Motion of the Mantle in the Translational Modes of the Earth and Mercury

Slichter modes refer to the translational motion of the inner core with respect to the outer core and the mantle. The polar Slichter mode is the motion of the inner core along the axis of rotation. Busse presented an analysis of the polar mode which yielded an expression for its period. Busse's analysis included the assumption that the mantle was stationary. This approximation is valid for planets with small inner cores, such as the Earth whose inner core is about 1/60 of the total planet mass. On the other hand, many believe that Mercury's core may be enormous. If so, the motion of the mantle should be expected to produce a significant effect. We present a formal framework for including the motion of the mantle in the analysis of the translational motion of the inner core. We analyze the effect of the motion of the mantle on the Slichter modes for a non-rotating planet with an inner core of arbitrary size. We omit the effects of viscosity in the outer core, magnetic effects, and solid tides. Our approach is perturbative and is based on a linearization of Euler's equations for the motion of the fluid and Newton's second law for the motion of the inner core. We find an analytical expression for the period of the Slichter mode. Our result agrees with Busse's in the limiting case of small inner core. We present the unexpected result that even for Mercury the motion of the mantle does not significantly change the period of oscillation.