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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. The burn however didn't go exactly as planned and the spacecraft missed the Moon by more than three times its radius and was sent into a heliocentric orbit.
The United States launched its first lunar impactor attempt, Ranger 3on January 26,which also failed to reach the Moon. This was followed by the first US success, Ranger 4on April 23, The first human-crewed mission to perform TLI was Apollo 8 on December 21,making them the first humans to leave the Earth's influence. This particular TLI burn lasted approximately seconds, providing 3.
In Japan launched its first lunar mission, using the Hiten satellite to fly by the Moon and place the Hagoromo microsatellite in a lunar orbit. Following that, it explored a novel low delta-v TLI method with a 6-month transfer time compared to 3 days for Apollo. The US Clementine spacecraft, designed to showcase lightweight technologies, used a 3 week long TLI with two intermediate earth flybys before entering a lunar orbit.
In Asiasat-3 became the first commercial satellite to reach the Moon's sphere of influence when, after a launch failure, it swung by the Moon twice as a low delta-v way to reach its desired geostationary orbit. After being launched into a Когда Поймёшь Умом - Константин Никольский - MP3 Stereo transfer orbit GTOit used solar powered ion engines for propulsion.
As a result of its extremely low delta-v TLI maneuver, the spacecraft took over 13 months to reach a lunar orbit and 17 months to reach its desired Translunar Injection - ST 37 - The Secret Society . China launched its first Moon mission inplacing the Chang'e 1 spacecraft in a lunar orbit. It used multiple burns to slowly raise its apogee to reach the vicinity of the Moon. India followed inlaunching the Chandrayaan-1 into a GTO and, like the Chinese spacecraft, increasing its apogee over a number of burns.
The soft lander Beresheet from 4 Voice Fantasia No. 9 In D Minor - Henry Purcell - The Fantasias For Viols Israel Aerospace Industriesused this maneuver inbut crashed on the Moon. Typical lunar transfer trajectories approximate Hohmann transfersalthough low-energy transfers have also been used in some cases, as with the Hiten probe.
A spacecraft performs TLI to begin a lunar transfer from a low circular parking orbit around Earth. The large TLI burnusually performed by a chemical rocket engine, increases Translunar Injection - ST 37 - The Secret Society spacecraft's velocity, changing its orbit from a circular low Earth orbit to a highly eccentric orbit.
As the spacecraft begins coasting on the lunar transfer arc, its trajectory approximates an elliptical orbit about the Earth with an apogee near to the radius of the Moon's orbit. The burn is timed so that the spacecraft nears apogee as the Moon approaches. Finally, the spacecraft No. 1: En El Generalife - Falla*, Gonzalo Soriano, Rodrigo*, Narciso Yepes, National Orchestra Of Sp the Moon's sphere of influencemaking a hyperbolic lunar swingby.
In some cases it is possible to design a TLI to target a free return trajectoryso that the spacecraft will loop around behind the Moon and return to Earth without need for further propulsive maneuvers. Such free return trajectories add a margin of safety to human spaceflight missions, since the spacecraft will return to Earth "for free" after the initial TLI burn.
The Apollos 8, 10 and 11 began on a free return trajectory,  while the later missions used a functionally similar hybrid trajectory, in which a midway course correction is required to reach the moon. TLI targeting and lunar transfers are a specific application of the n body problemwhich may be approximated in various ways. The simplest way to explore lunar transfer trajectories is by the method of Translunar Injection - ST 37 - The Secret Society conics.
The spacecraft is assumed to accelerate only under classical 2 body dynamics, being dominated by the Earth until it reaches the Moon's sphere of influence. Motion in a patched-conic system is deterministic and simple to calculate, lending itself for rough mission design and " back of the envelope " studies. More realistically, however, the spacecraft is subject to gravitational forces from many bodies.
Gravitation from Earth and Moon dominate the spacecraft's acceleration, and since the spacecraft's own mass is negligible in comparison, the spacecraft's trajectory may be better approximated as a restricted three-body problem.
This model is a closer Translunar Injection - ST 37 - The Secret Society but lacks an analytic solution,  requiring numerical calculation via methods such as Runge-Kutta.
More detailed simulation involves modeling the Moon's true orbital motion; gravitation from other astronomical bodies; the non-uniformity of the Earth's and Moon's gravity ; including solar radiation pressure ; and so on. Propagating spacecraft motion in such a model is numerically intensive, but necessary for true mission accuracy.
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