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:**

- Approximate the system model around a particular libration point by an infinite series, taking a finite number of terms
- Develop an analytical solution to the resulting system of differential equations
- Use the analytical solution to find a prediction for a "good" set of initial conditions
- 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