A family of zero-velocity curves in the restricted three-body problem

The equilibrium points and the curves of zero-velocity (Roche varieties) are analyzed in the frame of the regularized circular restricted three-body problem. The coordinate transformation is done with Levi-Civita generalized method, using polynomial functions of n degree. In the parametric plane, five families of equilibrium points are identified: (L_{i}^{1}, L_{i}^{2}, ldots, L_{i}^{n}) , (iin{ 1,2,ldots,5 }, n inmathbb{N}^{*}) . These families of points correspond to the five equilibrium points in the physical plane L ₁,L ₂,…,L ₅. The zero-velocity curves from the physical plane are transformed in Roche varieties in the parametric plane. The properties of these varieties are analyzed and the Roche varieties for n∈{1,2,…,6} are plotted. The equation of the asymptotic variety is obtained and its shape is analyzed. The slope of the Roche variety in (L_{1}^{1}) point is obtained. For n=1 the slope obtained by Plavec and Kratochvil (1964) in the physical plane was found.