The Period of the Chandler Wobble

Abstract

Realistic models of the Earth are known to possess a solid anelastic inner core, mantle and crust, and a fluid core and oceans. How might we go about calculating the theoretical free period of the Chandler wobble of such an Earth model? Let xi be a set of Cartesian axes with an origin at the center of mass, and let ωi be the instantaneous angular velocity of rotation of these axes with respect to inertial space. The net angular momentum is then Cijωj + hi, where Cij is the inertia tensor, and hi is the relative angular momentum. Let us affix the axes xi in the mantle and crust by stipulating that the relative angular momentum is that of the core and oceans alone, i.e., hi (mantle and crust) = 0; hi = hi (core and oceans). For an infinitesimal free oscillation of angular frequency σ, we can write ωi = Ω(δi3 + mi eiσt), Cij = A(δilδjl + δi2δj2) + Cδi3δj3 + cij eiσt, and hi = hi eiσt, where Ω is the mean rate of rotation and A and C are the mean equatorial and polar moments of inertia.