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Unit 11 · Lesson 11-3
Halo and Lissajous Orbits
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Halo and Lissajous Orbits

Linear motion near a collinear point, Richardson’s approximation, differential correction with the state-transition matrix, the monodromy matrix and the halo family

Stanford, 1966 — Cape Canaveral, 12 August 1978
Robert Farquhar was a graduate student when he worked out how to put a relay satellite behind the Moon. Apollo astronauts on the far side would be out of radio contact with the Earth; a satellite near Earth–Moon L2 could see both. But it could not sit at L2 — it would sit right behind the Moon as seen from the Earth, and L2 is unstable anyway. Farquhar's solution was a large loop around L2, tilted out of the orbital plane so that from the Earth it would always appear beside the Moon, never behind it, like a halo. He called it a halo orbit. NASA never flew the relay. Twelve years later the idea flew elsewhere. ISEE-3, launched on 12 August 1978, became the first spacecraft in a halo orbit, around Sun–Earth L1, watching the solar wind before it reached the Earth. In 1982 it was steered out of its orbit and on to a new mission; in September 1985, renamed ICE, it flew through the tail of comet Giacobini–Zinner.
Start with small motions about L1. Linearised, the motion has three parts: an exponentially growing and decaying pair (the saddle found in the lesson on the five Lagrange points), an in-plane oscillation, and an out-of-plane oscillation. Choose the starting state so the growing part is absent and the spacecraft oscillates.
For Sun–Earth L1 the first lab finds an in-plane period of 175.06 days (an ellipse 3.229 times as long in yy as in xx) and an out-of-plane period of 181.25 days. The two drift a full cycle apart every 14.0 years, so the path never closes: it traces a Lissajous figure, the same family of curves an oscilloscope draws from two sine waves of nearly equal frequency.
💡Where halo orbits come from
The linear frequencies are only the small-amplitude limit. As the orbit grows, nonlinear terms shift both frequencies — and at a large enough amplitude they become equal. Then in-plane and out-of-plane motion stay in step and the orbit closes: a three-dimensional periodic orbit, the halo. For Sun–Earth L1 that happens for in-plane amplitudes above about 200,000 km. Each size of halo comes in a northern and a southern version, mirror images in the xx–yy plane. David Richardson published a third-order analytic approximation in 1980. It is good, but not exact: the second lab flies Richardson's starting state for one period, and the orbit does not close.
💡Differential correction
A halo orbit is symmetric about the xx–zz plane: it crosses it perpendicularly, with x˙=z˙=0\dot x = \dot z = 0. So start on that plane with state (x0,0,z0,0,y˙0,0)(x_0, 0, z_0, 0, \dot y_0, 0), integrate to the next crossing, and require x˙=z˙=0\dot x = \dot z = 0 there. Keep z0z_0 fixed and solve for x0x_0 and y˙0\dot y_0 by Newton's method. The derivatives come from the state-transition matrix Φ(t)=∂x(t)/∂x0\Phi(t) = \partial \mathbf x(t)/\partial \mathbf x_0, integrated alongside the orbit from Φ˙=A(t)Φ\dot\Phi = A(t)\Phi with Φ(0)=I\Phi(0) = I. Because the crossing time also moves when the initial state changes, each Newton step uses Φ\Phi minus a correction for that shift:
(δx˙δz˙)=[(Φ41Φ45Φ61Φ65)−1y˙(x¨z¨)(Φ21Φ25)](δx0δy˙0)\begin{pmatrix}\delta\dot x\\ \delta\dot z\end{pmatrix} = \left[\begin{pmatrix}\Phi_{41} & \Phi_{45}\\ \Phi_{61} & \Phi_{65}\end{pmatrix} - \frac{1}{\dot y}\begin{pmatrix}\ddot x\\ \ddot z\end{pmatrix}\begin{pmatrix}\Phi_{21} & \Phi_{25}\end{pmatrix}\right]\begin{pmatrix}\delta x_0\\ \delta \dot y_0\end{pmatrix}

Set the left-hand side to minus the velocity errors at the crossing and solve for the corrections δx₀ and δẏ₀.

📐Worked Example: Aditya-L1’s orbit, computed
The second lab starts from Richardson's approximation for a Sun–Earth L1 halo with an out-of-plane amplitude parameter of 120,000 km. The velocity error at the crossing falls 56 → 4.4 → 0.027 → 2×10−62 \times 10^{-6} → 4×10−104 \times 10^{-10} m/s — Newton's quadratic convergence, the number of correct digits doubling each time. The result:
  • period 177.86 days;
  • semi-axes AxA_x = 208,913 km, AyA_y = 669,910 km, AzA_z = 119,358 km.
ISRO's published target for Aditya-L1 is AxA_x = 209,200 km, AyA_y = 663,200 km, AzA_z = 120,000 km, with a period of 177.86 days. A model with two point masses on circles, solved by Newton's method, lands within 1% of the real mission design.
💡How unstable is a halo orbit? The monodromy matrix
The state-transition matrix over one full period, M=Φ(T)M = \Phi(T), is the monodromy matrix. Its eigenvalues say what happens to a small error after each revolution. For the Aditya-like halo (third lab) they are 1729, four values equal to 1, and 1/1729:
  • 1729: one direction in which an error grows 1,729-fold per revolution — an e-folding time of 23.9 days, like L1 itself;
  • 1/1729: the matching direction in which errors decay — the stable direction, which the lesson on getting there and staying there will exploit;
  • the pairs equal to 1: moving along the orbit, and moving to a neighbouring halo of the family.
The determinant of MM is exactly 1: the flow of a Hamiltonian system preserves phase-space volume. Along the family the period hardly changes — 177.87 days at 100,000 km, 176.84 days at 600,000 km — which is why every Sun–Earth L1 halo takes almost exactly half a year.
Hill proposed to take a certain simple type of periodic orbit as a first approximation.
– Forest Ray Moulton, obituary of George William Hill, Popular Astronomy (1914)
🛰Halo or Lissajous?
Both are flown. Halo orbits are periodic, so the geometry repeats every revolution — simple for antennas, power and operations — and they can be sized to keep a spacecraft away from the solar disk as seen from the Earth, where the Sun's radio noise would swamp the downlink: ISEE-3, SOHO, Aditya-L1 at Sun–Earth L1, JWST at L2, Queqiao at Earth–Moon L2. Lissajous orbits are quasi-periodic, can be smaller and need no particular amplitude, but the geometry never repeats and they periodically line up with the Sun or Moon: ACE, Gaia and Genesis used them.
↓But there is a problem
A halo orbit exists, it can be computed to any accuracy, and it is unstable. Two questions remain before a spacecraft can use one: how do you get there, and how do you stay? The answer to the first came from a rescue. In 1990 a small Japanese lunar orbiter fell silent, and two engineers at JPL found a way to put its mother craft into lunar orbit with almost no propellant — by going the long way round.