# Beginning the Orbit Search

How to Find an Orbit

After locating the libration points, the next step is to search for an orbit. That is, we try to find a set of initial conditions (x,y,z,Vx,Vy,Vz) that will allow the spacecraft to enter into orbit around the desired libration point.

Here is the strategy:

1. Approximate the system model around a particular libration point by an infinite series, taking a finite number of terms
2. Develop an analytical solution to the resulting system of differential equations
3. Use the analytical solution to find a prediction for a "good" set of initial conditions
4. Take this guess and use the full set of system equations to numerically find a better guess.

The analytical approximation that we use in our project is given in detail in the paper Periodic Orbits About the L1 and L2 Collinear Points in the Circular Restricted Three Body Problem (1978) by D.L. Richardson.

One can now discuss the remaining part of the process, which is using the approximation to discover a set of initial conditions for an actual orbit. The mathematics of this step is discussed in the link below.

Computing the actual orbits and their stable and unstable manifolds