A simple analytic three-flavour description of the day-night effect in the solar neutrino flux

In the 3-flavour framework we derive a simple approximate analytic expression for the day-night difference of the flux of solar $\nu_e$ at terrestrial detectors which is valid for an arbitrary Earth density profile. Our formula has the accuracy of a few per cent and reproduces all the known analytic expressions for the Earth matter effects on the solar neutrino oscillations obtained under simplifying assumptions about the Earth's density profile (matter of constant density, 3 layers of constant densities, and adiabatic approximation). It can also be used for studying the Earth matter effects on the oscillations of supernova neutrinos. We also discuss the possibility of probing the leptonic mixing angle $\theta_{13}$ through day--night asymmetry measurements at future water Cherenkov solar neutrino detectors. We show that, depending on the measured value of the asymmetry, the current upper bound on $\theta_{13}$ may be improved, or even a lower bound on this mixing parameter may be obtained.