## 5 Conclusions and Discussion
See Equation (1), \(|r_{0}/r|<1\). As all the coefficients \(|C_{nm}|\) and \(|S_{nm}|\) are much less than 1, and for the normalized Legendre polynomial \(P_{nm}<\sqrt{4\pi}\), it is easy to understand that all the points with \(r>r_{0}\) converge. A rough proof for understanding the above comments is as follows. \[|U(r,\phi,\lambda)| < \left|\frac{GM}{r}\right[1+\sum_{n=1}^{\infty}\left(\frac{r_{0}}{r }\right)^{n}\sum_{m=0}^{n}|P_{nm}(\sin\phi)|\times\left[\sqrt{C_{nm}^{2}+S_{ nm}^{2}}\right]\right]\Bigg{|}\] (12) \[< \left|\frac{GM}{r}\right[1+\sum_{n=1}^{\infty}\left(\frac{r_{0}} {r}\right)^{n}\sum_{m=0}^{n}|P_{nm}(\sin\phi)|\Bigg{|}\Bigg{|}\] (13) \[< \frac{GM}{r}\right[1+\sum_{n=1}^{\infty}\left(\frac{r_{0}}{r} \right)^{n}\sum_{m=0}^{n}\sqrt{4\pi}\Bigg{]}\] (14) \[< \frac{GM}{r}\right[1+\sum_{n=1}^{\infty}\left(\frac{r_{0}}{r} \right)^{n}(1+2+...+n)\sqrt{4\pi}\Bigg{]}\] (15) \[= \frac{GM}{r}\right[1+\sum_{n=1}^{\infty}\left(\frac{r_{0}}{r} \right)^{n}(1+n)n\sqrt{\pi}\Bigg{]}\] (16)
5. Again, more clearly, if the radius is inside the Brillouin sphere, say \(r_{B}=0.8811\) km for Castalia, but larger than its mean reference radius, \(r_{0}=0.5431\) km, the spherical harmonic-expansion method still works, and the higher the degree and order that are chosen, the higher accuracy that can be obtained. This is an important conclusion reached in this paper, which is more specific than the previous discussions [8]. 6. For the gravity inside the reference radius, the polyhedral method without much modification can still apply both on and inside the surface of Castalia. This is another important conclusion obtained in this paper. 7. Based on the calculation of the minimum overall mean effective slope, we found that the density of Castalia could be more likely 2.9 g/cm\({}^{3}\) than the commonly assumed 2.1. This is the third special contribution of this paper. 8. In a word, the spherical harmonic expansion model only works at a position \(r>r_{0}\). When it is near \(r_{0}\), a higher-order model presents a better estimation of gravity. Based on the performance of the gravity fields analyzed in this paper for a typical prolate asteroid, it looks promising that the conclusions reached in this paper can be helpful for future engineers to calculate gravity in sample return missions or proximity operating spacecrafts, and the applications may even extend to the field of geodesy to improve the atmospheric/ocean transport systems, because this is a good example of a micro-gravity environment. Conceptualizaton, methodology, software, investigation, writing, visualization, W.H. ; writing--review and editing, subroutine of codes, T.F. ; methodology, subroutine of software, writing--review and editing, funding acquisition, C.L. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was partially supported by National Key R&D Plan (Grant No. 2020 YFC2004400), National Natural Science Foundation of China (Grant No. 62273324), Natural Science Foundation of Guangdong Province (Grant No. 2021A1515011678). **Data Availability Statement:** The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author. **Conflicts of Interest:** The authors declare no conflicts of interest.