Abstract—

The problem of expanding the potential of an almost circular elliptic Gaussian ring in a series in powers of eccentricity is posed and solved. The gravitational potential of the ring is represented by a power series up to the terms \({{e}^{4}}\) inclusively on the entire set of points of the principal plane of the ring. The main result: two sets of coefficients for the power series of the potential inside and outside the ring are obtained, which are expressed in terms of complete elliptic integrals of the first and second kind. To control the formulas, the Landen transforms are used. It is proved that at the point of active focus of the ring, the four coefficients of the first set vanish. The calculation results are used to construct the equipotentials of the Gaussian rings that simulate the orbits of the planets of the Solar System.

中文翻译：

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以偏心率为单位的一系列高斯环的共面势的分解

摘要-

提出并解决了在一系列偏心率中扩展几乎圆形的椭圆高斯环的潜力的问题。环的引力势由幂级数表示，直到项\({{e}^{4}}\)包含在环主平面的整个点集上。主要结果：得到了环内外位势幂级数的两组系数，分别用第一类和第二类完全椭圆积分表示。为了控制公式，使用了兰登变换。证明在环的主动焦点处，第一组的四个系数消失。计算结果用于构建模拟太阳系行星轨道的高斯环的等势线。