Total light deflection in the gravitational field of an axisymmetric body at rest with full mass and spin multipole structure

TL;DR

This paper derives formulas for light deflection caused by an axisymmetric body with complex mass and spin structures, providing practical upper limits for high-precision astrometry in the solar system.

Contribution

It introduces a method using Chebyshev polynomials to evaluate and limit light deflection from multipoles in the 1PN and 1.5PN approximations, enhancing astrometric accuracy.

Findings

Mass and spin multipoles up to l=10 and l=3 are sufficient for nano-arcsecond precision.

Derived upper limits help identify significant multipoles for light deflection.

Formulas are applied to the Sun and giant planets, confirming their relevance for high-precision measurements.

Abstract

The tangent vector of the light trajectory at future infinity and the angle of total light deflection in the gravitational field of an isolated axisymmetric body at rest with full set of mass-multipoles and spin-multipoles is determined in harmonic coordinates in the 1PN and 1.5PN approximation of the post-Newtonian (PN) scheme. It is found that the evaluation of the tangent vector and of the angle of total light deflection caused by mass-multipoles and spin-multipoles leads directly and in a compelling way to Chebyshev polynomials of first and second kind, respectively. This fact allows to determine the upper limits of the total light deflection, which are strictly valid in the 1PN and 1.5PN approximation. They represent a criterion to identify those multipoles which contribute significantly to the total light deflection for a given astrometric accuracy. These upper limits are used to determine the total light deflection in the gravitational field of the Sun and giant planets of the solar system. It is found that the first few mass-multipoles with l \le 10 and the first few spin-multipoles with l \le 3 are sufficient for an accuracy on the nano-arcsecond level in astrometric angular measurements.