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Nutational Damping Times in Solids of Revolution

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Nutational Damping Times in Solids of Revolution Mon. Not. R. Astron. Soc. 359, 79–92 (2005) doi:10.1111/j.1365-2966.2005.08864.x Nutational damping times in solids of revolution , Ishan Sharma,1 Joseph A. Burns1 2 and C.-Y. Hui1 1Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA 2Department of Astronomy, Cornell University, Ithaca, NY 14853, USA Accepted 2005 January 25. Received 2004 August 25; in original form 2003 October 17 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 mo- mentum 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 wob- bling, lose energy because internal stresses are associated with the accelerations caused by = µQ nutation. We find that Td D(h)( ρa23 ), where D(h)isaconstant of the order of a few times 102 that depends on the shape of the body with h being the (aspect) ratio of the lengths of axes to one another, µ is the elastic modulus, Q is a quality factor that describes the anelasticity of the material, ρ is the density of the body, a is its radius and is an angular velocity. This functional form of the damping time is consistent with previous results but is more soundly based. Coef- ficients 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. Keywords: techniques: radar astronomy – comets: general – minor planets, asteroids. angular momentum vectors are aligned with a body axis. The char- 1 INTRODUCTION acteristic interval over which this alignment occurs is the nutational In celestial rotation, regularity rules: most spinning objects are found damping time T d;itwill depend obviously on the energy loss rate in nearly pure spin (i.e. they rotate at a constant rate around a single which has to be estimated. axis, which is the axis of maximum moment of inertia). Such a Time-varying loads (stresses) lead to energy loss in isolated, rotation state is surprising because, over long times, bodies are being anelastic solid bodies; the former are a natural companion to the continually jostled about, either through collisions or during close nutational motion of the body, because this requires elements of the flybys. The pure spins that are so prevalent must therefore result body to undergo accelerations. Thus, in order to estimate the energy from an ongoing process. loss, we must first calculate the induced variable stresses, and then Energy and angular momentum are both conserved for an isolated we must model the energy loss associated with the time-varying rigid body because no external forces act nor are internal motions stresses. In each of these steps we must assume a model of material permitted. In general, rotation (described for biaxial bodies in Sec- behaviour. tion 5), such a body ‘wobbles’ or nutates (i.e. executes free Eulerian In the past, the nutational damping of asteroids as caused motion); this motion, which is specified once the angular momentum by internal energy dissipation was analysed approximately by J, kinetic energy Ek and moments of inertia of the body are spec- Prendergast (1958), Burns & Safronov (1973), McAdoo & Burns ified, can be quite complicated. The motions are usually described (1974), Efroimsky & Lazarian (2000) and Efroimsky (2000, 2001). in combinations of elliptic functions (Landau & Lifshitz 1960). Many other similar dynamical studies concern planetary wobble Internal dissipation, which occurs once internal motions are pos- decay (e.g. Peale 1973; Yoder & Ward 1979), including damping sible, however, leads to energy loss. This moves the body closer to of the Chandler wobble of Earth (e.g. Munk & MacDonald 1960; a state of pure rotation about its axis of maximum moment of iner- Lambeck 1980, 1987), interstellar dust grain alignment (e.g. Pur- tia as this state has the least energy for a fixed angular momentum cell 1979; Lazarian & Efroimsky 1999) and comets (e.g. Molina, (Landau & Lifshitz 1960). In this state, the angular velocity and the Moreno & Martinez-Lopez 2003). In most of these cases, the familiar Q model (Knopoff 1964; Burns 1977; Lambeck 1980) was assumed to estimate energy E-mail: [email protected] loss. Internal stresses were computed by either invoking Love C 2005 RAS 80 I. Sharma, J. A. Burns and C.