ORBITAL DYNAMICS IN THE VICINITY OF ASTEROID 216 KLEOPATRA

For a gravitational field and rotation state as stated above, Scheeres et al. (1998) presented the complete form of the equations of motion in the body-fixed frame as

$$\begin{equation} {\bf \ddot r} + 2{\bf{\bomega \times \dot r}} + {\bf \bomega \,\times }\left({{\bf \bomega \times r}} \right) = - \nabla U\left({\bf r} \right), \end{equation} \tag{ 1 }$$

where r is the body-fixed vector from the original point to the field point, ω is the angular velocity (a constant vector in the inertial frame), U(r) is the gravitational potential, −∇U(r) is the gravitational attraction, ∇ is the gradient operator, and the operators ${}^{\bm .}$ and ${^{\bm {..}}}$ represent the first- and second-order time derivatives with respect to the body-fixed-body frame.

A definition of efficient potential (2) was introduced in order to combine Equations (1) and (2) into Equation (3) (Scheeres et al. 1998):

$$\begin{equation} V\left({\bf r} \right) = - \frac{1}{2}\left({{\bf \bomega \times r}} \right)\left({{\bf \bomega \times r}} \right) + U\left({\bf r} \right), \end{equation} \tag{ 2 }$$

$$\begin{equation} {\bf \ddot r} + 2{\bf \bomega \times \dot r} = - \nabla V\left({\bf r} \right). \end{equation} \tag{ 3 }$$

Equation (3) is conserved because the Coriolis term $2{\bf \bomega \times \dot r}$ has no effect and the term −∇V(r) is potential force; thus, the divergence of Equation (3) is zero. Furthermore, it can be proved that Equation (3) describes a Hamiltonian system by means of Legendre transformation (Arnold 1989, p. 61); it retains a symplectic form and conserves phase volume in each extended phase space, resulting in unpredictability of long-term dynamical behaviors.

An integral of generalized energy exists due to the explicit time independence of the Hamiltonian function, which is expressed as

$$\begin{equation} H = \frac{1}{2}{\bf \dot r}\,{\bm\cdot}\,{\bf \dot r} + V({\bf r}). \end{equation} \tag{ 4 }$$

Therefore, the value of the Hamiltonian function is constant for all the motion with the same initial conditions, and the constant C is called the Jacobi integral.

The definition of the Jacobi integral enables a variety of initial conditions to lead to the same integral value; this serves well for the classification of the natural orbits around an asteroid. As reported in previous studies (Szebehely 1967, p. 164), Equation (4) establishes a well-defined region in the body-fixed frame for each of the orbit categories. The whole space is divided into an allowable region, V(r) ⩽ C, and a forbidden region, V(r) > C, based on the determined integral value. The boundary of these regions is governed by V(r) = C, which is known as the zero-velocity surface. The geometry and topology of these surfaces could have several intentional effects on the dynamical behaviors of the system. Figure 3 demonstrates the zero-velocity surfaces around Kleopatra with different values of the Jacobi integral.

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When C < 0, the vicinal initialized orbit may collide with and rebound off of the zero-velocity surface, suffering a final fate of conversion. On the other hand, when C ⩾ 0, the forbidden region shrinks and fades away, implying globally accessible exterior space. In detail, when C = C₁, the zero-velocity surface consists of two branches—an inner one that clings to and entirely wraps around the asteroid and an outer one that is cylindrical. Here, the allowable region is divided into two parts: orbits initialized in the outer branch will never approach the asteroid, whereas those initialized in the inner branch will end up colliding with the asteroid. These two branches begin to connect as the integral increases in value (C₁ < C < C₂); the allowable regions merge into one, and the outer and inner orbits move to the side through the neck area. The zero-velocity surface splits into two branches again around C = C₅, with the upper branch and lower branch dividing the forbidden region into two parts. This indicates that all orbits near the equatorial plane are probable at this time (C > C₅).

The equilibrium manifolds are investigated in this section in order to determine the motion around the equilibria and to estimate the complexity of the global structure of the orbits in general. The four equilibria all possess eigenvalues with positive, negative, and zero real parts at the same time; namely, they are non-hyperbolic. Thus, the local invariant manifolds in the neighborhood of these equilibria include not only the stable and unstable manifolds, but also the center manifolds. This combination of equilibrium manifolds adds new complexity to their behaviors, especially on the center manifolds, which enables the existence of dense periodic orbits around the equilibria, as discussed below.

