Can microlensing fold caustics reveal a second stellar limb-darkening   coefficient?
  Dense high-precision photometry of microlensed stars during a fold-caustic passage can be used to reveal their intensity profiles from which the temperature of the stellar atmosphere as function of fractional radius can be derived. While the capabilities of current microlensing follow-up campaigns such as PLANET allow a precise measurement of linear limb-darkening coefficients, the residual signal of a second coefficient characterizing square-root limb darkening is ~ 25 times smaller which prevents a proper determination except for unlikely cases of very high caustic-peak-to-outside relative magnifications with no adequate event being observed so far or for source stars passing over a cusp singularity. Although the presence of limb darkening can be well established from the data, a reliable measurement of the index of an underlying power-law cannot be obtained. 
M. Dominik