Numerical simulations of collisions and gravitational encounters in systems of non-identical particles

Numerical simulations of 200 mutually colliding non-identical particles indicate that the equipartition of random kinetic energy is possible only in systems having a narrow distribution of particle masses. Otherwise the random energy is concentrated on heavy particles. The form of the velocity distribution versus particle mass depends also on the elastic properties of the particles, and on the relative importance of the particle size. If the coefficient of restitution is a weakly decreasing function of impact velocity, a large difference in the equilibrium velocities of largest and smallest particles is possible. On the other hand, if the elasticity drops to a low level even in the small velocity regime, the dispersion of velocities is maintained by finite size and differential rotation, and the velocities of smallest particles are, at most, slightly larger than those of the largest ones. The results of simulations are consistent with the predictions of the collisional theory of non-identical particles (Hämeen-Anttila, 1984). The application to Saturn's rings indicates that the geometric thickness of cm-sized particles is of the order of 50 m in the rarefied regions of the rings. Without the gravitational encounters a thickness of about 30 m is derived. These estimations are made by using the latest measurements (Bridges et al., 1984) for the restitution coefficient of icy particles.     

Numerical simulations of collisions in self-gravitating systems

The influence of partially elastic collisions, gravitational encounters, and different gravitational potentials is studied in terms of computer simulations involving a ring of 200 identical particles. If the masses of the particles are small, their mutual gravitational attraction is found to produce a decrease in the dispersion of velocities below the collisional equilibrium value, but for heavier particles the dispersion again begins to increase. The vertical component of the force which is produced by the self-gravitation of the ring reduces its thickness. The results of the simulations are in good agreement with the predictions of the collisional theory.     

An improved and generalized theory for the collisional evolution of Keplerian systems

A calculation of collisional integrals with a higher accuracy yields excellent agreement between computer simulations and the collisional theory of Keplerian systems. Inclusion of axial rotation of particles modifies the evolution but does not introduce qualitatively new phenomena. Friction between the particles has a stabilizing influence, while deviations from an exactly spherical shape produce an opposite effect. The rotation of spherical or irregular bodies cannot prevent a final flattening of the system into a monolayer without also causing its disintegration. Computer simulations with a small number of particles do not represent the typical collisional evolution. They provide a test for the theory, but may sometimes lead to a misinterpretation of astronomical phenomena.     

Approximate calculations of collisional mean values for particulate systems

An improved analytical approximation for the dispersion of impact velocities is constructed. This leads to some modifications in the theory of collisional systems (Hämeen-Anttila et al., 1988) which is used as a basis for the calculations. A comparison with computer simulations shows that the new formulation improves the accuracy of theory.     

Velocity dispersion in particulate disks with size distribution

Quasi-equilibrium solutions for the pre-planetary disk are studied in terms of Hämeen-Anttila's theory (1984) of collisional, self-gravitating systems. The distribution of particle sizes is assumed to follow simple power-law distributions, with a power index in the range of 1.5–5.0. The treatment includes mutual impacts with a velocity dependent coefficient of restitution, as well as gravitational encounters with dynamical friction. The mean gravitational field of the disk is also taken into account. The results indicate that the energy(equi)-partition depends mainly on the index of size distribution, but is also affected by the optical thickness of the system, as well as on the vertical thickness as compared to the particle size. The vertical component of the gravitational field is found to be important, especially when the mass of the system is concentrated on the large particles.     

A revised flux vector for the collisional evolution of Keplerian orbits

In statistical Keplerian systems the disordered component of collisionally induced motion of matter introduces new terms into the flux vector. This contribution, which is calculated from a transport equation, tends to reduce the density gradient and causes the expansion which is observed in computer simulations of collisional systems. A quantitative comparison with Trulsen's (1972) simulations confirms the revised expression of the flux vector.     

Generalized theory of impacts in particulate systems

The theory of collisional systems is generalized for an arbitrary geometry and forces acting in the system, mixtures of different particle types, friction, small deviations from the ideal spherical form, axial rotation, finite size of particles and gravitational interactions. Terms for the formation of new particles and destruction of old ones are also included, and other unspecified parameters can be introduced. Although some approximations are made to simplify the basic equations and to avoid excessive numerical interactions, a comparison with computer simulations shows a good agreement. The tests were continued up to the optical thickness = 5.     

Random walk of magnetic field lines: analytical theory versus simulations

The analytical nonlinear theory of magnetic field line random walk predicts the existence of nondiffusive transport for certain forms of the turbulence spectrum. In the present article we use computer simulations to test these predictions made for well-established one- and two-dimensional models of magnetic field fluctuations. For the first time it is shown by using simulations, that for a whole family of spectra a non-diffusive behavior of field line wandering can be found.     

