A plane change borrowed from the Moon

A plane change by rocket costs twice the orbital speed times the sine of half the angle, which makes it the most expensive manoeuvre in routine spaceflight: turning a low orbit by thirty degrees costs as much as climbing most of the way to geostationary altitude. The cheapest place to turn is where the spacecraft is slowest, and the standard trick is to fly further out in order to turn — raise the apogee, turn at it where the speed is small, and come back. The further out the apogee, the cheaper the turn, and the dearer the trip out and back.

At the Moon’s distance the turn by engine is almost free, because a spacecraft arriving there on an orbit from near the Earth is moving at under two hundred metres a second. But at the Moon’s distance there is also the Moon, moving at a kilometre a second along its own orbit, and a spacecraft that arrives where the Moon is can be turned by the Moon’s gravity instead of its engine. A gravity assist changes the direction of a spacecraft’s velocity relative to the body it passes and not its size, and for a slow spacecraft passing a comparatively fast Moon, a change of direction relative to the Moon is a large change of direction relative to the Earth.

That makes the Moon a plane-change engine that burns no propellant. It also makes it an engine that does more than was asked of it, and the art of using it is in asking for the right amount.

How hard the Moon can turn #

The Moon is light — an eighty-first of the Earth’s mass — and it turns a fast spacecraft very little. It turns a slow one a great deal.

The turn angle of a hyperbolic flyby depends on the approach speed relative to the body and the closest approach distance through $\delta = 2\arcsin\left(1/(1 + r v_\infty^2/\mu)\right)$, where $\mu$ is the body’s gravitational parameter. For the Moon, with $\mu$ of 4,903 cubic kilometres per second squared, a spacecraft approaching at 0.8 kilometres a second and passing a hundred kilometres above the surface is turned by 108 degrees. At twenty thousand kilometres it is turned by thirty.

The speed relative to the Moon is the key, and it is low because of how the spacecraft gets there. A spacecraft on an orbit with its perigee near the Earth and its apogee at the Moon’s distance arrives at apogee moving at under two hundred metres a second relative to the Earth. The Moon is moving at 1.02 kilometres a second. Relative to the Moon, the spacecraft is therefore approaching at around nine hundred metres a second, mostly from ahead of the Moon in its orbit, because the Moon is overtaking it.

The geometry of the plane change #

The orbital plane is set by the direction of the velocity, and a flyby changes the velocity by rotating its Moon-relative part.

The spacecraft’s velocity relative to the Earth is the Moon’s velocity plus its velocity relative to the Moon. Before the flyby the second part is large and points backwards along the Moon’s orbit, slightly out of its plane by whatever inclination the spacecraft arrived with. The flyby rotates this relative velocity, keeping its length, about an axis chosen by where the spacecraft aims in the plane perpendicular to its approach. After the flyby the spacecraft’s velocity relative to the Earth is the Moon’s velocity plus the rotated relative velocity, and because the relative velocity is comparable in size to the Moon’s, rotating it swings the total through large angles in every direction.

The arriving inclination lives almost entirely in the spacecraft’s own small velocity, a couple of hundred metres a second inclined by fifty degrees. After the flyby the spacecraft’s velocity is the sum of two vectors of about a kilometre a second each, and its direction relative to the Moon’s orbit — which is what the new inclination is — can be almost anything the rotation allows. A spacecraft that arrived inclined by fifty degrees can leave nearly in the Moon’s plane, or more steeply inclined than it arrived.

One flyby, one menu #

What a flyby delivers is not one orbit but a family, one for each aim point, and the family has a shape that matters.

At a fixed flyby height the turn angle is fixed, and only its direction — the aim point’s position around the target plane — is free. As that direction goes round, the outgoing velocity traces a cone, and the orbit it leaves on traces a closed curve in the space of inclination and perigee. The figure draws three such curves.

The flyby a thousand kilometres up turns the relative velocity by over eighty degrees, and the curve shows what that buys: inclinations from about one degree to the forties, but with the perigee pushed out to two hundred thousand kilometres or more. A large turn of a large relative velocity changes the total velocity’s size as well as its direction, and a spacecraft leaving the Moon’s distance with more speed is on a larger orbit whose closest approach to the Earth is far out. The flyby ten thousand kilometres up turns less and leaves perigees of tens of thousands of kilometres; the one thirty thousand kilometres up turns least, and part of its curve brings the perigee below ten thousand kilometres, back towards the Earth.

Inclination and perigee are therefore not separate choices. Each aim point delivers one pair, and the pairs a single flyby offers are a one-dimensional curve in a two-dimensional space of wishes.

A close flyby spends its turn on energy #

The reason close flybys push the perigee out is that they change the spacecraft’s energy relative to the Earth, and a large enough turn changes it enough to leave.

