Radially anisotropic models of collisionless spherical stellar systems without central singularity

A new class of the simplest equilibrium two-parameter distribution functions for spherical stellar systems with a radially anisotropic stellar velocity distribution is investigated. The models under consideration are a less singular counterpart of the so-called generalized polytropes, which in the past were among the most popular models in works on the equilibrium and stability of gravitating systems. In contrast to the well-known generalized polytropes, the proposed models have finite density and potential at the center. The absence of a singularity is necessary for a proper consideration of the radial orbit instability, which is the most important instability of spherical stellar systems. The main observed parameters of the proposed models (potential, density, anisotropy) are compared with those in well-known equilibrium models.     

Undamped oscillations of collisionless stellar systems: Spheres,spheroids and discs

The collisionless Boltzmann equation governing self-gravitating systems such as galaxies has recently been shown to admit exact oscillating solutions with planar and spherical symmetry. The relation of the spherically symmetric solutions to the Virial theorem, as well as generalizations to non-uniform spheres, uniform spheroids and discs form the subject of this paper. These models generalize known families of static solutions. The case of the spheroid is worked out in some detail. Quasiperiodic as well as chaotic time variation of the two axes is demonstrated by studying the surface of section for the associated Hamiltonian system with two degrees of freedom. The relation to earlier work and possible implications for the general problem of collisionless relaxation in self gravitating systems are also discussed     

New adaptive time step symplectic integrator:an application to the elliptic restricted three-body problem

The time-transformed leapfrog scheme of Mikkola Aarseth was specifically designed for a second-order differential equation with two individually separable forms of positions and velocities.It can have good numerical accuracy for extremely close two-body encounters in gravitating few-body systems with large mass ratios,but the non-time-transformed one does not work well.Following this idea,we develop a new explicit symplectic integrator with an adaptive time step that can be applied to a time-dependent Hamiltonian.Our method relies on a time step function having two distinct but equivalent forms and on the inclusion of two pairs of new canonical conjugate variables in the extended phase space.In addition,the Hamiltonian must be modified to be a new time-transformed Hamiltonian with three integrable parts.When this method is applied to the elliptic restricted three-body problem,its numerical precision is explicitly higher by several orders of magnitude than the nonadaptive one's,and its numerical stability is also better.In particular,it can eliminate the overestimation of Lyapunov exponents and suppress the spurious rapid growth of fast Lyapunov indicators for high-eccentricity orbits of a massless third body.The present technique will be useful for conservative systems including N-body problems in the Jacobian coordinates in the the field of solar system dynamics,and nonconservative systems such as a time-dependent barred galaxy model in a rotating coordinate system.     

Effects of electromagnetic field on the collapse and expansion of anisotropic gravitating source

This paper is devoted to study the effects of electromagnetic on the collapse and expansion of anisotropic gravitating source. For this purpose, we have evaluated the generating solutions of Einstein–Maxwell field equations with spherically symmetric anisotropic gravitating source. We found that a single function generates the various anisotropic solutions. In this case every generating function involves an arbitrary function of time which can be chosen to fit several astrophysical time profiles. Two physical phenomenon occur, one is gravitational collapse and other is the cosmological expanding solution. In both cases electromagnetic field effects the anisotropy of the model. For collapse the anisotropy is increased while for expansion it deceases from maximum value to finite positive value. In case of collapse there exits two horizons like in case of Reissner–Nordström metric.     

Dynamics of tilted spherical star and stability of non-tilted congruence

This paper aims to study the dynamics of spherical star having anisotropic pressure, heat dissipation and shear viscosity with radial four-velocity. We formulate the field equations, equations of motion and equations for the Weyl tensor to study the inhomogeneity factor of the tilted congruence. An evolution equation for the shear is established to explore the stability of the non-tilted congruence. We conclude that the non-tilted congruence is stable for shear-free case.     

Reconstruction of A Non-Stationary Potential of Gravitating System on Given Evolving Orbits

We study the problem of the reconstruction of a non-stationary space symmetrical regular planar potential of the gravitating system on a family of evolving types of orbits being used in the dynamics of stationary stellar systems. An application of such an inverse problem to the dynamical evolution of stellar systems with variable masses is given. The general form of the evolving orbit which we use when writing out the differential equations for non-stationary potential may also be interpreted as an osculating orbit of the perturbed Keplerian motion. In this case we are making an additional transformation of the basic equation of the problem and demonstrating an appropriate example of the construction of a non-stationary potential of a gravitating system. In connection with the stellar dynamical character of our inverse problem, we also give a generalized form of its basic equation in a rotating coordinate system.     

