The Gauss–Legendre quadrature (GLQ) integration is a numerical method to calculate the gravitational field of a tesseroid and the approximation error of the GLQ integration increases as the tesseroid gets closer to the computation point. There are two ways to counterbalance this effect. One way is the subdivision of a tesseroid into smaller units and the other way is to increase the quadrature order. In this paper, we develop variable-order GLQ to model the gravitational field of a tesseroid based on the relation between the quadrature order and the node spacing. This algorithm uses a scalar, referred to distance-to-spacing ratio (RLS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ R_{\text{LS}} $$\end{document}), to control the order of the GLQ integration and the calculation accuracy without the subdivision strategy. We perform the approximation error analysis for the gravitational potential, attraction, and Marussi tensor components to examine the applicability of the variable-order GLQ. Numerical experiments show that the variable-order GLQ is suitable for the case that the horizontal dimensions of the tesseroids are larger than 0.5∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 0.5^\circ $$\end{document} and the computation height is over 50 km. We obtain the optimal values for the ratio RLS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ R_{\text{LS}} $$\end{document} based on the relation between the approximation errors and the ratio RLS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ R_{\text{LS}} $$\end{document} for the tesseroid with the horizontal dimensions 0.5∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 0.5^\circ $$\end{document} to 5∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 5^\circ $$\end{document}. Numerical results show that if the maximum tolerate approximation error is 10-3%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 10^{ - 3} \% $$\end{document}, the values of the ratio RLS=2.5,5.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ R_{\text{LS}} = 2.5, 5.5 $$\end{document} and 10.5 are required for the gravitational potential V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ V $$\end{document}, the vertical gravitational attraction Vz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ V_{z} $$\end{document}, and the vertical gravitational gradient Mzz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ M_{zz} $$\end{document}, respectively.

中文翻译：

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变阶 Gauss-Legendre 求积的 tesseroid 重力场

高斯-勒让德求积 (GLQ) 积分是一种计算 tesseroid 引力场的数值方法，随着 tesseroid 越来越接近计算点，GLQ 积分的近似误差会增加。有两种方法可以抵消这种影响。一种方法是将 tesseroid 细分为更小的单元，另一种方法是增加正交阶数。在本文中，我们开发了可变阶 GLQ 以基于正交阶数和节点间距之间的关系对 tesseroid 的引力场进行建模。该算法使用标量，指的是距离间距比（RLS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage {upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ R_{\text{LS}} $$\end{document})，控制 GLQ 积分的顺序和计算精度，无需细分策略。我们对引力势、引力和 Marussi 张量分量进行近似误差分析，以检查变阶 GLQ 的适用性。数值实验表明，变阶GLQ适用于tesseroids水平尺寸大于0的情况。