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:
- A circumlunar free-return trajectory around the Moon. The spacecraft passes behind the Moon. It moves there in a direction opposite to that of the Moon. If the craft start in a normal (west to east) direction near Earth, then it makes a figure 8.
- 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 is drawn back to Earth by Earth's gravity. (There is no real distinction between these trajectories and similar ones that never go beyond the moon's orbit, but these latter may not get very close to the moon so are not considered as relevant.)
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.
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. 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. In fact, within hours after the accident, Apollo 13 used the lunar module to maneuver from its planned trajectory to a free-return trajectory. 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).
A free-return transfer orbit to Mars is also possible. As with the moon, this option is mostly considered for manned missions. Robert Zubrin, in his book The Case for Mars, discusses various trajectories to Mars for his mission design Mars Direct. The naive Hohmann transfer orbit can be made free-return. It takes 250 days in the transit to Mars, and in the case of a free-return style abort without the use of propulsion at Mars, 1.5 years to get back to Earth, at a total delta-v requirement of 3.34 km/s. Zubrin advocates a slightly faster transfer, that takes only 180 days to Mars, but 2 years back to Earth in case of an abort. This route comes also at the cost of a higher delta-v of 5.08 km/s. Zubrin claims that even faster routes have a significantly higher delta-v cost and free-return duration (e.g. transfer to Mars in 130 days takes 7.93 km/s delta-v and 4 years on the free return), and thus advocates for the 180-day transfer even if more efficient propulsion systems, that are claimed to enable faster transfers, should materialize. A free return is also the part of various other mission designs, such as Mars Semi-Direct and Inspiration Mars.
NASA published the Design Reference Architecture 5.0 for Mars in 2009, advocating a 174-day transfer to Mars, which is close to Zubrin's proposed trajectory. It cites a delta-v requirement of approximately 4 km/s for the trans-Mars injection, but does not mention to the duration of a free return to Earth.
- Diagram of the free return
- Schwaninger, Arthur J. (1963). Trajectories in the Earth-Moon Space with Symmetrical Free Return Properties. Technical Note D-1833. Huntsville, Alabama: NASA / Marshall Space Flight Center.
- Hybrid trajectory diagram
- Wheeler, Robin (2009). "Apollo lunar landing launch window: The controlling factors and constraints". NASA. Retrieved 2009-10-27.
- Stephen Cass, "Apollo 13, We Have a Solution," IEEE Spectrum, APRIL 2005 (accessed August 6, 2012)
- Zubrin, Robert (1996). The case for Mars: the plan to settle the red planet and why we must. New York: Free Press. ISBN 978-0-684-83550-1.
- Human Exploration of Mars Design Reference Architecture 5.0