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 P 1 is dominant and is a source of radiation while the other two small primaries P 2 and P 3 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-ofplane 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.

The Motion of Out-of-Plane Equilibrium Points in the Elliptic Restricted Three-Body Problem at <i>J₄</i>

“…As evidenced in Tables 6-9, for each set of values, there exist at least one complex root with positive real part and hence in Lyapunov sense, the stability of the out-of-plane equilibrium points are unstable. This result affirmed with that of [1,4,10,11,12,14,15].…”

“…These points lie in the −plane symmetrically with respect to the − along the curve almost directly above and below the center of each oblate primary. These points are denoted by 6,7 [1,10,13,14,15]. Das et al [14] observed that, in the photo-gravitational circular restricted three-body problem, the out-of-plane equilibrium points are of a passive micro size particle when their stability in the field of radiating binary systems are considered.…”

“…Das et al [14] observed that, in the photo-gravitational circular restricted three-body problem, the out-of-plane equilibrium points are of a passive micro size particle when their stability in the field of radiating binary systems are considered. Singh and Amuda [15] studied the out-of-plane equilibrium points with Poynting-Robertson (P-R) drag, they found that due to the ratio of radiation to the gravitational force of the smaller primary and the expression of coordinate with oblateness of the bigger primary. The out-of-plane equilibrium points exist but its stability analysis remain the same and are unstable.…”

The motion of infinitesimal body in CR4BP with variable masses and Albedo effect

“…Papadouris et al [9,10] studied the existence, the location, stability and periodic orbits of the equilibrium points on and out of the orbital plane in the photo-gravitational R4BP. Singh and Vincent [11,12] studied equilibrium points in CR4BP with the effect of solar radiation pressure and also not considered the albedo effect.…”

The effect of perturbations on the circular restricted four-body problem with variable masses

“…Shrivastava et al in [46] evaluated the equilibrium points in the Robes restricted problem of three-bodies with effect of perturbations in the coriolis and centrifugal forces. Singh et al in [50][51][52][53][54][55][56][57][58][59] studied the restricted problem of three-bodies and four-bodies in circular and elliptic cases with different perturbations. Khanna et al in [29,30] explored the existence and stability of libration points in the restricted three-body problem when the smaller primary is a triaxial rigid body and the bigger one an oblate spheroid and observed that there are five libration points in which three collinear libration points are unstable and two triangular points are stable for the particular mass parameter.…”