## 4 Calculation Results and Analysis
### Gravitational Accelerations on Intersecting Planes  
These results inside the asteroid show the power of the polyhedral method for a typical highly non-spherical asteroid. Figure 27 shows the magnitude of gravitational acceleration on the plane \(z=0\), i.e., the equatorial cross-section when the asteroid Castalia does not rotate. The largest gravity  
Figure 24: Gravity acceleration by the potential of the fourth degree and order on the sphere 4 \(r_{0}\). The unit of the color bar is also mgal. Figure 26: Gravity acceleration based on the \(16\times 16\) model on a sphere of 2 \(r_{0}\). The unit of the color bar is also mgal. appears at the surface, as the white dot line indicates. The density is chosen to be a constant, \(\sigma=2.1\) g/cm\({}^{3}\), in all the gravity calculations, as is usually assumed. Figure 28 also shows the effective gravity on the plane \(z=0\), when the asteroid rotates. Castalia rotates about its \(Z\) axis with the maximum moment of inertia. This time, the force acts as a motionless dust, \(\dot{x}=\dot{y}=0\), which could be inside, outside, or on the surface of the asteroid, which arises from both the gravitational and centripetal accelerations. \(\omega^{2}x+U_{x}\), \(\omega^{2}y+U_{y}\), and \(U_{x},U_{y}\) are calculated using the polyhedral method. We can see five equilibrium points, one stable inside and four unstable outside. The rotation period of Castalia is still 4.07 h. The outside equilibrium points are compatible with the analysis in paper [3] with just two gravity harmonic parameters \(C_{20}\) and \(C_{22}\), which are all unstable. In the paper [3], we found four equilibria: the two on the x-axis are always unstable, and the two on the y-axis are sometimes stable and sometimes unstable. We also derived stability conditions. But we do not discuss the stability of the equilibrium inside the asteroid. These conclusions agree well with the calculations in this paper with a polyhedral model. We can see the direction of the gravitational acceleration arrow from Figure 29, showing that all four equilibria are unstable, and the one inside is stable. This can ensure that the dust on surface does not fly away. These numerical calculations validate the zero-velocity analysis in reference [1]. Also, the overall gravity near the surface is smaller than the gravity without rotation; see Figure 27, because the maximum gravity, which should appear on the surface, becomes smaller. Figure 27: The magnitude of gravitational acceleration on the plane \(Z=0\) without rotation. The unit is \(10^{-5}\) m/s\({}^{2}\). Figure 28: The effective gravitational acceleration on the plane \(z=0\), when the asteroid rotates with a period of \(T=4.07\) h.  
More specifically, Figure 29 shows the direction distribution of the effective acceleration in the equatorial cross section plane, \(z=0\), when the asteroid rotates. This is also dust moving in the direction without initial velocity in the rotating asteroid, corresponding to Figure 28. Inside the dark blue ring, where the acceleration value is 4.8, dust will move downward, whereas outside the dark blue ring, dust will move outward. The ring could be regarded as an outside protection boundary layer of dust on the surface of the asteroid. Inside the asteroid, we can see the deformation vectors on the cross-section where the white dashed line is the surface boundary. Figure 30 shows the gravitational acceleration on the plane \(y=0\) when the asteroid does not rotate. The dotted white lines represent the intersection between the asteroid's surface and the principle plane \(y=0\). On the plane \(x=0\), the acceleration is nothing special, so it is not shown here. Figure 31 shows the gravitational acceleration, also on the plane \(y=0\), but when the asteroid rotates, with three equilibrium points, one inside and two outside. The white dot outline represents the intersections of Castalia's surface with its principal planes, plane \(y=0\), i.e., the \(xz\) plane. Figure 30: The gravitational acceleration on the plane \(y=0\) without rotation.