Numerical simulations of collisions in Keplerian systems

Computer simulations which were carried out for Keplerian collisional systems of 250 frictionless particles with a ratio of particle radius to mean semi-major axis of 0.001, confirm the theoretically predicted evolution very well until the thickness of the system is a few times the particle radius and the mass-point approximation becomes invalidated. Before this happens, the collisional contraction of denserregions can be observed. The local dispersions of the perihelia and ascending nodes diminish if the local mean orbit is not too close to a circle with zero inclination. When the mass-point approximation ceases to be valid, the system begins to expand, but with parameter values of our standard system this process is much slower than the simultaneously observed evolution toward grazing collisions which do not affect the orbital elements. Therefore, such systems are not dispersed into the space. If the ratio of particle radius to semi-major axis is larger, the expansion becomes faster and the contraction ceases earlier. In late evolutionary phases the thickness of the system remains essentially constant. At the end of the longest simulation (70 000 impacts) the centres of the particles were in a layer of thickness twice the radius of the particles. The cross-section of the system is often wave-like or irregular and may even include detached parts with their own mean plane. Accordingly the thickness as derived from the root-mean-square inclination of the whole system exceeds the true local thickness. The local dispersion of eccentricities may also be considerably smaller than the root-mean-square eccentricity of the whole system.     

Numerical simulations of collisions in systems of non-identical particles

The numerical simulations of 200 mutually colliding, non-identical particles indicate that if elasticity depends on the impact velocity, an equipartition of the random kinetic energy is possible if either the particle masses are close to each other, or the number of small particles significantly exceeds that of large particles. On the other hand, if the large particles dominate, the velocities of the smaller particles are at most a few times greater than those of the large ones. In the case of a constant coefficient of restitution no equipartition is observed.     

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.     

On the entropy of self-gravitating systems

Leningrad State University; Kursk Institute for Improvement of Teachers. Translated from Astrofizika, Vol. 29, No. 3, pp. 595–601, November–December, 1988.     

Uniform density equilibrium model of self-gravitating stellar systems

A class of equilibrium solutions of the Vlasov equation for self-gravitating systems is discussed. The density and the potential are derived in form of Jacobi polynomials, which in a special case give rise to a model with uniform density.     

On the clustering phase transition in self-gravitating N-body systems

The thermodynamic behaviour of self-gravitating N -body systems has been worked out by borrowing a standard method from molecular dynamics. The link between dynamics and thermodynamics is made in the microcanonical ensemble of statistical mechanics. Through the computation of basic thermodynamic observables and of the equation of state in the     plane, the clustering phase transition appears to be of the second-order type. The dynamical–microcanonical averages are compared with their corresponding canonical ensemble averages, obtained through standard Monte Carlo computations. The latter seem to have completely lost any information about the phase transition. Finally, our results – obtained in a 'microscopic' framework – are compared with some existing theoretical predictions – obtained in a 'macroscopic' (thermodynamic) framework: qualitative and quantitative agreement is found, with an interesting exception.     

Remarks on the evaluation of critical parameters in self-gravitating systems

Using a simple model sketched by Thirring, thermodynamical quantities of a system of self-gravitating fermions are investigated. The onset of a core-halo structure is considered as a phase transition; free energy degeneracy and related critical parameters are found.     

Mean field fluctuations and the evolution of self-gravitating systems

This paper considers the nonlinear effects of collective fluctuations about a slowly-varying solution of the collisionless Boltzmann equation appropriate for a system of stars with a distribution of masses. It is argued that these effects should be manifest on a time-scalet _M intermediate between the violent and collisional relaxation times, inhibiting the natural progression towards equipartition induced by collisions, and favouring instead a well-mixed state, realized in numerical simulations, in which the mean square velocity is independent of the mass.     

Full numerical simulations of catastrophic small body collisions

The outcome of collisions between small icy bodies, such as Kuiper belt objects, is poorly understood and yet a critical component of the evolution of the trans-neptunian region. The expected physical properties of outer Solar System materials (high porosity, mixed ice-rock composition, and low material strength) pose significant computational challenges. We present results from catastrophic small body collisions using a new hybrid hydrocode to N-body code computational technique. This method allows detailed modeling of shock propagation and material modification as well as gravitational reaccumulation. Here, we consider a wide range of material strengths to span the possible range of Kuiper belt objects. We find that the shear strength of the target is important in determining the collision outcome for 2 to 50-km radius bodies, which are traditionally thought to be in a pure gravity regime. The catastrophic disruption and dispersal criteria, , can vary by up to a factor of three between strong crystalline and weak aggregate materials. The material within the largest reaccumulated remnants experiences a wide range of shock pressures. The dispersal and reaccumulation process results in the material on the surfaces of the largest remnants having experienced a wider range of shock pressures compared to material in the interior. Hence, depending on the initial structure and composition, the surface materials on large, reaccumulated bodies in the outer Solar System may exhibit complex spectral and albedo variations. Finally, we present revised catastrophic disruption criteria for a range of impact velocities and material strengths for outer Solar System bodies.     

