A way home that needs no engine

A trip to another world is usually designed around burns, each of which has to work. In one important case there is a path that does not require the second burn at all. A spacecraft that leaves the Earth with the right speed and in the right direction goes to the Moon, swings behind it, and returns to the Earth’s atmosphere with its engine off. Nothing has to light, or be aimed, or be working. The Moon’s gravity turns it round.

That property made the free return the natural first path to the Moon. When Apollo 8 left Earth orbit on 21 December 1968, its path was within a small correction of a free return, because if the service module’s engine had failed on approach the crew would simply have come home after about six days. The Luna 3 probe had used a version of it in 1959 to photograph the far side of the Moon. It was analysed by Arthur Schwaninger for the American programme in the early 1960s, and earlier by Yuri Kondratyuk and others in the Soviet Union, and it remains in use, for Artemis II for example, as the conservative choice for a first crewed flight.

Yet the free return is a poor fit for the usual first approximation to a trajectory. A conic stitched to a conic can describe the path in outline. It cannot describe it well enough to say where it ends. The return depends on a sensitivity that is much greater than in the case of the outbound leg, and it is a property of the whole three-body geometry.

The figure of eight #

Set up the restricted problem for the Earth and Moon, with a Moon on a circular orbit, in the frame that rotates with it. Leave a parking orbit at a height of 185 kilometres, and find the combination of departure speed and direction which goes behind the Moon and comes back.

Three features of the picture are surprising. First, it is drawn in a rotating frame, and in an inertial frame the path looks nothing like a figure of eight. In that frame the spacecraft goes out on a very long, nearly parabolic ellipse, and the Moon arrives at the top of it, passes in front of it, and gives it a change of direction: a flyby in which the Moon’s motion carries the spacecraft backward. So the shape depends on the frame, and the name reflects the view from the Earth–Moon line.

Second, the crossing lies a long way from the middle. Drawings of the path in popular accounts tend to put the crossing midway between the two bodies with equal loops on either side. To scale, the crossing is only about a twelfth of the way to the Moon, and the loop round the Earth is tiny compared with the loop round the Moon. The Moon’s turn is rather abrupt, a very sharp hook near the end of the long loop.

Third, the outbound leg lies well off the Earth–Moon line, about 80,000 kilometres at its widest. The Moon moves about 40 degrees along its orbit during the outward leg, and in a rotating frame that drift looks like a large sideways excursion. The whole shape is a matter of arranging for the Moon to be where the spacecraft arrives, and for it to move on at the right rate afterwards.

How little speed there is to choose from #

The departure speed in this problem is remarkably constrained. A body leaving at 185 kilometres needs about 10.93 kilometres a second merely to have an apogee at the Moon’s distance, by the vis-viva equation:

$$v^2 = \mu\left(\frac{2}{r_0}-\frac{2}{r_0+r_M}\right).$$

Escape, at the same height, needs 11.02 kilometres a second. So the whole of the region where a spacecraft reaches the Moon without escaping the Earth lies in a range of about 90 metres a second, less than one per cent of its speed. All the different ways to go to the Moon are distinguished by small fractions of a per cent, within that range.

The family of solutions in the figure makes this quite clear. Each departure speed fixes one perilune for a free return. The relation runs the other way from what might be expected, in that a slower departure, which takes longer on its way, goes round the Moon at a greater distance. A slower arrival is turned through a larger angle by the same Moon, so it needs to pass further out to receive the same amount of turning.

This matches the size of the Moon’s available deflection. A spacecraft arriving at the Moon on a speed that gives it the time needed has a hyperbolic excess speed of about 800 metres a second to a kilometre per second relative to the Moon. The turning angle it receives, $\delta$, follows the standard flyby formula:

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

For a passage at about 800 kilometres, at 900 metres a second, $\delta$ is about ninety degrees. In other words the Moon’s gravity is quite capable of doing this job, and the requirement is a matter of aiming. A patched-conic solution gets the speed right to the metre or two and the perilune within a few hundred kilometres. For the return leg it has no reliable answer.

A tenth of a metre a second #

The most striking number is what small changes do to the path as a whole.

