# Is a 2:1 “figure-eight” Lissajous orbit possible in the Circular Restricted Three Body Problem?

Proper halo orbits have the same period for their in-plane oscillations and out of plane oscillations, so they are closed orbits with roughy circular motion in the rotating frame, whereas Lissajous orbits are those where the two periods are unequal. They could be closed with a period ratio that is a rational number or be open, non-repeating orbits. Either way they will have "etch-a-sketch" or "squiggly" Lissajous-like motion. and links to What is the difference between halo orbits and Lissajous orbits?

This got me wondering if a 2:1 or "figure eight" Lissajous orbit is possible in the CR3BP This would require one period to be half of the other which seems pretty extreme to me. If such an orbit is possible, under what constraints is this so? I'm guessing there are limits on $$\mu = m_2/(m_1+m_2)$$ and the distance between the orbit and the Lagrange point to which it is associated.

Note that the figure eight could be vertical or horizontal.

Question: Is a 2:1 "figure-eight" Lissajous orbit possible in the Circular Restricted Three Body Problem?

note: this would be in the synodic (rotating) frame of course.

• From that distinction, every planar lissajous orbit is a halo orbit. The planar figure eight orbit is probably the easiest one to find. – SE - stop firing the good guys Aug 13 '20 at 10:58
• @SE-stopfiringthegoodguys if that's the answer, then that's the answer! :-) – uhoh Aug 13 '20 at 11:00