Motion and zero velocity curves of a dust grain around collinear libration points for the binary IRAS 11472-0800 and G29-38 with a triaxial star and variable masses

This paper investigates motion and zero velocity curves of a dust grain around collinear libration points (CLPs) of the circular restricted three-body problem (R3BP) when both stars IRAS 11472-0800 and G29-38 vary their masses according to the unified Mestschersky law (UML) and their motion is described by the Gylden-Mestschersky problem (GMP) with further assumption that the bigger body is a variable triaxial star. The non-autonomous equations of motion of the model are derived and transformed to the autonomized forms using the Mestschersky transformation (MT), the UML, the particular integral and solutions of the GMP, and a transformation we introduced which enable us to convert the time dependent triaxiality of the bigger star to constant triaxiality. The CLPs are examined and it is seen that when motion takes place on the plane joining the stars, there are three CLPs. To further validate the existence of the CLPs, we obtain three polynomial equations of degree seven for the three CLPs, respectively and we numerically explored the roots. It is seen that only three CLPs exists for all values of the mass parameters coupled with triaxiality of the bigger star. The stability of these points is studied and it is seen that CLPs$L_1$ and$L_2$ are unstable for all mass parameters throughout the interval $0<\kappa <\infty$, where$\kappa$ is a constant of the particular integral of the GMP and depicts the mass variation parameter of the stars. However, CLP$L_3$can be stable and unstable due to the triaxiality of the bigger star, mass parameters and the mass variation parameter. Further, the zero velocity curves (ZVCs) of the dust grain around the CLPs are investigated and the effects of the system parameters divulged. It is seen that, the mass parameters and mass variation parameter can increase or decrease the region where motion of the dust grain is dynamically forbidden, while triaxiality of the bigger star increases the region where motion is forbidden around the CLPs and it allows the appearance of a petal-shaped ZVC around the bigger star. Finally, the orbits of the dust grain around the CLPs is explored, and it is seen that the paths around each collinear point differs. The deviations in the paths are due to the perturbing effects of the mass parameters, triaxiality of the bigger star and the mass variation parameter of the stars. Our problem could have important physical applications when studying celestial bodies of triaxial shape with changing masses of the bodies. In particular, we can assume that the binary system may describe a host star along with an exoplanet, or a massive exoplanet with an exomoon whose masses vary with time.