# Free return trajectory

Sketch of a circumlunar free return trajectory (not to scale).

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").[1][2]

In the case of the Earth and the Moon it is necessary to distinguish between:[1]

• A circumlunar free-return trajectory around the Moon. The perilune is behind the Moon; the spacecraft moves there in a direction opposite to that of the Moon (the path is 8-shaped).
• A cislunar free-return trajectory. The spacecraft goes beyond the orbit of the Moon, returns to inside the Moon's orbit, moves in front of the Moon while being diverted by the Moon's gravity to a path away from the Earth to beyond the orbit of the Moon again, and returning to Earth by Earth's gravity. The perilune is in front of the Moon; the spacecraft moves there in the same direction as the Moon.

The flight time for a cislunar free-return trajectory is longer than for a circumlunar free-return trajectory, especially for trajectories with a small perilune radius (close approach of the Moon): flight time for a cislunar free-return trajectory decreases with perilune radius, while flight time for a circumlunar free-return trajectory increases with perilune radius.[1]

In the simplified model where the orbit of the Moon around the Earth is circular, the special cases of free-return trajectories in the plane of the orbit of the Moon are periodic: after passing the Earth the spacecraft would return to the Moon, etc.[1] The same applies of course for similar three-body problems; this problem is an example of a circular restricted three-body problem.

While in a true free-return trajectory no propulsion is applied, in practice there may be small mid-course corrections or other maneuvers.

A free-return trajectory may be the initial trajectory to allow a safe return in the event of a systems failure; this was applied in the Apollo 8, Apollo 10, and Apollo 11 lunar missions. In such a case a free return to a suitable reentry situation is more useful than returning to near the Earth, but then needing propulsion anyway to prevent moving away from it again. Since all went well these Apollo missions did not have to take advantage of the free return, and inserted into orbit upon arrival at the Moon.

Due to the landing site restrictions that resulted from constraining the launch to a free return that flew by the Moon, subsequent Apollo missions, starting with Apollo 12 and including the ill-fated Apollo 13, used a hybrid trajectory that launched to a highly elliptical Earth orbit that fell short of the Moon with effectively a free return to the atmospheric entry corridor. They then performed a mid-course maneuver to change to a trans-Lunar trajectory that was not a free return.[3] This retained the safety characteristics of being on a free return upon launch, and only departed from free return once the systems were checked out and the lunar module was docked with the command module, providing back-up maneuver capabilities.[4] In fact, within hours after the accident, Apollo 13 used the lunar module to maneuver from its planned lunar orbit insertion trajectory to a free-return trajectory.[5] Apollo 13 was the only Apollo mission to actually turn around the Moon in a free-return trajectory (however, two hours after perilune, propulsion was applied to speed the return to Earth by 10 hours and move the landing spot from the Indian Ocean to the Pacific Ocean).