Imperial College London > Talks@ee.imperial > Control and Power Seminars > Periodic Image Trajectories in Earth-Moon Space

Periodic Image Trajectories in Earth-Moon Space

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In the first part of this seminar, we discuss the trajectories of a spacecraft moving in the Earth-Moon space and in the Moon orbital plane (2D case). Use is made of a rotating Cartesian coordinate system Oxy with the origin at the center of gravity of the Earth-Moon system and the x-axis pointing to the center of the Moon at all time instants. In this 2D case, all the feasible trajectories can be grouped into pairs of trajectories, one going toward the Moon and one returning from the Moon, which are mirror images of one another with respect to the Earth-Moon axis (theorem of image trajectories, discovered half century ago). Next we look at each of the Lagrange points L1 to L5 as to a trajectory which has degenerated into a single point. In this frame, it is clear that the each of the Lagrange points must be consistent with the theorem of image trajectories. We show that this is precisely the case. Then, we look at the class of trajectories intersecting the Earth-Moon axis orthogonally. If there is only one orthogonal intersection on the far side of the Moon, we obtain the subclass of trajectories employed by NASA in the Apollo program (Missions Apollo 1 to Apollo 11): the free-return symmetric trajectories. If there there are two orthogonal intersections, one on the far side of the Moon and one on the far side of the Earth, we obtain another subclass of trajectories: that of trajectories which repeat themselves identically and indefinitely in time. Hence the spacecraft has now become a satellite of the Earth -Moon system. In the second part of this seminar, we focus on trajectories combining the above two orthogonal intersections with a number of nonorthogonal intersections. For a given initial perigee altitude (on the far side of the Earth), the numerical method employed in this research is composed of two steps: (i) generation of an original modified Poincaré map (with a specified number of nonorthogonal crossings of the x-axis), (ii) numerical solution of an optimization problem, aimed at finding the precise value of the initial velocity corresponding to a periodic orbit. The method at hand yields results for an odd number of nonorthogonal intersections (1,3,5) and for given perigee altitudes (hpg = 1000 km, 40000 km, 160000 km). The corresponding periselenum altitudes and velocities are determined from the computation. Several periodic orbits are found and occasionally exhibit a complex geometry. These orbits satisfy the mirror properties also in a particular inertial reference frame. The technique presented in this work is general and in principle is able to find periodic orbits with an arbitary perigee altitude (on the far side of the Earth) and an arbitary number of nonorthogonal x-axis crossings. Future work can focus on two issues: (i) extending the technique to three-dimensional trajectories, by employing the theorem of image trajectories in 3D, (ii) analyzing the stability of the periodic orbits for their use in mission analysis.

This talk is part of the Control and Power Seminars series.

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