Measuring Limb Darkening of Stars in high magnification Microlensing Events by the Finite Element Method

The finite-size effect in gravitational microlensing provides a possibility to measure the limb darkening of distant stars. We use the Finite Element Method (FEM) as an inversion tool for discretization and inversion of the magnification-limb darkening integral equation. This method makes no explicit assumption about the shape of the brightness profile more than the flatness of the profile near the center of the stellar disk. From the simulation, we investigate the accuracy and stability of this method and we use regularization techniques to stabilize it. Finally, we apply this method to the single lens, high magnification transit events of OGLE-2004-BLG-254 (SAAO-I), MOA-2007-BLG-233/OGLE-2007-BLG-302 (OGLE-I, MOA-R), MOA-2010-BLG-436 (MOA-R), MOA-2011-BLG-93 (Canopus-V), MOA-2011-BLG-300/OGLE-2011-BLG-0990 (Pico-I) and MOA-2011-BLG-325/OGLE-2011-BLG-1101 (LT-I) in which light curves have been observed with a high cadence near the peak \citep{Ch}. The recovered intensity profile of stars from our analysis for five light curves are consistent with the linear limb darkening and two events with the square-root profiles. The advantage of FEM is to extract limb darkening of stars without any assumption about the limb darkening model.