## 5 Conclusions and Discussion  
We have evaluated the gravitational field environment around a typical two-lobe configuration prolate asteroid, Castalia, on its surface and also outside or inside its surface, with different planes and spheres. The gravitational field expansions on the surface and on spheres intersecting or outside are compared against the polyhedral gravity field model constructed from a homogeneously distributed body. Also, the gravitational accelerations in cross-sectional planes show the internal gravity, which can only be calculated using the polyhedral method. This paper studied the gravity characteristics of Castalia in depth using various kinds of examples. The accuracy of the harmonic-expansion gravity field deteriorates for the disfigured asteroid Castalia inside the mean reference radius, and the potential and acceleration errors for a \(4\times 4\), \(6\times 6\), and \(10\times 10\) expansion exceed well over 2 or even 10 times their assumed true value from the polyhedral model. As the harmonic-expansion gravity field suffers from the convergence problem within the mean reference sphere, the exterior gravity field should not be used to model the gravitation near or on the surface of the body where \(r<r_{0}\). The surface gravitational acceleration on an asteroid can be divided into two convergence and divergence regions if calculated by harmonic expansions. Regions whose radius \(r\) are larger than the mean radius \(r_{0}\) of the asteroid are the convergence regions and vice versa. This radius is not the arbitrarily chosen reference radius for the harmonic expansions, but it is sometimes chosen as the mean radius. The polyhedral method can be used to calculate the gravitational acceleration outside, inside, and on the surface of an asteroid, but the calculation run time is much longer than the harmonic-expansion method. For a common personal computer, the CPU calculation time for the surface gravitational acceleration is a few minutes for the harmonic-expansion method but a few hours for polyhedral model with 4092 faces. Recently, some approximate and hybrid models were utilized to reduce the CPU runtime [24] at the cost of losing some accuracy, especially near the surface of the asteroid. Off course, there are some other researchers also address this issue [25; 26]. A few special qualitative comments can be summarized as follows. 1. If the calculated sphere's radius is larger than the maximum radius of the asteroid, for example, if the radius is greater than \(2\ r_{0}=1.0862\) km here, the gravity harmonics work as well as polyhedron method; the farther the better, and the higher the degree \(n\times n\) the better. The sphere with the maximum radius of the asteroid is the so-called Brillouin sphere [8]. The maximum radius for the asteroid Castalia is 0.851 km. See Equation (1), \(|r_{0}/r|<1\) for all the calculated spheres with the radius \(2\ r_{0}\), \(3\ r_{0}\), \(4\ r_{0}\) for both the polyhedral and spherical harmonic models. 2. For preliminary analysis, the 4 th-order and degree gravity model is good enough, especially for spheres with \(r\geq 2\ r_{0}\), and not much of an obvious difference can be obtained by using the 8 th- or 16 th-order model or the polyhedron method with less  
Figure 31: The effective gravitational acceleration on the plane \(y=0\) when the asteroid rotates. than 1% error. The farther from the asteroid, the lower the order of the coefficient dominating the gravity. See Figures 20, 25 and 26, etc. 3. On surface points where the radius is less than the reference radius and the points inside the asteroid, only polyhedron method can obtain reasonable results. 4. For the points outside the asteroid or on the surface of the asteroid, if its radius is greater than reference \(r_{0}\) or less than its maximum radius, the harmonic expansion model still works with relatively large errors but still with acceptable accuracy; i.e., it still converges with 5% error, and the higher order of the model of harmonic functions, the closer to the polyhedra model's results.