Free-return trajectory

Last updated September 05, 2022

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).^[1]

Earth–Moon

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.^[2] 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:^[2]

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, or at least slower than the Moon in the same direction. If the craft's orbit begins in a normal (west to east) direction near Earth, then it makes a figure 8 around the Earth and Moon when plotted in a coordinate system that rotates as the Moon goes around the Earth.
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 the latter may not get very close to the Moon, so are not considered as relevant.)

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.^[2]

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.^[3]

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.^[2]

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.

This kind of trajectory can occur 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. The atmospheric entry interface velocity upon return from the Moon is approximately 36,500 ft/s (11.1 km/s; 40,100 km/h; 24,900 mph)^[4] whereas the more common spacecraft return velocity from low Earth orbit (LEO) is approximately 7.8 km/s (28,000 km/h; 17,000 mph).

Due to the lunar 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.^[5] 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.^[6] In fact, within hours after the accident, Apollo 13 used the lunar module to maneuver from its planned trajectory to a circumlunar free-return trajectory.^[7] 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).

Earth–Mars

A free-return transfer orbit to Mars is also possible. As with the Moon, this option is mostly considered for crewed missions. Robert Zubrin, in his book The Case for Mars , discusses various trajectories to Mars for his mission design Mars Direct. The Hohmann transfer orbit can be made free-return. It takes 250 days (0.68 years) 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 writes that 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 so he advocates for the 180-day transfer.^[8] A free return is also the part of various other mission designs, such as Mars Semi-Direct and Inspiration Mars.

There also exists the option of two- or three-year free-returns that do not rely on the gravity of Mars, but are simply transfer orbits with periods of 2 or 1.5 years, respectively. A two-year free return means from Earth to Mars (aborted there) and then back to Earth all in 2 years.^[9] The entry corridor (range of permissible path angles) for landing on Mars is limited, and experience has shown that the path angle is hard to fix (e.g. +/- 0.5 deg). This limits entry into the atmosphere to less than 9 km/s. On this assumption, a two-year return is not possible for some years, and for some years a delta-v kick of 0.6 to 2.7 km/s at Mars may be needed to get back to Earth.^[10]

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.^[11] It cites a delta-v requirement of approximately 4 km/s for the trans-Mars injection, but does not mention the duration of a free return to Earth.

Related Research Articles

A trans-lunar injection (TLI) is a propulsive maneuver used to set a spacecraft on a trajectory that will cause it to arrive at the Moon.

In astronautics, the Hohmann transfer orbit is an orbital maneuver used to transfer a spacecraft between two circular orbits of different altitudes around a central body. Examples would be used for travel between low Earth orbit and the Moon, or another solar planet or asteroid. It is accomplished by placing the craft into an elliptical orbit that is tangential to both the initial and target orbits in the same plane. The maneuver uses two engine impulses: the first prograde impulse places it on the transfer orbit by raising the craft's apoapsis to the target orbit's altitude; and the second raises the craft's periapsis to match the target orbit. The Hohmann maneuver often uses the lowest possible amount of impulse to accomplish the transfer, but requires a relatively longer travel time than higher-impulse transfers. In some cases where one orbit is much larger than the other, a bi-elliptic transfer can use even less impulse, but with a greater penalty in travel time.

Zond was the name given to two distinct series of Soviet robotic spacecraft launched between 1964 and 1970. The first series, based on the 3MV planetary probe, was intended to gather information about nearby planets.

Gemini 11 was the ninth crewed spaceflight mission of NASA's Project Gemini, which flew from September 12 to 15, 1966. It was the 17th crewed American flight and the 25th spaceflight to that time. Astronauts Charles "Pete" Conrad Jr. and Richard F. Gordon Jr. performed the first direct-ascent rendezvous with an Agena Target Vehicle, docking with it 1 hour 34 minutes after launch; used the Agena rocket engine to achieve a record high-apogee Earth orbit; and created a small amount of artificial gravity by spinning the two spacecraft connected by a tether. Gordon also performed two extra-vehicular activities for a total of 2 hours 41 minutes.

