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In CR3BP, M1 called primary body is generally much bigger than the M2 called secandary body. Thus, disturbing effect of M2 on M1 is relativly less, and motion around M1 is much more like in clasical 2-Body motion. As it is shown in Poincaré Map section, Periodic Orbit (PO) around M1 can be easily found in surface of section method. So, Let’s show PO of Earth-Moon binary system for an example. All we need; integrating a bunch of trajectory around Earth (M1), then gather the point data on a chosen surface;

Then we get the upper plot for C=3.16, direct orbits of Earth-Moon binary system. As it seen, we have some island (circles inside circles) and ocean (chaoticly scattering point datas). And as you may guess, the islands give us the Periodic orbits, let’s see then;

It is so easy to use the Poincaré Map. In 2D-CR3BP, we have 4 dimensional phase space as [X, Y, Vx, Vy]. So the trajectories are actually surfing in this 4-D pahse space. All we do is to chose a surface for the map and gather all data points while trajectories are shooting this surface. In this example, we choose (X, Vx) surface while Y=0, and Vy is computed using Jacobi integral (V²=2U(x,y)-C). Then simply picking the data from this (X, Vx) surface map, and already Y=0, and Vy=sqrt(V²-Vx²); we can get initial conditions of PO, as these;

- X=-0.505, Y=0, Vx=0.00, Vy=sqrt(V²-Vx²), for 2 tips PO
- X=-0.183, Y=0, Vx=0.00, Vy=sqrt(V²-Vx²), for 3 tips PO
- X=-0.062, Y=0, Vx=0.00, Vy=sqrt(V²-Vx²), for 4 tips PO
- X=-0.310, Y=0, Vx=0.58, Vy=sqrt(V²-Vx²), for 5 tips PO
- X=-0.185, Y=0, Vx=1.60, Vy=sqrt(V²-Vx²), for 7 tips PO

C=3.16, and μ=0.01215 for Earth-Moon, of course …. ^_^

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