Evolution of Comet Nucleus Rotation

The secular evolution of comet nucleus rotation states subject to outgassing torques is studied. The dynamical model assumes that the nucleus inertia ellipsoid is axially symmetric. The outgassing torques acting on the surface are modeled using standard cometary activity formulae. The general rotational equations of motion are derived and separately averaged over the fast rotational dynamics terms and the comet orbit. Special cases where the averaging assumptions cannot be applied are evaluated separately. The modification of the comet orbit due to comet outgassing is neglected. Resulting from this analysis is a system of secular differential equations that describes the dynamics of the comet nucleus angular momentum and rotation state. We find that the qualitative secular evolution of the rotation state is controlled by a single parameter that combines parameters related to the comet orbit and parameters related to the nucleus surface geometry and activity. From this solution, we find qualitatively different evolutionary paths for comet nuclei whose entire surface is active, as compared to nuclei with only a single active region. For surface activity models between these extremes, we show that certain evolutionary paths are more likely than others. Additionally, our solution indicates that a comet nucleus' rotational angular momentum will tend to increase over time, potentially contributing to the observed phenomenon of comet nucleus splitting.     

The dynamics of globally active triaxial comets, with applications to asteroid rotation

It is demonstrated how globally distributed outgassing activity on a triaxial comet nucleus bridges the gap between the intuitive Sekanina model, used for comet orbit solutions, and the physics of the problem. In this activity and shape limit, it is shown how a recoil force component, which originates from a day-side restricted sublimation process, is necessary to describe the comet's rotational evolution. Modifications of the non-gravitational force cosines are suggested, with a fundamentally different interpretation than before. Applications to asteroid rotation yield that the ability of specular reflection, of solar photons on an asteroid's surface, to change the asteroid's rotation period and equatorial obliquity, is not dependent on the overall shape of the asteroid.     

Rotational dynamics of subsolar sublimating triaxial comets

A model with subsolar water sublimation on a triaxial, ellipsoidal comet nucleus is presented for the calculation of reactive torques. The resulting differential equations describing the comet's rotation are then Hamiltonian, and gravity-gradients are trivial to include. While effects derived from a weak perturbing function are neither able to change the rotational excitation nor the spin magnitude of the nucleus, it is shown how the spin orientation of comets can change significantly over an orbital run. However, of the four comets studied, 1P, 19P, 46P and the Rosetta target 67P, 19P and 46P were the only objects clearly exhibiting this feature, thereby confirming a technique used to derive the consequences of a more elaborate model of sublimation induced torques. In particular, the rotational parameters of 67P were seen to be very stable, indicating that a highly kinematical model of its rotation for the mapping of the comet's gravitational field during the Rosetta mission can be used. The model's hierarchy with 1P/Halley as the object with highest excitation probability, is consistent with observations.     

Comet 67P/Churyumov-Gerasimenko: Modeling of orientation and structure

The comet 67P/Churyumov-Gerasimenko is the current target of the mission Rosetta, initially planned to investigate comet 46P/Wirtanen. These two comets have similar orbits, except the distance to the Sun at perihelion, but different orbital histories and different masses. Thus, structures of the nuclei can be significantly different. The evolution of comet Wirtanen was simulated by several authors, while comet Churyumov-Gerasimenko became an object of high interest only recently and is not well investigated. In the present work we simulate the evolution of the nucleus, down to tens of meters below the surface, using an extended version of the model previously applied for comet Wirtanen [Kossacki et al., 1999. Comet 46P/Wirtanen: evolution of the subsurface layer. Icarus 142, 202-218.]. The model includes strengthening of the nucleus due to sintering of the ice grains. Simulations are performed for different latitudes, accounting for the evolution of the orbit and for changes of the nucleus orientation, as well as diurnal and seasonal changes of insolation. The calculated loss of water vapor from the comet is integrated over the nucleus surface and is compared with the observational data. We have found, that the sublimation through the dust mantle can be large enough to reproduce the profile of the total water production as a function of time from perihelion. The required dependence of thickness of the dust layer on latitude qualitatively matches present distribution of the absorbed solar flux. The non-gravitational acceleration in the comet motion together with the simulated sublimation flux are used in order to estimate the mass and the bulk density of the nucleus.     

