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Rotation of Cometary Nuclei 281 Samarasinha et al.: Rotation of Cometary Nuclei 281 Rotation of Cometary Nuclei Nalin H. Samarasinha National Optical Astronomy Observatory Béatrice E. A. Mueller National Optical Astronomy Observatory Michael J. S. Belton Belton Space Exploration Initiatives, LLC Laurent Jorda Laboratoire d’Astrophysique de Marseille The current understanding of cometary rotation is reviewed from both theoretical and ob- servational perspectives. Rigid-body dynamics for principal axis and non-principal-axis rota- tors are described in terms of an observer’s point of view. Mechanisms for spin-state changes, corresponding timescales, and spin evolution due to outgassing torques are discussed. Differ- ent observational techniques and their pros and cons are presented together with the current status of cometary spin parameters for a variety of comets. The importance of rotation as an effective probe of the interior of the nucleus is highlighted. Finally, suggestions for future re- search aimed at presently unresolved problems are made. 1. INTRODUCTION space missions to comets; for example, it allows the mis- sion planners to assess the orientation of the nucleus dur- Since the publication of the first Comets volume (Wil- ing a flyby. kening, 1982), our understanding of the rotation of comet- In the next section, we will discuss basic dynamical as- ary nuclei has evolved significantly, first due to research pects of cometary rotation, while section 3 deals with ob- kindled by the apparently contradictory observations of servational techniques and the current status of cometary Comet 1P/Halley, and more recently due to numerical mod- spin parameters. Section 4 addresses interpretations of ob- eling of cometary spin complemented by a slowly but servations and some of the current challenges. The final sec- steadily increasing database on cometary spin parameters. tion discusses suggestions for future research. Subsequent review papers by Belton (1991), Jewitt (1999), Jorda and Licandro (2003), and Jorda and Gutiérrez (2002) 2. ROTATIONAL DYNAMICS highlight many of these advances. In this chapter, we dis- cuss our current understanding of cometary rotation and the 2.1. Rigid-Body Dynamics challenges we face in the near future. Knowledge of the correct rotational state of a cometary The prediction from the icy conglomerate model of the nucleus is essential for the accurate interpretation of obser- nucleus (Whipple, 1950) and the subsequent spacecraft vations of the coma and for the determination of nuclear images of Comets 1P/Halley (e.g., Keller et al., 1986; Sag- activity and its distribution on the surface. The spin state, deev et al., 1986) and 19P/Borrelly (e.g., Soderblom et al., orbital motion, and activity of a comet are linked to each 2002) are consistent with a single solid body representing other. Accurate knowledge of each of these aspects is there- the nucleus. Therefore, in order to explain the rotational fore required in order to properly understand the others, as dynamics of the nucleus, to the first approximation, we con- well as to determine how they will evolve. As we will elab- sider it as a rigid body. Chapters in this book on splitting orate later, ensemble properties of spin parameters — even events (Boehnhardt, 2004), nuclear density estimates (Weiss- the spin rates alone — can be effectively used to understand man et al., 2004), and formation scenarios (Weidenschilling, the gross internal structure of cometary nuclei. This has di- 2004) are suggestive of a weak subsurface structure made up rect implications for understanding the formation of com- of individual cometesimals, and the question of any effects ets in the solar nebula as well as for devising effective mi- of non-rigid-body behavior are still on the table. tigation strategies for threats posed by cometary nuclei We use the terms “rotational state” and “spin state” inter- among the near-Earth-object (NEO) population. In addition, changeably, and they are meant to represent the entire rota- a priori knowledge of the spin state is necessary for effec- tional state. Similarly, “rotational parameters” and “spin tive planning and maximization of the science return from parameters” are used interchangeably. However, the term 281 282 Comets II Fig. 1. (a) Component rotations for short-axis modes (SAM) and long-axis modes (LAM). Component rotations are depicted in relation to the long axis. (b) Characteristics of different spin states as the energy of rotation for a given rotational angular momen- tum increases from the least-energetic state (rotation around the short axis, M2 = I ) to the most energetic state (rotation around 2E s the long axis, M2 = I ). Behavior of the component periods P and 2E l φ Pψ are indicated for spin states near principal-axis states. For SAM states, the amplitude of the oscillatory motion of the long