Exact computation of the gravitational field and gravitational gradient tensor for a general mass body is a core routine to model the density structure of the Earth. In this study, we report on the existence of closed-form solutions of the gravitational potential, gravitational field and gravitational gradient tensor for a general polyhedral mass body with a polynomial density function of arbitrary non-negative integer orders that can simultaneously vary in both horizontal and vertical directions. Our closed-form solutions of the gravitational potential and the gravitational field are singularity-free, which implies that the observation sites can have arbitrary geometric relationships with polyhedral mass source bodies. However, weak logarithmic singularities exist on the edges of polyhedra for the gravitational gradient tensor. A simple prismatic mass body with polynomial density contrast varying in the vertical direction and a complicated dodecahedral mass body with quartic-order density contrasts were tested to verify the accuracy of the newly derived closed-form solutions. For the gravitational potential, gravitational fields and gradient tensors, our closed-form solutions are in excellent agreement with previously published analytical solutions and Gaussian numerical quadrature solutions.

中文翻译：

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具有任意非负整数阶多项式密度对比的多面体引力场和梯度张量的递归解析公式

精确计算一般质量体的引力场和引力梯度张量是模拟地球密度结构的核心程序。在这项研究中，我们报告了具有任意非负整数阶多项式密度函数的一般多面体质量体的引力势、引力场和引力梯度张量的闭式解的存在，这些函数可以同时在两个水平方向上变化。和垂直方向。我们的引力势和引力场的闭式解是无奇点的，这意味着观测点可以与多面体质量源体具有任意的几何关系。然而，引力梯度张量的多面体边缘存在弱对数奇点。测试了具有多项式密度对比度在垂直方向上变化的简单棱柱质量体和具有四阶密度对比度的复杂十二面体质量体，以验证新导出的封闭形式解的准确性。对于引力势、引力场和梯度张量，我们的闭式解与先前发布的解析解和高斯数值正交解非常吻合。