光传播公式的数值精度与解析精度
Numerical versus analytical accuracy of the formulas for light propagation
摘要：
对史瓦西度规中光传播微分方程的数值积分表明，在某些与实际观测有关的情况下，著名的后牛顿光传播解的误差可达16微弧秒。这项工作的目的是证明这一事实，确定这一误差的原因，并推导出一个精确到1微弧秒水平的分析公式，以满足高精度天体测量项目的需要（例如，盖亚）。在PPN和后线性参数$β$、$γ$和$ε$增广的Schwarzschild度规下，给出了光传播柯西问题和边值问题的后后牛顿解析解.使用分析上估计的每一个术语，我们调查后后牛顿条款可能发挥作用的观察员在太阳系中的1微弧秒的水平，并得出结论，只有一个后后牛顿条款仍然是重要的数值精度。这样，光传播的边界问题的解析解推导。该解包含后牛顿和后后牛顿阶的项，但对给定的1微弧秒的数值水平有效。推导出的解析解已被验证使用的结果的高精度数值积分的微分方程的光传播和发现是正确的，在水平远低于1微弧秒的任意观察者位于太阳系内。此外，起源后后牛顿条款相关的微弧秒的精度进行了说明。我们证明，这些条款的结果从一个不适当的选择碰撞参量在标准的后牛顿公式。
英文摘要：
Numerical integration of the differential equations of light propagation in the Schwarzschild metric shows that in some situations relevant for practical observations the well-known post-Newtonian solution for light propagation has an error up to 16 microarcsecond. The aim of this work is to demonstrate this fact, identify the reason for this error and to derive an analytical formula accurate at the level of 1 microarcsecond as needed for high-accuracy astrometric projects (e.g., Gaia). An analytical post-post-Newtonian solution for the light propagation for both Cauchy and boundary problems is given for the Schwarzschild metric augmented by the PPN and post-linear parameters $β$, $γ$ and $ε$. Using analytical upper estimates of each term we investigate which post-post-Newtonian terms may play a role for an observer in the solar system at the level of 1 microarcsecond and conclude that only one post-post-Newtonian term remains important for this numerical accuracy. In this way, an analytical solution for the boundary problem for light propagation is derived. That solution contains terms of both post-Newtonian and post-post-Newtonian order, but is valid for the given numerical level of 1 microarcsecond. The derived analytical solution has been verified using the results of a high-accuracy numerical integration of differential equations of light propagation and found to be correct at the level well below 1 microarcsecond for arbitrary observer situated within the solar system. Furthermore, the origin of the post-post-Newtonian terms relevant for the microarcsecond accuracy is elucidated. We demonstrate that these terms result from an inadequate choice of the impact parameter in the standard post-Newtonian formulas.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