That figure holds the essential trouble with the idea of a safe path. On the way out, a change of 0.2 metres a second is not visible in the path at all. It is a part in fifty thousand of the speed and moves the arrival by about 150 kilometres at the Moon. That sensitivity by itself is already large, and it comes from the long lever of the time of flight. Then the encounter multiplies it by a further factor of ten or twenty. A flyby acts as a lens on any aiming error, since a slightly different distance at closest approach gives a slightly different turn, and that small change of direction has three days of return flight in which to grow into a large displacement.

The size of the required target makes this serious. The re-entry corridor for a capsule returning from the Moon at about 11 kilometres a second has a height of only about 40 kilometres, centred at a perigee near 40 to 50 kilometres. If it is too shallow the capsule skips out, on an orbit from which it may not return in time. If too steep the deceleration will be beyond what a crew or a heat shield can stand. So at a perilune of 700 kilometres, the perilune must be right to better than a couple of kilometres, and the speed at departure to within a few centimetres per second, if no correction is to be made at all.

No spacecraft has that accuracy after a burn. The figure of eight, therefore, is not a strictly passive way home. It requires a correction of a metre or a few metres a second on the way back, using small thrusters. What the free return guarantees is that the main engine is not needed. The reaction-control jets, a guidance system, and enough power to run them for a few days are all still necessary. That is a smaller thing than the name suggests, and it was to turn out to be an important one.

Why Apollo stopped using it #

There is a difficulty with a free return as a way to land anywhere. The whole path, including its perilune and the plane it lies in, is fixed by the requirement that it comes home. That leaves very little choice of arrival. The plane and height of arrival at the Moon are tied to the free-return condition, and a lunar landing has to get from that arrival to orbit over the landing site. In practice, the free-return arrivals restricted the possible landing sites to a band near the lunar equator.

Apollo 11 stayed on a free return until it braked into lunar orbit. From Apollo 12 onward the missions left it deliberately, because later sites such as Fra Mauro, Hadley–Apennine and Taurus–Littrow needed something else. The “hybrid” trajectory left Earth on a free return. Then, at about a day into flight, a small midcourse burn moved it onto a path that would not come home unaided, but which allowed the arrival geometry, time of arrival, and lighting required for the chosen site. As long as the burn was made while the main engine and the lunar module’s descent engine both had a large margin, the crew retained an abort option.

This is a good example of a choice with the safety built into a different place. A free return is safe because it depends on nothing further, and a hybrid is safe because it depends on one of two independent engines. Which was the better bet was a judgement call, and it was tested on the very next flight after Apollo 12.

Thirteen #

Apollo 13 made its hybrid burn at about 30 hours and 40 minutes into flight, for its landing at Fra Mauro. Its perigee on return with no further burn was then several thousand kilometres above the Earth’s surface; the crew would have missed the atmosphere. The oxygen tank failed at about 56 hours. The service module’s engine could no longer be trusted, and it was connected too closely to the damage to be risked.

The lunar module’s descent engine was all that remained. At 61 hours and 30 minutes it was fired for about 35 seconds, adding 38 feet per second or about 11.6 metres per second. That was enough to return the path to a free return, one whose perigee came within the corridor, and it saved the mission. A second burn after the Moon shortened the return by about half a day, again with the descent engine. The planar model can show why so small a burn was worth so much.

The leverage of a day before the Moon is large because the effect of a burn at that moment accumulates through the encounter: tens of metres a second change the perilune by hundreds of kilometres, and then the return by tens of thousands. That means a small engine can reach most of the choices that a free return’s rigid family has to offer, provided the burn is done before the encounter and not after it. Its accuracy turned out to be excellent, and only small further corrections were needed; for one of them, debris from the explosion made star sightings impossible, and the crew aligned the spacecraft by hand on the Earth’s terminator and the Sun.

So the outcome supports both sides of the Apollo judgement. The free return would have needed nothing, and it was not available because the mission had given it up. But it took only one small working engine, used at a moment when the geometry gave it great leverage, to get it back. It is a nice reversal of the expected relation between a flexible design and a safe one.

