Equilibrium points and their linear stability analysis under the effect of dark matter halo

The motion of a test particle is studied under the influence of the gravitational force from the primaries and the force due to dark matter halo. To investigate the motion, we have modified the restricted three-body problem in which an additional potential from the dark matter halo is considered. We have obtained the locations of collinear and triangular equilibrium points semi-analytically. The linear stability of triangular equilibrium points is studied. The critical mass is obtained which is important for the stability analysis. The variations of critical mass parameter are shown with respect to the different parameters. Moreover, we have obtained the orbits around the triangular equilibrium points and it is found that the motion around the triangular points is oscillatory and bounded.