Analytical solution for light propagation in Schwarzschild field having an accuracy of 1 micro-arcsecond

TL;DR

This paper derives an extended analytical solution for light propagation in the Schwarzschild metric with 1 micro-arcsecond accuracy, improving upon standard formulas for high-precision astronomical observations like Gaia.

Contribution

It identifies the relevant post-post-Newtonian terms and provides a simplified analytical formula accurate at the 1 micro-arcsecond level for the boundary problem in light propagation.

Findings

Standard post-Newtonian formula can have errors up to 16 micro-arcseconds.

Only one post-post-Newtonian term significantly affects accuracy at 1 micro-arcsecond level.

The derived analytical solution matches high-accuracy numerical integrations within the required precision.

Abstract

Numerical integration of the differential equations of light propagation in the Schwarzschild metric shows that in some extreme situations relevant for practical observations (e.g. for Gaia) the well-known standard post-Newtonian formula for the boundary problem has an error up to 16 \muas. The aim of this note is to identify the reason for this error and to derive an extended formula accurate at the level of 1 \muas as needed e.g. for Gaia. The analytical parametrized post-post-Newtonian solution for light propagation derived by \citet{report1} gives the solution for the boundary problem with all analytical terms of order $\OO4$ taken into account. Giving an 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 \muas. We conclude that only one post-post-Newtonian term remains important for this numerical accuracy and derive a simplified analytical solution for the boundary problem for light propagation containing all the terms that are indeed relevant at the level of 1 \muas. 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 \muas for arbitrary observer situated within the solar system.