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

Sven Zschocke, S. A. Klioner|ArXiv.org|Apr 23, 2009

Geophysics and Sensor Technology3 references3 citations

TL;DR

This paper derives a compact, analytical solution for light propagation in the Schwarzschild gravitational field with an accuracy of 1 micro-arcsecond (𝜇as), identifying and retaining only the post-post-Newtonian terms that matter at this precision level. It corrects a 16 𝜇as error in the standard post-Newtonian formula by showing that a single 𝛾-dependent post-post-Newtonian term—originating not from the metric’s post-post-Newtonian terms but from the boundary problem transformation—dominates the error and must be retained for microarcsecond astrometry, as validated by high-accuracy numerical integration.

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.

Motivation & Objective

• To identify the source of a 16 𝜇as error in the standard post-Newtonian formula for light deflection in the Schwarzschild field, relevant for microarcsecond astrometry.
• To determine which post-post-Newtonian terms are numerically significant at the 1 𝜇as accuracy level for observers within the solar system.
• To derive a simplified analytical solution for the boundary problem of light propagation that retains only terms relevant at 1 𝜇as precision.
• To verify the derived solution against high-accuracy numerical integration of the geodesic equations.

Proposed method

• Derives analytical upper bounds for all individual post-post-Newtonian terms in the light propagation solution using the parametrized post-post-Newtonian framework from Klioner & Zschocke (2007).
• Performs numerical magnitude estimates of terms in the transformation between direction vectors k, σ, and n, focusing on terms of order 𝒪(c⁻⁴) and 𝒪(m²/d²).
• Identifies that only one post-post-Newtonian term—proportional to (1+𝛾)m/d²² and dependent on the PPN parameter 𝛾—dominates the error and must be retained.
• Constructs a simplified analytical solution (Eqs. 66–67) that includes the critical 𝛾-dependent term and excludes all other post-post-Newtonian terms deemed negligible at 1 𝜇as.
• Validates the derived solution by comparing it with high-precision numerical integration of the null geodesic equations across a wide range of observer and source positions.
• Confirms that the solution achieves sub-1 𝜇as accuracy for all observers within the solar system, except within 5 angular radii of the Sun.

Experimental results

Research questions

• RQ1 Why does the standard post-Newtonian formula for light deflection in the Schwarzschild field exhibit an error of up to 16 𝜇as in extreme astrometric scenarios?
• RQ2 Which post-post-Newtonian terms in the light propagation solution are numerically significant at the 1 𝜇as accuracy level for solar system observers?
• RQ3 Can a simplified analytical solution be derived that retains only the terms necessary for 1 𝜇as precision, without including negligible higher-order terms?
• RQ4 Is the derived analytical solution consistent with high-accuracy numerical integration of the geodesic equations?

Key findings

• The 16 𝜇as error in the standard post-Newtonian formula arises not from the post-Newtonian approximation itself, but from the analytical transformation used to convert the initial value problem into the boundary problem solution.
• Only one post-post-Newtonian term—proportional to (1+𝛾)m/d²² and dependent on the PPN parameter 𝛾—contributes significantly to the error and must be retained for 1 𝜇as accuracy.
• All other post-post-Newtonian terms, including 'native' terms from the metric tensor, are estimated to be negligible at the 1 𝜇as level, except within 5 angular radii of the Sun.
• The derived analytical solution (Eqs. 66–67) achieves numerical accuracy well below 1 𝜇as when compared to high-accuracy numerical integration across the entire solar system.
• The solution is valid for sources at distances greater than 1 pc and can be used with 𝝈 = 𝝆 for microarcsecond astrometry, though Eqs. (56)–(57) offer slightly better accuracy for very close stars when parallax is known.
• The dominant error term is not of the form 𝒪(m²/d²) but arises from the transformation process, highlighting a disconnect between analytical order and numerical significance.