Collision mechanics and the structure of planetary ring edges

A numerical simulation of collisional evolution, originally developed to model planetary accretion processes, is applied to a hypothetical ring with parameters modeled after Saturn's rings in order to study changes in radial structure near ring edges. The tendency of rings to spread so as to conserve angular momentum while energy is dissipated in collisions is confirmed if random motion is in equilibrium. Even with no energy loss (coefficient of restitution in velocity ε = 1), spreading occurs becase random motion is increasing. With a moderately side-scattering collisional model, characteristic of collisions of nonrotating spheres (the slippery “billiard-ball” model), random motion increases for ε > 0.63, in agreement with analytical models. For isotropic scattering, which may be more realistic given particle rotation, damping dominates for ε up to 0.83. As long as random motion is damped, ring edges may contract rather than spread, producing concentrations of material just inside the ring edges reminiscent of results of earlier stimulation which did not precisely conserve angular momentum.     

Fragment ejection velocities and the collisional evolution of asteroids

We have applied the algorithm developed by Petit and Farinella (Celest. Mech. 57, 1–28, 1993) to model the outcomes of impacts between asteroids of different sizes, to show that a crucial feature of these models is the assumed relationship between velocity and mass of fragments ejected after a shattering impact. Not only how the mean velocity depends upon mass is important to determine the extent of fragment reaccumulation, but also the distribution of velocities about the mean values. The available experimental evidence on this issue is still sparse, and does not constrain the collisional models well enough to allow us to make reliable predictions on the outcomes of impacts between bodies of size much larger than the laboratory targets. As a consequence, when the collisional outcome models are used as an input for simulations of the asteroid collisional history since the origin of the solar system, the results show a strong sensitivity to the assumed velocity vs mass relationship. This sensitivity is stronger in the diameter range (a few tens to a few hundreds of km) where the self-gravitational reaccumulation of fragments is most effective, but may also extend to much smaller sizes.     

Exploring local N-body simulations of Saturn's rings

The dynamical behavior of low and moderately high optical depth regions of Saturn's ring system of discrete, mutually gravitating, and inelastically colliding particles is studied by simplified local N-body simulations in Hill's linearized equations context. The focus is on a statistical analysis of time-evolution of fine-scale structures seen in the simulations and the comparison between theoretical predictions and computer experiments. Prospects for the Cassini spacecraft mission are briefly summarized.     

Scalable Dynamics of High Energy Relativistic Electrons: Theory, Numerical Simulations and Experimental Results

Similarity theory, which is necessary in order to apply the results of laboratory astrophysics experiments to relativistic astrophysical plasmas, is presented. The analytical predictions of the similarity theory are compared with PIC numerical simulations and the most recent experimental data on monoenergetic electron acceleration in diluted plasmas and high harmonic generation at overdense plasma boundaries. We demonstrate that similarity theory is a reliable tool for explaining a surprisingly wide variety of laboratory plasma phenomena the predictions of which can be scaled up to astrophysical dimensions.     

On the 'coarse-grained' evolution of collisionless stellar systems

We have suggested in a previous article that the coarse-grained evolution of a collisionless stellar system could be viewed as a diffusion process in velocity space compensated by an appropriate friction. Using a quasi-linear theory, we calculate the diffusion coefficient associated with this evolution. This provides a new self-consistent relaxation equation for f , the locally averaged distribution function. This equation bears some resemblance to the conventional Fokker–Planck equation of collisional systems but the friction term is non-linear in f (accounting for degeneracy effects) and the relaxation time is much smaller (in agreement with the concept of 'violent relaxation'). Under the condition that the diffusion current vanishes identically at equilibrium, we recover Lynden-Bell's distribution function; but if we allow stars to escape from the system at a constant rate, we can derive a truncated model which coincides with Lynden-Bell's solution in the core but provides a depletion of high-energy stars in the halo. This distribution function has a finite mass and is the generalization of the Michie–King model to the case of (possibly degenerate) collisionless stellar systems.     

A tree code for planetesimal dynamics: comparison with a hybrid direct code

We present a tree code for simulations of collisional systems dominated by a central mass. We describe the implementation of the code and the results of some test runs with which the performance of the code was tested. A comparison between the behaviour of the tree code and a direct hybrid integrator is also presented. The main result is that tree codes can be useful in numerical simulations of planetary accretion, especially during intermediate stages, where possible runaway accretion and dynamical friction lead to a population with a few large bodies in low-eccentricity and low-inclination orbits embedded in a large swarm of small planetesimals in rather excited orbits. Some strategies to improve the performance of the code are also discussed.     

