WHY CAN THE MAJOR PLANETS HAVE THEIR SATELLITES MOVING IN DIRECT AND RETROGRADE ORBITS AS WELL?

Now we use the Jacobian integral of circular restricted three-body problem toestablish a testing function of the stability of satellites.This method of criterion may beapplied to the stability problem of satellites when the six elements of the instantaneous orbitof the satellite with respect to its parent planet are known.By means of an electronic computer,we can find the stable region of a satellite with aquasi-circular orbit.The boundary surface of this region is a nearly oblate ellipsoid.Thevolume of this enclosed space is much smaller than that of binding by Hill surface and thatof“sphere of action”.As the expressions of relative kinetic energy of a satellite with respect to its parentplanet have the same form for the direct as well as the retrograde orbits,they can coexist inthe same region at the same time.