Revealing the existence and stability of equilibrium points in the circular autonomous restricted four-body problem with variable mass

We have numerically investigated the circular autonomous restricted four-body problem where the fourth particle of variable mass is moving under the gravitational influence of three bodies known as primaries. Moreover, these primaries move in circular orbit around their common center of mass in such a way that their configuration remains an equilateral triangle configuration. The effect of the parameter alpha on the existence as well as on the locations of the libration points are investigated. The parametric variation of the positions of the libration points and zero velocity curves are also revealed when the parameter alpha (which occurs in Jeans' law) increases. Moreover, the Newton-Raphson basins of convergence corresponding to the libration points are unveiled numerically when the parameter alpha increases. The obtained results strongly suggest that the study of the evolution of the attracting domains of the proposed dynamical system is worth studying in spite of their complexity.