The Existence and Stability of Equilibrium Points in the Robe's Restricted Three-Body Problem

Abstract

The existence of all the equilibrium points, their location and stability in the Robe's (1977) restricted three—body problem have been studied. It is seen that the center of the first primary is always an equilibrium point, whatever be the values of the density parameter K, eccentricity parameter eand mass parameter μ. The other equilibrium points exist only when K ≠ 0 and e = 0 that is when the second primary, a mass point, moves around the first, a spherical shell filled with fluid, in a circular orbit. When K > 1, there is one additional equilibrium point lying on the line joining the center of the first primary and the second primary. When K + μ = 1, there are infinite number of equilibrium points in the x—yplane lying on a circle of radius one and center as the second primary, provided the points are inside the spherical shell. When K 0, there are two more equilibrium points lying in the x—z plane forming triangles with the center of the shell and the second primary. Results of the stability of the equilibrium point (−μ,0,0), center of the first primary are the same as those given by Robe (1977). Circular points and triangular points are always unstable. The equilibrium point collinear with the center of the shell and the second primary is stable provided μ and Ksatisfy the inequality $$16\mu (K - 1)^2 < (\mu + \sqrt {\mu ( - 4 + 4K + \mu )} )^3$$ . Thus, Robe's elliptic restricted three-body problem has only one equilibrium point for all values of the parameters Kand μ and Robe’s circular restricted three-body problem can have two, three or infinite number of equilibrium points depending upon the values of Kand μ. This is contrary to the classical-restricted problem where there are five equilibrium points, which are finite in number. Further, only the points collinear with the center of the shell and the second primary are stable in the Robe's problem where as in the classcial problem collinear points are unstable and triangular points are stable.