On stability of triangular points of the restricted relativistic elliptic three-body problem with triaxial and oblate primaries

Abstract

The importance of the Lagrangian points as possible locations for large space stations, which can be utilized in interplanetary navigation, much increases as time advances. This requires an accurate analysis of locations and linear stability of these points. The linear stability of triangular points was examined in several studies [see Musielak and Quarles (2014) for a review]. Bhatnagar and Hallan (1998) studied the linear stability of relativistic triangular points. They found that these points are unstable for the range of mass ratio 0 ≤ μ ≤ 0.5, despite the fact that the non-relativistic triangular points are stable for μ < μ0 = 0.03852, where μ0 is the Routh critical mass ratio. The same problem was revisited by Douskos and Perdios (2002) and Ahmed et al. (2006) whose results showed that the relativistic triangular points are linearly stable in the range of mass ratios less than a critical value μc, i.e. 0 ≤ μ < μc. This critical value was estimated by Douskos and Perdios (2002) to be μc = μ0 − 17 √ 69/486c while Ahmed et al. (2006) calculated it to be μc = 0.03840. Palit et al. (2009) analyzed the stability of circular orbits in the Schwarzschild-de Sitter spacetime. Yamada and Asada (2010) computed the relativistic corrections to the Sun-Jupiter libration points. Also, Yamada and Asada (2011) continued their work and investigated collinear solutions to the general relativistic three-body problem. They proved the uniqueness of the configuration for given system parameters (the masses and the end-to-end length). Ichita et al. (2011) investigated the postNewtonian effects on Lagrange’s equilateral triangular solution for the three-body problem. For three finite masses, they found that the equilateral triangular configuration satisfies the post-Newtonian equation of motion in general relativity if and only if all three masses are equal. The post-Newtonian effects on Lagrange’s equilateral triangular solution for the three-body problem were re-examined by Yamada