Position and velocity sensitivities at the triangular libration points in the restricted problem of three bodies when the bigger primary is an oblate body

M R Hassan,K B Bhatnagar,H M Antia

doi:10.1007/s10509-013-1448-8

Abstract

In this paper we have examined the stability of triangular libration points in the restricted problem of three bodies when the bigger primary is an oblate spheroid. Here we followed the time limit and computational process of Tuckness (Celest. Mech. Dyn. Mech. 61, 1–19, 1995) on the stability criteria given by McKenzie and Szebehely (Celest. Mech. 23, 223–229, 1981). In this study it was found that in comparison to other studies the value of the critical mass μ c has been reduced due to oblateness of the bigger primary, i.e. the range of stability of the equilateral triangular libration points reduced with the increase of the oblateness parameter I and hence the order of commensurability was increased.

Highlights

It is well established that the libration points in the Restricted three-body problem are infinitesimally stable or linearly stable for the values of the mass ratio μ < μc =
Leontovich (1962) established the non-linear stability of the triangular libration points and proved that the triangular libration points L4 and L5 are stable for all values of μ < μc except for a set of measure zero, where μc is the Rouths critical mass of the Restricted three-body problem
Breakwell and Pringle (1965) discussed the motion of the third body around L4 in the Earth-Moon system using Von Zeipel’s method with an additional effect of the Sun’s gravitation. He did a detailed study of the stability of the triangular libration points

Summary

Introduction

McKenzie and Szebehely (1981) introduced a new idea to test the stability of the equilateral triangular libration point L4 They defined the maximum velocity and maximum displacement envelopes within which the third body remains for a long time starting from the suitable initial conditions so that the third body may not cross the x-axis. By using all the stability criteria of McKenzie and Szebehely (1981), Tuckness (1995) excellently, investigated numerically the sensitivities of the third body around L4 when it is given positional and velocity deviations away from L4 with a suitable initial conditions He used Poincare’s surface of sections to compare the periodic, quasi-periodic and stochastic regions to the trajectories with the definitions of stability given by McKenzie and Szebehely (1981). All the other conditions and criteria will remain the same as those chosen by McKenzie and Szebehely (1981) and Tuckness (1995)

Mathematical formulation for critical mass

Numerical integration

Time limit for numerical integration

Conclusion