Free return trajectory

A free-return trajectory is a trajectory of a spacecraft traveling away from a primary body (for example, the Earth) where gravity due to a secondary body (for example, the Moon) causes the spacecraft to return to the primary body without propulsion (hence the term "free").

Free return trajectories were introduced by Arthur Schwaniger of NASA in 1963 with reference to the Earth-Moon system. Limiting the discussion to the case of the Earth and the Moon, if the trajectory at some point crosses the line going through the centre of the earth and the centre of the moon, then we can distinguish between:

For trajectories in the plane of the moon's orbit with small periselenum radius (close approach of the Moon), the flight time for a cislunar free-return trajectory is longer than for the circumlunar free-return trajectory with the same periselenum radius. Flight time for a cislunar free-return trajectory decreases with increasing periselenum radius, while flight time for a circumlunar free-return trajectory increases with periselenum radius.

Using the simplified model where the orbit of the Moon around the Earth is circular, Schwaniger found that there exists a free-return trajectory in the plane of the orbit of the Moon which is periodic: after returning to low altitude above the Earth (the perigee radius is a parameter, typically 6555 km) the spacecraft would return to the Moon, etc. This periodic trajectory is counter-rotational (it goes from east to west when near the earth). It has a period of about 650 hours (compare with a sidereal month, which is 655.7 hours, or 27.3 days). Considering the trajectory in an inertial (non-rotating) frame of reference, the perigee occurs directly under the moon when the moon is on one side of the earth. Speed at perigee is about 10.91 km/s. After three days it reaches the moon's orbit, but now more or less on the opposite side of the earth from the moon. After a few more days the craft reaches ifs (first) apogee and begins to fall back toward the earth. But then the moon comes along and attracts the craft. The craft passes on the near side of the moon at a radius of 2150 km (410 km above the surface) and is thrown back outwards where it reaches a second apogee. It then falls back toward the earth, goes around to the first side, and goes through another perigee close to where the first perigee had taken place.

There will of course be similar trajectories with periods of about two sidereal months, three sidereal months, and so on. In each case, the two apogees will be further and further away from Earth. These were not considered by Schwaniger.

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