What the plane hides #

The real problem has three dimensions. Everything in the figures is in the Moon’s orbital plane. A real free return is inclined to it, by amounts that depend on launch site, date and the required arrival geometry, and a spacecraft’s plane relative to the Earth’s equator and the Moon’s orbit has to be arranged, as well as the distance. Plane-change costs are the main part of the choice of launch date.

The Moon’s orbit is taken to be circular. Its real orbit has an eccentricity of about 0.055 and is tilted 5 degrees to the ecliptic. That shifts the required speed by tens of metres per second over a month, which is comparable with the whole family drawn. Each launch date has a different solution. The real departures also come from a parking orbit of a different height, and the 185-kilometre value is standard but not universal.

The Sun is missing. Its tide is not important over six days, but it is not nothing: it amounts to a shift of some tens of kilometres at perigee, greater than the corridor. Real free-return computations include it, along with the Earth’s oblateness at departure.

A negative perigee is a calculation artefact below the surface. The integration goes through the Earth as if it were a point mass, and it should be read as “strikes the Earth”. It is the point in the curve at which the spacecraft enters the atmosphere too steeply, and everything below that is a matter of how the numbers look.

An old idea on a longer road #

Free returns exist for Mars too. A path leaving Earth at a suitable date can go round Mars and return after about 1.5 years, or a longer way after about 2 years, with no burn at all. The former is more demanding and requires a high departure speed, and both are constrained by the same condition: the arrival geometry cannot be chosen freely, given the requirement to return. The 2018 proposal for a crewed flyby called Inspiration Mars was built around a 501-day free return using a rare alignment and a Venus-assisted fallback, and it was notable that the proposal needed a heat shield for an Earth entry much faster than that from the Moon.

The 2-year form has a connection with a much more ambitious idea. If a spacecraft could return to the Earth with a free return every synodic period, then it could remain in such an orbit indefinitely, with crew getting on and off at each encounter. Such an orbit is called a cycler. In that case it would be a mass that never needs a major burn. The difficulty is that a free return is a solution for one encounter, and successive ones do not join up without a further change in direction.

Why the first guess is not good enough #

A design for such a path is an instructive application of the way a trajectory gets corrected onto the real one. The patched-conic estimate is reasonable for the outbound leg. It treats the Earth alone until the sphere of influence, about 66,000 kilometres from the Moon, and it then treats the Moon alone. That approximation hides a discontinuity at the seam, which is comparatively unimportant for one crossing of the sphere and very important for two. A free return has a crossing on the way in and another on the way out, with a long leg afterwards. An error of a kilometre per hour in the velocity at the second crossing changes the perigee by hundreds of kilometres. In practice, the conic gives a starting point, and then a shooting method searches the true problem for the path that satisfies the two conditions at each end.

The search is well behaved, thanks to the speed of the arrival. A spacecraft takes less than a day to get through the sphere of influence, so the Moon’s pull acts for a short time, and the arrival depends smoothly on the departure. This is quite unlike the low-energy transfers near the Lagrange points, where the spacecraft moves slowly near the Moon for weeks and Newton’s method fails. In that sense, the free return has about the best numerical properties a three-body path can have, while still having a sensitivity large enough to threaten the mission.

Still open: how far a free return can be stretched #

Today’s crewed lunar missions use free returns on their first flights and more complicated trajectories afterwards, and interest has moved to high orbits near the Moon, which have a different kind of sensitivity connected with the exponential divergence near a Lagrange point. Those are orbits where a small burn at the wrong moment is amplified rather than corrected, and there is usually no natural way home from them. The question that remains is how to compare the protection given by a path like this one, which depends on nothing, with an operational safety that relies on a large and reliable system with many alternate modes.

There is no easy way to put numbers on a path’s safety that takes into account engine failure, tank failure, loss of guidance and timing. The Apollo programme’s approach was largely to reason through particular cases. A more complete analysis of the risk from a sequence of abort options, over the whole mission, would need probabilities that are not known for a vehicle that has only flown a few times. For this path, the mathematics has been finished for half a century. What the mathematics cannot say is how much of the safety was in the path and how much in the people who had the numbers ready when the path was gone.