Stability regions of equilibrium points in restricted four-body problem with oblateness effects

Abstract

In this paper, we extend the basic model of the restricted four-body problem introducing two bigger dominant primaries $m_1$ and $m_2$ as oblate spheroids when masses of the two primary bodies ($m_2$ and $m_3$) are equal. The aim of this study is to investigate the use of zero velocity surfaces and the Poincar\'{e} surfaces of section to determine the possible allowed boundary regions and the stability orbit of the equilibrium points. According to different values of Jacobi constant $C$, we can determine boundary region where the particle can move in possible permitted zones. The stability regions of the equilibrium points expanded due to presence of oblateness coefficient and various values of $C$, whereas for certain range of $t$ ($100 \le t \le 200$), orbits form a shape of cote's spiral. For different values of oblateness parameters $A_1~ (0<A_1<1)$ and $A_2~ (0<A_2<1)$, we obtain two collinear and six non-collinear equilibrium points. The non-collinear equilibrium points are stable when the mass parameter $\mu$ lies in the interval ($0.0190637,~0.647603$). However, basins of attraction are constructed with the help of Newton Raphson method to demonstrate the convergence as well as divergence region of the equilibrium points. The nature of basins of attraction of the equilibrium points are less effected in presence and absence of oblateness coefficients $A_1$ and $A_2$ respectively in the proposed model.