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Does the orbital period of halo/Lissajous orbits around L1/2 points increase, decrease or stay the same when increasing the (average) orbital radius?

A function period(radius,m1,m2) would be cool, although I guess for Lissajous orbits it would require additional parameters.

At some large radii the orbits would probably become unstable.

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As the halo orbits grow out of plane, their orbital periods generally decrease. You can see a plot of the period (in days) of the L1 and L2 halo families at the Moon as a function of the closest approach distance from the Moon (rp) in Figure 2 of this paper: https://engineering.purdue.edu/people/kathleen.howell.1/Publications/Conferences/2017_AAS_DavPhiHow.pdf

In the Earth-Moon system, the range from around 2 weeks for nearly-planar halos to less than a week for the smallest NRHOs at L2.

The orbits actually get more stable as the families grow out of plane. The flatter halos are unstable.

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