 degrees. Any orbit that meets the Earth at intervals of $S$ has to be somewhere on the Earth’s orbit, 48.7 degrees on from the previous meeting, after that time.

The simplest attempt fails. An orbit with a period equal to $S$ would return to the same point in space after $S$, by which time the Earth has moved on 48.7 degrees. So the spacecraft’s path has to be made to take in those extra degrees. Between meetings it goes once round the Sun, and then a further 48.7 degrees: 408.7 degrees in all.

Geometry fixes how to do that. The two meeting points have to be at the same distance from the Sun, and on a single ellipse those are symmetric about the line of apsides. A route of one lap and a short arc through perihelion from true anomaly $-\nu$ to $+\nu$ does it, with $2\nu=48.7^\circ$. The spacecraft crosses the Earth’s orbit 24 degrees before perihelion, goes round, and comes back to the same place, then goes through perihelion and meets the Earth at 24 degrees after it.

That gives two conditions, on a single ellipse: it passes through 1 AU at $\nu=24.35^\circ$, and a lap plus the arc takes $S$. The two unknowns are the semi-major axis and the eccentricity, and there is no remaining freedom.

The result that follows is the surprising connection in the whole business. The Earth-to-Earth condition makes no reference to Mars’s orbit other than through its period, which fixes $S$. The aphelion that results lies well past Mars’s distance, so the orbit crosses that of Mars. That is not a fact about the distance of Mars, which was never used in the calculation, but about Kepler’s third law. A planet with a period near 1.88 years has a synodic period of about two years; an orbit around two years in period has a semi-major axis of about 1.6 AU; and that reaches further than a planet with semi-major axis 1.52. Had the planet orbited further out, its period would be longer, the synodic period shorter, and so on. The reason the crossing arises is that both come from the same period.

The rosette #

An orbit that takes a lap plus an arc, starting on the inbound side and arriving on the outbound side, cannot just continue. Its next lap has to begin on the inbound side once more, 48.7 degrees further round. So each ellipse is the previous one, turned by that angle.

Seven of those cycles nearly fit into a circle, which gives the familiar rosette drawing, and they are a key to why it works at all. In that drawing the ship’s orbit goes through each successive position in turn. An outbound crossing of Mars’s orbit occurs about 146 days after leaving the Earth. That is a short trip, compared with the roughly 260 days of a Hohmann transfer, because the spacecraft’s speed at this stage is large. Mars must be there at the right moment, which is fixed by the same synodic rhythm: every 2.135 years, there is a planetary alignment. The ship and the planet arrive together.

The return trip is the problem. A ship on this orbit takes well over a year and a half to come back to the Earth from Mars, because it goes all the way out past Mars to aphelion and returns. So a single Aldrin cycler serves one direction well: it gets quickly from Earth to Mars, and very slowly back. A second cycler in a mirrored orbit is needed to make the fast trip from Mars to Earth in about 146 days. Two ships on the same orbit shape, one turned round the Sun from the other, make a two-way service.

The turn the Earth has to give #

To get from one ellipse to the next, the ship must change direction when it meets the Earth. The incoming path crosses the Earth’s orbit outbound, at a certain small flight-path angle. The outgoing path must be inbound by the same angle. The hyperbolic excess speed relative to the Earth has to be turned from one to the other.

The speed relative to the Earth comes from vis-viva: at this orbit’s crossing, about 6.4 kilometres a second. This is a large speed for a flyby of the Earth. The turn it requires comes from the geometry of the velocities and is about 80 degrees. The largest turn an Earth flyby can give at a safe height comes from the same formula used for any flyby:

$$\sin\frac{\delta}{2}=\frac{1}{1+r_p v_\infty^2/\mu_E}.$$

So the shortfall is modest but it is not zero. That is the central fact about this orbit. The Earth gets the ship seven-eighths of the way. The missing seven degrees has to be supplied by rockets. Published analyses of the Aldrin cycler, by Byrnes, Longuski and Aldrin in 1993, and by Rogers, Landau and others, consistently show a required $\Delta v$ on the order of a few tenths of a kilometre per second per cycle. That is much better than a kilometre, because the burn can be made in deep space, far from the flyby and with an optimal geometry that gains from the fact that a shift in arrival direction is much cheaper than rotating the velocity at closest approach. A few hundred metres a second, every 2.135 years, is reasonably small for a 100-tonne ship. Over a thirty-year lifetime it is enough to be serious, though it might be handled by ion engines or solar sails.

