The increasing interest towards exploring small bodies in the Solar system has paved the way for the investigation of different gravity field models allowing robust orbit design and dynamical environment characterization. Among these, the spherical harmonics representation and the polyhedral gravity model are the most employed and studied—the former for historical reasons and the latter due to the fact that it is an exact and closed-form representation of the gravity field arising from a constant-density polyhedron. The exact algorithm computing the spherical harmonics coefficients from a given polyhedron is available in the literature. Unfortunately, little to no insight into the uncertainty in the spherical harmonics coefficients is provided alongside their computed value, as the polyhedron is customarily considered as a deterministically known solid. During orbit design, spacecraft operations and small body characterization, it is crucial to account for the uncertainty in the spherical harmonics coefficients and how it influences the overall mission architecture. This paper provides the analytical derivation of the partial derivatives of the transformation between the constant density polyhedron and the spherical harmonics coefficients with respect to the vertices of the polyhedron. This derivation allows for the quantification of the spherical harmonics coefficients’ uncertainties when a stochastic polyhedral shape is considered, i.e., a shape whose vertices do not have a deterministic position. As a result, the analytical expressions of the uncertainty in the gravity potential and acceleration are assembled to rigorously quantify the influence of a stochastic polyhedron on the surrounding dynamical environment. A brief explanation of the implementation procedure and the polyhedron vertices covariance matrix definition is provided to the reader. The final result of the paper is a set of numerical simulations validating the proposed model and demonstrating its capability to provide the uncertainties in the spherical harmonics coefficients, the gravity potential and acceleration arising from a stochastic polyhedral shape.

中文翻译：

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由随机多面体形状引起的重力球谐系数的不确定性

对探索太阳系中小天体的兴趣日益浓厚，为研究不同的重力场模型铺平了道路，从而实现了稳健的轨道设计和动态环境表征。其中，球谐函数表示法和多面体引力模型是使用和研究最多的——前者是历史原因，后者是由于它是重力场的精确和封闭形式的表示，由常数-密度多面体。从给定的多面体计算球谐系数的精确算法可在文献中找到。不幸的是，除了计算值之外，几乎没有提供对球谐函数系数的不确定性的洞察，因为多面体通常被认为是确定性已知的固体。在轨道设计、航天器运行和小天体表征过程中，考虑球谐系数的不确定性及其对整体任务架构的影响至关重要。本文提供了等密度多面体与球谐系数关于多面体顶点变换的偏导数的解析推导。当考虑随机多面体形状，即顶点没有确定性位置的形状时，该推导允许对球谐系数的不确定性进行量化。因此，组合重力势和加速度不确定性的分析表达式，以严格量化随机多面体对周围动力环境的影响。向读者提供了实现过程和多面体顶点协方差矩阵定义的简要说明。论文的最终结果是一组数值模拟，验证了所提出的模型，并证明了其提供球谐系数、重力势和随机多面体形状引起的加速度的不确定性的能力。