# Motions in CR3BP

Because CR3BP has non-linear dynamics, there is no close and general analytical solution. Therefore, some particular solutions such as linearization or perturbation method can give some restricted analytical solutions. But some complicated motions (trajectories and periodic orbits) are computed using some numerical methods, integrators. Also for finding these complicated motions, especially advantageous trajectories and periodic orbits, Some scheme, which is supported with integrators, is used, such as; differential corrector or mapping methods. Also, the motions in CR3BP can be classified in many aspect.

In first classification, the motion in CR3BP can be simply classify in 3 main groups;

• Bounded (so stable); Trajetories are bounded in particular area in effect of gravitational forces. They are always being and wandering in same neighborhood either making same motion repeatedly as in periodic or quasi-periodic orbits, or making almost similar motion chaoticly but always in same neighborhood. There are numerous sets and families of bounded orbits according to their shape, or size (width) and place of neighborhood. Also note that, every periodic orbits are not stable, some of them are unstable, and are dynamicly very sensitive that they can easily escape their  neighborhood, unless artifical control mechanism. So they can be also counted as unbounded, if any artifical control mechanism is not applied to keep them stable.
• Unbounded (so unstable); Trajetories are not bounded in particular area, and they can go any where energy curves allow. They are real ramblers :P; they are not only wandering in particular neighborhood, and they are quite chaotic that they can hardly estimated where they go.
• Impact (crash to either M1 or M2); Trajectories crash either M1 or M2, sooner or later. They are actually either bounded (periodic) or unbounded trajectores when M1 or M2 are thought as point mass; but they become impact trajectories if trajectory’s route passes through the body (volume) of M1 or M2.

Also, as it’s known, the CR3BP is studied in rotation frame, so that trajectories can be also classified against to rotation or not.

• If the motion around bodies (both bigger and smaller mass) is in the same direction with rotated axis, it is called ‘Direct motion or orbit’.
• If the motion is inversed relative to rotated axis, it is called ‘Retrograde motion or orbit’.

Retrograde motion around primaries (in around both M1 and M2) is much more stable than direct one. Because centrifugal effect is reduced in retrograde motion, so third body (spacecraft or comet) can be staying more stable orbit. Retrograde orbits are generally “stable periodic motion” around primary (for both bigger and second mass) mass even in high energy level. But the direct orbits are not as stable as retrograde orbits, and they can be quite chaotic in high energy levels. Also they tend to make motion around bigger mass and tend to escape from the smaller mass in high energy level. Although the direct orbits make too chaotic and unstable motion, they are quite suitable for transfer (capture or escape) trajectories.

Morover, motion of trajectories can be classified around either M1 or M2. In CR3BP, one body (M1) is generally much bigger than another (M2). So Gravitational force of M1 become more dominant, and even it effects the Motion around M2 very much. Thus, trajectories are quite stable around M1 either they are direct or retrograde; and they can be easily determimed with Poincaré Map. But for around M2, trajectories are only stable in special circumstances, otherwise they become quite unstable and chaotic. Especially capture problem around M2 is very challenging, but also interesting. Also some these circumstances, which makes motions around M2 stable, are that retrograde trajectories, and motions in low-energy level. Also the “inclinited” orbit aroun M2 are tend to be more stable, which will explaine with special map, later. ;)

Furthermore, Motion around equilibrium points are quite intersting and promising for space mission design. As it is explined in equilibrium points and energy surface section, CR3BP has 3 unstable (L1, L2 and L3) and 2 stable (L4 and L5) equilibrium points (EP).

• Motion around Stable EP; the motions around L4 and L5 are bounded and stable. They can be very perfect in parking or stationary space mission types. Such as Trojan asteroids are sitting in L4 and L5 equilibrium points of Sun-Jupiter system in quite stable.
• Motion Around Unstable EP; L1, L2 and L3 equilibrium points are in saddles in energy surface and Trajectories around there are unstable. But using linearization method, we get some basic idea how motion in there can be , that’s 4 main types;

—◊—

As asigned in colurs;

Black  → unstable periodic orbit

Green → asymptotic orbit

Red     → transit orbit

Blue   → nontransit orbit

—◊—

Unstable periodic orbit around unstable equilibrium points are called sometimes as Halo Orbits, if its orbit is in halo shape; and sometimes it is called Lissajous orbits, if it is in a quasi-periodic motion. In fact, there are many types of unstable periodic orbit in there, and they can be used for science orbits; such as SOHO and Genesis.

Asymptotic orbits are simply asymptotic to this unstable periodic orbit. But the ones, which asymptotic to Halo orbits have spacial importance; they all together form the tubes called invariant manifold tubes, which are widely and effectively used in many space mission design and studies. They are one of the most important components in Low Energy Transfer, sometimes referred to as the Interplanetary Transport Network. (Such as Hiten Mission is very good example to this type of transport)

Note that; actually all periodic orbits around unstable equilibrium points can be used for constructing invariant manifold tubes, but the tubes shapes would be diversification according to shape of peridoic orbits, of course. But because Halo orbit has relatively simple shape (like a simple circle), its invariant manifold tube appear as simple as cylinder tube.

Transit orbits can pass from one realm to another inside the asymptotic tube, while non-transit orbits are stay in same realm. Both transit and non-transit trajectories have their own advantageous feature. Such as transit trajectories can be used transfer for traveling one region to another region in the boundaries of zero velocity surfaces. Also they can be used for capture or escape trajectories as well. However the non-transit trajectories are always stay in same region and they never leave their region, so they can be used station keeping orbits or some kind of science orbits.

Because my research is particularly about capture dynamics around M2, so I mainly focus the direct trajectories around M2, which are the most promising ones for low-cost capture (used very very low fuel) scheme, which can be very useful for various rescue or science missions. I’ll later give further information and plots of these trajectories …. ^_^