We present a mathematical model that predicts the variation of illuminance during a solar eclipse, considering continuous effects of limb darkening. We assume that (1) the Sun and the Moon constitute perfect spheres, (2) the Moon crosses the Sun with a constant apparent velocity, and (3) sunspots, prominences, and coronae can be neglected. We compare predictions of this model with actual measurements made by M$ddot{o}$llmann & Vollmer (2006) during a total solar eclipse in Turkey, and with predictions of existing models. The new model is shown to describe the actual phenomenon more accurately than existing models.