Accurate analytic formula for light bending in Schwarzschild metric

¹ Department of Physics and Astronomy, University of Turku, 20014 Turku, Finland
e-mail: juri.poutanen@utu.fi
² Space Research Institute of the Russian Academy of Sciences, Profsoyuznaya Str. 84/32, 117997 Moscow, Russia
³ Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, Sweden

Received: 9 January 2020
Accepted: 24 June 2020

Abstract

We propose new analytic formulae describing light bending in the Schwarzschild metric. For an emission radii above the photon orbit at the 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°. 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.