 2. Examples of the restricted three-body problem (in two frames of reference) This section allows to observe the same three-body motions as in the previous section simultaneously in the inertial centre-of-mass frame of reference and the non-inertial frame associated with the heavier massive body ("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. Ratio of the masses 1:0.5:0. The motion can continue indefinitely. 2. Satellite orbiting a moon that elliptically orbits a planet. Ratio of the masses 1:0.5:0. This motion can also be stationary and continue indefinitely. (Click also here to better 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 (ratio of the masses 1:0.5:0) 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 (ratio of the masses 1:0.75:0) 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 (ratio of the masses 1:0.25:0) 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 (ratio of the masses 1:0.275:0) traces an almost closed orbit moving synchronously with the other component of the binary system. (Click also here to see the applet.) 7. Satellite in 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 – ratio of the masses m/M = 0.33. (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. (Ratio of the masses 1:0.5:0.) 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. (Ratio of the masses 1:0.5:0.) 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. (Ratio of the masses 1:0.5:0.) 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.) Back to top Author's home page | Overview | Contents | Previous section | Next section Collection of remarkable three-body motions – 2 of 7