<p>For the orbit propagations using analytical and semi-analytical methods, the computation of the perturbed orbit solutions due to the Earth’s non-spherical gravitational perturbation is constrained by the necessity of calculating a large number of the inclination functions, along with an inherently high number of terms involved. Based on the modified Gooding’s method, its associated algorithm is improved to address non-singularity solutions (this enhanced method is referred to as the modified Gooding’s non-singularity method), and an interpolation method is proposed to solve the inclination functions in the short-term orbit propagation in this paper. Taking the modified Gooding’s non-singularity method as the reference, the errors in calculating the inclination functions and the corresponding derivatives with the interpolation method are computed and analyzed, as well as the maximum errors of orbital position changes caused by the higher-order short-period solutions due to the Earth’s non-spherical gravitational perturbation up to different truncation degrees. The interpolation method demonstrates high computational accuracy. For a near-circular low Earth orbit object, the maximum error of orbital position changes caused by the higher-order short-period terms is less than 0.01 m with two interpolation nodes and an interpolation interval of 0.1<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$^{\circ }$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mrow /> <mo>∘</mo> </mmultiscripts> </math></EquationSource> </InlineEquation>, when the truncation degree of the Earth’s non-spherical gravitational perturbation is set to 80. Furthermore, the interpolation method significantly enhances computational efficiency. Specifically, the computational times of the inclination functions and the corresponding derivatives are reduced by 97.8%, 96.8%, and 95.9% using the interpolation method with two, three, and four interpolation nodes compared to the modified Gooding’s non-singularity method, respectively, when the truncation degree of the Earth’s non-spherical gravitational perturbation is set to 80. In conclusion, the interpolation method proposed in this paper not only ensures computational accuracy but also substantially improves computational efficiency, effectively supporting the computation of the inclination functions in short-term orbit propagations.</p>

A fast method to calculate the inclination functions for non-singularity solutions

期刊论文
发表日期：

2026年4月16日

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Ying-Ji Yuan,
Ming-Jiang Zhang,
Jian-Ning Xiong

摘要

For the orbit propagations using analytical and semi-analytical methods, the computation of the perturbed orbit solutions due to the Earth’s non-spherical gravitational perturbation is constrained by the necessity of calculating a large number of the inclination functions, along with an inherently high number of terms involved. Based on the modified Gooding’s method, its associated algorithm is improved to address non-singularity solutions (this enhanced method is referred to as the modified Gooding’s non-singularity method), and an interpolation method is proposed to solve the inclination functions in the short-term orbit propagation in this paper. Taking the modified Gooding’s non-singularity method as the reference, the errors in calculating the inclination functions and the corresponding derivatives with the interpolation method are computed and analyzed, as well as the maximum errors of orbital position changes caused by the higher-order short-period solutions due to the Earth’s non-spherical gravitational perturbation up to different truncation degrees. The interpolation method demonstrates high computational accuracy. For a near-circular low Earth orbit object, the maximum error of orbital position changes caused by the higher-order short-period terms is less than 0.01 m with two interpolation nodes and an interpolation interval of 0.1 $^{\circ }$ $^{\circ}$ , when the truncation degree of the Earth’s non-spherical gravitational perturbation is set to 80. Furthermore, the interpolation method significantly enhances computational efficiency. Specifically, the computational times of the inclination functions and the corresponding derivatives are reduced by 97.8%, 96.8%, and 95.9% using the interpolation method with two, three, and four interpolation nodes compared to the modified Gooding’s non-singularity method, respectively, when the truncation degree of the Earth’s non-spherical gravitational perturbation is set to 80. In conclusion, the interpolation method proposed in this paper not only ensures computational accuracy but also substantially improves computational efficiency, effectively supporting the computation of the inclination functions in short-term orbit propagations.