Relaxation of viscoelastic tumblers with application to 1I/2017 (‘Oumuamua) and 4179 Toutatis

<jats:title>ABSTRACT</jats:title> <jats:p>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 time-scale 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 time-scale longer by a factor of 1010 and higher than the time-scales found in the works relying on the $\, Q$-factor approach, while the latter yields a time-scale shorter by 107 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 $0.2{{\ \rm per\ cent}}$, when compared to numerical integration, for close to non-dissipative bodies and $0.003{{\ \rm per\ cent}}$ for moderately to highly energy dissipating rotators.</jats:p>