Stability Analysis in the Restricted Four Body Problem With Oblatness and Radiation Pressure.

Abstract

In the present work, the canonical form of the differential equations is derived from the Hamiltonian function H which is obtained for the system of the four-body problem. This canonical form is considered as the equations of motion, the equilibrium points of the restricted four-body problem are studied under the effects of radiation pressure and oblatness Lyapunov function is used to provide a method for showing that equilibrium points are stable or asymptotically stable. If the system has an equilibrium point conditionally the eigenvalues of the system contain negative real parts, the scalar potential function is positive definite, then The Lyapunov center's theorem is used to analyze the stability and periodicity of the motion of orbits about these equilibrium points of the restricted four-body problem. From this theorem, the Lyapunov function is found. Also, the stability regions are studied by using The Poincare maps, an analytical and numerical approach had been used. A cod of Mathematica is constructed to truncate these steps. The periodic orbits around the equilibrium points are investigated for the Sun-Earth-Moon system.

Effects of Radiation Pressure on The Elliptic Restricted Four Body Problem

“…Additionally, let the value of the orbital angular momentum with respect to the motion of the primaries m 1 and m 2 as unity (see Zebehely [17] and Ismail et al [18,19]).…”

Effects of Radiation Pressure on the Elliptic Restricted Four-Body Problem

“…To introduce the dimensionless system, let the distance between m 1 and m 2 equals the unity [18], that means…”

“…Additionally, let the value of the orbital angular momentum with respect to the motion of the primaries m 1 and m 2 as unity (see Zebehely [17] and Ismail et al [18,19]).…”

Stability Analysis of the Rhomboidal Restricted Six-Body Problem

“…Liu et al [15] studied the fourbody problem and found that the boundaries of possible motions obey the change in parameter c 2 E, that is, if the value of c 2 E is less than or equal to a critical value (c 2 E) cr , then the system is stable. Ismail et al [16] studied the fourbody problem by considering the effects of radiation pressure and oblateness and used the Lyapunov function to show the stability of equilibrium points. Wang and Gao [17] did a numerical study of the restricted five-body problem regarding the zero velocity surface and transfer trajectory by considering four equal masses (primaries) forming a regular tetrahedron configuration and the fifth (infinitesimal) mass moving under the gravitational influence of the four primaries.…”