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Quasi-periodic orbits about the translunar libration point

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Analytical solutions for quasi-periodic orbits about the translunar libration point are obtained by using the method of Lindstedt-Poincaré and computerized algebraic manipulations. The solutions include the effects of nonlinearities, lunar orbital eccentricity, and the Sun's gravitational field. For a small-amplitude orbit, the orbital path as viewed from the Earth traces out a Lissajous figure. This is due to a small difference in the fundamental frequencies of the in-plane and out-of-plane oscillations. However, when the amplitude of the in-plane oscillation is greater than 32 379 km, there is a corresponding value of the out-of-plane amplitude that will produce a path where the fundamental frequencies are equal. This synchronized trajectory describes a ‘halo orbit’ of the Moon.

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Farquhar, R.W., Kamel, A.A. Quasi-periodic orbits about the translunar libration point. Celestial Mechanics 7, 458–473 (1973).

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