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

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

16 pages, 3 figures

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