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Free-return trajectory
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Free-return trajectory
In orbital mechanics, 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).
Many free-return trajectories are designed to intersect the atmosphere; however, periodic versions exist which pass the moon and Earth at constant periapsis, which have been proposed for cyclers.
The first spacecraft to use a free-return trajectory was the Soviet Luna 3 mission in October 1959. It used the Moon's gravity to send it back towards the Earth so that the photographs it had taken of the far side of the Moon could be downloaded by radio.
Symmetrical free-return trajectories were studied by Arthur Schwaniger of NASA in 1963 with reference to the Earth–Moon system. He studied cases in which the trajectory at some point crosses at a right angle the line going through the center of the Earth and the center of the Moon, and also cases in which the trajectory crosses at a right angle the plane containing that line and perpendicular to the plane of the Moon's orbit. In both scenarios we can distinguish between:
In both the circumlunar case and the cislunar case, the craft can be moving generally from west to east around the Earth (co-rotational), or from east to west (counter-rotational).
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.
The speed at a perigee of 6555 km from the centre of the Earth for trajectories passing between 2000 and 20 000 km from the Moon is between 10.84 and 10.92 km/s regardless of whether the trajectory is cislunar or circumlunar or whether it is co-rotational or counter-rotational.
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 start over on the same trajectory. 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 3 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 its (first) apogee and begins to fall back toward the Earth, but as it approaches the Moon's orbit, the Moon arrives, and there is a gravitational interaction. 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 other side, and goes through another perigee close to where the first perigee had taken place. By this time the Moon has moved almost half an orbit and is again directly over the craft at perigee. Other cislunar trajectories are similar but do not end up in the same situation as at the beginning, so cannot repeat.
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Free-return trajectory
In orbital mechanics, 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).
Many free-return trajectories are designed to intersect the atmosphere; however, periodic versions exist which pass the moon and Earth at constant periapsis, which have been proposed for cyclers.
The first spacecraft to use a free-return trajectory was the Soviet Luna 3 mission in October 1959. It used the Moon's gravity to send it back towards the Earth so that the photographs it had taken of the far side of the Moon could be downloaded by radio.
Symmetrical free-return trajectories were studied by Arthur Schwaniger of NASA in 1963 with reference to the Earth–Moon system. He studied cases in which the trajectory at some point crosses at a right angle the line going through the center of the Earth and the center of the Moon, and also cases in which the trajectory crosses at a right angle the plane containing that line and perpendicular to the plane of the Moon's orbit. In both scenarios we can distinguish between:
In both the circumlunar case and the cislunar case, the craft can be moving generally from west to east around the Earth (co-rotational), or from east to west (counter-rotational).
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.
The speed at a perigee of 6555 km from the centre of the Earth for trajectories passing between 2000 and 20 000 km from the Moon is between 10.84 and 10.92 km/s regardless of whether the trajectory is cislunar or circumlunar or whether it is co-rotational or counter-rotational.
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 start over on the same trajectory. 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 3 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 its (first) apogee and begins to fall back toward the Earth, but as it approaches the Moon's orbit, the Moon arrives, and there is a gravitational interaction. 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 other side, and goes through another perigee close to where the first perigee had taken place. By this time the Moon has moved almost half an orbit and is again directly over the craft at perigee. Other cislunar trajectories are similar but do not end up in the same situation as at the beginning, so cannot repeat.
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