Relaxation of Viscoelastic Tumblers with Application to 1I/2017 (`Oumuamua) and 4179 Toutatis (2005.02747v1)

Abstract: Motivated by the observation of comets and asteroids rotating in non-principal axis (NPA) states, we investigate the relaxation of a freely precessing triaxial ellipsoidal rotator towards its lowest-energy spin state. Relaxation of the precession arises from internal dissipative stresses generated by self-gravitation and inertial forces from spin. We develop a general theory to determine the viscoelastic stresses in the rotator, under any linear rheology, for both long-axis (LAM) and short-axis (SAM) modes. By the methods of continuum mechanics, we calculate the power dissipated by the stress field and the viscoelastic material strain which enables us to determine the timescale of the precession dampening. To illustrate how the theory is used, we apply our framework to a triaxial 1I/2017 (`Oumuamua) and 4179 Toutatis under the Maxwell regime. For the former, employing viscoelastic parameters typical of very cold monolithic asteroids renders a dampening timescale longer by a factor of $10^{10}$ and higher than the timescales found in the works relying on the $\,Q$-factor approach, whilst the latter yields a significantly shorter timescale as a consequence of including self-gravitation. We further reduce our triaxial theory to bodies of an oblate geometry and derive a family of relatively simple analytic approximations determining the NPA dampening times for Maxwell rotators, as well as a criterion determining whether self-gravitation is negligible in the relaxation process. Our approximations exhibit a relative error no larger than $0.2\%$, when compared to numerical integration, for close to non-dissipative bodies and $0.002\%$ for highly energy dissipating rotators.