axis, Aψ (as well as the nodding amplitude of the long axis, Aθ), in- creases as the spin state becomes more energetic. For LAM states, the mean value of the angle θ (as well as the amplitude of the nodding motion of the long axis, Aθ) decreases as the kinetic energy of the rotational state increases. “spin vector” specifically refers to the instantaneous angular ertia around the three principal axes satisfy the condition ≥ ≥ velocity vector. Is Ii Il. Spin states with kinetic energies in between are The most stable rotational state of a rigid body is de- characterized by two independent periods. These spin states fined by the least-energetic state for a given rotational an- are known as non-principal-axis (NPA) states (also called gular momentum (this does not mean that the rotational complex rotational states, or tumbling motion, with the latter angular momentum is fixed, but for each given rotational term primarily used by the asteroid community). It should angular momentum of the rigid body, there is a correspond- be stressed that, unlike for PA states, the spin vector and ing lowest-energy spin state). This lowest-energy spin state M are not parallel to each other for NPA states. A nucleus is represented by a simple rotation around the short princi- having a spin state other than the PA rotation around the pal axis of the nucleus occurring at a constant angular ve- short axis is in an excited rotational state. locity. (Note that in this chapter, unless specified otherwise, Figure 1 shows the basic characteristics of different rigid long, intermediate, and short axes, denoted by l, i, and s body rotational states. The short-axis-mode (SAM) and respectively, refer to the mutually orthogonal principal axes long-axis-mode (LAM) states (Julian, 1987) differ from of the nucleus as determined by the inertia ellipsoid. De- each other depending on whether the short or the long axis pending on the shape and the internal density distribution, “encircles” the rotational angular momentum vector, M, these axes may have offsets from the physical axes defined which is fixed in the inertial frame. The SAM states are less by an ellipsoidal fit to the physical shape.) For this spin energetic than the LAM states. The component periods can state, the rotational angular momentum, M, and the rota- be defined in terms of Euler angles θ, φ, and ψ (Fig. 2). M2 tional kinetic energy, E, are given by 2E = Is, where Is is Most cometary nuclei are elongated (i.e., closer to prolates the moment of inertia around the short axis. Since the nu- than oblates), as implicated by large lightcurve amplitudes cleus is only rotating around the short axis, this is called a and spacecraft images. From an observational point of view, principal-axis (PA) spin state. Other PA spin states include component rotations defined in terms of the long axis are the dynamically unstable rotation around the intermediate easier to discern than those defined with respect to the short M2 axis (e.g., Landau and Lifshitz, 1976) corresponding to 2E = axis. Therefore, following Belton (1991), Belton et al. (1991), Ii and that around the long axis, which characterizes the and Samarasinha and A’Hearn (1991), we adopt the system M2 θ most energetic rotational state at 2E = Il. Moments of in- of Euler angles where defines the angle between M and Samarasinha et al.: Rotation of Cometary Nuclei 283 the long axis, ψ defines the rotation around the long axis tional angular momentum vector, which requires two pa- itself, and φ defines the precession of the long axis around rameters; and the reference values for the Euler angles φ M as depicted in Fig. 2. [Many textbooks (e.g., Landau and and ψ at a given time. Lifshitz, 1976) use a different system of Euler angles appro- The rate of change of M of the rigid body in an inertial priate for flattened objects such as Earth, as opposed to frame (with the origin at the center of mass) is given by elongated objects; see Jorda and Licandro (2003) for a comparison between the two systems.] In Fig. 1, P refers θ dM to the period of the nodding/nutation motion of the long = N (1) dt axis, Pψ to the period of oscillation (in the case of SAM) inertial or rotation (in the case of LAM) of the long axis around itself, and Pφ to the mean period of precession of the long where N is the external torque on the body. Since the rate axis around M (for an asymmetric rotator, the rate of change of change of M in the inertial and in the body frames are of φ is periodic with Pψ/2 and therefore we consider the related by time-averaged mean value for Pφ). The period Pθ is equal to P for SAM or exactly P /2 for LAM, leaving only two ψ ψ dM dM independent periods Pφ and Pψ. In general, Pφ ≠ Pψ and there- = + Ω × M (2) fore for a random NPA spin state, even approximately the dt inertial dt body same orientations of the rotator in the inertial frame are rare.
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