Collisional processes in stellar systems

The theory of collisional relaxation in stellar systems is discussed in terms of an expansion in powers of 1/N, the inverse of the total number of stars. The results are expressed in terms of the concept of gravitational polarization.     

Starn's rings and bimodality of Keplerian systems

The correction terms which are introduced by non-zero size of the particles into the mechanics of Keplerian systems can be replaced by relatively simple approximations which agree with computer simulations. The theory of finite particles confirms the bimodality of collisional systems which has previously been discussed in terms of the mass-point approximation. In Saturn's rings the ringlets correspond to the degenerate mode while the matter which fills the gaps is in the non-degenerate state. The predicted volume density of the ringlets (the fraction of space which is occupied by the particles), 0.2, is much higher than the conventional value which follows from the theory of mutual shadowing. Therefore, the opposition effect of Saturn's rings must originate in the particles themselves. The transition from one mode to the other which is needed to create a dense ring in a cloud of small particles follows from the growth of mass in the central body. This may be a recently-formed planet; but, more probably, the transition occurs in a loose pre-planetary disc.     

Velocity dispersions of dwarf spheroidal galaxies: dark matter versus MOND

We present predictions for the line-of-sight velocity dispersion profiles of dwarf spheroidal galaxies and compare them to observations in the case of the Fornax dwarf. The predictions are made in the framework of standard dynamical theory of spherical systems with different velocity distributions. The stars are assumed to be distributed according to Sérsic laws with parameters fitted to observations. We compare predictions obtained assuming the presence of dark matter haloes (with density profiles adopted from N -body simulations) with those resulting from Modified Newtonian Dynamics (MOND). If the anisotropy of velocity distribution is treated as a free parameter, observational data for Fornax are reproduced equally well by models with dark matter and with MOND. If stellar mass-to-light ratio of 1 M_⊙/L_⊙ is assumed, the required mass of the dark halo is     , two orders of magnitude larger than the mass in stars. The derived MOND acceleration scale is     . In both cases a certain amount of tangential anisotropy in the velocity distribution is needed to reproduce the shape of the velocity dispersion profile in Fornax.     

Galactic scale gas flows in colliding galaxies: 3-Dimensional,N-body/hydrodynamics experiments

We present some results from three dimensional computer simulations of collisions between models of equal mass galaxies, one of which is a rotating, disk galaxy containing both gas and stars and the other is an elliptical containing stars only. We use fully self consistent models in which the halo mass is 2.5 times that of the disk. In the experiments we have varied the impact parameter between zero (head on) and 0.9R (whereR is the radius of the disk), for impacts perpendicular to the disk plane. The calculations were performed on a Cray 2 computer using a combined N-body/SPH program. The results show the development of complicated flows and shock structures in the direction perpendicular to the plane of the disk and the propagation outwards of a density wave in both the stars and the gas. The collisional nature of the gas results in a sharper ring than obtained for the star particles, and the development of high volume densities and shocks.     

Using the transition to action variables of the newtonian problem in the numerical solution of the N-body problem

We propose an approach for overcoming the problem of close encounters in collisional systems, globular and open star clusters. As is well known, the numerical integration step in such systems, for example, during the formation of close binary stars, begins to fragment and the rate of calculations goes down to a complete stop. We show that using the perturbation theory in the proposed approach, one can isolate the singularity and to increase considerably the integration step without losing the physical effects that affect significantly the evolution of star clusters.     

Low-velocity collisions between centimeter-sized snowballs: Porosity dependence of coefficient of restitution for ice aggregates analogues in the Solar System

Understanding the collisional behavior of ice dust aggregates at low velocity is a key to determining the formation process of small icy bodies such as icy planetesimals, comets and icy satellites, and this collisional behavior is also closely related to the energy dissipation mechanism in Saturn’s rings. We performed head-on collision experiments in air by means of free-falling centimeter-sized sintered snowballs with porosities from 44% to 80% at impact velocities from 0.44 m s^?1 to 4.12 m s^?1 at ?10 °C. In cases of porosity larger than 70%, impact sticking was the dominant collision outcome, while bouncing was dominant at lower porosity. Coefficients of restitution of snow in this velocity range were found to depend strongly on the porosity rather than the impact velocity and to decrease with the increase of the porosity. We successfully measured the compaction volume of snowballs after the impact, and it enabled us to estimate the dynamic compressive strength of snow with the assumption of the energy conservation between kinetic energy and work for deformation, which was found to be consistent with the upper limit of static compressive strength. The velocity dependence of coefficients of restitution of snow was analyzed using a Johnson’s model, and a diagram for collision outcomes among equal-sized sintered snowballs was successfully drawn as a function of porosity and impact velocity.