Periodic orbits in the gravity field of a fixed homogeneous cube

In the current study, the existence of periodic orbits around a fixed homogeneous cube is investigated, and the results have powerful implications for examining periodic orbits around non-spherical celestial bodies. In the two different types of symmetry planes of the fixed cube, periodic orbits are obtained using the method of the Poincar'e surface of section. While in general positions, periodic orbits are found by the homotopy method. The results show that periodic orbits exist extensively in symmetry planes of the fixed cube, and also exist near asymmetry planes that contain the regular Hex cross section. The stability of these periodic orbits is determined on the basis of the eigenvalues of the monodromy matrix. This paper proves that the homotopy method is effective to find periodic orbits in the gravity field of the cube, which provides a new thought of searching for periodic orbits around non-spherical celestial bodies. The investigation of orbits around the cube could be considered as the first step of the complicated cases, and helps to understand the dynamics of orbits around bodies with complicated shapes. The work is an extension of the previous research work about the dynamics of orbits around some simple shaped bodies, including a straight segment, a circular ring, an annulus disk, and simple planar plates.

💡 Research Summary

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The paper investigates the existence and characteristics of periodic orbits of a test particle moving in the gravitational field of a fixed homogeneous cube. Although a cube is geometrically simple, its three‑dimensional gravity field is highly non‑spherical, making the dynamics rich and analytically challenging. The authors first derive the exact gravitational potential of the cube using the polyhedral (Werner–Scheeres) method, scaling the half‑edge length to unity and setting the product of the gravitational constant and the density to one (G·σ = 1) for simplicity.

Two families of symmetry planes are examined: (i) planes parallel to the faces (e.g., the xy‑plane) and (ii) diagonal planes that cut the cube through opposite edges. In each case the potential is symmetric with respect to the plane, so the motion reduces to a two‑degree‑of‑freedom conservative system. By integrating many trajectories with the same Jacobi constant (E = –1.2) and constructing Poincaré surfaces of section (plotting (x, ẋ) each time the orbit crosses the x‑axis), a central fixed point is identified, indicating a stable periodic orbit. Using the coordinates of that fixed point as an initial condition (for example x≈3.34, ẋ≈1.543), the authors integrate the equations of motion for about 100 revolutions and confirm the orbit’s periodicity and symmetry about both the x‑ and y‑axes. The same procedure applied to the diagonal plane (after a suitable rotation of the coordinate axes) yields another periodic orbit with initial condition x₁≈3.3375, ẋ₁≈1.5478. Linear stability is assessed via the monodromy matrix; in both cases all six eigenvalues lie on the unit circle, confirming linear stability.

The main difficulty arises when searching for periodic orbits in non‑symmetry (asymmetry) planes, especially those containing a regular hexagonal cross‑section of the cube. The potential in these planes lacks the simplifying symmetries, rendering direct Poincaré analysis ineffective. To overcome this, the authors introduce a homotopy continuation method. They define a homotopy H(ε)= (1‑ε)U_sphere + ε U_cube that continuously deforms the simple Kepler problem (ε = 0) into the full cube problem (ε = 1). Starting from a Keplerian ellipse (semi‑major axis a = 4 or 5, eccentricity e = 0, inclination varied from 0° to 90°, and fixed values for Ω, ω, f), they increment ε by 0.01. At each step the previous periodic solution is used as the initial guess for a Gauss–Newton correction that enforces the periodicity condition (the state after one period must match the initial state within 10⁻⁸). When ε reaches 1, a genuine periodic orbit of the cube field is obtained.

Applying this continuation to several initial Keplerian configurations yields three distinct families of periodic orbits:

Family A – Orbits lying in planes parallel to the cube faces (essentially the same as those found by the Poincaré method). They are nearly circular, symmetric about both coordinate axes, and all six monodromy eigenvalues lie on the unit circle, indicating stability.

Family B – Orbits located near the asymmetry planes that contain a regular hexagonal cross‑section. These are three‑dimensional, with the out‑of‑plane amplitude (z₂) roughly three orders of magnitude smaller than the in‑plane motions (x₂, y₂). Despite the lack of planar symmetry, the monodromy eigenvalues are again on the unit circle, so the orbits are linearly stable.

Family C – Orbits confined to diagonal planes (e.g., the plane spanned by i – k and j). They are symmetric about the x‑axis, and like the other families they are linearly stable.

Table 1 in the paper lists six representative periodic orbits (two from each family) with their initial Cartesian positions and velocities, periods (≈ 24.9 s), and stability classification (all “S” for stable). Figures 8–10 illustrate the three‑dimensional shapes and their projections onto the principal coordinate planes, confirming the geometric descriptions.

The study’s contributions are threefold. First, it provides the first systematic catalog of periodic orbits around a fixed homogeneous cube, demonstrating that such orbits are abundant both in symmetry planes and in more general, non‑symmetric orientations. Second, it validates the homotopy‑Gauss‑Newton continuation as an effective numerical tool for locating periodic solutions in highly non‑linear, non‑spherical gravity fields where traditional symmetry‑based methods fail. Third, the stability analysis shows that all discovered families are linearly stable, suggesting practical relevance for mission design around irregular bodies (e.g., small‑body orbiters, lander trajectories).

Finally, the authors argue that a cube can serve as a basic building block for modeling the gravity of arbitrarily shaped asteroids or comet nuclei by decomposing the body into a collection of cubes (or tetrahedra). The present work therefore constitutes a foundational step toward understanding and exploiting the dynamics around complex, irregular celestial bodies.