Determining the Neutrino Mass Hierarchy with Atmospheric Neutrinos

The possibility to determine the type of neutrino mass hierarchy by studying atmospheric neutrino oscillations with a detector capable to distinguish between neutrino and antineutrino events, such as magnetized iron calorimeters, is considered. We discuss how the ability to distinguish between the neutrino mass spectrum with normal and inverted hierarchy depends on detector characteristics like neutrino energy and direction resolutions or charge miss-identification, and on the systematical uncertainties related to the atmospheric neutrino fluxes. We show also how the neutrino mass hierarchy determination depends on the true values of $ heta_{13}$ and $ heta_{23}$, as well as on the type of the true hierarchy. We find that for $mu$-like events, an accurate reconstruction of the energy and direction of the neutrino greatly improves the sensitivity to the type of neutrino mass spectrum. For $sin^22 heta_{13} cong 0.1$ and a precision of 5% in the reconstruction of the neutrino energy and $5^circ$ in the neutrino direction, the type of neutrino mass hierarchy can be identified at the 2$sigma$~C.L. with approximately 200 events. For resolutions of 15% for the neutrino energy and $15^circ$ for the neutrino direction roughly one order of magnitude larger event numbers are required. For a detector capable to distinguish between $ u_e$ and $ar u_e$ induced events the requirements on energy and direction resolutions are, in general, less demanding than for a detector with muon charge identification.