## 4 Calculation Results and Analysis
### Gravity on Spheres Intersecting or Outside the Asteroid  
There are a few reasons to calculate the gravity on a sphere. First, since all the radius distances are the same, one degree of freedom is removed, and many kinds of comparisons become easy. Second, many real spacecraft missions such as satellite gravity survey, navigation, and remote sensing tasks are designed on circular or near circular orbits. The third reason is that it can be used as a kind of visualization strategy for the gravity experienced by a spacecraft. We calculate the gravitational magnitude for different radius, \(r_{0}\), \(2\)\(r_{0}\), \(3\)\(r_{0}\), by the 4 th, 8 th, 16 th order and degree harmonics models and the polyhedral model in this section. Based on all these calculations, we can answer the frequently asked questions by application engineers about which model is the best choice when we calculate the gravity of a typical asteroid under different conditions. Figure 17 shows the gravitational acceleration obtained by the polyhedron model for Castalia at the sphere of the r\({}_{0}\) = 0.5431 km radius. The gravitational acceleration unit here is mgal, where 1 mgal = \(10^{-5}\) m/s\({}^{2}\) = \(10^{-8}\) km/s\({}^{2}\). Special attention should be given to the fact that the blue boundary on the red edge is on the surface of the asteroid. Inside the boundary, the gravity is inside the asteroid, which is smaller than the corresponding place with the same radius on the surface. This is physically correct. So we can visualize the gravity inside the asteroid, which is helpful to understanding the evolution of the asteroid. This sphere includes both the inside and outside parts of the asteroid. Figure 18 shows the gravitational acceleration by the potential of the fourth degree and order for Castalia on the sphere r\({}_{0}\). The unit here is mgal. The axes are the longitude and latitude, respectively. The sphere with radius r\({}_{0}\) intersects the asteroid. Some part of the sphere is outside the asteroid, and the other part is inside the asteroid. We see the gravitational acceleration at the points where the surface radius of the asteroid is larger than \(r_{0}\) cannot be acceptable. Actually, the sphere at these part is inside the asteroid. Comparison can be made to Figure 17.  
Figures 19 and 20 show the gravitational acceleration of polyhedron model and by the potential of the fourth degree and order for Castalia at the distance of \(2\)\(r_{0}\), which is just outside the Brillouin sphere. The unit is \(10^{-5}\) m/s\({}^{2}\), on average, approximating \(8\)\(\times\)\(10^{-5}\) m/s\({}^{2}\) acceleration. The unit here is mgal (1 mgal = \(10^{-8}\) km/s\({}^{2}\) = \(11\) milligals = \(10^{-5}\) m/s\({}^{2}\) = \(10^{-2}\) cm/s\({}^{2}\)). The axes are longitude and latitude, respectively. Figures 21 and 22 show the gravitational acceleration by the polyhedron model and by the potential of fourth degree and order for Castalia at a distance of \(r=3\)\(r_{0}\). The unit is \(10^{-5}\) m/s\({}^{2}\). On average, the overall mean value approximates 3.6 \(\times\)\(10^{-5}\) m/s\({}^{2}\). The axes are longitude and latitude, respectively. The difference between these figures becomes smaller compared to the case when \(r=2\)\(r_{0}\) ba  
Figure 17: The gravitational acceleration by polyhedron model at the sphere with an \(r_{0}\) = 0.5431 km radius. The unit here is \(10^{-5}\) m/s\({}^{2}\). The blue lines is on the surface of the asteroid. Figure 23 and 24 show the gravitational acceleration of the polyhedron model and by the potential of the fourth degree and order for Castalia at a distance of 4 \(r_{0}\). The unit is \(10^{-5}\) m/s\({}^{2}\). On average, the value approximates 2.0 \(\times\)\(10^{-5}\) m/s\({}^{2}\) in acceleration. The axes are longitude and latitude, respectively. At this radius, radial gravitational acceleration dominates in the whole magnitude of acceleration. Figure 19: The gravitational acceleration by the polyhedron model at the distance of 2 \(r_{0}\) sphere. The unit is \(10^{-5}\) m/s\({}^{2}\).