A flyby cannot change the speed relative to the Moon, but by turning it, it changes the speed relative to the Earth, which is the vector sum of the Moon’s velocity and the relative velocity. Turned to point forwards along the Moon’s motion, the relative velocity adds to the Moon’s, and the spacecraft leaves at up to 1.57 kilometres a second — above the 1.44 kilometres a second needed to escape the Earth from the Moon’s distance. Turned to point backwards, it subtracts, and the spacecraft leaves slower than it arrived and falls back towards the Earth. Every intermediate direction gives an intermediate speed.

The upper edge of that band is the lunar gravity assist used to leave the Earth–Moon system cheaply: two spacecraft launched together in 2006 to observe the Sun from either side used lunar flybys to separate onto orbits ahead of and behind the Earth. The lower edge sends spacecraft back inward. A plane change sits in the middle, asking the flyby to rotate the velocity out of one plane and into another without changing its size much — which needs a turn that is mostly sideways rather than forwards or backwards, and a turn modest enough not to add a large change of speed as a side effect.

The window of gentle flybys #

The opening figure puts this together. Asked only for a low inclination, a flyby can deliver it from almost any height closer than tens of thousands of kilometres: the dashed curve sits near zero across most of the range. Asked also to return the spacecraft to an orbit whose perigee is between the atmosphere and twenty thousand kilometres — an orbit from which it can reach geostationary altitude, or anywhere else near the Earth — the close flybys fail entirely. Every aim point that flattens the orbit also throws it outward.

Only from about 9,500 kilometres above the Moon does a returning orbit become possible. Between that height and about 18,000 kilometres a single flyby can both bring the spacecraft back and leave its orbit nearly in the Moon’s plane. Further out, the turn is too small to remove fifty degrees of inclination, and the best achievable inclination rises steeply.

The plane change wants a gentle flyby, and the window is narrow. That is the opposite of the instinct a gravity assist usually encourages, where a closer pass gives a bigger effect, and it explains why the real missions that used the Moon to change their plane did so in two flybys rather than one: the first to set up a geometry, the second to finish the job, each at a height chosen for its turn rather than for the largest turn available.

Aiming at a point in a plane #

The menu figure’s free parameter is the aim point, and choosing it is a navigation problem with its own geometry.

A flyby is targeted not at a closest-approach distance but at a point in the plane through the Moon perpendicular to the spacecraft’s incoming relative velocity — the target plane in which every flyby is aimed, because there the miss distance responds linearly to a course correction. The distance of the aim point from the Moon’s centre sets the turn angle; its direction around the centre sets which way the relative velocity is turned. A plane change wants the turn directed mostly perpendicular to the Moon’s orbital plane, so the aim point lies above or below the Moon as seen from the approaching spacecraft, and its distance sets how much of the inclination comes off.

The sensitivity of the outcome to the aim is large. A plane change of tens of degrees from a flyby at fifteen thousand kilometres depends on hitting the aim point to within tens of kilometres, and an error at one flyby grows into a larger error at the next, so missions that use two lunar flybys plan a correction between them.

What the aim cannot change is the speed relative to the Moon. That speed is fixed by the arriving orbit, and it is the invariant a flyby cannot alter: every orbit in the menu has the same speed relative to the Moon at the moment of the flyby, and so every one of them, if it comes back to the Moon’s distance at the right time, meets the Moon at the same relative speed. A sequence of lunar flybys is a walk across the menus of successive encounters, all sharing that one number, and returning to the same body several times is how a sequence reaches outcomes no single flyby offers.

Two flybys instead of one #

The window of gentle flybys explains the shape of the missions that used the Moon for a plane change. One flyby inside the window can, in the idealised geometry drawn here, both flatten the orbit and return the spacecraft. In practice the Moon’s orbit is tilted to the equator, the arriving geometry is not ideal, and the spacecraft must also end up with a particular perigee and period. One flyby rarely satisfies all of that.

Two flybys give two aim points, and the outcome space becomes two-dimensional: the first flyby can be used mainly to change the orbit’s energy and timing, so that the spacecraft returns to the Moon at a chosen moment and relative geometry, and the second to make the plane change from that better starting point. The rescued satellite of 1998 did exactly this, and so did later missions that needed unusual orbits.

Beyond a few flybys lies a different regime, in which the Sun’s gravity matters as much as the Moon’s and the spacecraft can be steered through the region where the two balance rather than past the Moon at a single moment. Those trajectories take months and can do almost anything with the orbit’s plane and size for very little propellant, at the cost of time and of a sensitivity to navigation that only modern tracking makes manageable.

When the trip is worth making #

The flyby is free, but the trip to the Moon’s distance is not, and the comparison decides when the method is worth using.

