Fast Computational Convolution Methods For Extended Source Effects In Microlensing Lightcurves.

Extended source effects can be seen in gravitational lensing events when sources cross critical lines. Those events probe the stellar intensity profile and could be used to measure limb darkening coefficients to test stellar model predictions. A data base of accurately measured stellar profiles will be needed to correctly subtract the stellar flux in planetary transient events. The amount of data that is being and will be produced in current and future microlensing surveys, from the space and the ground, requires algorithms that can quickly compute light curves for different source-lens configurations. Based on the convolution method we describe a general formalism to compute those curves for single lenses. We develop approximations in terms of quadratures of elliptic integrals that we integrate by solving the associated first order differential equations. We construct analytic solutions for a limb darkening and, for the first time, for a parabolic profile that are accurate at the $\sim 1-3\%$ and $0.5\%$ level, respectively. These solutions can be computed orders of magnitude faster than other integration routines. They can be implemented in pipelines processing large data sets to extract stellar parameters in real time.