Several ideas have been proposed to explain why the irregular radial structure is formed and maintained against viscous diffusion due to collisions and gravitational scattering of ring particles.

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They claimed this instability to be analogous to the well-known viscous mechanism that converts Maclaurin spheroids to Jacobi ellipsoids. The size of the assemblies is limited by tidal forces from Saturn and collisional erosion by impacting particles. In this process the disc self-gravity is essential, while interparticle collisions play a modest role. Esposito already pointed out that the rings are a kinetic system, where the deviations from perfect circular, equatorial motion can be considered as random velocities in a viscous fluid gas.

In contrast to all previous studies, a mathematical formalism in the approximation of weak turbulence a quasi-linearization of the Boltzmann kinetic equation is developed, which is a direct analogy with the plasma quasi-linear or weakly non-linear; e. This allows us to study the reaction of the oscillations on the equilibrium parameters of the system.

We establish a connection between several plasma physics phenomena and the dynamics of Saturn's rings. Such a connection deepens our understanding of the nature of this phenomenon and broadens the reader audience.

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Conditions for the applicability of quasi-linear theory of Jeans-type gravitational instability in Saturn's rings are given in the Appendix. Generally speaking, the quasi-linear approach, i.

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Arbitrary disturbances can be expressed as a superposition of eigenmodes, with each eigenmode evolving independently. The theory of strong turbulence is still far from complete. As shown in the present paper, the quasi-linear theory can be applied to aperiodic Jeans-type instabilities we are interested in, provided that in the initial state the system parameters deviate little from the critical values at which instability sets in. The results of the analysis are applied to Saturn's rings, composed of rock and ice: Recent direct N -body simulations of the Saturnian ring system confirmed the basic prediction of local computer experiments Griv et al.

In Franklin et al. Accordingly, the presence of wakes causes the effective area covered by particles and hence the brightness of the ring to vary at different longitudes: This explanation, which requires some degree of self-gravitation between nearby orbiting bodies, accounts both for the presence of the azimuthal brightness variations in Saturn's ring A and for their absence in ring B. A thin rapidly rotating disc of identical particles is often taken as a model though idealized of the Saturnian ring system. In the spirit of Griv et al. We expect that the waves and their instabilities propagating in the disc plane have the greatest influence on the development of structures in the system Shu This justifies the two-dimensional treatment of the main part of a rapidly rotating disc of randomly colliding particles.

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Limiting ourselves to the case of an infinitesimally thin disc simplifies the algebra without introducing any fundamental changes in the physical results. Thirdly, the Boltzmann form for the collision integral is based on an assumption that the duration of a collision is much less than the time between collisions — instantaneous collisions are considered. The Boltzmann integral for elastic collisions vanishes if the distribution function is Maxwellian Alexandrov et al. We assume that there are only binary physical collisions, and momentum is conserved in collisions.

There is no correlation in motion between the colliding species, i.

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Boltzmann's hypothesis of molecular chaos is adopted cf. In addition, the most interesting case of rare interparticle collisions is considered: Then the collisions cause only small corrections of the distribution function and in the zeroth-order approximation of the theory all dissipative effects may be ignored. This is just the opposite of the procedure in ordinary gas dynamics, where collisions are the dominant effect.

This approach is valid for high random velocity—temperatures and low densities, when the mean potential between neighbouring particles is small compared with the thermal energy. Collisions are to be relatively unimportant and hence the form of the collision integral can be grossly approximated. At these densities the applicability of Goldreich and Tremaine's standard model we are interesting in becomes questionable Araki a. This paper is organized as follows. First, in Section 2 the basic equations of the theory, the collision integral and the equilibrium are presented.

In Section 3 the basic ideas of the quasi-linear theory are described. In Sections 4 and 5 , respectively, the generalized Lin—Shu-type dispersion relation and the quasi-linear diffusion equations are obtained and solved analytically. Section 6 contains a summary of the results of the work. Finally, in the Appendix the conditions for the applicability of the quasi-linear theory are derived.

This set of equations is called the set of equations with a self-consistent field. The combined system of equations 1 and 2 is the counterpart of the system of Boltzmann and Maxwell equations in electromagnetic plasmas. The Boltzmann kinetic equation says that the phase space density changes because of the collisions, following the phase space trajectories. It is this equilibrium that is examined for stability: The investigations carried out in the linear approximation allow us to determine only the spectrum of the excited oscillations and their growth rates during the first stages of the excitation.

Other problems, which can be treated within the framework of non-linear theory, are account of the reaction of the oscillations on the equilibrium parameters of the system and the determination of the amplitude of the oscillations that are produced.

Apparently, Kulsrud first discussed the inverse effects of different instabilities of gravity oscillations on the averaged velocity distribution function of particles by collective interaction by considering a collisionless stellar disc of spiral galaxies. These properties would be manifested in the behaviour of small gravity perturbations arising against the equilibrium background. Collective processes are completely analogous to two-body collisions, except that one particle collides not with another one but with many that are collected together by some coherent process such as a wave.

The collective processes are random, and usually much stronger than the ordinary two-body collisions and leads to a random walk of the particles that takes the complete system towards a thermal quasi-steady state. Thus, relaxation in collisionless almost collisionless particulate systems could occur without binary particle—particle collisions through the influence of collective motions of the particulate gas upon the particle distribution.

