Asymptotic Expansions and Amplification of a Gravitational Lens Near a Fold Caustic

TL;DR

This paper develops advanced analytical methods to improve the accuracy of gravitational lensing models near fold caustics, enabling better interpretation of observational data and light curve fitting.

Contribution

It introduces 'post-linear' corrections to the lens equation near fold caustics, extending previous approximations to fourth order for more precise amplification calculations.

Findings

Derived analytical expressions for amplification with extended source models.

Reduced chi^2 by 30% when fitting light curves of Q2237+0305.

Highlighted importance of corrections for accurate source model comparisons.

Abstract

We propose two methods that enable us to obtain approximate solutions of the lens equation near a fold caustic with an arbitrary degree of accuracy. We obtain "post-linear" corrections to the well-known formula in the linear caustic approximation for the total amplification of two critical images of a point source. In this case, in order to obtain the nontrivial corrections we had to go beyond the approximation orders earlier used by Keeton et al. and to take into account the Taylor expansion of the lens equation near caustic up to the fourth order. Corresponding analytical expressions are derived for the amplification in cases of the Gaussian and power-law extended source models; the amplifications depend on three additional fitting parameters. Conditions of neglecting the correction terms are analysed. The modified formula for the amplification is applied to the fitting of light curves of the Q2237+0305 gravitational lens system in a vicinity of the high amplification events (HAEs). We show that the introduction of some "post-linear" corrections reduces chi^2 by 30% in the case of known HAE on the light curve of image C (1999). These corrections can be important for a precise comparison of different source models with regard for observational data. Key words: gravitational lensing: micro - quasars: individual (Q2237+0305) - gravitational lensing: strong - methods: analytical