Rotation rate and velocity dispersion of planetary ring particles with size distribution: I. Formulation and analytic calculation

We derive an equation for the evolution of rotational energy of Keplerian particles in a dilute disk due to mutual collisions. Three-dimensional Keplerian motion of particles is taken into account precisely, on the basis of Hill's approximation. The Rayleigh distribution of particles' orbital eccentricities and inclinations, and the Gaussian distribution of their rotation rates are also taken into account. Performing appropriate variable transformation, we show that the equation can be expressed with two terms. The first term, which we call collisional stirring term, represents energy exchange between rotation and random motion via collisions. The second term, which we call rotational friction term, tends to equalize the mean rotational energy of particles with different sizes. The equation can describe the evolution of rotational energy of Keplerian particles with an arbitrary size distribution. We analytically evaluate the rates of stirring and friction for the random kinetic energy and rotational energy due to inelastic collisions, for non-gravitating particles in a dilute disk. Using these results, we discuss equilibrium states in a disk of spinning, non-gravitating Keplerian particles.     

短周期彗星表面水冰升华分布研究

彗星是太阳系遗留的原始星子,研究彗星彗核的演化对理解太阳系其他天体的形成和演化历史具有重要意义.在太阳的辐射作用下,彗星携带的挥发性成分会发生升华,并带动尘埃运动,造成彗核物质的损失.因此,彗核的升华活动对其表面形貌甚至整体形状演化都会产生影响.从IAU (International Astronomical Union) MPC (Minor Planet Center)获取轨道数据,并考虑了彗核的自转以及进动,利用MONET (Mass lossdriven shape evolution model)形状演化模型对短周期彗星做数值模拟,计算得到了短周期彗星1P/Halley、9P/Tempel 1、 19P/Borrelly、 67P/C-G (Churyumov-Gerasimenko)、 81P/Wild 2和103P/Hartley 2在一个轨道周期内的太阳辐射能量以及表面侵蚀深度的分布,结合其动力学参数讨论了自转、进动和公转等特性对其表面水冰升华分布的影响以及造成南北侵蚀差异的可能性.     

Modelling of Shape Changes of the Nuclei of Comets C/1995 O1 Hale–Bopp and 46P/Wirtanen Caused by Water Ice Sublimation

The aim of this modelling work is to assess shape changes of cometary nuclei caused by sublimation of ices. The simplest possible model is assumed with the nucleus being initially spherical and its thermal conductivity being neglected. We have calculated the time-dependent sublimation flux versus cometographic latitude. If the rotation axis of the comet is inclined to the orbital plane, then sublimation leads to non-symmetrical changes of the nucleus shape. Calculations were performed for the nuclei of comets Hale–Bopp and Wirtanen.     

Energy balance and the gas flux from the surface of comet 46P/Wirtanen

Understanding the power balance at the surface of the nucleus is essential to study the chemical and physical evolution of a comet. Therefore, we present a detailed energy budget analysis for the surface of a model comet in the orbit of 46P/Wirtanen, target comet of the European space craft mission Rosetta, for a variety of parameters and assumptions. We will show that for a fast spinning Jupiter-family comet such as 46P/Wirtanen with a rotation period of about 6 h, a fast rotator approximation underestimates the effective energy input. This yields lower gas fluxes from the surface. For an 100% active, non-dust covered surface we obtain a water gas flux on the order of about 1.5×10²⁸ molecules s⁻¹ at perihelion, assuming a radius of 600 m. The calculated gas flux of water is within the order of measured values for comet 46P/Wirtanen. But our calculated values are maximum gas fluxes at noon—not averaged over one cometary day or taking the lesser insolation at the polar areas into account. Therefore, we conclude that either the radius of comet 46P/Wirtanen may be much larger than the accepted value of 600 m. A radius in the order of 2 km seems more likely to explain the measurements. Or, an other possibility could be that water-ice particles are blown off from the surface like dust particles. This may also increase the effective surface area of sublimation.     

The rotation period of comet 29P/Schwassmann-Wachmann 1 determined from the dust structures (Jets) in the coma

The results of the photometric observations of comet 29P/Schwassmann-Wachmann 1 are analyzed. The comet demonstrates substantial activity at heliocentric distances larger than 5 AU, i.e., outside the water ice sublimation zone. The CCD images of the comet were obtained in wideband R filters at the 6-meter telescope of the Special Astrophysical Observatory of the Russian Academy of Sciences (SAO RAS) and at the 2-meter Zeiss-2000 telescope of the Peak Terskol Observatory. The processing of the images with special digital filters allowed the active structures (jets) to be distinguished in the dust coma of the comet. With the cross-correlation method, the rotation period of the cometary nucleus was determined as 12.1 ± 1.2 and 11.7 ± 1.5 days for the observations taken in December 2008, and February 2009, respectively. The probable causes of the difference in the estimates of the rotation period of the cometary nucleus obtained by different authors are discussed.     

