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Patched Conic Approximation

Simplifies complex multi-body dynamics by treating the trajectory in distinct gravitational regimes (e.g., Earth-centered, Sun-centered) sequentially.
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The statement of the theorem

Let r(t)\mathbf{r}(t) be the trajectory. The approximation models the motion as a piecewise solution governed by distinct gravitational parameters μi\mu_i within defined spheres of influence (SOI). The total trajectory is defined by the concatenation of solutions ri(t)\mathbf{r}_i(t):\n\nr(t)={r1(t)for t∈[t0,t1](Governed by μ1)r2(t)for t∈[t1,t2](Governed by μ2)⋮\mathbf{r}(t) = \begin{cases} \mathbf{r}_1(t) & \text{for } t \in [t_0, t_1] \quad (\text{Governed by } \mu_1) \\ \mathbf{r}_2(t) & \text{for } t \in [t_1, t_2] \quad (\text{Governed by } \mu_2) \\ \vdots & \end{cases}\n\nWhere the boundary conditions ensure continuity of position and velocity at the transition times tit_i: ri(ti)=ri+1(ti)\mathbf{r}_i(t_i) = \mathbf{r}_{i+1}(t_i) and vi(ti)=vi+1(ti)\mathbf{v}_i(t_i) = \mathbf{v}_{i+1}(t_i).