# Fast and Efficient Lunar Finite Element Gravity Model
## 1 Introduction  
On 4 March 2022, at approximately 4:25 a.m. PST, a rocket component was reported to have crashed into the far side of the Moon [1]. Although the specific rocket component is still not verified, losing control over spacecraft during lunar orbit missions has been an ongoing issue since the launch of the Luna 1 rover in 1959. Moreover, scientists have noticed a peculiar phenomenon about the lunar surface gravity; as satellites and probes orbited the Moon over certain craters and impact basins, they periodically veered off course and sometimes even plummeted toward the lunar surface. This is a particular problem when observing these effects on the far side of the Moon (See Figure 1). It was not until the Apollo 15 and 16 missions that scientists described the Moon's surface gravity as \"lumpy\" due to the differing mass concentrations caused by the geology. These fluctuations in the lunar surface gravity have always puzzled scientists, which led to a high demand for precise lunar gravity data in the hopes of avoiding these crashes. Additionally, from the Lunar Reconnaissance Orbiter (LRO) [2] to the recent Artemis program [3], the field of lunar-based orbit propagation research has been increasingly growing over the decades. As a result of this, the importance of modeling the lunar surface gravity anomaly to aid in future Moon missions has become crucial in predicting the trajectories of spacecraft orbiting around the lunar surface. Using high-resolution gravity data, many lunar surface gravity models, such as NASA's Gravity Recovery and Interior Laboratory's (GRAIL) GRGM1200B model [4] and the GSFC LP-100k lunar potential Model [5], have been created as an attempt to constrain lunar surface gravity anomalies in response to this demand. Although sufficient enough to model the lunar surface gravity, these existing models rely on the global spherical harmonic (SH) series to solve Laplace's equation for gravity. The SH method for solving the equation of gravity was initially implemented as a way to model the geopotential through the use of Earth-based satellites recognized in the late 1950s [7]. Recent projects that implement gravitational acceleration modeling about the Moon can be viewed in the following literature: [8; 9; 10]. However, the SH approach proves to be computationally demanding, with three main issues arising [11; 12; 13]:  
1. At polynomial orders \(n>2\), convergence proves to be inefficient and slow, which requires tens of thousands of terms to obtain accuracy for higher-order gravity representation. 2. In any astronomical surface gravity modeling, the north and south poles represent singularities in SH, which result in poor approximations at the poles. For further examinations on obstacles in SH gravity approximations, see [14; 15; 16]
3. SH is only accurate with spherical-shaped astronomical bodies. This means that the gravity of objects such as comets and oblate bodies cannot be modeled using this method. The primary aim is to introduce a Finite Element Model (FEM) local gravity representation of the higher-order perturbation to offset the issue of slow convergence of the gravity models at polynomial orders greater than 2. Specifically, lower-order functions are used to sufficiently model the gravity perturbations about the lunar surface. This is highly motivated by the literature, where the projects propose similar advancements to the classical SH approach, such as in [PERSON] [17] and, more importantly, in recent investigations by the following: [18; 19; 20]. To establish an efficient resolution to the above computational challenges, this paper details a new approach to modeling surface gravity anomaly, increasing computational speed while maintaining a conservative error. Specifically, the paper proposes replacing the  
Figure 1: (**Left**): The Moon’s near side. (**Right**): The Moon’s far side. Image credit: [6].