Exit basins for the Sitnikov problem with variable mass

Abstract

In the present paper, we perform a numerical study of the Sitnikov problem aiming to characterize the orbits of a variable mass particle (e.g., comet, rocket, asteroid or spacecraft) and determine the uncertainty in the prediction of the final state of the test particle. The classification of final states was done through the well-known exit basins, while the determination of the uncertainty was calculated using a new tool named Basin entropy. It is found that for small values of the initial mass of the test particle, the number of initial conditions leading to bounded orbits gets increased, thus reducing the uncertainty in the final states. The same behavior in uncertainty is observed for increasing values of the exponent in Jeans law for the variation of the mass. Our results allow us to conclude that: i) an accelerated fuel consumption in the initial stages of stabilization of a satellite can keep the object in an oscillatory state around the primaries and ii) if the mass of the satellite is less than one hundredth of the mass of each primary, it is possible to predict with a very high certainty the final state of the satellite, regardless of the accuracy in the initial conditions of the system.