Dynamical Environment in the Vicinity of Asteroids with an Application to 41 Daphne

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.

systems [2]. 41 Daphne is a large asteroid from the main belt [3,4]. Matter et al. [3] applied the convex 3-D shape model with the data from lightcurves and images and derived the volume equivalent diameter of asteroid (41) Daphne between 194 and 209 km.
Conrad et al. [4] estimated the size of 41 Daphne to be 239×183×153 km with observation data. 41 Daphne has a moonlet S/2008 (41) 1 with the size smaller than 2 km [4]. The diameter ratio [3,4] for the moonlet and the primary in the binary system 1 has the most extreme size ratio among known binary asteroid systems in the Solar system [3][4][5][6]. Thus we investigate the dynamics in the gravitational potential of (41) Daphne; the results are also useful to understand the dynamical behaviors in other large-size-ratio binary asteroid systems.
The polyhedron model can deal with the gravitational potential generated by an irregular-shaped asteroid with constant density [7][8][9][10]. Previous studies used the polyhedron model to analyze the dynamical environments around asteroids 21 Lutetia [11], 216 Kleopatra [12,13], 433 Eros [14,15], etc. Wang et al. [16] used the irregular shape model of asteroids generated by observed data and calculated the positions and topological cases of several contact binary asteroids, including 1996 HW1, 4769 Castalia, 25143 Itokawa, etc. Bosanac et al. [17] used the restricted three-body problem (CR3BP) to model the gravitational force of the primary in the large mass ratio binary system, and investigated the stability of motion of a massless moonlet in the potential of the primary. Our interest is to analyze the dynamical environments around the primary of large-size-ratio binary asteroid systems with considering the gravitational potential generated by the irregular shape of the primary. This paper is split into the following sections. Section 2 deals with the polyhedron model, zero-velocity curves, and equilibrium points for 41 Daphne, and calculated eigenvalues, Jacobi integral, Hessian matrix, and topological cases of equilibrium points for 41 Daphne. Section 3 focuses on the variety of topological classification, stability, and index of inertia of the equilibrium points of the body during the shape variety from 41 Daphne to a sphere. Section 4 covers the simulation of orbits of the moonlet in the gravitational potential of 41 Daphne. The gravitational potential of 41 Daphne is computed using the polyhedron model with observation data.
Finally, the Conclusion section presents a brief review of the results.

Daphne Dynamical Environment: Numerical Study
Let ω be the rotation velocity of the asteroid, and the unit vector z e be the z axis in the body-fixed frame. z e is defined by   z ω e . Denote

 
,, U x y z as the gravitational potential of the asteroid. The body-fixed frame is defined as the coordinate system of principal axis of inertia. The z, y, and x are principal axes of largest, intermediate, and smallest moment of inertia.
The gravitational potential and force [7] of the body can be calculated using the where ω is the rotation speed of the asteroid, which is the norm of ω.
The effective potential is defined as The Jacobi integral is The Jacobi integral is a constant if  is time invariant. The physical meaning of the Jacobi integral is the relative energy of the particle.
Then the linearised equations [18] of motion for a massless particle relative to the equilibrium point can be written as where  represents the eigenvalues of the equilibrium point.
In Figure 1, we plot the zero-velocity curves and equilibrium points around asteroid 41 Daphne. The zero-velocity curves and equilibrium points are showed in the equatorial plane of Daphne. From Figure 1, one can see that there are five equilibrium points in the potential of Daphne. Figure 2 shows projection of zero-velocity curves in different planes.
To show the values of positions and Jacobi integrals of equilibrium points numerically, we also calculated these values and presented the relative quantity to help to analyze. Table 1 presents positions and Jacobi integrals of equilibrium points around 41 Daphne, while Table 2 presents relative distance of the equilibrium point position and coordinate axis. In Table 2, x axis equilibrium points represent equilibrium points which are near x axis; the same for y axis equilibrium points.
Several previous literatures used symmetric simple-shaped bodies to model the gravitational field of asteroids. These symmetric simple-shaped bodies include cube model [19], dumbbell-shaped model [20], ellipsoid-sphere model [21,22], etc. Feng et al. [21] used the ellipsoid-sphere body to model the gravitational field of contact binary asteroid 1996 HW1, and calculated the locations of outside equilibrium points around 1996 HW1. A contact binary asteroid is also a single asteroid, the shape of a contact binary asteroid consists of two parts, and the two parts are connected by a neck. Liang et al. [22] also used the ellipsoid-sphere body to model the gravitational field of 1996 HW1, they also presents the locations of outside equilibrium points around 1996 HW1. Using the ellipsoid-sphere body to calculate, the locations of equilibrium points around asteroid 1996 HW1 are symmetric relative to one axis of The distribution of eigenvalues of the equilibrium point can confirm the topological cases and stability of the equilibrium point. Table 3 gives eigenvalues of these equilibrium points. Table 4 gives the topological classification, stability, and index of inertia of equilibrium points of asteroid 41 Daphne. Table A1 in Appendix A presents Hessian matrix of the equilibrium points around Daphne. The positive/negative index of inertia for an equilibrium point is defined as the positive/negative index of inertia of the Hessian matrix of the equilibrium points. The topological cases [16,18] are defined by distributions of eigenvalues, i.e. Case 1:

