Neutrino Tomography of the Earth: the Earth Total Mass, Moment of Inertia and Hydrostatic Equilibrium Constraints

We investigate the implications of the constraints following from the precise knowledge of the total Earth mass, $M_\oplus$, and moment of inertia, $I_\oplus$, and from the requirement that Earth be in hydrostatic equilibrium (EHE), in the neutrino tomography studies of the Earth density structure. In order to estimate the sensitivity of a given neutrino detector to possible deviations of the inner core (IC), outer core (OC), core (IC + OC) and mantle Earth densities from those obtained using geophysical and seismological data and described by the preliminary reference Earth model (PREM), in the statistical analyses performed within the neutrino tomography studies one typically varies the density of each of these structures. These variations, however, must respect the $M_\oplus$, $I_\oplus$ and EHE constraints. Working with PREM average densities we derive the $M_\oplus$, $I_\oplus$ and EHE constraints on the possible density variations when one approximates the Earth density structure with i) three layers - mantle, outer core and inner core, and ii) four layers - upper mantle, lower mantle, outer core and inner core. We get drastically different results in the two cases.