Existence and stability of triangular points in the restricted three-body problem with numerical applications

Abstract

In this paper, we prove that the locations of the triangular points and their linear stability are affected by the oblateness of the more massive primary in the planar circular restricted three-body problem, considering the effect of oblateness for J 2 and J 4 . After that, we show that the triangular points are stable for 0 < μ < μ c and unstable when μ c ≤ μ ≤ 1 2 , where μ c is the critical mass parameter which depends on the coefficients of oblateness. On the other hand, we produce some numerical values for the positions of the triangular points, μ and μ c using planets systems in our solar system which emphasis that the range of stability will decrease; however this range sometimes is not affected by the existence of J 4 for some planets systems as in Earth-Moon, Saturn-Phoebe and Uranus-Caliban systems.

On the libration collinear points in the restricted three – body problem

“…For instance, the existence of libration points, their stability and the periodic orbits in the proximity of these points under the oblateness, triaxialty of the primaries or the effect of photogravitational force or combination of them are studied by Sharma [3], Singh and Ishwar [4], Sharma et al [5,6], Singh and Mohammed [7], Abouelmagd and El-Shaboury [8], Abouelmagd [9], Abouelmagd [10,11], Abouelmagd et al [12], Abouelmagd and Sharaf [13], Abouelmagd et al [14,15], Abouelmagd et al [1,16,17,18], Abouelmagd and Mostafa [19], Abouelmagd et al [20], Abouelmagd and Guirao [21].…”

Periodic orbits around the collinear libration points

“…They also provided the periodic orbits around the triangular points. In addition, it was proved in [2] that the locations for the triangular points as well as their linear stability depend on the parameters regarding the first two even zonal harmonic of the more massive primary in the planar circular restricted three-body problem. Moreover, it was also showed therein that the triangular points become stable for 0 < µ < µ c , and unstable provided that µ c ≤ µ ≤ 1/2, where µ c is the critical mass parameter, which depends on the coefficients of zonal harmonic.…”

The effect of zonal harmonic coefficients in the framework of the restricted three-body problem

“…In the range of linear stability under the effects of the perturbed forces of Coriolis and centrifugal, it is deduced that long and short periodic orbits exist around these points and are stated their periods, orientation and eccentricities affected by the non sphericity and the perturbations in the Coriolis and centrifugal forces. Abouelmagd (2012) studies the effects of oblateness coefficients J 2 and J 4 of the bigger primary in the planar restricted three-body problem on the locations of the triangular points and their linear stability. It was found that these locations are affected by the coefficients of oblateness.…”