Very Restricted Four-Body Problem.

Qualitative behavior of artificial satellites in the earth- moon-sun system has been discussed in terms of a very restricted four-body problem of masses m1, m2, m3, and m such that m1 m2+m3 m. If the separation between m2 and m3 is very much smaller than the distance of their center of mass from m1, we can first idealize a state of motion (with approximation) of the three bodies m3, m2, and m3 in such a way that m2 and m3 revolve around each other in circular orbits and furthermore the center of mass 0' of m2 and m3 revolves around m, in a circular orbit too. By choosing a coordinate system with the origin at 0' and revolving together with m2 and m3, we derive an integral which is the counterpart to Jacobi's integral in the restricted three-body problem. The position of m1 enters into the integral through the angle subtended at 0' by m1 and m. Therefore no stationary zero-velocity surface can be defined in the present problem. However, we can introduce the conception of osculating zero-velocity surfaces which provide the instantaneous limiting surfaces that m of a given initial condition cannot pass through. Double points of the zero-velocity surfaces have also been studied. In the restricted three-body problem these points are fixed in the rotating frame of reference and are the solution of the problem. It is not so in the very restricted four-body problem studied here. The double points rotates in the xy plane as m3 moves with respect to the m2 - m3 system. A study of the motion of these points leads to an interesting result that under certain conditions depending upon the mass of m1 and its distance from 0', the critical zero-velocity surfaces which pass through two double points become degenerated into one single surface. On both sides of degeneracy, the zero-velocity surfaces have fundamentally different appearances. It can be shown from these surfaces that any artificial satellite around the moon has to be close, in order to be stable.