Accurate analytic formula for light bending in Schwarzschild metric

TL;DR

This paper introduces new analytic formulas for light bending in Schwarzschild spacetime, achieving high accuracy for emission points outside the photon orbit, useful for astrophysical modeling and code testing.

Contribution

The authors present novel analytic approximations for light bending angles and lensing factors in Schwarzschild metric with high accuracy, simplifying calculations for astrophysical applications.

Findings

Accuracy better than 0.2% for bending angle

Accuracy better than 3% for lensing factor

Applicable to emission points above the black hole horizon

Abstract

We propose new analytic formulae describing light bending in Schwarzschild metric. For emission radii above the photon orbit at 1.5 Schwarzschild radius, the formulae have an accuracy of better than 0.2% for the bending angle and 3% for the lensing factor for any trajectories that turn around a compact object by less than about 160 deg. In principle, they can be applied to any emission point above the horizon of the black hole. The proposed approximation can be useful for problems involving emission from neutron stars and accretion discs around compact objects when fast accurate calculations of light bending are required. It can also be used to test the codes that compute light bending using exact expressions via elliptical integrals.