## 4 Calculation Results and Analysis
### On the Asteroid Surface
#### 4.1.2 Harmonic-Expansion Method  
With the polyhedral results of the gravitational acceleration on the surface of the asteroid, we can evaluate the harmonic expansion computation with different order and degrees. We give the calculation with the order and degree of 4 \(\times\) 4, 6 \(\times\) 6, 10 \(\times\) 10. This give us a direct physical sense to choose the order and degree of harmonic expansion. Nevertheless, the polyhedral method's computation cost is too much higher than that of the harmonic-expansion method. If the probe's radius is a little bit larger than the reference radius, a 10 \(\times\) 10 order and degree harmonic expansion is good enough, and for a rough analysis, 4 \(\times\) 4 is enough. If the probe is far enough, say, 4\(r_{0}\) from the center of mass, all methods will obtain similar results. This means the 4 \(\times\) 4 harmonic expansion is best because it is fast and simple. For all the comparisons here, the homogeneous-density polyhedral gravity field is treated as the truth model since it works anywhere outside, inside, and on surface of an asteroid. Figure 7 shows the gravity field of the sixth degree and order for Castalia on the surface. For the \"land areas\" where its radii are larger than the reference radius \(r_{0}\), the calculated gravitational accelerations are acceptable (within 5% error), whereas on the \"ocean floor,\" where their radius is less then that of reference \(r_{0}\), which can be compared with the \"sea level\" of the Earth, the gravity calculated by harmonic expansion is much amplified by up to 6 to 8 times larger than the corresponding value from the polyhedral method, which is assumed as a true value. Hence, in this case, the results are completely meaningless. Apparently, we can choose a different reference radius to change the convergence problem. Actually, this radius is better determined by the inherent mean radius, which correspondsto a sphere's radius, which has the same mass as Castalia. Hence, it is not an arbitrarily chosen value. This is a fundamental invariant limitation for the spherical harmonic-expansion method, no matter which number is chosen as the reference radius because any number can be transformed to this value with a ratio and cancelled by corresponding harmonic coefficients. This means the larger the \(r_{0}\) chosen, the smaller the values of \(C_{nm}\) and \(S_{nm}\). Interested readers can find a detailed comparison in the contour map Figure 2; i.e., in deep valleys of the ocean, the contours are quite similar, which means that the gravitational accelerations are mainly determined by the radius at these places, where the radius is much smaller than the reference radius. It can be seen that the maximum acceleration at the high-latitude region, where the radius is at a minimum, approximates to the value near the crevice between the two Castalia's lobes. This is all because the harmonic-expansion method diverges in these regions. The exterior gravity field is roughly valid outside the reference radius \(r_{0}=0.5431\). For the part inside the reference radius, the acceleration errors are larger than what is acceptable. The average mean acceleration on the surface of the asteroid is about 30 \(\times\)\(10^{-5}\) m/s\({}^{2}\). At the north and south pole and near the crevice, the calculated value is high at up to 200. Figure 8 shows the gravitational acceleration of the fourth degree and order for Castalia on surface, and Figure 9 shows the gravitational acceleration of the 10 th degree  
and order for Castalia on the surface. This is very close to the polyhedron results in Figure 6 at the corresponding surface where \(r>r_{0}\), i.e., inside the thick red boundaries, with only about 1-3% difference. And the higher-order model can help to increase the accuracy on the surface when its radius is larger than \(r_{0}=0.5431\) km. It is important to note that the large errors are contained only in regions where \(r<r_{0}=0.5431\) on the surface.