The problem of seeking a true ballistic cycler has received much attention. Using Mars as well as the Earth for turns, and allowing longer cycles of many synodic periods, has given a whole taxonomy of cycles of varying ratios, developed especially by Russell and Ocampo. Some of these have no need of propulsion at all, within the circular coplanar model, but have much longer trip times or very close flybys of Mars. Mars’s gravity is much weaker and it gives less turn for a given speed, which limits how much it can contribute. So the Aldrin is a compromise among the many, with very short transit times but at the cost of some steady thrust.

What the free ride charges at the door #

Consider how passengers board the ship. The whole point of a cycler is that it does not stop. A taxi has to leave low Earth orbit at the speed of the cycler relative to the Earth, a hyperbolic excess speed of 6.4 kilometres a second.

The numbers are large, and it is a matter of which mass is being charged. Boarding at the Earth costs a little more than the Hohmann value, and that extra 1.3 kilometres a second is a noticeable cost for the small taxi, but it applies only to the taxi. The massive part, including shielding and supplies, stays on board, and it does not need to be accelerated every time. The ratio of mass to payload favours the arrangement greatly if the ship is heavy enough to be worth not launching again.

Mars is the difficulty. A cycler passes it at 9.7 kilometres a second, a very high speed, greater than the velocity of escape from Mars at its surface by almost twice. An unpowered taxi would need a very substantial heat shield to survive direct entry at that speed, or else a large propulsive burn to get into Mars orbit. Aerocapture, using the Martian atmosphere to slow down, has been studied, with a deceleration that would be severe by any standard. The fuel cost is multiplied by the same equation that was meant to be avoided. That is the most frequently cited drawback, and the reason that some designs prefer other orbits with lower Mars speeds at the cost of a much longer transit.

There is also a timing problem, not visible in the numbers. A rendezvous at 6.4 kilometres a second relative to the Earth, in deep space, has to match in position and speed to within very fine limits, and it happens only once in 780 days. The taxi cannot wait. If its engine does not fire correctly, there is no second chance for 26 months. Passengers have to carry enough to survive a miss; and the failure mode of a missed rendezvous is a ride around the Sun on a trajectory that may or may not come back, perhaps a free return. That is a good reason for care in mission design, and a recurring issue is that the same strict timing that makes the system possible makes each step of it unforgiving.

The orbit it is meant to meet is not a circle #

Everything so far has assumed that Mars’s orbit is circular. It is not, by a large amount.

That is the same 15-year pattern that makes launch costs vary from one window to the next. An ideal rosette cannot actually meet Mars at the same time each cycle, because Mars is at different distances. An aphelion at 2.22 AU leaves room for this, since the orbit crosses all of the range, though the crossing happens at different times along the path. And so a real cycler is not a simple rotation of one ellipse but a sequence of slightly different ones, designed from an ephemeris with a set of small burns. Likewise, Mars’s orbit is inclined 1.85 degrees to the ecliptic, and that adds a further cost to do with plane, as is true for every other Mars trajectory.

Those adjustments can sometimes be helpful. Over the fifteen-year cycle the flight times and flyby speeds vary a good deal, and some cycles require less propulsion than the circular model suggests. Published ephemeris calculations over multiple cycles have found total propulsive costs similar to the circular estimate on average, with large differences between individual cycles.

A system nobody has built, and why #

No cycler has been flown, and the reason lies in how the costs are spread. A ship on such an orbit has to be built and sent on its way. Its own departure from Earth costs as much as one normal expedition. Then each trip saves mass only once there are frequent trips. A permanent settlement on Mars would have a use for such a service, while a programme of occasional expeditions does not. The return on the investment comes after perhaps five or ten trips, spread over twenty years. Hardly any human spaceflight project has been planned on such a timescale.

