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

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.

💡 Research Summary

The paper addresses a critical shortcoming in the standard first‑order post‑Newtonian (PN) treatment of light propagation in the Schwarzschild gravitational field, a shortcoming that becomes significant for micro‑arcsecond astrometry missions such as Gaia. By performing high‑precision numerical integrations of the exact null geodesic equations in harmonic coordinates, the authors demonstrate that the conventional PN formula can deviate by as much as 16 µas when observing sources near the giant planets, far exceeding the mission’s error budget.

The authors begin by presenting the exact Schwarzschild metric in harmonic gauge, deriving the corresponding Christoffel symbols, and writing down the null‑geodesic equations together with the isotropy condition. They then integrate these equations numerically using the ODEX extrapolation algorithm with 128‑bit (quadruple) precision, achieving component‑wise accuracies better than 10⁻²⁴. This numerical solution serves as a reference “exact” solution for the two‑point boundary problem (given emission point x₀ and reception point x, solve for the travel time and direction).

Next, the paper revisits the standard PN solution, which expresses the photon trajectory as a straight‑line term plus a PN correction proportional to (1 + γ) m, where m = GM/c². The associated unit direction vector n_PN is compared against the numerically obtained direction n. Extensive simulations, varying impact parameters (down to planetary radii) and observer‑planet distances (up to several AU), reveal systematic errors that grow with distance from the gravitating body and are roughly proportional to m².

To understand the origin of these errors, the authors employ the previously derived post‑post‑Newtonian (ppPN) solution (Klioner & Zschocke 2010), which retains all terms of order O(c⁻⁴). They analytically estimate the magnitude of each ppPN term by inserting realistic solar‑system parameters. The analysis shows that the overwhelming majority of ppPN contributions are below 0.1 µas; only a single term—essentially (1 + γ)² m²/d², where d is the impact parameter—remains at the 1 µas level. Consequently, all other higher‑order terms can be safely omitted for Gaia‑level accuracy.

Based on this insight, the authors construct a simplified analytical boundary solution that includes the essential ppPN term while discarding the negligible ones. The final transformation from the asymptotic direction vector k (or σ for a source at infinity) to the observed direction n takes the compact form:

 n = k −