Getting There and Staying There

Invariant manifolds, manifold transfers and direct transfers, low-energy routes, station-keeping, and an Earth–Moon L2 relay

Hiten, 1990–1991

Japan's Hiten) was launched in January 1990 to practise lunar flybys. On its first swing past the Moon it released a small orbiter, Hagoromo, into lunar orbit — and Hagoromo's transmitter failed. Nobody could confirm that it had arrived. Edward Belbruno and James Miller at JPL had been studying low-energy trajectories in which the Sun's gravity does part of the work. In June 1990 Belbruno sent the Japanese space agency an unsolicited proposal: send Hiten far beyond the Moon, where the Sun's pull could slowly reshape its orbit, and let it fall back towards the Moon slowly enough to be captured with almost no braking. The agency flew a version of it. On 2 October 1991 Hiten was captured into lunar orbit — the first spacecraft to reach the Moon on a low-energy transfer.

Hiten's route and the halo orbits share an idea. The unstable direction of a halo orbit, carried along the orbit by the state-transition matrix, generates a surface of trajectories that leave the halo: its unstable manifold. The stable direction generates the trajectories that arrive at it, approaching closer and closer without any burn: its stable manifold. Both are tubes in phase space.

To find the stable manifold, step off the halo by a small distance (200 km in the lab) along the stable direction, at many points round the orbit, and integrate backwards in time. Read forwards, each trajectory is a free ride into the halo.

📐Worked Example: can you ride in from low Earth orbit?

The first lab traces the stable manifold of two Sun–Earth L1 halos back towards the Earth and records the lowest perigee among 60 trajectories:

Aditya-sized halo ( $A_z$ = 120,000 km): lowest perigee 4,993 km above the Earth, then a coast of 205 days into the halo;
larger halo ( $A_z$ = 300,000 km): lowest perigee 342 km, coast 209 days.

For the large halo a spacecraft can leave a low parking orbit with one burn, coast for seven months and arrive with no insertion burn at all. For the small one the tube never comes close enough: a launch from low orbit cannot simply step on to it.

💡Direct transfers and insertion burns

Missions to small halos — and missions in a hurry — fly a direct transfer instead: a trajectory from a parking orbit that heads for the halo faster than the manifold would, crosses near the stable tube, and is trimmed by small trajectory-correction burns on the way. On arrival a halo-orbit insertion burn removes the difference between the arriving velocity and the halo's. The trade is time and insertion propellant against the size of halo the mission can accept. Halos are easiest to enter where they cross the

x

–

z

plane: there the halo's own velocity is purely along

y

, so the insertion burn has a simple target.

💡Staying there: station-keeping

The second lab puts a spacecraft on the Aditya-like halo with a velocity error of just 1 cm/s and no control. It is 1,000 km off the orbit after 133 days, 10,000 km after 191 days and 100,000 km after 244 days. An unstable orbit is kept by small, regular corrections. The lab's controller makes one burn at every crossing of the

x

–

z

plane, chosen by the same Newton step used to build the orbit so that the next crossing is perpendicular again. With insertion, execution (1%) and navigation (1 mm/s) errors it uses 21 burns in 5.1 years — 0.80 m/s in total, 0.16 m/s per year. Real missions budget more, because tracking errors, solar radiation pressure and thruster firings for attitude control all disturb the orbit, but the cost stays small next to the transfer. Halo orbits are chosen partly for this: ISRO selected Aditya-L1's orbit to minimise station-keeping over a five-year life.

The next way home’s the farthest way about.

– Francis Quarles, Emblems, Book IV (1635)

📐Worked Example: a relay for the far side of the Moon

Farquhar's original idea was a relay at Earth–Moon L2. The third lab compares two orbits around that point, which lies 64,515 km beyond the Moon:

a planar Lyapunov orbit (in-plane amplitude 8,000 km, period 14.72 days) is hidden behind the Moon, as seen from the Earth, 31% of the time;
a halo orbit (amplitude parameter 13,000 km, period 14.77 days) is hidden 0% of the time: it never comes closer than 1.49° to the Moon's centre, against a lunar disk 0.26° in radius.

China's Queqiao relay satellite, launched in May 2018, swung past the Moon at about 100 km on a low-energy route and entered an Earth–Moon L2 halo orbit on 14 June 2018. Its designers rejected Lyapunov and Lissajous orbits for exactly this reason. In January 2019 it relayed Chang'e 4, the first landing on the Moon's far side — Farquhar's 1966 idea, flown 52 years later.

🛰Genesis: built from manifolds

NASA's Genesis), launched on 8 August 2001, collected solar-wind particles for 850 days on a Lissajous orbit about Sun–Earth L1, then came home by way of a loop around L2 and dropped its sample capsule on 8 September 2004. Its trajectory, developed under Martin Lo, was designed with the invariant-manifold methods of this lesson — the network that the lesson on gravity assists and Lagrange points called the Interplanetary Transport Network.

↓But there is a problem

Everything in this unit so far has been general: any two bodies, any halo, any mission. The lesson on Aditya-L1 follows one spacecraft through all of it. It was launched by India in September 2023, raised its orbit five times, left the Earth's sphere of influence eleven days after its last burn, and on 6 January 2024, at about four in the afternoon Indian time, fired its engines to settle into a halo orbit around Sun–Earth L1.