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Lagrangian Relative Motion Orbital Mechanics
The Circular Restricted Three-Body Problem
Equations of Motion for the Circular Restricted Three-Body Problem (CR3BP)
Below are the equations of motion for the Circular Restricted Three-Body Problem (CR3BP). These are a form of relative motion where the primary body is near the center of the frae shifted to the left by a value of pi2r12, and the secondary body is in the positive radial direction shifted to the right of the frame by a value of pi1r12.
Locations of the Lagrange Points
The first three Lagrange points are found by solving for x in the equation below. Note that the equation below solves for their x component on the axis of the two bodies. The y components for the first three Lagrange points are all 0 since they all lie on the axis of the two bodies.
The following Figures show the Lagrange points for the Sun-Earth, Earth-Moon, and Saturn-Titan two-body systems.
Figure 1-1: The Lagrange points of the Earth-Moon system.
Figure 1-2: The Lagrange points of the Sun-Earth system.
Figure 1-3: The Lagrange points of the Earth-Moon system.
Figure 1-4: The Lagrange points of the Saturn-Titan system.
2. Lagrange Points of an Eccentric Two-Body System
When analyzing the CR3BP and comparing it to a real two-body system, the obvious difference is that in reality, there are not perfectly circular two-body systems. The most significant two-body systems are the Earth-Moon and Sun-Earth systems. Figure 2-1 shows the Earth-Moon system in the rotating synodic frame, and the Moon's eccentricity becomes apparent as it causes the Lagrange points to ebb and flow.
The Moon has an eccentricity of 0.1124 where its perigee is 36,000 km and its apogee is 405,000 km. This presents a problem when maneuvering a spacecraft to a Lagrange point that is constantly changing. A spacecraft needs to maneuver to a desired Lagrange point, such as L1 for example, and alter its eccentricity to match the Moon's. Then the spacecraft will ebb and flow along with L1 and the Moon and be able to enter into a halo orbit around the Lagrange point or position itself at the Lagrange point.
Figure 2-1: The Earth-Moon rotating synodic frame with Moon perigee and apogee of 360,000 km and 405,000 km, respectively.
Figure 2-2 below shows the Earth-Moon rotating synodic system where the rotating frame stays fixed on the Moon as it orbits Earth in an eccentric orbit. The Lagrange points follow the stable valley points of L4 and L5, as well as the unstable saddle points of L1, L2, and L3.
Figure 2-2: The dynamic Lagrange points of the eccentric Earth-Moon system.
From Figure 2-2, it appears that there is a significant energy difference between the Moon's perigee and apogee. When the rotating synodic motion is sped up, it appears to ebb and flow where the energy required to traverse through cislunar space past the Moon, L1, and L2 is significantly less during perigee.
3. Lagrange Points of an Eccentric Three-Body System
Figure 3-1: The Jacobi constant visualized in an eccentric three-body system.
Figure 4-1: The periodic orbit about L1 when the normalized radial velocity is 0.810.
Figure 4-2: The periodic orbit about L1 when the normalized radial velocity is 0.820.
Figure 4-3: The periodic orbit about L1 when the normalized radial velocity is 0.830.
Figure 4-4: The periodic
4. Periodic Orbit around Lagrange Point L2
Figure 4-1: The periodic orbit about L2 when the normalized readial velocity is 1.090.
