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 brightness profiles, from which the temperature of the stellar atmosphere as a function of fractional radius can be derived. While the capabilities of current microlensing follow-up campaigns such as PLANET allowed for several precise measurements of linear limb-darkening coefficients, all attempts to reveal a second limb-darkening coefficient from such events have failed. It is shown that the residual signal of a second coefficient characterizing square-root limb darkening is similar to25 times smaller, which prevents a proper determination except for unlikely cases of very high caustic-peak-to-outside magnification ratios 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.