Precession relaxation of viscoelastic oblate rotators

Abstract

Perturbations of all sorts destabilise the rotation of a small body and leave it in a non-principal spin state. In such a state, the body experiences alternating stresses generated by the inertial forces. This yields nutation relaxation, i.e., evolution of the spin towards the principal rotation about the maximal-inertia axis. Knowledge of the timescales needed to damp the nutation is crucial in studies of small bodies' dynamics. In the literature hitherto, nutation relaxation has always been described with aid of an empirical quality factor Q introduced to parameterise the energy dissipation rate.Among the drawbacks of this approach was its inability to describe the dependence of the relaxation rate upon the current nutation angle. This inability stemmed from our lack of knowledge of the quality factor's dependence on the forcing frequency. In this article, we derive our description of nutation damping directly from the rheological law obeyed by the material. This renders us the nutation damping rate as a function of the current nutation angle, as well as of the shape and the rheological parameters of the body. In contradistinction from the approach based on an empirical Q -factor, our development gives a zero damping rate in the spherical-shape limit. Our method is generic and applicable to any shape and to any linear rheological law. However, to simplify the developments, here we consider a dynamically oblate rotator with a Maxwell rheology.Key words: minor planets, asteroids: general -celestial mechanics -methods: analytical 1 MOTIVATION Prendergast (1958) and, later, Burns (1971 drew attention to the fact that alternating dissipative forces emerging in a precessing rotator must reduce its kinetic energy without affecting its angular momentum. With no external torques acting on the body, its nonprincipal rotation must relax to motion about the maximal-inertial axis -a spin state corresponding to the minimal energy with a fixed angular momentum. The then available statistics on asteroids contained only singly-periodic curves, warranting that all the observed asteroids were in the final spin state.In the turbulent antecedents of the solar system, there were aeons when collisions and disruptions were regular, so asteroids (and their smithereens) were set into wobble often. Even in the present epoch, there happen events capable of kicking asteroids out of principal spin. Such events include occasional collisions, as well as tidal interactions during asteroids' close flybys near planets. Small bodies can be ⋆ Contact e-mail: julien.frouard.ctr@navy.mil † Contact e-mail: michael.efroimsky@navy.mil driven into nonprincipal axis (NPA) spin also by the YORP effect -see, e.g., Vokrouhlický et al. (2007) and Breiter, Rożek & Vokrouhlický (2011) and references therein. Wobble of comets is impelled mainly by jetting. Gradual outgassing, too, may contribute to the effect, because a rotator goes into tumbling when it changes its principal axes through a partial loss or redistribution of the material.In an excited...

Simulations of wobble damping in viscoelastic rotators

“…Acceleration in the body frame depends on the precession frequency. We outline notation for computation of this frequency for ellipsoids of rotation following Sharma et al (2005); Frouard & Efroimsky (2018). In the body frame, our coordinate system is defined by the three principal axes of inertia with coordinates x, y, z; and with unit vectors e x , e y , e z .…”

“…If a body has three different moments of inertia (e.g., is a triaxial ellipsoid), the spin vector in the body frame expanded in Fourier series contains multiple frequency components. The frequency dependence of the energy dissipation rate affects the wobble damping timescale so triaxial bodies are more difficult to model than prolate or oblate ellipsoids of revolution (Efroimsky & Lazarian 2000;Sharma et al 2005;Breiter et al 2012;Frouard & Efroimsky 2018). The paucity of asteroids above a spin limit of 2.2 hours and distribution of body axis ratios at each spin period, requiring shear strength but little tensile strength, has lead to a granular aggregate or rubble pile interpretation of asteroid composition (Walsh 2018).…”

“…Going beyond a Q quality factor description of energy dissipation, Frouard & Efroimsky (2018) have calculated the wobbling damping, or nutation damping timescale for homogeneous oblate ellipsoids that obey a Maxwell viscoelastic rheology. They found that the energy dissipation rate and associated tumbling damping timescale is primarily a function of viscosity η with 1/η replacing the ratio of a spin frequency to the shear modulus times the quality factor, µQ, in the predicted dissipation rate.…”

“…Thus mass-spring N-body models are an attractive method for simulating rotational dynamics. Because our mass-spring model adds both spring damping forces and spring elastic forces on pairs of particles, our model approximates a Kelvin-Voigt viscoelastic rheology (Frouard et al 2016) rather than the Maxwell rheology studied by Frouard & Efroimsky (2018). Dissipation rates as a function of frequency can be compared between the two viscoelastic models using the complex compliances for each model, relating stress to strain as a function of frequency (e.g., see appendix D by Frouard & Efroimsky 2018).…”

“…We discuss numerical measurements of energy dissipation rates for homogeneous ellipsoids undergoing NPA rotation. In section 3, we modify the predictions by Frouard & Efroimsky (2018); Breiter et al (2012) for our simulated viscoelastic rheology and compare them to our numerically measured dissipation rates. In section 4, we discuss numerical experiments on inhomogeneous bodies.…”