Available online at www.isr-publications.com/jmcs J. Math. Computer Sci., 17 (2017), 365–377 Research Article Journal Homepage: www.tjmcs.com - www.isr-publications.com/jmcs The effect of perturbations on the circular restricted four-body problem with variable masses Abdullah A. Ansari, Ziyad A. Alhussain, Rabah Kellil ∗ College of Science at Al-Zulfi, Majmaah University, KSA. Abstract This paper presents a new investigation of the circular restricted four body problem under the effect of any variation in coriolis and centrifugal forces. Here, masses of all the bodies vary with time. This has been done by considering one of the primaries as oblate body and all the primaries are placed at the vertices of a triangle. Due to the oblateness, the triangular con- figuration becomes an isosceles triangular configuration which was an equilateral triangle in the classical case. After evaluating the equations of motion, we have determined the equilibrium points, the surfaces of the motion, the time series and the basins of attraction of the infinitesimal body. We note that, when we increase both the coriolis and centrifugal forces, the curves, surfaces of motion, and the basins of attraction are shrinking except when we fix the centrifugal force and increase the value of coriolis force, the curves are expanding and the equilibrium points are away from the origin. The behavior of the surfaces of motion and the basins of attraction in the last case (fixing the centrifugal force and increasing the value of coriolis force) will be studied next. In all the present study, we found that all the equilibrium points are unstable. c 2017 All rights reserved. Keywords: Circular restricted four body problem, isosceles triangular configuration, coriolis and centrifugal forces, oblateness, variable mass, basins of attraction, unstable. 2010 MSC: 70F15, 85A20, 85A99, 70F05, 70F07. 1. Introduction Since many decades scientists performed many mathematical models in the celestial mechanics as two- body, three-body, four-body, and n-body problems. The restricted problem was also an interesting topic for them. Many scientists have studied the restricted three-body and the restricted four-body problems. Many related topics have been studied. For example, in [39], Moulton evaluated the equations of motion in the four-body problem and demonstrated that the finite bodies can be placed by twenty-eight different ways. On the other hand, Jeans in [27] discussed the two-body problem with variable mass and Meˇ sˇ cerski˘ ı in [37] investigated the mechanics of the bodies with variable mass. For their work, Sharma et al. in [45] investigated numerically the location of the collinear libration points in the restricted three-body problem when the primaries are oblate spheroids and observed that these equilibrium points are unstable. To be complete, we give below an important list of works close to our investigation. ∗ Corresponding author Email addresses: a.ansari@mu.edu.sa (Abdullah A. Ansari), z.alhussain@mu.edu.sa (Ziyad A. Alhussain), r.kellil@mu.edu.sa (Rabah Kellil) doi:10.22436/jmcs.017.03.03 Received 2017-03-16