On the photo-gravitational restricted four-body problem with variable mass

Abstract

This paper deals with the photo-gravitational restricted four-body problem (PR4BP) with variable mass. Following the procedure given by Gascheau (C. R. 16:393–394, 1843) and Routh (Proc. Lond. Math. Soc. 6:86–97, 1875), the conditions of linear stability of Lagrange triangle solution in the PR4BP are determined. The three radiating primaries having masses $m_{1}$, $m_{2}$ and $m_{3}$ in an equilateral triangle with $m_{2}=m_{3}$ will be stable as long as they satisfy the linear stability condition of the Lagrangian triangle solution. We have derived the equations of motion of the mentioned problem and observed that there exist eight libration points for a fixed value of parameters $\gamma (\frac{m \ \text{at time} \ t}{m \ \text{at initial time}}, 0<\gamma\leq1 )$, $\alpha$ (the proportionality constant in Jeans’ law (Astronomy and Cosmogony, Cambridge University Press, Cambridge, 1928), $0\leq\alpha\leq2.2$), the mass parameter $\mu=0.005$ and radiation parameters $q_{i}, (0< q_{i}\leq1, i=1, 2, 3)$. All the libration points are non-collinear if $q_{2}\neq q_{3}$. It has been observed that the collinear and out-of-plane libration points also exist for $q_{2}=q_{3}$. In all the cases, each libration point is found to be unstable. Further, zero velocity curves (ZVCs) and Newton–Raphson basins of attraction are also discussed.