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 $L_{6,7\kappa }$ and $L_{8,9\kappa }$, 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 $L_{8,9\kappa }$ are closer to the primaries than $L_{6,7\kappa }$. 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.