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 Mollmann & 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.

The sun is not equally bright over the whole sphere, but rather is darkened toward the limb. This effect is well-known as limb darkening. The limb darkening coefficient is defined by the ratio of the center intensity to limb intensity. In this study, we calculate the limb darkening coefficient using the photospheric intensity estimated from solar images taken by solar and helispheric observatory (SOHO) and solar dynamics observatory (SDO). The photospheric intensity data cover almost two solar cycles from May 1996 to December 2016. The limb darkening coefficient for a size of 0.9 diameter is about 0.69 and this value is consistent with solar limb darkening. The limb darkening coefficient estimated from SOHO shows a temporal increase at solar maximum and a gradual increase since the solar minimum of 2008. The limb darkening coefficient estimated from SDO shows a constant value of about 0.65 and a decreasing trend since 2014. The increase in the coefficient reflects the effect of weakened solar activity. However, the decrease since 2014 is caused by the aging effect.