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1. Restricted three-body problem a satellite in the binary planet system

The motion of a three-body system is the subject of a restricted three-body problem if the mass of one of the bodies is negligible compared to the masses of the other two bodies. In this case we can ignore the influence of the light body on the other two, which means that the two massive bodies move synchronously under mutual gravitation along conic sections just as they do in the two-body system. However, the motion of the light third body under the gravitational forces, exerted on it by the other two bodies, can be very complicated, though the motion of these massive bodies is rather simple. This part of the collection presents several examples of such motions of a satellite in the system of a binary planet (like the earth - moon system, but with an arbitrary masses of the components). The motions can be displayed either in the inertial centre-of-mass frame or in the geocentric frame.

Examples:

1. Satellite orbiting a moon that circularly orbits a planet. This example shows a stationary motion of the satellite around the heavier component in the binary system whose components move in circles about the centre of mass. The motion can continue indefinitely.

2. Satellite orbiting a moon that elliptically orbits a planet. This motion can also be stationary and continue indefinitely. (Click also here to see the applet.)

3. Satellite orbiting a moon and a planet by turn. At certain conditions the satellite in the binary system can change its primary from time to time. In this example the satellite ends this "game of space basketball" by crashing against the planet. (Click also here to see the applet.)

4. Wandering satellite orbiting a moon and a planet by turn. In this example the satellite makes several transitions from orbiting the moon and the planet, and eventually hits the moon. (Click also here to see the applet.)

5. Periodic multi-petalled orbit of a satellite in the double planet system. The satellite of the heavier component can trace an almost closed orbit in the inertial plane moving synchronously with the other component of the binary system. (Click also here to see the applet.)

6. Periodic orbit of a retrograde satellite in the double planet system. In this example the satellite of the lighter component traces an almost closed orbit moving synchronously with the other component of the binary system. (Click also here to see the applet.)

7. Satellite at the triangular libration point. This example illustrates regular elliptical motion of a satellite at the triangular Lagrange point. The motion is unstable because of the large mass of the moon in this example. (For the Earth - Moon system m/M = 0.0123 < 0.04, so that the triangular libration point is stable.) (Click also here to see the applet.)

8. Satellite at the outer collinear libration point. This example illustrates regular motion of a satellite at the outer collinear Lagrange point for the case of circular motions. The motion of the satellite is unstable: soon it leaves this point and orbits irregularly one of the bodies. (Click also here to see the applet.)

9. Satellite at the inner collinear libration point. This example illustrates regular motion of a satellite at the inner collinear Lagrange point for the case of circular motions. The motion of the satellite is unstable: soon it leaves this point and orbits irregularly one of the bodies. (Click also here to see the applet.)

10. Elliptical orbit at the outer collinear libration point. This example illustrates regular motion of a satellite at the inner collinear Lagrange point for the case of elliptical motions of the bodies. The motion of the satellite is unstable: soon it leaves this point and orbits irregularly one of the bodies. (Click also here to see the applet.)

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Collection of remarkable three-body motions 1 of 7