Averaged rotational dynamics of an asteroid in tumbling rotation under the YORP torque

The secular effect of YORP torque on the rotational dynamics of an asteroid in non-principal axis rotation is studied. The general rotational equations of motion are derived and approximated with an illumination function expanded up to second order. The resulting equations of motion can be averaged over the fast rotation angles to yield secular equations for the angular momentum, dynamic inertia and obliquity. We study the properties of these secular equations and compare results to previous research. Finally, an application to several real asteroid shapes is made, in particular we study the predicted rotational dynamics of the asteroid Toutatis, which is known to be in a non-principal axis state.     

The morphology and surface processes of Comet 19/P Borrelly

The flyby of the nucleus of the Comet 19P/Borrelly by the Deep Space 1 spacecraft produced the best views to date of the surface of these interesting objects. It transformed Borrelly from an astronomical object shrouded in coma of gas and dust into a geological object with complex surface processes and a rich history of erosion and landform evolution. Based on analysis of the highest resolution images, stereo images, photometry, and albedo we have mapped four major morphological units and four terrain features. The morphological units are named dark spots, mottled terrain, mesas, and smooth terrain. The features are named ridges, troughs, pits, and hills. In strong contrast to asteroids, unambiguous impact craters were not observed on Borrelly's surface. Because of the relatively short period of this comet, surface erosion by volatile sublimation is, in geologic terms, a very active process. The formation and the morphologies of units and features appear to be driven by differential rates of sublimation erosion. Erosional rates across the comet are probably controlled by solar energy input and the location of the subsolar point during perihelion. Differences in energy input may produce different varieties of sublimation erosional landforms. The terrains on Borrelly suggest that solar energy input could map directly into erosional processes and landforms.     

Some properties of a two-body system under the influence of the Galactic tidal field

In this paper the effect of the Galactic tidal field on a Sun–comet pair will be considered when the comet is situated in the Oort cloud and planetary perturbations can be neglected. First, two averaged models were created, one of which can be solved analytically in terms of Jacobi elliptic functions. In the latter system, switching between libration and circulation of the argument of perihelion is prohibited. The non-averaged equations of motion are integrated numerically in order to determine the regions of the ( e ,  i ) phase space in which chaotic orbits can be found, and an effort is made to explain why the chaotic orbits manifest in these regions only. It is evident that for moderate values of semimajor axis, a ∼50 000 au , chaotic orbits are found for ( e <0.15 , 40°≤ i ≤140°) as determined by integrating the evolution of the comet over a period of 10⁴ orbits. These regions of chaos increase in size with increasing semimajor axis. The typical e-folding times for these orbits range from around 600 Myr to 1 Gyr and thus are of little practical interest, as the time-scales for chaos arising from passing stars are much shorter. As a result of Galactic rotation, the chaotic regions in ( e ,  i ) phase space are not symmetric for prograde and retrograde orbits.     

Coma expansion and photometry of comet Bowell (1980b)

Optical and infrared observations of comet Bowell are presented. The optical observations indicate that the solid grain coma is expanding at only 0.9 ± 0.2 m sec^?1. This is two orders of magnitude slower than the local gas sound speed and may suggest that gas drag is not responsible for stripping the grains from the nucleus. The hypothesis of “electrostatic snap-off” is tentatively advanced to account for the ejection of the grains. Alternatively, the grains may have an unusual size distribution. The extrapolated motion of the grains suggests that the bulk of the coma was formed when the comet was at a heliocentric distance $\text{R ? 10}\text{AU}$. Any water ice in the nucleus would be too cold to give rise to the observed grain coma by equilibrium sublimation at this R. Further evidence against the production of the grain coma by equilibrium sublimation of the nucleus is provided by broadband (J) photometric observations. Almost all of the observed photometric variations of comet Bowell can be ascribed to geometric effects. Simple models indicate that the total grain cross section has been nearly constant since the time of the earliest observations. The present observations, which suggest that water ice sublimation does not control either the optical morphology or the near infrared photometric behavior of comet Bowell, are contrasted with reported high OH production rates. It is concluded that the grain coma may be largely a relic of activity occurring on the nucleus at $\text{R ? 10}\text{AU}$ while the OH may indicate sublimation from the nucleus near perihelion and from coma grains near $\text{R ? 4.6}\text{AU}$.     

