Measuring stellar limb darkening by gravitational microlensing

Observations of microlensing transit events can be used to measure the limb darkening of the lensed star. We discuss the advantages and drawbacks of several microlensing light-curve inversion methods. The method of choice in this work is inversion by means of decomposition of the stellar surface brightness profile. We construct an ideal basis by principal component analysis of brightness profiles obtained from model atmosphere calculations. Limb-darkening approximations using such a basis are superior to those using standard power-law limb-darkening laws. We perform a full analysis of simulated single-lens microlensing transit events including a detailed error analysis of the method. In realistic events with a low impact parameter the brightness profile of the source can be recovered with a relative accuracy of 2% from the center of the source disk to 0.9 of the disk radius. We show that in the particular case of the observed MACHO Alert 9530 event the intrinsic complex variability of the lensed red giant hinders efforts to recover its surface features.