Existence and stability of equilibrium points in the photogravitational restricted four-body problem with Stokes drag effect

Abstract

The restricted four-body problem consists of an infinitesimal particle which is moving under the Newtonian gravitational attraction of three finite bodies, $m_{1}$, $m_{2}$ and $m_{3}$. The three bodies (called primaries) are moving in circular orbits around their common centre of mass fixed at the origin of the coordinate system. Moreover, according to the solution of Lagrange, these primaries are fixed at the vertices of an equilateral triangle. The fourth body does not affect the motion of the three bodies. In this paper, we deal with the photogravitational version of the problem with Stokes drag acting as a dissipative force. We consider the case where all the primaries are sources of radiation and that two of the bodies, $m_{2}$ and $m_{3}$, have equal masses ($m _{2} = m_{3} = \mu$) and equal radiation factors ($q_{2} = q_{3} = q$) while the dominant primary body $m_{1}$ is of mass $1 - 2\mu$. We investigate the dynamical behaviour of an infinitesimal mass in the gravitational field of radiating primaries coupled with the Stokes drag effect. It is found that under constant dissipative force, collinear equilibrium points do not exist (numerically and of course analytically) whereas the existence and positions of the non-collinear equilibrium points depend on the parameters values. The linear stability of the non-collinear equilibrium points ($L_{i},i = 1,2, \ldots ,8$) is also studied and it is found that they are all unstable except $L_{1}$, $L _{7}$ and $L_{8}$ which may be stable for a range of values of $\mu$ and various values of radiation factors. Finally, we justify the relevance of the model in astronomy by applying it to a stellar system (Ross 104-Ross775a-Ross775b), for which all the equilibrium points have been seen to be unstable.