Impacts of variable mass disk on region and stability of motion around triangular equilibrium points of the R3BP with variable mass binaries Maxi J1659-152 and Kepler-16

The paper explores the impacts of variable mass disk on region and stability of motion of a dust grain around the triangular equilibrium points (TEPs) of the circular restricted three-body problem (R3BP), with variable mass binaries Maxi J1659-152 and Kepler-16. The masses of the binaries vary with time under the Mestschersky unified law (MUL) and their motion is described by the Gylden-Mestschersky problem (GMP), with the assumption that the system is enclosed by a non-discrete disk of dust whose mass is also assumed to vary with time under the description of the MUL. Both secondary bodies in the binaries are assumed to be radiation emitters. The time varying system of equations are deduced and transformed to the autonomized equations with constant coefficients. The TEPs of the autonomized and non-autonomous systems are investigated and it is seen that there exists a pair of TEPs defined by the mass parameter$\upsilon$, mass of disk${\mu }_d$and the mass variation parameter$\kappa$, for the autonomized system while those of the non-autonomous system additionally are defined by a function of time$t$. It is seen that the radial and vertical scale lengths of the disk affects the locations of the TEPs but this impact depends largely on the mass of the disk. Further, the impacts of the varying disks masses on the zero velocity curves (ZVC) of the dust grain around the TEPs, is investigated, and it is seen that, as the disk mass is increasing, the energy level decreases and the region where motion of the dust grain is forbidden increases. Additionally, the regions where motion is allowed around the binaries and the TEPs are determined by the disks masses. Next, we investigate the linear stability of the TEPs of the autonomized system and found it to be stable in some interval of the masses of the disks for both binaries. The TEPs of the non-autonomous system are unstable due to time$t$. To validate the linear stability, we explore the orbital numerical integration around the TEPs and calculated the Lyapunov exponent indicator (LEI) for different masses of the disks. It was observed that the orbits are stable in the interval of the linear stability and their corresponding LEI tend to zero. Finally, we analyze the Poincaré surface of sections (PSS) around the TEPs for the binaries and found that in the range of linearly unstable TEPs, chaos exists while periodic, quasi-periodic trajectories and islands exist in the range of the linearly stable TEPS. This model can be used to analyze the long-term motion of satellites and planets in binary systems with mass variations of disk in the environment. This model can be used to analyze the long-term motion of satellites and planets in binary systems with mass variations of disk in the environment.