Dynamical Environment with the Shape Change of Daphne
Considering asteroid 41 Daphne has a highly irregular shape, and the deviations of equilibrium points relative to the coordinate axes of the principal inertia frame are large. We change the shape of this asteroid to see the variety of zero-velocity curves and equilibrium points.
We use the homotopy method to generate the shape variety of the body. The new shape of the body is calculated by where K represents the shape of asteroid 41 Daphne, S represents the shape of a sphere,  changes from 0 to 1. The radius of the sphere is set to be 130.71 km.
When  equal 0, the new shape is a sphere; while  equal 1, the new shape and the shape of asteroid 41 Daphne are the same. Figure 4 shows zero-velocity curves and equilibrium points with the shape variety of the body.
When  =1.0, the shape has no change relative to the shape of Daphne, Table 3 gives the eigenvalues of the equilibrium points while Table 4 gives the topological classification, stability, and index of inertia of the equilibrium points.
In Figure 4, we present four cases, i.e.  =0.8, 0.6, 0.4, and 0.2. Table A2 in Appendix A presents positions and Jacobi integrals of the equilibrium points of the body during the shape variety. From Figure 4 and Eigenvalues of the equilibrium points of the body during the shape variety have been presented in Table A3 in Appendix A. Comparing Table A3 with The forms of eigenvalues of equilibrium points for  =0.6, 0.4, and 0.2 are the same. Table A4 in Appendix A gives the topological classification, stability, and index of inertia of the equilibrium points of the body during the shape variety. From Table A2

Simulation of Orbits of the Moonlet
Asteroid (41) Daphne has a moonlet S/2008 (41) 1. The semi-major axis of the moonlet relative to the equatorial inertial coordinate system of (41) Daphne is 443 km [4]. The rotation period of (41) Daphne is 5.98798h [3]. The diameter of the moonlet S/2008 (41) 1 is smaller than 2 km [4]. Using the shape model of (41) Daphne, we calculated the size of (41) Daphne to be 261.42×198.38×181.19 km.   Chanut et al. [23] studied the stability of 3D plausible orbital stability in the gravitational potential of 216 Kleopatra, which is the primary of a triple asteroid system. They give an example of 3D orbit that keep stable in the gravitational potential of 216 Kleopatra after 1000h. The radius of the orbit is 250km and the eccentricity is 0.2, thus they conclude that the stable orbits exist at a periapsis radius of 250 km and the eccentricity of 0.2. Our work indicates that the stable orbit exist in the gravitational potential of 41 Daphne with the semi-major axis the same as the moonlet's. Considering the irregular shape of 41 Daphne, the effects of the irregular shape to the motion of the particle is significant when the distance between the particle and the mass center of 41 Daphne is not very large. From the figure of effective potential shown in Section 2 and 3, one can see that the value of effective potential varies significantly when the particle is moving near the equilibrium points.
Thus if we want the perturbation of the irregular shape of 41 Daphne to the orbit of the particle become small, the particle should be far away from the equilibrium points.
Likewise, if we want the sudden change of the mechanical energy of the orbit become unconspicuous, the particle should also be far away from the equilibrium points.

Conclusion
In this paper, we used the polyhedron model with observational data of the irregular shape to generate the 3D asymmetric convex shape model of asteroid ( The orbit for the moonlet in the potential of (41) Daphne has been simulated using the gravitational field computed by the polyhedron model with the observation data of the irregular shape. The mechanical energy of the orbit changes periodically, the Jacobi integral of the orbit keeps conservative.