A Study of the Day - Night Effect for the \SK Detector: III. The Case of   Transitions into Sterile Neutrino
  Using the results of a high precision calculation of the solar neutrino survival probability for Earth crossing neutrinos in the case of MSW $\nu_e --> \nu_s$ transition solution of the solar neutrino problem, we derive predictions for the one-year averaged day-night (D-N) asymmetries in the deformed recoil-$e^-$ spectrum and in the energy-integrated event rate due to the solar neutrinos, to be measured with the Super - Kamiokande detector. The asymmetries are calculated for three event samples, produced by solar $\nu_e$ crossing the Earth mantle only, the core, and the mantle only + the core (the full night sample). The effects of the uncertainties in the value of the bulk matter density and in the chemical composition of the Earth core on the predictions for the D-N asymmetries are investigated. The dependence of the D - N effect related observables on the threshold recoil - e- kinetic energy, TeTh, is studied. It is shown, in particular, that for $\sin^2 2\theta_V \leq 0.030$ the one year average D-N - asymmetry in the sample of events due to the core-crossing neutrinos is larger than the asymmetry in the full night sample typically by a factor $\sim (2 - 4)$. We find, however, that at small mixing angles $\sin^2 2 \theta_V \ltap 0.014$, the D-N asymmetry in the case of solar $\nu_e --> \nu_s$ transitions is considerably smaller than if the transitions were into an active neutrino, $\nu_e --> \nu_{\mu(\tau)}$. In particular, a precision better than 1% in the measurement of any of the three one year averaged D-N asymmetries considered by us would be required to test the small mixing angle nonadiabatic $\nu_e --> \nu_s$ solution at $\sin^2 2\theta_V ~\ltap ~0.01$. 
M. Maris (OAT - Trieste - Italy), S. T. Petcov (SISSA - Trieste -   Italy)