Understanding measured water rotational temperatures and column densities in the very innermost coma of Comet 73P/Schwassmann–Wachmann 3 B

Direct sublimation of a comet nucleus surface is usually considered to be the main source of gas in the coma of a comet. However, evidence from a number of comets including the recent spectacular images of Comet 103P/Hartley 2 by the EPOXI mission indicates that the nucleus alone may not be responsible for all, or possibly at times even most, of the total amount of gas seen in the coma. Indeed, the sublimation of icy grains, which have been injected into the coma, appears to constitute an important source. We use the fully-kinetic Direct Simulation Monte Carlo model of Tenishev et al. (Tenishev, V.M., Combi, M.R., Davidsson, B. [2008]. Astrophys. J., 685, 659?677; Tenishev, V.M., Combi, M.R., Rubin, M. [2011]. Astrophys. J., 732) to reproduce the measurements of column density and rotational temperature of water in Comet 73P-B/Schwassmann–Wachmann 3 obtained with a very high spatial resolution of ～30 km using IRCS/Subaru in May 2006 (Bonev, B.P., Mumma, M.J., Kawakita, H., Kobayashi, H., Villanueva, G.L. [2008]. Icarus, 196, 241?248). For gas released solely from the cometary nucleus at a heliocentric distance of 1 AU, modeled rotational temperatures start at 110 K close to the surface and decrease to only several tens of degrees by 10–20 nucleus radii. However, the measured decay of both rotational temperature and column density with distance from the nucleus is much slower than predicted by this simple model. The addition of a substantial (distributed) source of gas from icy grains in the model slows the decay in rotational temperature and provides a more gradual drop in column density profiles. Together with a contribution of rotational heating of water molecules by electrons, the combined effects allow a much better match to the IRCS/Subaru observations. From the spatial distributions of water abundance and temperature measured in 73P/SW3-B, we have identified and quantified multiple mechanisms of release. The application of this tool to other comets may permit such studies over a range of heliocentric and geocentric distances.     

Structure and density of cometary nuclei

Abstract— Understanding the nature of the cometary nucleus remains one of the major problems in solar system science. Whipple's (1950) icy conglomerate model has been very successful at explaining a range of cometary phenomena, including the source of cometary activity and the nongravitational orbital motion of the nuclei. However, the internal structure of the nuclei is still largely unknown. We review herein the evidence for cometary nuclei as fluffy aggregates or primordial rubble piles, as first proposed by Donn et al. (1985) and Weissman (1986). These models assume that cometary nuclei are weakly bonded aggregations of smaller, icy‐conglomerate planetesimals, possibly held together only by self‐gravity. Evidence for this model comes from studies of the accretion and subsequent evolution of material in the solar nebula, from observations of disrupted comets, and in particular comet Shoemaker‐Levy 9, from measurements of the ensemble rotational properties of observed cometary nuclei, and from recent spacecraft missions to comets. Although the evidence for rubble pile nuclei is growing, the eventual answer to this question will likely not come until we can place a spacecraft in orbit around a cometary nucleus and study it in detail over many months to years. ESA's Rosetta mission, now en route to comet 67P/Churyumov‐Gerasimenko, will provide that opportunity.     

Comet Tempel-Tuttle and the Leonid meteors

The distribution of dust surrounding periodic comet Tempel-Tuttle has been mapped by analyzing the associated Leonid meteor shower data over the 902–1969 interval. The majority of dust ejected from the parent comet evolves to a position lagging the comet and outside the comet's orbit. The outgassing and dust ejection required to explain the parent comet's deviation from pure gravitational motion would preferentially place dust in a position leading the comet and inside the comet's orbit. Hence it appears that radiation pressure and planetary perturbations, rather than ejection processes, control the dynamic evolution of the Leonid particles. Significant Leonid meteor showers are possible roughly 2500 days before or after the parent comet reaches perihelion but only if the comet passes closer than 0.025 AU inside or 0.010 AU outside the Earth's orbit. Although the conditions in 1998–1999 are optimum for a significant Leonid meteor shower, the event is not certain because the dust particle distribution near the comet is far from uniform. As a by-product of this study, the orbit of comet Tempel-Tuttle has been redetermined for the 1366–1966 observed interval.     