Nozomi was a Mars orbiter that failed to reach Mars due to electrical failures. The mission was terminated on December 31, 2003.

In astrodynamics and aerospace, a delta-v budget is an estimate of the total change in velocity (delta-v) required for a space mission. It is calculated as the sum of the delta-v required to perform each propulsive maneuver needed during the mission. As input to the Tsiolkovsky rocket equation, it determines how much propellant is required for a vehicle of given empty mass and propulsion system.

Artificial gravity is the creation of an inertial force that mimics the effects of a gravitational force, usually by rotation. Artificial gravity, or rotational gravity, is thus the appearance of a centrifugal force in a rotating frame of reference, as opposed to the force experienced in linear acceleration, which by the equivalence principle is indistinguishable from gravity. In a more general sense, "artificial gravity" may also refer to the effect of linear acceleration, e.g. by means of a rocket engine.

A Moon landing is the arrival of a spacecraft on the surface of the Moon. This includes both crewed and robotic missions. The first human-made object to touch the Moon was the Soviet Union's Luna 2, on 13 September 1959.

Spacecraft flight dynamics is the application of mechanical dynamics to model how the external forces acting on a space vehicle or spacecraft determine its flight path. These forces are primarily of three types: propulsive force provided by the vehicle's engines; gravitational force exerted by the Earth and other celestial bodies; and aerodynamic lift and drag.

The Ares V was the planned cargo launch component of the cancelled NASA Constellation program, which was to have replaced the Space Shuttle after its retirement in 2011. Ares V was also planned to carry supplies for a human presence on Mars. Ares V and the smaller Ares I were named after Ares, the Greek god of war.

Lunar orbit Orbit of an object around the Moon

In astronomy, lunar orbit is the orbit of an object around the Moon.

A low-energy transfer, or low-energy trajectory, is a route in space that allows spacecraft to change orbits using significantly less fuel than traditional transfers. These routes work in the Earth–Moon system and also in other systems, such as between the moons of Jupiter. The drawback of such trajectories is that they take longer to complete than higher-energy (more-fuel) transfers, such as Hohmann transfer orbits.

The Saturn C-2 was the second rocket in the Saturn C series studied from 1959 to 1962. The design was for a four-stage launch vehicle that could launch 21,500 kg (47,300 lb) to low Earth orbit and send 6,800 kg (14,900 lb) to the Moon via Trans-Lunar Injection.
The C-2 design concept was for a proposed crewed circumlunar flight and the Earth orbit rendezvous (EOR) missions. It was initially considered for the Apollo lunar landing at the earliest possible date (1967).

A trans-Earth injection (TEI) is a propulsion maneuver used to set a spacecraft on a trajectory which will intersect the Earth's sphere of influence, usually putting the spacecraft on a free return trajectory.

The Lunar Atmosphere and Dust Environment Explorer was a NASA lunar exploration and technology demonstration mission. It was launched on a Minotaur V rocket from the Mid-Atlantic Regional Spaceport on September 7, 2013. During its seven-month mission, LADEE orbited around the Moon's equator, using its instruments to study the lunar exosphere and dust in the Moon's vicinity. Instruments included a dust detector, neutral mass spectrometer, and ultraviolet-visible spectrometer, as well as a technology demonstration consisting of a laser communications terminal. The mission ended on April 18, 2014, when the spacecraft's controllers intentionally crashed LADEE into the far side of the Moon, which, later, was determined to be near the eastern rim of Sundman V crater.

In orbital mechanics, a circumlunar trajectory, trans-lunar trajectory or lunar free return is a type of free return trajectory which takes a spacecraft from Earth, around the far side of the Moon, and back to Earth using only gravity once the initial trajectory is set.