In the linear theory, one can select one of the Fourier harmonics: The solution in such a form represents a smooth spiral wave with m arms. The resonances, however, have only a limited radial extent. In this work only the main part of the disc is considered, which lies sufficiently far from the resonances: The dispersion relation 31 is valid only sufficiently far from resonant circumferences and must be substituted by another equation near them e. The non-axisymmetric kinetic WKB approximation, explored in this paper, seems to be an acceptable method for exploring instabilities in particulate discs along with the various hydrodynamic treatments that pervade the literature.

According to equation 36 , rare interparticle collisions lead to the weak damping of both Jeans-unstable and Jeans-stable modes. This is physically obvious: The Jeans instabilities are fastest in the weakly collisional limit, in the sense that their growth rates are slowed down for increasing interparticle collisions. In turn, Jeans-stable perturbations will weakly decay as a result of rare collisions: Therefore, as a result of collisions, a Jeans-stable wave tends to be damped on a time-scale of the order of the mean time between collisions.

Although Shu and Stewart's use of the Krook equation was restricted to the unperturbed axisymmetric state, Shu et al. Probably, the linearized limit of that treatment is consistent with our discussion above of the damping of Jeans perturbations. The free kinetic energy associated with the differential rotation is one possible source for the growth of the average wave energy. The fierce Jeans instabilities are generated by almost all the particles of the phase space.

Thus, as a result of the Jeans instability of spiral gravity perturbations, the low and moderately high optical depth region of the system is subdivided into numerous spiral wakes with size of the order of m or even less. To repeat ourselves, the wavelength the size of spirals and the spacing between them of the most unstable perturbations is proportional to the mean size of an epicycle: The gradient perturbations are damped as a result of collisions and are independent of the stability of the Jeans modes.

According to equation 42 , the critical Toomre's velocity dispersion c T should stabilize only axisymmetric radial perturbations of the Jeans type. The differentially rotating and spatially inhomogeneous disc is still unstable against non-axisymmetric spiral Jeans perturbations. It would be natural to suppose that as a disc of mutually gravitating particles evolves it should arrive at a state near the limit of stability, so that the mean particle radial velocity dispersion would come close to the modified critical value c crit as given by equation Maxwell made a correction that, in such a system, the azimuthal force resulting from azimuthal displacements was more important in determining the stability than was the radial force resulting from radial displacements cf.

Julian and Toomre's calculations showed that even a stable stellar system possessing a velocity dispersion more than sufficient for local axisymmetric gravitational stability to be remarkably responsive in a spiral-like manner to localized forcing. These forced spiral waves are not to be confused, of course, with Lin and Shu's fully self-consistent density wave proposal explored in the present paper. See also Morozov , b , Griv et al. Our analysis is based on the microscopic kinetic description of the properties and processes of a particulate disc.

A relatively simple macroscopic hydrodynamical model can also be used to investigate the Jeans instability e.

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Clearly, also our model has nothing to do with viscous overstability of the densest central portions of Saturn's B ring with very frequent interparticle collisions e. Interestingly, analysis of the Voyager 2 data Lane et al. This could be indicative of a self-gravitation mechanism for generating spiral fine-scale structure in Saturn's rings. Although equation 31 can be analysed directly and even solved analytically in the case of low-frequency oscillations as just shown above, a graphical representation of the roots is much more convenient. A general impression of how the spectrum of non-axisymmetric Jeans perturbations behaves in a homogeneous non-uniformly rotating disc can be gained from Fig.

The latter is in good agreement with the approximate solution given by equation In contrast to our study, however, Nakamura et al. The dashed curves represent the imaginary part of the dimensionless wavefrequency of low-frequency vibrations. Equations 43 and 44 form the closed system of weakly non-linear equations for Jeans oscillations of the rotating inhomogeneous disc, and describe a diffusion of particles in configuration space.

The spectrum of oscillations and their growth rate are determined from equations 31 and 37 , respectively, and the stability criterion is given by equation Evidently, the unstable fluid-like Jeans oscillations must influence the distribution function of the main, non-resonant part of particles in such a way as to hinder the wave excitation, i. Finally, in the disc there can be established a quasi-stationary distribution so that the Jeans-unstable perturbations are completely vanishing.

By local N -body simulations, Salo already suggested that such an increase of the velocity dispersion would come from scatterings by the collective wakes. It was also suggested that the systematic motion of particles in the wakes the systematic motion of stars in the field of spiral arms of galaxies; Lin et al.

Also, this interaction of particles with the gravitationally unstable waves increases the radial spread of the disc. This will cause the disc matter to move inwards and outwards. The diffusion in configuration space is due entirely to the growth of the almost aperiodic Jeans-unstable modes in a self-gravitating system subject to a time-dependent potential. The non-resonant particles acquire mechanical oscillatory energy associated with the Jeans-unstable perturbations. This mechanism increases both the random velocity spread and the spatial spread of particles in the Saturnian ring system.

Subsequently, sufficient velocity dispersion and the radial scale of the spatial inhomogeneity prevents the Jeans instability from occurring equation Thus, diffusion presumably tapers off as Jeans stability is approached: Such a situation naturally implies a finite lifetime for the fine-scale structure. We conclude by noting that the distribution is modified so that it is everywhere marginally Jeans-stable. Stability is achieved for a sufficiently large thermal velocity spread.

In Saturn's rings such a cooling mechanism is actually operating: That is, mechanical energy is converted into thermal energy or the energy of smaller fragments, and is essentially lost. Finding Paradise Romance Boxed Set. The Deadly Series Bundle. If You Believe in Me. The Parasol Protectorate Boxed Set. The Phantom Series Boxed Set. Marblestone Mansion, Omnibus, Books 1 - 3. Dragon Chalice Series Box Set. Psychic Visions Set Falling For Her Fiance.

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