Models of P/Wirtanen nucleus: active regions versus non-active regions

From the current understanding we know that comet nuclei have heterogeneous compositions and complex structures. It is believed that cometary activity is the result of a combination of physical processes in the nucleus, like sublimation and recondensation of volatile ices, dust grains release, phase transition of water ice, depletion of the most volatile components in the outer layers and interior differentiation.The evolution of the comet depends on the sublimation of ices and the release of different gases and dust grains: the formation of a dust crust, the surface erosion and the development of the coma are related to the gas fluxes escaping from the nucleus. New observations, laboratory experiments and numerical simulations suggest that the gas and dust emissions are locally generated, in the so-called active regions. This localized activity is probably superimposed to the global nucleus activity. The differences between active and inactive regions can be attributed to differences in texture and refractory material content of the different areas.In this paper we present the results of numerical models of cometary nucleus evolution, developed in order to understand which are the processes leading to the formation of active and non-active regions on the cometary surface. The used numerical code solves the equations of heat transport and gas diffusion within a porous nucleus composed of different ices—such as water (the dominant constituent), CO₂, CO- and of dust grains embedded in the ice matrix.By varying the set of physical parameters describing the initial properties of comet P/Wirtanen, the different behaviour of the icy and dusty areas can be followed.Comet P/Wirtanen is the target of the international ROSETTA mission, the cornerstone ESA mission to a cometary nucleus. The successful design of ROSETTA requires some knowledge of comet status and activity: surface temperatures, amount of active and inactive surface areas, gas production rate and dust flux.     

The provenance and evolution of comets

The process of comet formation through the hierarchical aggregation of originally submicron-sized interstellar grains to form micron-sized particles and then larger bodies in the protoplanetary disc, culminating in the formation of planetesimals in the disc extending from Jupiter to beyond Neptune, is briefly reviewed. The ‘planetesimal’ theory for the origin of comets implies the existence of distinct cometary reservoirs, with implications for the immediate provenance of observed comets (both long-period and short-period) and their evolution as a result of planetary perturbations and physical decay, for example splitting and sublimation. The principal mode of cometary decay and collisional interaction with the terrestrial planets is through the formation and evolution of streams of cometary debris and hitherto undiscovered ‘families’ of cometary asteroids. Recent dynamical results, in particular the sungrazing and sun-colliding end-state for short-period comet and asteroid orbits, are briefly discussed.     

Synchronous spin-orbital resonance locking of large planetary satellites

A numerical investigation of the chaotic rotation of large planetary satellites before their synchronous spin-orbital resonance locking with regard to tidal friction is carried out. The rotational dynamics of seven large satellites greater than 1000 km in diameter and with known inertial parameters (Io, Europa, Ganymede, Callisto (J1–J4), Tethys (S3), Iapetus (S8), and Ariel (U1)) in the epoch of synchronous resonance locking is modeled. All of these satellites have a small dynamic asymmetry. The planar case is considered in which the satellite’s axis of rotation is orthogonal to the plane of orbit. The satellites possessing an initial rapid rotation pass through various resonant states during the tidal evolution. Here, the probability of their locking into these states exists. The numerical experiments presented in this paper have shown that, with a rather high arbitrariness in the choice of initial states, the satellites during the course of the tidal evolution of their rotational motion have passed without interruption through the regions of the 5: 2, 2: 1, and 3: 2 resonances in the phase space and are locked into the 1: 1 resonance. The estimate for the tidal deceleration time is obtained both theoretically and on the numerical experimental basis.     

Exact analytic solution for the rotation of a rigid body having spherical ellipsoid of inertia and subjected to a constant torque

The exact analytic solution is introduced for the rotational motion of a rigid body having three equal principal moments of inertia and subjected to an external torque vector which is constant for an observer fixed with the body, and to arbitrary initial angular velocity. In the paper a parametrization of the rotation by three complex numbers is used. In particular, the rows of the rotation matrix are seen as elements of the unit sphere and projected, by stereographic projection, onto points on the complex plane. In this representation, the kinematic differential equation reduces to an equation of Riccati type, which is solved through appropriate choices of substitutions, thereby yielding an analytic solution in terms of confluent hypergeometric functions. The rotation matrix is recovered from the three complex rotation variables by inverse stereographic map. The results of a numerical experiment confirming the exactness of the analytic solution are reported. The newly found analytic solution is valid for any motion time length and rotation amplitude. The present paper adds a further element to the small set of special cases for which an exact solution of the rotational motion of a rigid body exists.