Editing Trans-lunar injection (section)

== Modeling ==
[[File:Constellation trans-lunar injection.jpg|thumb |Artist's concept of NASA's [[Constellation program|Constellation]] stack performing the trans-lunar injection burn]]

=== Patched conics ===
TLI targeting and lunar transfers are a specific application of the [[N-body problem|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|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 (astrodynamics)|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 calculation|back of the envelope]]" studies.

=== Restricted circular three body (RC3B) ===
More realistically, however, the spacecraft is subject to [[Gravitational force|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 [[Euler's three-body problem|restricted three-body problem]].  This model is a closer approximation but lacks an analytic solution,<ref>[[Henri Poincaré]], ''Les Méthodes Nouvelles de Mécanique Céleste'', Paris, Gauthier-Villars et fils, 1892-99.</ref> requiring numerical calculation.<ref>[[Victor Szebehely]], ''Theory of Orbits, The Restricted Problem of Three Bodies'', Yale University, Academic Press, 1967.</ref>

=== Further accuracy ===
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 [[Gravitational field|gravity]]; including [[Solar wind|solar radiation pressure]]; and so on.  Propagating spacecraft motion in such a model is numerically intensive, but necessary for true mission accuracy.