Patched conic approximation is a method used in astrodynamics and orbital mechanics for modeling the trajectory of an object (like a spacecraft) that is moving through space, especially when it is influenced by the gravitational fields of multiple celestial bodies. The approach is particularly useful for simplifying complex trajectories that involve multiple gravitational interactions, such as a spacecraft traveling between planets.
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The patched conic approximation models different portions of a spacecraft trajectory by separate two-body Kepler orbits, matching position and velocity where one attracting body gives way to another. For a surface departure from a nonrotating planet, conservation of energy gives , where is surface launch speed, is the planet's surface escape speed, and is the planet-relative hyperbolic excess speed. Adding that excess velocity vector to the planet's orbital velocity supplies the heliocentric initial condition. It neglects the simultaneous full gravitational action of both bodies near the matching region.