A geostationary transfer orbit’s apogee is at 42,000 kilometres, where the spacecraft moves at 1.6 kilometres a second, and turning there costs twice that times the sine of half the angle. Raising the apogee to the Moon’s distance costs about 0.68 kilometres a second at perigee, and lowering it again the same, a fixed price of 1.36 for the round trip. Turned by engine at the Moon’s distance, the turn adds only a little more, because the spacecraft is moving at under two hundred metres a second. Turned by a flyby, it adds nothing.

The fixed price of the trip beats the turn at transfer apogee beyond about fifty degrees. That is a large angle, larger than any routine launch needs, which is why lunar plane changes are not routine. But it is almost exactly the angle between a transfer orbit launched from a high-latitude site such as Baikonur, at 51.6 degrees, and the equator — and that is the case in which the method was first used commercially, to rescue a satellite.

The comparison is of pure plane changes. A real transfer to geostationary orbit combines the turn with the burn that circularises the orbit at apogee, which makes the direct route cheaper than drawn, and the lunar route pays in corrections and in weeks of waiting for the Moon to be in the right place. The figure shows where the crossover lies before those refinements, not where any particular mission would put it.

What was actually measured #

A stranded satellite. A communications satellite launched from Baikonur in December 1997 was left in a transfer orbit inclined at 51.6 degrees when its upper stage failed, with no way to reach a useful geostationary orbit on its own propellant by a direct plane change. Its manufacturer took it over and sent it to the Moon instead, raising its apogee to the Moon’s distance and using two lunar flybys in May and June 1998 to rotate its plane and raise its perigee, ending in a day-long orbit with most of the inclination removed — the first lunar gravity assist used on a commercial satellite, and done with propellant the satellite already carried.

Exploration missions. A Japanese spacecraft in 1990 and 1991 made repeated lunar flybys to test the technique, and a planet-hunting telescope launched in 2018 used a single lunar flyby to reach a highly elliptical orbit with a period exactly half the Moon’s, tilted far out of the ecliptic, at a fraction of the propellant a direct insertion would have needed.

The general practice. Lunar flybys are now a standard tool for missions leaving the Earth or reaching unusual Earth orbits. Each is designed by searching the space of aim points and flyby heights for the outcome wanted, which is the menu drawn here with the corrections and the Sun’s perturbation added.

Where the patched conics stop #

The Moon on a circle in the reference plane. The Moon’s orbit is drawn circular and in the reference plane, so inclinations are measured from the Moon’s orbit rather than from the Earth’s equator. The Moon’s orbit is inclined to the equator by between 18 and 29 degrees, depending on the 18.6-year cycle of its node, and a real plane change to the equator must account for that tilt and choose a time when it helps.

Patched conics. The flyby is computed as an instantaneous rotation at the Moon’s position, joining a geocentric ellipse to another. Near the Moon’s distance the Sun’s gravity and the Earth’s matter too, and real trajectories are computed with all three; the discontinuities a patched-conic model hides are largest exactly in this regime.

Arrival at the node. The spacecraft is assumed to meet the Moon exactly at the line where its orbit crosses the Moon’s. Getting there requires phasing — waiting in the transfer orbit for the Moon to come round — which costs time and small corrections.

No atmosphere, no limit on perigee. The returning orbits are required only to have perigees above the atmosphere and below twenty thousand kilometres; a real mission’s constraints are tighter and specific to its destination.

A turn that comes with a change of speed #

The general lesson is about what a gravity assist actually changes. In the body’s frame it changes only a direction. In any other frame it changes both a direction and a speed, in proportions set by how the body’s velocity compares with the spacecraft’s relative velocity. When they are comparable, as they are at the Moon for a spacecraft arriving from near the Earth, a rotation in the body’s frame is an arbitrary mixture of a turn and an acceleration in the planet’s frame, and asking for one without the other restricts the choice of rotation severely.

A free manoeuvre is free only in the frame where it is defined; in every other frame it is a package, and the package cannot be split. The skill in using a flyby is choosing a rotation whose package contains what is wanted and little else — which for a plane change means a gentle flyby, and for an escape a close one.

Still open: the plane change done in a spiral #

Every plane change so far has been made at a point: by an engine firing briefly, or by a flyby that takes hours. A spacecraft with an electric engine turns its plane gradually, thrusting a little out of the plane on every orbit over months, and the cost of a plane change done that way follows a different formula, with a factor of $\pi/2$ that no impulsive analysis produces. How the thrust should be pointed through such a spiral — not along the velocity, and not always perpendicular to the orbit, but along a direction that changes with position in a way that has to be derived rather than guessed — is the question of how a low-thrust trajectory is found rather than priced.