Hill Four-Body Problem with Oblate Bodies: An Application to the Sun–Jupiter–Hektor–Skamandrios System

Abstract

We consider a restricted four-body problem, with a precise hierarchy between the bodies: two larger bodies and a smaller one, all three of oblate shape, and a fourth, infinitesimal body, in the neighborhood of the smaller of the three bodies. The three heavy bodies are assumed to move in a plane under their mutual gravity, and the fourth body to move in the three-dimensional space under the gravitational influence of the three heavy bodies, but without affecting them. We first find that the triangular central configuration of the three heavy oblate bodies is a scalene triangle (rather than an equilateral triangle as in the point mass case). Then, assuming that these three bodies are in such a central configuration, we perform a Hill approximation of the equations of motion describing the dynamics of the infinitesimal body in a neighborhood of the smaller body. Through the use of Hill’s variables and a limiting procedure, this approximation amounts to sending the two larger bodies to infinity. Finally, for the Hill approximation, we find the equilibrium points for the motion of the infinitesimal body and determine their stability. As a motivating example, we identify the three heavy bodies with the Sun, Jupiter, and the Jupiter’s Trojan asteroid Hektor, which are assumed to move in a triangular central configuration. Then, we consider the dynamics of Hektor’s moonlet Skamandrios.

Appendices

Appendix A: Expressions for the Eigenvectors of the Matrix M in (5.25)

Below we provide the expressions of the eigenvectors $v_1$, $v_2$ associated with the eigenvalues (5.26), respectively. Denote

$$\begin{aligned} \Theta :=\sqrt{-u^4-v^4+2 u^2+2 v^2+2 u^2 v^2-1}.\end{aligned}$$

We have

$$\begin{aligned} \begin{aligned} v_1=&\left[ -\left( (1-\mu )(-v^7+(1+u^2)v^5)-\mu (-u^7+(1+v^2)u^5)\right) \Theta ,\right. \\&\quad (1-\mu )(v^9-2(1+u^2)v^7+v^5(1+u^4))\\&\quad +\mu (u^9-2(1+v^2)u^7+u^5(1+v^4))\\&\quad \left. - \sqrt{2}u^2v^2\sqrt{((1-\mu )v^3+\mu u^3)^2-\mu (1-\mu )uv\Theta ^2}\right] ,\\ v_2=&\left[ -\left( (1-\mu )(-v^7+(1+u^2)v^5)-\mu (-u^7+(1+v^2)u^5)\right) \Theta ,\right. \\&\quad (1-\mu )(v^9-2(1+u^2)v^7+v^5(1+u^4))\\&\quad +\mu (u^9-2(1+v^2)u^7+u^5(1+v^4))\\&\quad \left. + \sqrt{2}u^2v^2\sqrt{((1-\mu )v^3+\mu u^3)^2-\mu (1-\mu )uv\Theta ^2}\right] \end{aligned} \end{aligned}$$

(A.1)

Appendix B: Existence of ‘Out-of Plane’ Equilibria

The existence of ‘out-of-plane’ equilibria near an oblate asteroid, such as the z-equilibrium points found in Sect. 6.2, does not agree with the physical intuition, as it seems that the combined gravitational force acting on the infinitesimal mass must be pointing toward the $\{z=0\}$ plane.

Such ‘out-of-plane’ equilibria appear due to the $J_2$-approximation of the gravitational potential. The $J_2$-approximation is a truncation of the spherical harmonic series expansion of the gravitational potential. Such expansion is known to be convergent outside the Brillouin sphere (which is the smallest sphere that contains the body), while the nature of the series inside the Brillouin sphere is unknown in general. (For certain shapes, e.g., for ellipsoids, the series is divergent inside the Brillouin sphere).

Nan et al. (2018) shows analytically that ‘out-of-plane’ equilibrium points do not physically exist in the restricted three-body problem when one primary is a rotational ellipsoid. They note that the same conclusion can be drawn if both primaries are rotational ellipsoids. Their argument can also be carried out for the Hill four-body problem when all heavy bodies are rotational ellipsoids.

However, we shall note that for non-convex shapes, ‘out-of-plane’ equilibria are physically possible. We show a ‘rubble pile’-model that has true ‘out-of-plane’ equilibria. The model consists of six balls, with four identical larger balls of radius R and two identical smaller balls of radius r, arranged as in the left side of Fig. 5. The centers of the larger balls are at $(\pm 1,0, \pm R)$, and the centers of the smaller balls are at $(\pm r,0,0)$. The condition that the balls in the configuration are tangent is $r = 1/(2(1 + R))$. The right side of Fig. 5 represents the plot of the gravitational force along the z-axis, computed by direct numerical integration. The intersection points with the horizontal axis in this plot correspond to the z-values of the ‘out-of-plane’ equilibria. We note that such ‘out-of-plane’ equilibria exist only for certain ranges of values of R, and disappear through a saddle-node bifurcation. We plan to study families of such configurations in future works.

Fig. 5

Left: Six-ball ‘rubble-pile’ model. Right: the sgravitational force along the z-axis

Many small bodies in the solar system are believed to be ‘rubble piles’, consisting of smaller elements separated by voids. Moreover, many known asteroids have highly irregular shapes. Hence, the study of ‘out-of-plane’ equilibria for asteroids is an interesting problem, with possible applications to missions targeting asteroids.