Selecting Spherical Harmonic Gravity Model Truncation Degree with Regards to Model Uncertainty

Abstract Accurate modeling of the orbital motions of lunar spacecraft is necessary for planning satellite and manned missions to the Moon, especially when operating near the lunar surface. Conventional two-body trajectories are inadequate to capture the spacecraft dynamics in lunar orbits due to the significant variability of the mass distribution. To address this, the spherical harmonic gravity (SHG) model is used to predict the dynamics of the spacecraft. This paper introduces a novel method for selecting the appropriate model fidelity when implementing the SHG model, considering local model uncertainty. Methods to assess SHG model uncertainty are discussed. A quantitative approach to identify models of equivalent complexity is developed to rationalize the truncation of the infinite term SHG model for practical applications. A scalar measure based on the Mahalanobis distance is developed for this purpose. The study evaluates the efficacy of reduced-order models based on acceleration errors and orbital state errors. Numerical examples demonstrate the utility of the model reduction tools to design missions for lunar orbits. The study evaluates the efficacy of reduced-order models based on acceleration and orbital state errors. For example, using the GRAIL lunar gravity model, the method finds that maximum truncation degrees of 85, 48, and 29 are sufficient to reduce acceleration errors to two orders of magnitude below the model uncertainty at altitudes of 500 km, 1000 km, and 2000 km, respectively.