# Fast and Efficient Lunar Finite Element Gravity Model
## 5 Conclusions  
This research introduced a new approach to modeling the Moon's surface gravity anomaly, improving approximation methods for Laplace's differential equations related to gravitational forces. The orthogonal FEM was used to approximate a gravitational acceleration model, utilizing data from NASA's Lunar Prospector mission up to 100 spherical harmonic degrees. The FEM model was then evaluated against the traditional SH solution in terms of accuracy and computational cost. The findings demonstrated the efficiency of orthogonal approximation techniques, including Kronecker-factorable regression matrices for higher dimensions using algebraic identities on one-dimensional orthogonal least-squares operators. The use of Chebyshev polynomials enabled efficient selection of orthogonal basis functions, supporting an effective sampling mechanism. Two radial sampling techniques are also evaluated, maintaining a tolerance of \(10^{-9}\) m s\({}^{-2}\) on the resulting disturbance acceleration and saving the corresponding radially adaptive coefficient values spanning from the lunar surface to 200 km above. For instance, with a polynomial order N = 18, 361 orthogonal polynomial terms are deduced on the lunar surface. As a result, an average surface gravity acceleration value of \(1.624\pm 7.626\ \times\ 10^{-9}\) m s\({}^{-2}\) was found for a polynomial order of 18 and at 200 km above the surface of the Moon, an average value of \(0.5017\pm 1.36595\ \times\ 10^{-9}\) m s\({}^{-2}\). A significant highlight was the computational speedup achieved with the FEM approach. The results indicate that the FEM model offers a five-order-of-magnitude reduction in computational time compared to the original SH method. Consequently, there is a growing interest in applying these techniques to precise long-term orbit propagation for spacecraft and satellites in lunar missions. Three numerical orbital tests were conducted for a satellite orbiting the Moon. In all three tests, the coefficient values produced from the original FEM model were used to approximate the satellite's trajectory as it goes across a (4 \(\times\) 4) square shell 100 km above the surface. These accelerations showed small errors [10\({}^{-6}\)-10\({}^{-10}\) m s\({}^{-2}\)] across all three tests with a four-order-of-magnitude speedup in computation time. A long-span trajectory was also achieved using FEM for one of the tests, displaying promising results in the interpolation of  
Figure 18: Long Span Satellite Orbit Trajectory Approximated by FEM for 100 km above the surface. The blue line indicates the satellite Trajectory within the cell. The redline is complete orbit. orbital mechanics representing the two-body problem. These findings are crucial for precise orbit propagation in various astrodynamics simulations. This research lays the ground-work for future advancements in orthogonal approximation theory and orbit propagation. Potential directions include code parallelization for computational efficiency, FEM-based machine learning benchmarks, \(n\)-body orbital propagation for satellites, and FEM modeling of the entire lunar surface for improved precision in gravitational disturbance. This approach could also be applied to model the surface gravity of other astronomical bodies like Mars, Europa, and comets. Overall, the method shows great promise for advancing astronomy and astrodynamics. Conceptualization, A.B.Y. ; Methodology, A.B.Y. ; Software, G.N. ; Validation, G.N. ; Formal analysis, G.N. ; Investigation, A.B.Y. ; Resources, A.B.Y. ; Data curation, G.N. ; Writing--original draft, G.N. and A.B.Y. ; Writing--review and editing, A.B.Y. and A.A.; Supervision, A.B.Y. ; Project administration, A.B.Y.All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. No new data were created or analyzed in this study. Data sharing is not applicable to this article. We would like to thank [PERSON] for his insight as he pioneered FEM gravity modeling.