A Trans Lunar Injection (TLI) is a propulsive maneuver used to set a spacecraft on a trajectory which will arrive at the Moon.
Typical lunar transfer trajectories approximate Hohmann transfers, although low energy transfers have also been used in some cases, as with the Hiten probe. For short duration missions without significant perturbations from sources outside the Earth-Moon system, a fast Hohmann transfer is typically more practical.
A spacecraft performs TLI to begin a lunar transfer from a low circular parking orbit around Earth. The large TLI burn, usually performed by a chemical rocket engine, increases the 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 TLI burn is sized and timed to precisely target the moon as it revolves around the Earth. The burn is timed so that as the spacecraft nears apogee when the Moon is near. Finally, the spacecraft enters the Moon's sphere of influence, making a hyperbolic lunar swingby.
TLI targeting and lunar transfers are a specific application of the n body problem, which may be approximated in various ways. The simplest way to explore lunar transfer trajectories is by the method of patched 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.
Restricted circular three body (RC3B)
In reality, though, the spacecraft is subject to gravitational forces from many bodies. Since the 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 provides enhanced accuracy but lacks an analytic solution, requiring numerical calculation via methods such as Runge-Kutta.
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