-Y. Hui numbers1 (Munk & MacDonald 1960; Burns & Safronov 1973; accounts for energy dissipation while a suitable choice of the axes Peale 1973; Yoder & Ward 1979; Lambeck 1980) or by directly at- of the ellipsoid would set the correct dynamics and would approx- tempting to solve the full three-dimensional (3D) elasticity problem imate most physical bodies reasonably well. As a first step toward for the body (Prendergast 1958; Purcell 1979; Denisov & Novikov the general solution for an anelastic triaxial ellipsoid, we have de- 1987; Lazarian & Efroimsky 1999; Efroimsky & Lazarian 2000; rived the full 3D solution for any elastic solid of revolution (i.e. for Efroimsky 2000; Washabaugh & Scheeres 2002; Molina et al. 2003). a prolate or an oblate spheroid). The energy dissipated is calculated When computing stresses, the body was usually taken to be elastic or using a Q model (described below) as the loss mechanism. at best viscoelastic (McAdoo & Burns 1974; Lambeck 1980, 1987); The first step given in the outline (Section 3) does not pose seri- with the latter model, only nearly spherical bodies were considered. ous difficulties because the acceleration field for a general triaxial Lastly, we should mention that Denisov & Novikov (1987) managed ellipsoid undergoing free Eulerian motion is available (Landau & to solve for the stresses inside a tumbling, symmetric, linear-elastic Lifshitz 1960). However, step 2 requires that the full 3D elasticity ellipsoid. However, we came to know of this reference only after problem be solved for a given body. This is a formidable task due to: we had completed the calculations for the stresses. (a) the (likely) anelasticity of the material, (b) the absence of body All the above authors derive the same functional form for the forces derivable from a potential and (c) the likely complex shape damping time (as could have been anticipated from dimensional of the body. analysis alone); see Section 9. However, the numerical coefficients When deriving his eponymous numbers, Love (1946) assumed in these published results are claimed to differ by factors of the or- that the body forces are derivable from a potential and further that der of 10. This disagreement could be attributed to: (a) the different the body is linear-elastic spherical (or near-spherical) in shape. Nei- body shapes that were used when stresses were estimated or, more ther of these assumptions is necessarily appropriate for our situation. fundamentally, (b) the alternate approaches that were employed to First, we will demonstrate below the non-potential nature of body estimate stresses or, more prosaically, (c) errors in analysis or in- forces in our problem. Secondly, we often wish to consider aspher- correct assumptions. To complicate matters further for astronomers ical objects and any restrictions on the shape of the body limits the or geophysicists who might wish to use these results when estimat- usefulness of the result. These two failings have encouraged us to ing actual damping times, the various authors often chose different not employ Love numbers when estimating internal stresses. values for the physical constants that appeared in their expressions. An anelastic triaxial ellipsoid (with suitable choices for the The calculation to be presented below should be useful because, anelasticity model and the axes of the body) is a very versatile model as an exact elasticity solution, it provides the best estimate of the that can approximate many real objects fairly well. Furthermore, this internal stresses employed in this and earlier works, and because is probably the most general model permitting an analytical solu- the character of the spheroid can be varied (from flattened discs to tion. Unfortunately, this general model remains algebraically cum- narrow cigars) in order to explore how shape influences damping. bersome and hence we have decided to simplify it. To start with, in order to ease the algebra while still preserving our ability to vary the 2 OUTLINE OF THE CALCULATION shape of the body significantly, we restrict ourselves to ellipsoids of revolution. The construction of the elasticity solution for the gen- Before proceeding further, we outline the broad steps involved in eral triaxial case is fairly straightforward but involves much more our calculation. Questions as to why the problem is tackled in this algebra.2 manner and/or why a possible effect is neglected will be addressed Next we take the material to be homogeneous, linear and elas- in the following sections. tic. Energy dissipation can be broadly classified into three cat- The general philosophy is that nutational motion of the body egories: (a) frictional (rate-independent) dissipation; (b) viscous introduces time-varying internal stresses which cause energy loss. (rate-dependent) dissipation (as in fluids); and (c) a combination of This dissipation then tends to align the angular velocity vector, the rate-independent and dependent mechanisms, an example of which angular momentum vector and the axis of maximum moment of would be furnished by granular aggregates (which most asteroids inertia.
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