First, the structure of the invariant subspaces induced by the linear part of the system (6) is determined. Specifically, the eigenvalues demonstrated in Table 2 with positive, zero, and negative real parts determine the stable (E ^s), central (E ^c), and unstable (E ^u) subspaces, respectively. These subspaces are expanded by the eigenvectors affiliated with the corresponding eigenvalues, satisfying R⁶ = E^s⊕E^c⊕E^u. The subspaces are described in the following manner:

$$\begin{equation} \begin{array}{l} E^s = {\rm span}\left\{ {{\bf v}_1^s ,{\bf v}_2^s , \ldots ,{\bf v}_{n_s }^s } \right\} \\\ms E^u = {\rm span}\left\{ {{\bf v}_1^u ,{\bf v}_2^u , \ldots ,{\bf v}_{n_u }^u } \right\} \\\ms E^c = {\rm span}\left\{ {{\bf v}_1^c ,{\bf v}_2^c , \ldots ,{\bf v}_{n_c }^c } \right\}, \\ \end{array}\end{equation} \tag{ 8 }$$

where v^s is the eigenvector of the eigenvalue with a negative real part, v^u is the eigenvector of the eigenvalue with a positive real part, and v^c is the eigenvector of the purely imaginary eigenvalue. The complex conjugate vector bases are replaced by the real and imaginary parts of these vectors.

Since the local manifolds are tangent to their corresponding invariant subspaces at the equilibria, an iterative method was applied in order to approximate the local invariant manifolds around the equilibria that were initialized at the solutions of the linearized system. The center manifold theorem ensures the unique existence of local stable and unstable manifolds of the same dimensions as E^s and E^u, and the existence of a local center manifold with the same dimension as E^c.

For equilibria E₁ and E₂, the dimension of subspaces E ^s, E ^c, and E ^u are respectively 1, 4, and 1, based on the number of corresponding bases. Figure 5 illustrates the one-dimensional local stable and unstable manifolds of E₁ and E₂, which are tangent to the hyperbolic subspaces and form a saddle manifold. Nearby orbits will approach E₁ and E₂ logarithmically or displace them exponentially along their respective manifolds. For E₁, the characteristic times of the stable and unstable manifolds are both 0.74 hr; for E₂, these values are both 0.66 hr.

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Mini periodic orbits around E₁ and E₂ are guaranteed to exist by the local manifolds, which are not unique objects and do not possess corresponding dimensions. Two basic quasi-elliptical families of periodic orbits in close vicinity to E₁ and E₂ are obtained by direct prolongation of the linearized periodic solutions affiliated with the minimum center subspaces; here, the subspaces are expanded by a pair of eigenvectors that correspond to purely imaginary eigenvalues. These local center manifolds are two-dimensional and are tangent to their corresponding minimum center subspaces, as Figure 6 illustrates. Calculations show that the exact values of the periods are quite close to the periods of the corresponding linearized systems, which are 4.11 hr (blue) and 4.22 hr (red) for E₁, and 3.74 hr (blue) and 4.22 hr (red) for E₂.

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Figure 7 shows the local stable and unstable manifolds of E₃ and E₄, which are two-dimensional objects tangent to the invariant subspaces and constitute spiral manifolds. The red surfaces in Figure 7 represent the unstable manifolds, in which the trajectories move exponentially away from the equilibrium in a spiral pattern. The blue surfaces represent the stable manifolds, in which the trajectories exponentially approach the equilibrium in a spiral pattern. The characteristic times for E₃ and E₄ are 1.38 hr and 1.37 hr, respectively, and the spiral period is about 5.7 hr.

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Unlike E₁ and E₂, only one pair of purely imaginary eigenvalues exists for E₃ and E₄; this imaginary pair gives two-dimensional center subspaces with the linearized periods 5.41 hr and 5.34 hr, respectively. As Figure 8 illustrates, corresponding periodic orbit families are obtained close to these equilibria, forming two-dimensional center manifolds tangent to the center subspaces; here, the values of the periods are approximately equal to the linearized periods.

