Out-of-plane Equilibrium Points in the Photogravitational Restricted Four-body Problem with Oblateness

Abstract

The restricted four-body problem consists of an infinitesimal particle which is moving under the Newtonian gravitational attraction of three massive bodies, called primaries. The three bodies are moving in circles around their common centre of mass fixed at the origin of the coordinate system, according to the solution of Lagrange, where they are always at the vertices of an equilateral triangle. The fourth body does not affect the motion of the primaries. We consider that the primary body P1 is dominant and is a source of radiation while the other two small primaries P2 and P3 modeled as oblate spheroids have equal masses and oblateness coefficients. The out of equilibrium points of the problem are sought and we found that such critical points exist. These points lie in the − xz plane in symmetrical positions with respect to −xy plane. We investigate numerically the effects of radiation and oblateness on the positions of out-of-plane equilibrium points, their stability, as well as the regions allowed to motion of the infinitesimal body as determined by the zero-velocity surface. It is found that radiation and oblateness have strong effects on the positions of the critical points. We examined the stability of these points and found that the out of plane equilibrium points are unstable