The precise determination of the arrival direction of cosmic rays is a fundamental prerequisite for the search for sources or the study of their anisotropies on the sky. One of the most important aspects to achieve an optimal measurement of these directions is to properly take into account the measurement uncertainties in the estimation procedure. In this article we present a model for the uncertainties associated with the time measurements in the Auger surface detector array. We show that this model represents well the measurement uncertainties and therefore provides the basis for an optimal determination of the arrival direction. With this model and a description of the shower front geometry it is possible to estimate, on an event by event basis, the uncertainty associated with the determination of the arrival directions of the cosmic rays.