Stability of libration points in the restricted four-body problem with variable mass

Abstract

We have investigated the stability of the Lagrangian solutions for the restricted four-body problem with variable mass. It has been assumed that the three primaries with masses $m_{1}$ , $m_{2}$ and $m_{3}$ form an equilateral triangle, wherein $m_{2}=m_{3}$ . According to Jeans’ law (Astronomy and Cosmogony, Cambridge University Press, Cambridge, 1928), the infinitesimal body varies its mass $m$ with time. The space–time transformations of Meshcherskii (Studies on the Mechanics of Bodies of Variable Mass, GITTL, Moscow, 1949) are used by taking the values of the parameters $q=1/2$ , $k=0$ , $n=1$ . The equations of motion of the infinitesimal body with variable mass have been determined. The equations of motion of the current problem differ from the ones of the restricted four-body problem with constant mass. There exist eight libration points, out of which two are collinear with the primary $m_{1}$ and the rest are non-collinear 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), $0\leq\alpha\leq2.2$ ) and $\mu=0.019$ (the mass parameter). All the libration points are found to be unstable. The zero velocity surfaces (ZVS) are also drawn and regions of motion are discussed.