###### Abstract  
For the convenience of comparison with previous literature, the gravity calculations are revisited for the Asteroid 4769 Castalia, but with extensions on its surface and on intersecting planes and spheres around it, using the polyhedron and harmonic expansion methods with different order and degree for different cases, especially including the gravitational accelerations inside the asteroid, which did not appear at all before. In these evaluations, a few different facts of the these methods and results are revealed, such as the fact that gravity diverges when the position radius is less than the mean radius from harmonic-expansion method, and the maximum gravity is not at the deep valley and mountain top. For a surface that intersects the asteroid, the maximum gravity on it is at the intersection lines between the asteroid surface and the spheres or planes. This means that on the sphere and the plane, the gravities inside and outside the asteroid are smaller than the gravity on the intersection, i.e., on the surface. Some analyses of these conclusions are given with many examples with different radii of the sphere and with different order and degree harmonic expansion models for the above asteroid surface and surrounding spheres. It is interesting to note that very few researchers know that the polyhedral method can also be used to calculate the gravity inside an asteroid with just some modifications of the code. Some special gravity figures on surface and planes inside the asteroid Castalia are computed and made for the first time. The calculations also include tangential gravity, potential, and gravitational slope on surface. Specifically, we find that the overall mean gravitational slope could be one kind of indicator of the density of an asteroid. The minimum overall mean slope happens when the asteroid density is about 2.9 g/cm\({}^{3}\), which is much larger than a usually assumed value between 1.7 and 2.5 for asteroid Castalia when its period is 4.07 h, since rotation period should be a more accurate parameter than its estimated density. These conclusions about this typical prolate-like asteroid could be a benchmark for analyzing other similar asteroids. gravitational potential and acceleration; on asteroid surface gravity; harmonic expansion; polyhedral method; Asteroid 4769 Castalia +
Footnote †: journal: Applied  
0000-0001-5002-7999][PERSON] 0000-0001-5002-2273][PERSON] 0000-0002-0002-3330][PERSON] 0000-0002-0002-3331-1988][PERSON] 0000-0002-0002-3330-3303][PERSON] 0000-0002-3333-3303][PERSON] 0000-0002-333-33303][PERSON] 0000-0002-333-3303  
shows the figures of gravity on the surface and on the planes and spheres inside the asteroid Castalia. Other researchers utilize the triaxial model [10; 11] and the masscon method [12; 13] to calculate the gravity of an asteroid. Recently, the gravity of asteroids has also caught the attention of researchers outside of aerospace engineering, such as those in astronomy, astrophysics [14; 15], geology, geophysics [16; 17], and others. These researchers are particularly interested in the internal gravity of asteroids and have begun using CFD with a variable density approach. However, the most commonly used methods are still the polyhedral and spherical harmonic expansion due to its precision and speed, respectively. In this paper, the gravities on the surface of the asteroid and on spheres around the asteroid, as well as cross-sectional planes, including the internal gravity, are calculated using the classical polyhedral method and the harmonic-expansion method. For the convenience of comparison to previous literatures, the asteroid Castalia is used again as an example. Many interesting conclusions are given. Some special physical perspectives are revealed. The remainder of this article is organized as follows. Section 2 introduces the method harmonic expansion and polyhedral method briefly with some comments about their special applications in this paper.