Flipping particles with nodal circulations by general relativity under the presence of an inner perturber

From a secular Hamiltonian up to the quadrupole level with general relativity (GR), we study nodal circulations with orbital flips of test particles of the Habitable Zone (HZ) around a solar-mass star, which are perturbed by an inner planetary-mass companion. Nodal circulations with orbital flips of an HZ test particle with eccentricity e2 are possible for any mass m1 and eccentricity e1 of the inner perturber and a suitable inclination i2. In particular, the greater the values of m1 and e2 , the smaller the minimum extreme inclination i2 capable of producing nodal circulations with orbital flips for each e1. As long as nodal librations with orbital flips are not possible for any i2, the greater the values of m1, e1, and e2, the larger the region of the plane (Ω2, i2) associated with nodal circulations with orbital flips. If nodal librations with orbital flips occur, the region of the plane (Ω2, i2) referred to nodal circulations with orbital flips increases with a decrease in m1 and e2, and with an increase in e1. We observe very good agreements between the analytical criteria and the N-body experiments for m1 ranging from Earth-mass planets to super-Jupiters, and small and moderate e2. The main discrepancies are found for high e2, which are more evident with an increase in m1 and e1.