# How stable are Lissajous orbits?

Now that the Gaia Space Telescope is on it's way to the Sun-Earth L2 Lagrangian point (SEL2), I start wondering about the stability of Gaia's orbit there. The Planck Telescope is already there, as was Wilkinson Microwave Anisotropy Probe (WMAP) and other probes, and from Wikipedia I learned that:

In practice, any orbit around Lagrangian points L1, L2, or L3 is dynamically unstable, meaning small departures from equilibrium grow exponentially over time.

Gaia has some kind of Orbital Maneuvering System (to borrow a Space Shuttle term) and some propellant on board, so has Planck, however I wonder how deterministic these orbits are and if both Planck and Gaia have automatic corrections and collision detection in their flight computers; L2 is "only" 1.5 million km (or about 5 light-seconds) away so surely there's time for manual correction.

Does anyone know a source which tells how different Gaia's and Planck's orbits are, if there are intersections between their orbital planes or even how likely the need for an unplanned orbital correction is? I know Lissajou-shapes from maths classes and I know how much the projected trail can differ depending on the precision of data types used in calculations (e.g. float vs. double). How does ESA/NASA handle this, now that SEL2 seems it will become a crowded place?

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Good question and I was thinking of asking a similar one after watching the launch webcast yesterday. But since it's more about management of orbits of spacecraft, procedures, attitude control, and similar than their intended function as space observatories, I think it would be a better fit for Space Exploration. It's not strictly off-topic on Astronomy though, so I'd leave it to you. If you agree that it should be migrated, please either state so in a comment, or flag your question for moderator's attention. Cheers! –  TildalWave Dec 20 '13 at 9:38
Oh, I didn't know about Space Exploration, thanks for the pointer. As of the question of migrating the question, I don't know. It would probably fit in both places, but astronomy is looking for good questions to pull it out of beta, as I read somewhere. If no one complains, I'd keep it here; if no good answer pops up, we could still migrate it if that's OK for everyone. –  Alexander Janssen Dec 20 '13 at 10:04
This paper (institution access or payment required) draws some interesting parallels with asteroids instead. Unfortunately I don't know enough about the subject to write an answer that does the question justice. –  Moriarty Dec 20 '13 at 23:09
I found one interesting one : arc.aiaa.org/doi/abs/10.2514/6.2002-4528 (Formation Flying Satellite Control Around the L2 Sun-Earth Libration Point ) It has almost exactly what you need, but I ain't got a full access right now. –  Cheeku Dec 21 '13 at 15:17

## 1 Answer

Gaia's original science and technology report (see page 221, see also the summary) gives an analysis of the Lissajous orbit. From what I understand Gaia will be placed in a small amplitude Lissajous orbit, giving it an orbital radius of $\sim400000$ km away from $\sim100000$ km along the Sun-Earth axis.

In addition to the fact that this orbit is intrinsically unstable, variable radiation pressure from the sun causes stochastic destabilization of the orbit. The prediction is that once in orbit, a small velocity correction of $\sim1$ m/s is needed about once a month in order to maintain the orbit.

From a summary on Planck's orbit it seems that this applies to Planck as well.

That the SEL2 point is being 'crowded' doesn't seem like such a big problem to me. Once the satellites are turned off (both Planck and Herschel are no longer active) their orbit will rapidly destabilize, effectively removing then from the region. But more importantly, the orbital radius is comparable to the distance to the moon, this really is a tremendously large amount of space. Since these satellites are only a few metres across, the probability of a collision is likely negligible.

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That is a lot to read, thank you. –  Alexander Janssen Dec 22 '13 at 8:07