Stability of triangular points in the elliptic restricted three-body problem with oblateness up to zonal harmonic J4 of both primaries

Abstract

In this paper, we study the locations and stability of triangular points in the elliptic restricted three-body problem when both primaries are taken as oblate spheroids with oblateness up to J4. The positions of the triangular points are seen to be affected by oblateness of the primaries and the eccentricity of their orbits. The triangular points are conditionally stable for \(0 < \mu < \mu_{c}\) and unstable for \(\mu_{c}\le \mu \le \frac{1}{2}\), where \(\mu_{c}\) is the critical mass parameter depending on the oblateness coefficients \(J_{2i}\) (i =1,2) and the eccentricity of the orbits. We further observe that both coefficients J2 and J4, semi-major axis and the eccentricity have destabilizing tendencies resulting in a decrease in the size of the region of stability with an increase in the parameters involved. Knowing that, in general, the triangular equilibrium points are stable for \(0 < \mu < \mu_{c}\), in particular systems (Alpha Centauri, \(X_{1}\) Bootis, Sirius and Kruger 60) this does not hold and such points are unstable.