Light propagation in 2PN approximation in the monopole and quadrupole field of a body at rest: Initial value problem

TL;DR

This paper derives the 2PN approximation for light propagation in the gravitational field of a resting body with monopole and quadrupole structures, highlighting the significance of enhanced terms for high-precision astrometry.

Contribution

It provides a closed-form solution for 2PN light deflection including monopole and quadrupole effects, with implications for ultra-precise astrometric measurements.

Findings

2PN monopole and quadrupole terms contribute less than 1 nano-arcsecond

Enhanced 2PN terms can reach up to 0.95 micro-arcseconds for Jupiter

Quadrupole effects are significant for sub-micro-arcsecond astrometry

Abstract

The light trajectory in the gravitational field of one body at rest with monopole and quadrupole structure is determined in the second post-Newtonian (2PN) approximation. The terms in the geodesic equation for light rays are separated into time-independent tensorial coefficients and four kind of time-dependent scalar functions. Accordingly, the first and second integration of geodesic equation can be reduced in each case to only four kind of scalar master integrals. These integrals can be solved in closed form by recurrence relations. The 2PN terms of monopole and quadrupole contribute less than $1$ nano-arcsecond to the total light deflection. There are, however, enhanced terms in the 2PN light deflection, both in case of monopole and quadrupole. These enhanced 2PN terms are caused by the use of an impact vector which is indispensable for modeling of real astrometric measurements. In case of grazing light rays at Jupiter and Saturn, the enhanced 2PN terms, caused by the quadrupole structure of the body, amount up to 0.95 micro-arcseconds and 0.29 micro-arcseconds, respectively. Thus, the 2PN quadrupole terms are relevant for high-precision astrometry on the sub-micro-arcsecond scale of accuracy.