Patched conic approximation

Definition

Consider a spacecraft of negligible mass moving under the gravity of the Sun and of planets, a case of the n-body problem. Let a planet of mass $m$ move at distance $a$ from the Sun of mass $M$. Its sphere of influence is the ball about the planet of radius $$r_{\mathrm{SOI}} = a \left( \frac{m}{M} \right)^{2/5}.$$ The patched conic approximation replaces the trajectory by a sequence of arcs such that:

1. inside the sphere of influence of a planet, the arc is a Kepler orbit about that planet alone;
2. outside all spheres of influence, the arc is a Kepler orbit about the Sun alone;
3. at the boundary of a sphere of influence, consecutive arcs share the position; their velocities differ by the planet's velocity relative to the Sun.