Day-night effect in solar neutrino oscillations with three flavors

We investigate the effects of a nonzero leptonic mixing angle ${\theta }_{13}$ on the solar neutrino day-night asymmetry. Using a constant matter density profile for the Earth and well-motivated approximations, we derive analytical expressions for the ${\nu }_e$ survival probabilities for solar neutrinos arriving directly at the detector and for solar neutrinos which have passed through the Earth. Furthermore, we numerically study the effects of a nonzero ${\theta }_{13}$ on the day-night asymmetry at detectors and find that they are small. Finally, we show that if the uncertainties in the parameters ${\theta }_{12}$ and $\Delta m^2$ as well as the uncertainty in the day-night asymmetry itself were much smaller than they are today, this effect could, in principle, be used to determine ${\theta }_{13}.$