## 4 Calculation Results and Analysis
### On the Asteroid Surface
#### 4.1.3 Gravity Tangential Acceleration and Slopes  
Figure 10 shows the surface tangential acceleration when the asteroid does not rotate. We see the largest accelerations appear at the crevice between the two lobes, around a bottom crater (\(-40\), \(-80^{\circ}\)), and around a top crater (\(150\), \(70^{\circ}\)) in a circle form in the longitude vs. latitude space. Figure 11 shows the contour map of the surface tangential gravitational acceleration when the asteroid does not rotate. It is noted that the overall pattern of figure is quite similar to the gravitational slope: as the surface gravity does not change much, then the tangential gravity is roughly proportional to the slope since the slope is not large. If the tangential gravity is zero or small, i.e., in the dark blue region, then the attraction and normal force vectors are anti-parallel. Dust near these places can exist, because when the tangential gravity is small, the motion of a loose particle is controlled by the coefficient of friction between the particle on the asteroid surface. If there is a place whose tangential gravity is large, there is no loose particle here, it is probably a rocky place without dust. Note that the two lobes are widely separated at yellow crevice region, a longitude of \(-100^{\circ}\), and a latitude from \(-40\) to \(40^{\circ}\); hence, an influx of a regolith may not be adequate to fill in this crevice, even over long time spans. Hence, in the yellow regions, a dust cannot exist; it must be stony in these areas. These calculations can help us to discriminate which parts of the asteroid is dusty or stony. Figure 12 shows the surface tangential gravitational acceleration when the asteroid rotates at a period of \(4.07\) h. This time, the acceleration acting on a dust at each location on the surface of the asteroid arises from both the gravitational and the centripetal accelerations. We see that the overall gravity is obviously smaller than the gravity without rotation. Dust motion is reduced. Dust only appears at the place where the tangential gravity is small. This can help us to choose a landing site where tangential gravity should be small, since the place is rocky where its tangential gravity is large. When an asteroid rotates slowly, the gravitational attraction between the particles can hold them together. But if it rotates fast enough, the centrifugal forces would even overwhelm the gravitational pull and rip the rocks apart. Figure 13 shows the contour of the surface's tangential gravitational acceleration when the asteroid rotates at \(T=4.07\) h. The largest value here is \(14\) mgal, whereas its value is \(18\) for case with no rotation, though the overall pattern is similar. See Figure 11.  
Figure 9: The gravitational acceleration of the 10 th degree and order for Castalia on the surface. The unit of the color bar is also \(10^{-5}\) m/s\({}^{2}\). Figure 14 shows the surface's effective slope when the period is 4.07 h for asteroid Castalia. Here, the density is 2.1 g/cm\({}^{3}\). It is the angle between effective gravity and the negative local surface normal, i.e., the angle between the local surface horizon and the local gravity horizon. This angle is a practical definition to analyze the dust motion on surface. If the value is small, the dust can be motionless because of friction. If the value is large, say 33\({}^{\circ}\), ([22]), the friction cannot resist the tangential force to move the dust. Figure 15 shows the 2D contour of the slope, which is completely different from results in FIG.14(P.80) in reference [1]. This is also a reason we revisit the gravity of Castalia in this paper. At the crevice where the longitude is about \(-90\) deg between the two lobes of Castalia, the slope is large. The overall pattern is quite similar to the tangential acceleration. See Figure 13, as the total surface gravitational acceleration does not change much.