Approximation of Light-Ray Deflection Angle and Gravitational Lenses in the Schwarzschild Metric. I. Derivation and Quasar Lens

Astronomy and Astrophysics – Astronomy

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Stars: Binaries: Close, Gravitation, Cosmology: Gravitational Lensing, Relativity

Scientific paper

We calculate a second-order approximation for the light-ray deflection in a Schwarzschild metric. The approximation is based on Darwin's exact solution for the deflection angle, which is written in terms of elliptic integrals. Even if the derived equation is almost as simple as the usual Einstein deflection angle, it follows the Darwin solution more faithfully. This enables relatively simple calculations of gravitational lensing in axially symmetric conditions with a small value of the impact parameter b. We demonstrate the effects of second-order accuracy by modeling the lensed quasar QSO B0957+561A, B. Our simple toy model leads to a 4% difference in mass estimate for the deflector, when comparing the Einstein solution with the approximation. The naive point mass model is certainly inadequate when modeling a realistic quasar lens, but it demonstrates clearly the effects of the second-order accuracy and could, nevertheless, be used for some supermassive black hole models.

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