Evaluation and visualization of a rectangle’s gravity on its surface, and on spheres inside, outside, or intersecting it - Hu, Weiduo; Fu, Tao; Wang, Yue (Frontiers in Space Technologies, 2024)
First, for comparison convenience and physical meaning clearness, the gravity on surface of a homogeneous cube and on spheres inside, outside and intersecting about it is calculated by polyhedral or harmonic expansion methods. Secondly, a rectangle's spherical harmonics gravity coefficients are both derived analytically and evaluated numerically, where only 5 terms are nontrivial up to the order 4, which are somewhat unexpected when we first obtained them. There are some similarities about these coefficients to an ellipsoid for the terms C 20 , C 22 , C 42 , but they are much different in the terms C 40 , C 44 . After that, a few special gravity characteristics are revealed or visualized, for example, it is shown that the maximum gravity appears at the sphere intersecting the cube as expected, but maximum surface gravity which is at the center of a mid-plane of a rectangle's surface which is different from the gravity on an ellipsoid at the end of its short axis. Based on above results, an orbit around a cube is integrated by polyhedron method, and its secular motion analysis by averaging theory is investigated, the numerical and analytic results fit very well. Finally, a few special trajectories on a surface plane of a cube are simulated, and the physical meaning is quite clear and some insights are shown, such as why a natural celestial body in the shape of a rectangle with sharp corners is rarely found because of its surface gravity distribution. All the gravity calculations are visualized on 3D figures both for a cube or a rectangle. Additionally, examples of an asteroid and an ellipsoid are shown that the techniques discussed here can be adopted to analyze the gravity of other shape directly.