A high-degree gravitational potential and gradient calculation method without singularities

doi:10.11947/j.AGCS.2025.20250181

Abstract

Abstract:

The spherical harmonic model of the Earth's gravitational field holds significant application value in areas such as precise orbit determination for very low earth orbit (VLEO) satellites and high-precision inertial navigation systems (INS). The Cunningham recurrence algorithm is a Cartesian coordinate-based spherical harmonic recurrence method capable of singularity-free computation of gravitational potential, acceleration, and gradients to any degree globally. It is primarily applied for gravitational calculations in satellite dynamic orbit determination. However, as the degree of gravitational field models continues to increase, the recurrence relations in this algorithm suffer from numerical overflow issues due to factorial terms. This study introduces a novel scaling factor to optimize the recurrence relations, thereby controlling the growth of factorial terms within the recurrence functions and mitigating the numerical overflow problem. The improved algorithm, implemented in Cartesian coordinates using double-precision floating-point arithmetic, enables computation up to degree 1000 without generating numerical overflow. Compared to existing mainstream spherical harmonic recurrence methods, the computational efficiency for single gravitational potential and acceleration calculations is increased by 16.8% and 8.0%, respectively.

Key words: associated Legendre functions, recursion optimization, high-degree gravity potential, Cartesian coordinate system, numerical stability

CLC Number:

P223

Zhen LI, Zhenghang HE, Chuang SHI. A high-degree gravitational potential and gradient calculation method without singularities[J]. Acta Geodaetica et Cartographica Sinica, 2025, 54(9): 1572-1582.

Figures/Tables 15

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计算内容	格网数据U	格网数据	轨道数据U	轨道数据
改进CM算法	1.51	2.59	1.47	2.66
跨阶次递推法	1.76	2.88	1.82	2.83
CM算法	2.42	3.71	2.44	3.65
标准列递推法	2.37	3.96	2.43	4.01