The Circular-Restricted Three-Body Problem (CRTBP)

In the circular-restricted three-body model, there are two massive bodies in orbit about their mutual barycenter, which is depicted in Figure 1. To simplify the model, each body orbits the barycenter in the same plane in perfectly circular orbits. A spacecraft with infinitesimal mass experiences forces due to the gravitational influence of both bodies simultaneously, and the two bodies are approximated as point masses. The more massive body is labeled P1 and the other is P2. The coordinate frame has its origin at the barycenter and rotates with the two bodies so that P1 and P2 are always on the x-axis, with the positive x-direction going from P1 to P2. The positive y-axis is parallel to the velocity vector of P2.

The three-body gravitational parameter is called μ and is equal to m2/(m1 + m2), where m1 is the mass of P1 and m2 is the mass of P2. The mass of the primaries is nondimensionalized so that the mass of P2 is defined to be μ and the mass of P1 is 1 - μ. One nondimensional length unit (LU) is equal to the distance between the two primaries, so the distance along the x-axis from the origin to P1 is -μ LU and from the origin to P2 is 1 - μ LU. The time unit (TU) is defined such that P2 orbits around P1 in 2π TU.
Figure 1. The Circular-Restricted Three-Body Problem
Five equilibrium points exist in the CRTBP when using a rotating coordinate frame. A spacecraft placed on an equilibrium point with zero velocity will remain at that equilibrium point indefinitely. These points are called Lagrange points or libration points, and they are shown in Figure 2. Three of the equilibrium points are on the x-axis. L3 is on the far side of the larger body, L1 lies between the two bodies, and L2 is on the far side of the smaller body. L4 and L5 are in the x-y plane and each forms an equilateral triangle with the two bodies.
Figure 2. Equilibrium (Lagrange) points in the CRTBP.

The colinear Lagrange points (L1, L2, and L3) are considered unstable, but the equilateral Lagrange points (L4 and L5) are stable. A 3D surface plot of the potential in the CRTBP is shown in Figure 3, with a μ of 0.05. The colors indicate the zero-velocity Jacobi Constant at each point, and show that the energy required to transfer to L1 and L2 is quite low. Spacecraft can arrive and depart from these two Lagrange points with very little ΔV. Spacecraft near L1 or L2 can also use satellite crosslink measurements to perform autonomous orbit determination.

Figure 3. Potential in the CRTBP with a μ of 0.05, color coded with the zero-velocity Jacobi Constant.