Numerical versus analytical accuracy of the formulas for light propagation

TL;DR

This paper compares numerical and analytical formulas for light propagation in the Schwarzschild metric, demonstrating that high-precision astrometry requires post-post-Newtonian corrections to achieve microarcsecond accuracy.

Contribution

It derives an analytical post-post-Newtonian solution for light propagation that achieves 1 microarcsecond accuracy, improving upon standard post-Newtonian formulas.

Findings

Standard post-Newtonian formulas have errors up to 16 microarcseconds.

Only one post-post-Newtonian term significantly affects accuracy at the microarcsecond level.

The 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 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 $\beta$, $\gamma$ and $\epsilon$. 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.