This work aims to present an analytical study on the dynamics of a third body in the restricted three-body problem. We study this model in the context of the third body having variable-mass changes according to Jeans' law. The equation of motion is constructed when the variation of the mass is non-isotropic. We find an appropriate approximation for the locations of the out-of-plane equilibrium points in the special case of a non-isotropic variation of the mass. Moreover, some graphical investigations are shown for the effects of the parameters which characterize the variable mass on the locations of the out-of-plane equilibrium points, the regions of possible and forbidden motions of the third body. This model has many applications, especially in the dynamics behavior of small objects such as cosmic dust and grains. It also has interesting applications for artificial satellites, future space colonization or even vehicles and spacecraft parking.Keywords Restricted three-body problem · Variable mass · Out of plane equilibrium points B E.I. Abouelmagd
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.
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