# Fast and Efficient Lunar Finite Element Gravity Model
## 4 FEM Application in Astrodynamics: Satellite Trajectory Propagation
#### Trajectory Propagation Using FEM Versus LP (100 \(\times\) 100) SH Gravity Model
It follows to start off with converting \(\phi_{0}=0^{o}\), \(\lambda_{o}=0^{o}\), and \(R\) into the satellite's initial Cartesian coordinate: \(x_{0}\), \(y_{0}\), and \(z_{0}\). Unlike the previous two approaches, in this test, the satellite's final position is found by converting \(\phi_{final}=4^{o}\), \(\lambda_{final}=4^{o}\), and \(R\) into the satellite's final Cartesian coordinate: \(x_{final}\), \(y_{final}\), and \(z_{final}\). All that is left to do is to find the satellite's positions during \(t=0\)s and \(t=t_{final}\). For all three approaches, the initial ranges of \(\phi\) and \(\lambda\) and \(R\) are used to generate satellite accelerations using the FEM routine and the LP (\(100\times 100\)) SH routine. In the FEM analysis, the coefficients saved from the original model at \(R=100\) km \(+R_{m}\) were multiplied by the Chebyshev polynomials approximated at each elevation and azimuth. These results are then compared to SH results for computational time and error. Figure 15 displays the computational time for both approaches for each of the three tests for each of the three radii. For all three tests, FEM timing is three to four orders of magnitude smaller than that of SH. For all three tests, the FEM model's error is calculated by subtracting it from the \"truth\" value, specified before, to be the LP (\(100\times 100\)) model. The resulting errors are illustrated in Figure 16. In Tests 1 and 2, the resulting FEM error from all three radii is in the order of \(10^{-10}\) m s\({}^{-2}\). With the previously mentioned increase in computational speedup, these two approaches produced above-adequate results for approximating the acceleration produced in the (\(4\times 4\)) square degree shell 100 km above the lunar surface. However, for the Monte Carlo simulation (Test 2), there was a higher order of points in the iteration compared to Test 1, which is reflected in the computational time required for each of the tests. Figure 17 displays the initial trajectories created from these three tests over the (\(4\times 4\)) square degree section. In Test 1, the trajectory is smooth and laminar, which is expected from a small spacecraft trajectory, while Test 2 displays a randomized path, which is representative of a typical Monte Carlo simulation. It is good to note that the trajectory found in Test 2 does not suggest that the model failed to produce a correct spacecraft orbit. This proves that the model was able to approximate a disorderly trajectory, as apparent in the small error. The error found in Test 3 is in the order of \(10^{-9}\) m s\({}^{-2}\) across all three radii. It is also good to note that the error is lowest at the \"end\" points of the calculation due to the initial values set at each starting position. As illustrated in Figure 17c, the trajectory across the (\(4\times 4\)) square degree shell is similar to that of the trajectory for Test 1; however, this test produced more data points. Finally, a long space orbit propagation was carried out for all three tests. For this, a time span of 360 min was used. Figure 18 shows the long-span predicted trajectory of the satellite, as apparent from Test 3 for a radius of 100 km above the lunar surface as an example. This long-span trajectory is shown in red, while the initial small shell trajectory is in blue. The shape of this conic section is representative of typical trajectories found with two-body applications: [56; 57; 58]. This application of FEM shows promising results and demonstrates a potential path to significant advancements in computational speedup and precision for differential equations in astrodynamics. Figure 15: Computational time for the three tests for FEM and LP (100 \(\times\) 100) SH approaches for each radius above the lunar surface. (**a**) Order of CPU time (s) for Test 1. (**b**) Order of CPU time (s) for Test 2. (**c**) Order of CPU time (s) for Test 3. Figure 16: FEM error for three orbital propagated initial positions tests. (a) 100 km. (b) 150 km. (c) 200 km.