Investigation of motion around out-of-plane points in the restricted three-body problem with variable shape and masses

Abstract

The paper examines out-of-plane equilibrium points (OEPs) of the restricted three-body problem with variable masses and shape. The bigger primary varies it shape as the lengths of the semi-axes vary with time. For the autonomized system, two pair of OEPs L6,7κ and L8,9κ, are obtained and differ from those of the non-autonomous system due to time t. The stability of OEPs of both systems is found to be unstable. Further, numerical illustrations is provided when variations in shape of the bigger primary is, a triaxial prolate, a sphere and a triaxial oblate shape. The positions, stability and zero velocity curves (ZVC) of the particle around the OEPs are explored. It is seen that when the bigger primary is a triaxial prolate body, the OEPs L8,9κ are closer to the primaries than L6,7κ. However, the converse happens when it is a triaxial oblate body. Also, when the bigger primary is a triaxial oblate body, the OEPs are farther away from the primaries than when it is a triaxial prolate. In the case of the ZVC, it is seen that when the bigger primary is a triaxial prolate body, there is a petal around it, and region of allowed motion of the particle increases, while the region reduces when the bigger primary evolves from a sphere to a triaxial oblate body. This study can be used to describe motion of a dust grain in the vicinity of Betelgeuse, a red giant star whose mass and shape changes with time and its stellar companion.