  We studied the dynamical environment in the vicinity of the primary of the
binary asteroid. The gravitational field of the primary is calculated by the
polyhedron model with observational data of the irregular shape. The
equilibrium points, zero velocity surfaces, as well as Jacobi integral have
been investigated. The results show that the deviations of equilibrium points
are large from the principal axes of moment of inertia. We take binary asteroid
41 Daphne and S 2008 41 1 for example. The distribution of topological cases of
equilibrium points around 41 Daphne is different from other asteroids. The
topological cases of the outer equilibrium points E1 E4 are Case 2, Case 5,
Case 2, and Case 1. The topological case of the inner equilibrium point E5 is
Case 1. Among the four outer equilibrium points E1 E4, E4 is linearly stable
and other outer equilibrium points are unstable. Considering the shape variety
of the body from Daphne to a sphere, we calculated the zero velocity surfaces
and the locations as well as eigenvalues of equilibrium points. It is found
that the topological case of the outer equilibrium point E2 change from Case 5
to Case 1, and its stability change from unstable to linearly stable. Using the
gravitational force acceleration calculated by the polyhedron model with the
irregular shape, we simulated the orbit for the moonlet in the potential of 41
Daphne.
