On the efficient computation of the quadrupole light deflection

TL;DR

This paper derives simplified formulas for quadrupole light deflection that are accurate to 1 micro-arcsecond, enabling faster computations in astrometric missions like Gaia.

Contribution

It introduces the simplest possible formulas for quadrupole light deflection with high accuracy, facilitating efficient calculations in practical astrometric applications.

Findings

Derived simplified formulas with 1 micro-arcsecond accuracy

Provided upper estimates for quadrupole deflection

Enabled quick decision on when detailed computation is necessary

Abstract

Although the formulas for the light deflection due to quadrupole gravitational field of deflecting bodies are well known, the formulas are rather complicated, so that massive computations of quadrupole light deflection (e.g., in the framework of astrometric survey missions like Gaia) are time-consuming. Considering an observer situated within a few million kilometers from the Earth (clearly the most practical case), we derive the simplest possible form of the relevant formulas still having numerical accuracy of 1 micro-arcsecond. This form leads to simple upper estimates for the quadrupole light deflection in various cases allowing one to relate the magnitude of the actual quadrupole deflection with the corresponding monopole deflection due to the same body. These upper estimates can be used to decide if, for a given configuration, the actual quadrupole deflection should be computed for a given accuracy goal.