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We investigated the dynamical behavior in the potential of the primary in the triple asteroid (45) Eugenia with the calculation of the full gravitational potential caused by its 3D irregular shape. We presented the whole structure of the gravitational potential and the effective potential of (45) Eugenia in the coordinate planes, and showed the surface height, surface gravitational force accelerations, and the surface effective potential. The surface gravitational environment has been discussed. The zero-velocity curves and the position of external equilibrium points are calculated and showed relative to the 3D shape of the asteroid to help compare the relationship of the characteristic of the gravitational potential and the shape of the asteroid. There are five equilibrium points in the gravitational potential (45) Eugenia. We presented the positions, eigenvalues, topological cases, and stability of these equilibrium points. To analyze the variety of the orbital parameters close to (45) Eugenia, we computed two different orbits and compared the results. The mechanical energy, the semi-major axis, and the eccentricity have two different periods: the long period and the short period. The inclination have three different periods, an intermediate period is occurred. The longitude of the ascending node and the argument of periapsis not only have two periodic terms, but also have a secular term.

The dynamical environment of the primary in the triple asteroid (45) Eugenia