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The descriptions of the local manifolds around the equilibria E₁–E₄ suggest the decomposability of their orbital motion; the component in stable manifolds approaches the equilibria logarithmically, the component in unstable manifolds departs from the equilibria exponentially, and the component in center manifolds repeats periodic motion along a quasi-elliptical orbit. These descriptions also indicate that the instability of motion around the equilibria is due to the unstable manifold, wherein even slight perturbations propagate exponentially in time. In practice, the motion initialized in the stable or center manifolds may fail at any time due to random and regular perturbations in space, perturbations that are uncertain due to the multiplicity of nearby unstable branches. Nevertheless, the four non-hyperbolic equilibria E₁–E₄ can be classified into two types by the dimensions of their local manifolds, which are related to the different modes of the orbits in the vicinity of the asteroid. First, the unstable spiral manifolds at E₃ and E₄ dominate the orbits near the equatorial plane, which spiral in the opposite direction of the asteroid's rotation. Second, the orbits near E₁ and E₂ are dominated by the direct unstable manifolds and thus gain greater departing speed. Loopy Lissajous orbits exist in the center manifolds of E₁ and E₂ because of the hybrid periodic orbit families about, while around E₃ and E₄, periodic orbits are monotonous and quasi-elliptic.

Differing from the equilibria, periodic orbits in the body-fixed frame have limited periods and non-point trajectories, which enable quantitative evaluation of the stability. Generally, periodic orbits in the inertial frame are not periodic orbits in the body-fixed frame, except in a resonance situation (Scheeres et al. 2000); thus, the majority of periodic orbits in the inertial frame are instantaneous and temporary, even without perturbations. Hu & Scheeres (2004) revealed a close relationship between the stability of equilibria and that of their nearby periodic orbits under planar second-degree and -order gravity fields. In this section, the periodic orbits around the equilibria are searched in a wider context, which may not exactly represent the local center manifolds, but these orbits are still delimited by the linear regime. These critical orbits are obtained by continuously searching the center subspaces stated above, which are initialized with periodic solutions of variable sizes. The initial iterative value is chosen as kv^c, where k is the size of orbit and v^c is the basis of the corresponding center subspaces. Figure 9 illustrates the six families of periodic orbits emerging from the equilibria E₁–E₄, as obtained through the search with eigenvectors.

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The period ranges of these orbits are listed in Table 3 with the characteristic times of their corresponding linear subspaces; the two times are quite similar for all the equilibria, as shown below.

These six orbit families generally expand from their respective center manifolds at the equilibria, but there are a few exceptions. As mentioned above, loopy periodic orbits are probably found around E₁ and E₂, leading to a hybrid of the two modes of vibration, while in reality, these two natural periods are so close that it is difficult to reduce the ratio to a simple value. In fact, the ideal Lissajous orbit would possess extremely long periods and would be extremely difficult to form due to the unstable manifolds. In addition, modes of vibration exist for the four equilibria which are roughly perpendicular to the equatorial plane. Last, the shape of the periodic orbits around E₁ and E₂ are quasi-elliptic, while the periodic orbits around E₃ and E₄ have a three-dimensional bean-like shape with the concavity toward the asteroid.

The Floquet theorem (Perko 1991) is applied to determine the stability of these periodic orbits. This theorem deals with a periodic coefficient linear system, which is obtained by linearizing the original system in Equation (3) along a specific orbit (Hu & Scheeres 2008). Here, the periodic coefficient matrix A(t) is six-dimensional, thus the criterion for the planar case will not work. The monodromy matrix M = Φ(t₀ + T, t₀) is defined as the solution to the state transition matrix for one period (Montagnier et al. 2004), which is solved in this section tracking every periodic orbit; it indicates the propagation characteristic of the perturbations along an orbit. The stability of the periodic orbits is determined by calculating the eigenvalues of M; if the maximum norm of the eigenvalues is greater than one, ||λ||_max > 1, the orbit is unstable; otherwise, if ||λ||_max < 1, the orbit is stable. Finally, ||λ||_max = 1 represents the critical case.

Table 4 lists the ranges of the exact values of ||λ||_max for the six orbit families, which shows that all six families of periodic orbits are unstable. Moreover, these values measure the instability of these orbits, which largely rests with the instability of the equilibria: the periodic orbits around E₃ and E₄ show less instability than those around E₁ and E₂.