Nutational damping times in solids of revolution

Abstract

We derive the characteristic nutational damping time T d for a linear, anelastic ellipsoid of revolution. Our calculation is based on the well-known idea that energy loss within an isolated spinning body causes the axis of maximum inertia of the body to align with its angular momentum vector, leading to pure spin. Energy loss occurs within an anelastic material whenever internal stresses are time variable; thus even freely rotating bodies in space, if they are wobbling, lose energy because internal stresses are associated with the accelerations caused by nutation. We find that T d = D(h)(μQ/ρa 2 Ω 3 ), where D(h) is a constant of the order of a few times 10 2 that depends on the shape of the body with h being the (aspect) ratio of the lengths of axes to one another, 11 is the elastic modulus, Q is a quality factor that describes the anelasticity of the material, p is the density of the body, a is its radius and Ω is an angular velocity. This functional form of the damping time is consistent with previous results but is more soundly based. Coefficients in past expressions vary between various authors, leading to predicted damping times that can differ by factors of the order of 10. To estimate damping times for typical asteroids, we choose values for the various parameters in this expression. We conclude that the extent of energy dissipation was over, rather than underestimated, in previous treatments. None the less, we argue that asteroids will generally be found in pure rotation, unless objects are small, spinning slowly and recently excited.