Artemis 2 is the second scheduled mission of NASA's Artemis program, and the first scheduled crewed mission of NASA's Orion spacecraft, currently planned to be launched by the Space Launch System (SLS) in May 2024. The crewed Orion spacecraft will perform a lunar flyby test and return to Earth. This is planned to be the first crewed spacecraft to travel beyond low Earth orbit since Apollo 17 in 1972. Formerly known as Exploration Mission-2 (EM-2), the mission was renamed after the introduction of the Artemis program. Originally, the crewed mission was intended to collect samples from a captured asteroid in lunar orbit by the now canceled robotic Asteroid Redirect Mission.

A distant retrograde orbit (DRO), as most commonly conceived, is a spacecraft orbit around a moon that is highly stable because of its interactions with two Lagrange points (L₁ and L₂) of the planet-moon system.

The Lunar Gateway, or simply Gateway, is a planned small space station in lunar orbit intended to serve as a solar-powered communication hub, science laboratory, short-term habitation module for government-agency astronauts, as well as a holding area for rovers and other robots. It is a multinational collaborative project involving four of the International Space Station partner agencies: NASA, European Space Agency (ESA), Japan Aerospace Exploration Agency (JAXA), and Canadian Space Agency (CSA). It is planned to be both the first space station beyond low Earth orbit and the first space station to orbit the Moon.

CAPSTONE is a lunar orbiter that will test and verify the calculated orbital stability planned for the Lunar Gateway space station. The spacecraft is a 12-unit CubeSat that will also test a navigation system that will measure its position relative to NASA's Lunar Reconnaissance Orbiter (LRO) without relying on ground stations. It was launched on 28 June 2022, and will spend over six months flying around the Moon.

References

↑ Diagram of the free return Archived 2016-03-08 at the Wayback Machine .
1 2 3 4 Schwaniger, 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.
↑ Schwaniger, Fig. 9, p. 16.
↑ Entry Aerodynamics at Lunar Return Conditions Obtained from the Fliigh of Apollo 4, Ernest R. Hillje, NASA, TN: D-5399, accessed 29 December 2018.
↑ Hybrid trajectory diagram Archived 2013-01-18 at the Wayback Machine .
↑ 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.
↑ Paul Wooster; et al. (Aug 2006). "Trajectory Options for Human Mars Missions" (PDF). AIAA/AAS Astrodynamics Specialist Conference and Exhibit. doi:10.2514/6.2006-6308. ISBN 978-1-62410-048-2. Archived (PDF) from the original on December 2, 2017.
↑ Wooster et al., op. cit., Table 2.
↑ Human Exploration of Mars Design Reference Architecture 5.0.

External links

Gravitational n-body simulation of a lunar free-return trajectory

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Diagram of the free return Archived 2016-03-08 at the Wayback Machine .

[traj-2] 1 2 3 4 Schwaniger, 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.

[3] Schwaniger, Fig. 9, p. 16.

[4] Entry Aerodynamics at Lunar Return Conditions Obtained from the Fliigh of Apollo 4, Ernest R. Hillje, NASA, TN: D-5399, accessed 29 December 2018.

[5] Hybrid trajectory diagram Archived 2013-01-18 at the Wayback Machine .

[6] Wheeler, Robin (2009). "Apollo lunar landing launch window: The controlling factors and constraints". NASA . Retrieved 2009-10-27.

[7] Stephen Cass, "Apollo 13, We Have a Solution", IEEE Spectrum, APRIL 2005 (accessed August 6, 2012).

[Zubrin_1996_p.-8] 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.

[Wooster-9] Paul Wooster; et al. (Aug 2006). "Trajectory Options for Human Mars Missions" (PDF). AIAA/AAS Astrodynamics Specialist Conference and Exhibit. doi:10.2514/6.2006-6308. ISBN 978-1-62410-048-2. Archived (PDF) from the original on December 2, 2017.

[10] Wooster et al., op. cit., Table 2.

[11] Human Exploration of Mars Design Reference Architecture 5.0.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]