论文标题
环形的外部重力电势
The exterior gravitational potential of toroids
论文作者
论文摘要
我们执行轴对称绿色功能的双变量泰勒膨胀，以确定带有圆形截面的静态细圆形壳的外部电势，如拉普拉斯方程所给出的。在本节的中心进行的这种扩展包括一个无限的序列，其壳的小半径比率$ e $ $。它适用于固体，均匀的圆环以及不均匀的身体（考虑了核心分层的情况）。我们表明，主术语与具有相同主半径和相同质量|的循环的潜力相同。该“相似性”显示出在$ {\ cal o}（e^2）$ order中保存。该系列的收敛非常好，尤其是靠近环形的表面，在$ e \！ = \！0.1 $在零订单下，低至二阶$ 10^{ - 6} $。按顺序满足laplace方程{\ em恰好}，因此不会通过截断引起额外的密度。重力加速在动力学研究中很重要，以相同的精度再现。该技术还适用于在陆地和天体物理等离子体中遇到的方位角产生的磁电势和场。
We perform a bivariate Taylor expansion of the axisymmetric Green function in order to determine the exterior potential of a static thin toroidal shell having a circular section, as given by the Laplace equation. This expansion, performed at the centre of the section, consists in an infinite series in the powers of the minor-to-major radius ratio $e$ of the shell. It is appropriate for a solid, homogeneous torus, as well as for inhomogeneous bodies (the case of a core stratification is considered). We show that the leading term is identical to the potential of a loop having the same main radius and the same mass | this "similarity" is shown to hold in the ${\cal O}(e^2)$ order. The series converges very well, especially close to the surface of the toroid where the average relative precision is $\sim 10^{-3}$ for $e\! = \!0.1$ at order zero, and as low as a few $10^{-6}$ at second order. The Laplace equation is satisfied {\em exactly} in every order, so no extra density is induced by truncation. The gravitational acceleration, important in dynamical studies, is reproduced with the same accuracy. The technique also applies to the magnetic potential and field generated by azimuthal currents as met in terrestrial and astrophysical plasmas.
