## 4 Calculation Results and Analysis
### On the Asteroid Surface
#### 4.1.3 Gravity Tangential Acceleration and Slopes
To  
Figure 11: The contour map of the surface tangential gravitational acceleration corresponding Figure 10. The unit of the color bar is \(10^{-5}\) m/s\({}^{2}\). Figure 12: The surface’s effective tangential gravitational acceleration when the asteroid rotates at a period of 4.07 h. The unit of the color bar is \(10^{-5}\) m/s\({}^{2}\). Figure 10: The surface tangential acceleration with respect to its gravity horizon when the asteroid does not rotate. The unit of the color bar is \(10^{-5}\) m/s\({}^{2}\). easily describe the overall pattern of the slope, we assume the shape is a rectangle. On the top and bottom surfaces, i.e., where the latitude is greater than 25 deg and less than \(-30\) deg, the whole slope is about 10 deg, except fortwo craters, one at (170, 80) and one at (\(-40\), \(-80\)). On the side, in the middle area, i.e., the equatorial strip, the slope is very small, less than 5 deg. In the regions up near the edge, i.e., where the latitude is about 20 deg and and \(-25\), the slope is large, about 25 deg. This slope distribution is quite different from the slope of the asteroid Bennu [23], which is a nearly circular oblate asteroid. Figure 16 shows the histogram of the acceleration slope. In the shape model, there are a total of 2048 vertices and 4092 faces for the asteroid Castalia. The vertical axis is the number of the faces, and the horizontal axis is the surface slope with respect to local gravity  
Figure 14: The surface’s effective slope when the period is 4.07 h for the asteroid Castalia. Here, the density is 2.1 g/cm\({}^{3}\). The unit of the color bar is degrees. Figure 13: The contour of the surface’s effective tangential gravitational acceleration corresponding to the 3D Figure 12. The unit of the color bar is also \(10^{-5}\) m/s\({}^{2}\). Figure 15: The 2D contour of the slope, corresponding to Figure 9:0 The unit of the color bar is degrees. horizon direction for each face. In the figure, the number of faces vs. the gravitational slope for rotation (brown) and no rotation (blue) case are presented. We see that the number of faces shifts to left because of the rotation, which means the overall gravitational slope is smaller for the rotation case. The overall mean slope is 12.89 deg for the rotation case when its period is 4.07 h. The overall mean slope is 15.28 deg for the case with no rotation. More calculations show the mean slope for a different rotation period. The results are as follows: slope(period), 81.26 (2 h), 29.50 (3 h), 12.89 (4 h), 11.48 (5 h), 12.25 (6 h), 12.98 (7 h), 15.28 (infinity), and (rho = 2.1). Here, we choose a density of 2.1 g/cm\({}^{3}\); actually, the density could be between 1.7 and 2.5. From our analysis, the density should be larger than 2.1. A more reasonable value is 2.9, because the minimum mean slope appears when the period is 5 h in this calculation. More tests show that when the density is 2.9, the overall mean slope is at its minimum when the period is still 4.07 h, i.e., the mean effective slope (density) 12.57 (2.1), 11.75 (2.4), 11.46 (2.7), 11.43 (2.8), 11.32 (2.9), 11.43 (3.0), 11.53 (3.3). Since the period is estimated relatively more accurately using a light curve, the density could be higher than the usually assumed value of 2.1 g/cm\({}^{3}\). Hence, the overall mean slope could be used to estimate the density inversely. This could be an important conclusion of this paper. Since the asteroid Castalia's shape is bifurcated into two distinct lobes (stones) separated by a crevice, its porosities could be more stony irons of 60 percent than the ordinary chondrites of 40 percent. Previous studies have assumed a maximum slope of 33 deg before slipping occurs ([PERSON] et al., 1994) ([PERSON] et al., 1994). All the slope values computed here are less than 33 deg when the the case period is 4.07 h and the density is 2.1 g/cm\({}^{3}\), except when there are five faces.