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 P_{1} and the other is P_{2}. The coordinate frame has its origin
at the barycenter and rotates with the two bodies so that P_{1} and P_{2} are always
on the x-axis, with the positive x-direction going from P_{1} to P_{2}. The positive
y-axis is parallel to the velocity vector of P_{2}. The three-body gravitational parameter is called μ and is equal to m_{2}/(m_{1} + m_{2}), where m_{1} is the mass of P_{1} and m_{2} is the mass of P_{2}. The mass of the primaries is nondimensionalized so that the mass of P_{2} is defined to be μ and the mass of P_{1} 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 P_{1} is -μ LU and from the origin to P_{2} is 1 - μ LU. The time unit (TU) is defined such that P_{2} orbits around P_{1} 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. L_{3} is on the far side of the larger body, L_{1} lies between the two bodies, and L_{2} is on the far side of the smaller body. L_{4} and L_{5} 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 (L_{1}, L_{2}, and L_{3}) are considered unstable, but the equilateral Lagrange points (L_{4} and L_{5}) 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 L_{1} and L_{2} is quite low. Spacecraft can arrive and depart from these two Lagrange points with very little ΔV. Spacecraft near L_{1} or L_{2} 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. |