We present a detailed study of the star formation history (SFH) of the Tucana dwarf spheroidal galaxy. High quality, deep HST/ACS data, allowed us to obtain the deepest color-magnitude diagram to date, reaching the old main sequence turnoff (F814 ~ 29) with good photometric accuracy. Our analysis, based on three different SFH codes, shows that Tucana is an old and metal-poor stellar system, which experienced a strong initial burst of star formation at a very early epoch (~ 13 Gyr ago) which lasted a maximum of 1 Gyr (sigma value). We are not able to unambiguously answer the question of whether most star formation in Tucana occurred before or after the end of the reionization era, and we analyze alternative scenarios that may explain the transformation of Tucana from a gas-rich galaxy into a dSph. Current measurements of its radial velocity do not preclude that Tucana may have crossed the inner regions of the Local Group once, and so gas stripping by ram pressure and tides due to a close interaction cannot be ruled out. On the other hand, the high star formation rate measured at early times may have injected enough energy into the interstellar medium to blow out a significant fraction of the initial gas content. Gas that is heated but not blown out would also be more easily stripped via ram pressure. We compare the SFH inferred for Tucana with that of Cetus, the other isolated LG dSph galaxy in the LCID sample. We show that the formation time of the bulk of star formation in Cetus is clearly delayed with respect to that of Tucana. This reinforces the conclusion of Monelli et al. (2010) that Cetus formed the vast majority of its stars after the end of the reionization era implying, therefore, that small dwarf galaxies are not necessarily strongly affected by reionization, in agreement with many state-of-the-art cosmological models. [abridged]
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