Impact of dark matter halo and albedo effect on the out-of-plane motion in the generalized photogravitational RTBP

This research investigates the impact of a dark matter halo and albedo effect in the context of existence and linear stability analysis of out-of-plane equilibrium points under the frame of a generalized photogravitational restricted three body problem. The system considers the out-of-plane motion of an infinitesimal mass under the combined influence of the dark matter halo, radiation pressure force due to radiating primary and albedo from oblate secondary. To compute the out-of-plane equilibrium points, an approximation method in the form of power series expansion and the classical Newton–Raphson method are utilized followed by linear stability test is performed. A significant influence of the dark matter halo is observed in the context of positions of out-of-plane equilibrium points as well as of the stability region. Two symmetric out-of-plane equilibrium points are identified in the $xz$-plane and are examined across the defined ranges of assumed perturbing parameters $\gamma$, $\kappa$, $\rho$ and $M_h$ except for $\rho =\frac{\mu }{\kappa (1-\mu )}$ and $\kappa =\frac{\mu }{\rho (1-\mu )}$. Moreover, these points either shift their respective quadrants or align to the $z$-axis at critical values of $\rho$ and $\kappa$ and vanish under certain conditions. Further, linear stability analysis indicates that these equilibrium points are unstable for all values of perturbing parameters (including mass parameter $\mu$) within their defined ranges. The obtained results will help to study similar problem with different perturbations.