# Fast and Efficient Lunar Finite Element Gravity Model
## 4 FEM Application in Astrodynamics: Satellite Trajectory Propagation  
In this section, we discuss how the FEM can be applied to orbit trajectory propagation, and then we examine a case study to demonstrate how the coefficients produced from the FEM lunar model can aid in satellite trajectory propagation. Finally, we compare this result to an SH routine to analyze computational time and precision.  
Modeling spacecraft trajectories has been one of the most important fields in astrodynamics and space guidance. Satellite orbit propagation predicts how an object moves based on its current state. The satellite's position is defined by a group of approximate equations of motion where the degree of approximation is dependent on orbital information. A trajectory is the path a spacecraft follows through space as it is influenced by the gravitational force of one or more bodies, creating acceleration. The two-body model describes of a satellite with negligible mass orbiting a larger astronomical body. The equations of motion for this model can be solved, resulting in different conic sections depending on their distance away from the larger body. Although not represented in the two-body model, it is worth noting that when considering a satellite orbiting the Moon, we would need to consider the gravity of Earth. Introducing an \"n-th\" body example is to be investigated in future works.  
In the equation of motion, the larger astronomical body is treated as fixed within the two-body frame. We then consider the body as a uniform point mass to further simplify the equation of motion. As stated previously, the spacecraft has negligible mass compared to the main body, which means its acceleration is only due to the gravity of the main body.  
Figure 13: Average gravitational acceleration (m s\({}^{-2}\)) from the lunar surface to 200 km above surface.  
The following equation represents the two-body problem implemented using the FEM coefficients (represented by the disturbance acceleration):  
\[\ddot{\vec{r}}=-\frac{\mu}{r^{3}}\,\vec{r}+\sum_{l=0}^{k}G_{d}^{i} \tag{33}\]  
where \(\mu\) is the gravitational constant of the Moon provided by the original 'LP100K' model: \(\mu=4.902800238\times 10^{12}\,\mathrm{m}\,\mathrm{s}^{-2}\) alongside the disturbance acceleration where \(G_{d}^{i}\) is the \(i\)th coefficient of \(k\) gravity spherical harmonic perturbation terms. These terms were previously saved from the original FEM model.  
As mentioned previously, the position of a spacecraft at any point in its orbit is approximated as long as we have at least one reference point to use as a starting approximation. Numerical methods of orbit propagation, also termed special perturbations, compute and approximate solutions of general equations of motion [50; 51; 52]. Figure 14 illustrates a general approach to satellite trajectory propagation provided initial conditions. In the following sections, it follows to implement the coefficients acquired from the FEM routine performed on a small patch of \((4\times 4)\) square degrees on various radii on and from the surface of the Moon to interpolate the trajectory of a point source satellite orbiting the Moon. Finally, the accuracy and computational time are checked with respect to that of the original LP (\(100\times 100\)) SH model. Orbital propagation using FEM has been implemented in a series of projects, some of which can be seen here: [53; 54; 55].  
Figure 14: Representation of orbit determination.