Instability of In-plane Equilibrium Points of the Restricted Three-Body Problem with P-R Drag and Variable Masses

Abstract This paper studies instability of the in-plane equilibrium points (EPs) which lies on the line joining the main bodies in the frame of the restricted three-body problem with Poynting-Robertson (P-R) drag force and variable masses. The dynamical structures of the primaries and their mass variation are governed by the Gylden-Mestschersky problem (GMP) and the Mestschersky combined law (MCL), respectively, under the conditions that both primaries are radiation emitters with the Doppler shift and the absorption and subsequent re-emission of the incident radiation, considered. It is seen that there can be three, four and even up to six Collinear In-plane Equilibrium Points (CIEPs). Some are independent of the P-R drag while others depend on the mass parameter and P-R drag of the primaries. The stability of the CIEPs is investigated analytically and numerically for a dust grain particle in the gravitational field of all binary systems which covers most astronomical systems, and it is seen that all the CIEPs are unstable. Additionally, it is seen that the zero velocity curves (ZVC) around the CIEP L 0 , for all mass parameter do not exist as the P-R drag effects are annulled in the force function. Consequently, the resulting CIEP will be located on same point as the bigger primary, which in turns means that the force function will be undefined in this case. The stability of the CIEPs of the non-autonomous system which differ from those of the autonomized system by a function of time, are unstable as they do not converge as time tends to infinity. Our problem can be applied to the description of a dust grain in the gravitational field of stellar systems with radiation force and variable masses.