A note on the stability of one-dimensional self-gravitating isothermal systems

One-dimensional self-gravitating isothermal systems are stable with respect to one-dimensional perturbations.     

Numerical modelling of heating in porous planetesimal collisions

Collisions between planetesimals at speeds of several kilometres per second were common during the early evolution of our Solar System. However, the collateral effects of these collisions are not well understood. In this paper, we quantify the efficiency of heating during high-velocity collisions between planetesimals using hydrocode modelling. We conducted a series of simulations to test the effect on shock heating of the initial porosity and temperature of the planetesimals, the relative velocity of the collision and the relative size of the two colliding bodies. Our results show that while heating is minor in collisions between non-porous planetesimals at impact velocities below 10 km s⁻¹, in agreement with previous work, much higher temperatures are reached in collisions between porous planetesimals. For example, collisions between nearly equal-sized, porous planetesimals can melt all, or nearly all, of the mass of the bodies at collision velocities below 7 km s⁻¹. For collisions of small bodies into larger ones, such as those with an impactor-to-target mass ratio below 0.1, significant localised heating occurs in the target body. At impact velocities as low as 5 km s⁻¹, the mass of melt will be nearly double the mass of the impactor, and the mass of material shock heated by 100 K will be nearly 10 times the mass of the impactor. We present a first-order estimate of the cumulative effects of impact heating on a porous planetesimal parent body by simulating the impact of a population of small bodies until a disruptive event occurs. Before disruption, impact heating is volumetrically minor and highly localised; in no case was more than about 3% of the parent body heated by more than 100 K. However, heating during the final disruptive collision can be significant; in about 10% of cases, almost all of the parent body is heated to 700 K (from an initial temperature of ∼300 K) and more than a tenth of the parent body mass is melted. Hence, energetic collisions between planetesimals could have had important effects on the thermal evolution of primitive materials in the early Solar System.     

Numerical simulations of jet propagation

Astrojet simulations have been directly motivated for 20 years by the observations: from light jets to heavy jets, from hot to cold, from ionised to molecular, and adiabatic to supercooled. This is, coincidently, a sequence of increasing computational difficulty due to the increasing range of time and length scales. The progress in several directions is reviewed here, in terms of the effects of cooling, instabilities and the magnetic field. To simulate molecular jets from protostars now provides a great challenge. The contrast between physical and detectable quantities stresses the need for predicting both spectral and spatial distributions. These and some further outstanding problems are discussed.     

High-resolution simulations of stellar collisions between equal-mass main-sequence stars in globular clusters

We performed high-resolution simulations of two stellar collisions relevant for stars in globular clusters. We considered one head-on collision and one off-axis collision between two 0.6-M_⊙ main-sequence stars. We show that a resolution of about 100 000 particles is sufficient for most studies of the structure and evolution of blue stragglers. We demonstrate conclusively that collision products between main-sequence stars in globular clusters do not have surface convection zones larger than 0.004 M_⊙ after the collision, nor do they develop convection zones during the 'pre-main-sequence' thermal relaxation phase of their post-collision evolution. Therefore, any mechanism which requires a surface convection zone (i.e. chemical mixing or angular momentum loss via a magnetic wind) cannot operate in these stars. We show that no disc of material surrounding the collision product is produced in off-axis collisions. The lack of both a convection zone and a disc proves a continuing problem for the angular momentum evolution of blue stragglers in globular clusters.     

Power distributions for self-gravitating astrophysical systems based on nonextensive Tsallis kinetics

The long-time development of self-gravitating gaseous astrophysical systems (in particular, the evolution of the protoplanet accretion disk) is mainly determined by relatively fast processes of the collision relaxation of particles. However, slower dynamical processes related to force (Newton or Coulomb) interactions between particles should be included (as q-collisions) in the nonextensive kinetic theory as well. In the present paper, we propose a procedure to include the Newton self-gravity potential and the centrifugal potential in the near-equilibrium power-like q-distribution in the phase space, obtained (in the framework of nonextensive statistics) by means of the modified Boltzmann equation averaged with respect to an unnormalized distribution. We show that if the power distribution satisfies the stationary q-kinetic equation, then the said equation imposes clear restrictions on the character of the long-term force field and on the possible dependence of hydrodynamic parameters of the coordinates: it determines those parameters uniquely. We provide a thermodynamic stability criterion for the equilibrium of the nonextensive system. The results allow us to simulate the evolution of gaseous astrophysical systems (in particular, the gravitational stability of rotating protoplanet accretion disks) more adequately.