Furthermore, a ship’s habitat exposed in deep space for decades has to last, with maintenance only at very short visits. Radiation exposure in a thick shelter may be the strongest argument in favour of a cycler, since mass for shielding is the expensive part that a one-off craft cannot afford. There may be a few centimetres of water on an ordinary ship, whereas a cycler might have a metre or two, because the shielding need only be launched once. Galactic cosmic rays would then be reduced to a manageable level. That is an argument that the figures do not reach, and it may be the most important of them.

Buzz Aldrin has kept the concept in public discussion, and studies have been done for NASA at Purdue and for private groups. It has not moved past design. A related idea, which puts propellant depots in position along a route in the same spirit, has received rather more attention. The notion that spacecraft should stop being expendable has made more progress with launch vehicles, for which the case is straightforward.

How far the circular model can be trusted #

The planets are on circles in one plane. Real orbits break that as described, and the actual calculation has to use ephemerides. The figures show a limiting case in which there is exactly one solution.

The flyby is treated as a point event. The turn is computed as an impulse at the Earth’s position. A real flyby lasts about a day inside the Earth’s sphere of influence, during which the ship changes position by hundreds of thousands of kilometres. The errors that this introduces are small compared with those of the circular approximation.

The shortfall’s cost is a bound and not a design. The 0.77 kilometres a second in the figure is what a burn at closest approach would need for the whole rotation of the velocity, a simple estimate. A better placed deep-space burn is much less, and the literature’s figures vary from about 0.25 to 0.5 kilometres a second, depending on the model.

A taxi’s trip is not in the figures. The Hohmann comparison is only a reference, for a spacecraft leaving parking orbit on the slowest transfer. Real missions go faster, and their cost lies between the two.

How the idea relates to a flyby tour #

A cycler is a limit of something spacecraft already do. A planetary tour uses repeated flybys of one planet to change an orbit without fuel. Cassini did that at Titan more than a hundred times, in a sequence designed to take the orbit through just the changes the mission wanted. Each flyby turns the hyperbolic excess vector by a limited angle, and the speed relative to the moon stays the same between encounters. A tour therefore moves along a constant-speed contour, a map of which is what the Tisserand parameter provides. The same contour exists for the Earth and for a cycler: its speed relative to the Earth of 6.4 kilometres a second is a value that repeated flybys can preserve for ever.

The difference between the two is only in the goal. A tour is designed to change the orbit and go somewhere. A cycler is designed to keep it, in a pattern that returns to the same state, rotated, every synodic period. That demand for a repeating state is what makes the orbit unique, and what brings in the 80-degree turn. It’s also why the arrangement cannot be improved by flying a bit lower, since the turn the Earth gives increases only slightly: a flyby at 200 kilometres is already near the practical limit, and $\delta$ depends only weakly on height near it.

The arrangement has one more interesting connection, to a wall across the launch chart. Transfers that go almost exactly half-way round the Sun require a huge plane change. A cycler’s transfer from Earth to Mars sweeps through much less than that, roughly 90 degrees of heliocentric longitude in five months. So it never approaches that expensive geometry, and Mars’s inclination enters only as a small correction. Fast transfers have that as an incidental benefit, and it is one less thing to pay for on a route that must be flown decade after decade.

Still open: whether a system can be justified before there is traffic #

The physics is settled. Such orbits exist and their costs are fairly well known. The remaining question is one of timing, and it is not a question of orbital mechanics. An entire cycler system pays for itself only when there are many passengers, and there will be many passengers only if transport is cheap. Which comes first? In the early history of railways, and of shipping lines, public subsidy or speculative investment closed this circle. Whether anything like that applies to Mars is beyond what any trajectory calculation can say.

There are technical questions too, on a smaller scale. A Mars taxi must decelerate at Mars from almost ten kilometres a second, and no vehicle has been demonstrated at that entry speed with the payload a human landing requires. The best-understood case, Martian entry on a Hohmann-like path, comes in at about half that speed. It may turn out that another cycler orbit with lower Mars speeds, even at a cost in time, gives a better overall system. Each of the many cycler families in the literature has its own balance, and the choice between them cannot be made until the mass of a real taxi and a real heat shield are known. All of them, however, depend on one number the Solar System has fixed: the 48.7 degrees the Earth moves past its starting point every time it meets Mars.