The Expansion of the Distance Between Two Arbitrary Gravitating Points as Applied to Specific Models of Celestial Bodies

For the complete system of biharmonic functions a suitable representation in spheroidal coordinates is found. It is used for expanding the distance between two arbitrary gravitating points and its inverse quantity in appropriate series. Such expansions can be of interest and useful in a number of problems of celestial mechanics and stellar dynamics.     

Three-dimensional numerical simulation of a nonstationary gravitating <Emphasis Type="Italic">N</Emphasis>-body system with gas

We developed a three-dimensional numerical model to investigate nonstationary processes in gravitating N-body systems with gas. We used efficient algorithms for solving the Vlasov and Poisson equations that included the evolutionary processes under consideration, which ensures rapid convergence at high accuracy. We give examples of the numerical solution of the problem on the growth of physical instability in the model of a flat rotating disk with a gaseous component and its three-dimensional dynamics under various initial conditions including a nonzero velocity dispersion along the rotation axis.     

A three-dimensional numerical method for modelling weakly ionized plasmas

Astrophysical fluids under the influence of magnetic fields are often subjected to single- or two-fluid approximations. In the case of weakly ionized plasmas, however, this can be inappropriate due to distinct responses from the multiple constituent species to both collisional and non-collisional forces. As a result, in dense molecular clouds and protostellar accretion discs, for instance, the conductivity of the plasma may be highly anisotropic leading to phenomena such as Hall and ambipolar diffusion strongly influencing the dynamics.
Diffusive processes are known to restrict the stability of conventional numerical schemes which are not implicit in nature. Furthermore, recent work establishes that a large Hall term can impose an additional severe stability limit on standard explicit schemes. Following a previous paper, which presented the one-dimensional case, we describe a fully three-dimensional method which relaxes the normal restrictions on explicit schemes for multifluid processes. This is achieved by applying the little-known Super TimeStepping technique to the symmetric (ambipolar) component of the evolution operator for the magnetic field in the local plasma rest frame, and the new Hall Diffusion Scheme to the skew-symmetric (Hall) component.     

Scaling up tides in numerical models of galaxy and halo formation

The purpose of this article is to show that when dynamically cold, dissipationless self-gravitating systems collapse, their evolution is a strong function of the symmetry in the initial distribution. We explore with a set of pressureless homogeneous fluids the time evolution of ellipsoidal distributions and map the depth of potential achieved during relaxation as function of initial ellipsoid axis ratios. We then perform a series of N -body numerical simulations and contrast their evolution with the fluid solutions. We verify an analytic relation between collapse factor and particle number N in spherical symmetry, such that  ∝ N ^1/3  . We sought a similar relation for axisymmetric configurations, and found an empirical scaling relation such that  ∝ N ^1/6  in these cases. We then show that when mass distributions do not respect spherical or axial symmetry, the ensuing gravitational collapse deepens with increasing particle number N but only slowly: 86 per cent of triaxial configurations may collapse by a factor of no more than 40 as   N →∞  . For   N ≈10⁵  and larger, violent relaxation develops fully under the Lin–Mestel–Shu instability such that numerical N -body solutions now resolve the different initial morphologies adequately.     

Thermal evaluation of shear-free charged compact object

This paper is devoted to the thermal evolution of shear-free charged gravitating compact objects which undergoes exhausted its inner heat during gravitational collapse. The evaluation equations have been formulated by applying the perturbation of first order to Einstein–Maxwell equations and Catteno-type heat transportation equation. We show that the temperature gradient (induced in the system owing to external effects), sufficiently depends upon multiple of relaxation time and total time during which the gravitating system oscillates. By taking some particular examples of nuclear matter, it has been noted that at a particular value of luminosity, the change in existing thermal profile is greater corresponding to higher values of times during which a system comes to rest position.     

Poincarè Chaos and the Dynamical Evolution of Systems of Gravitating Bodies

Some general laws of evolution of a system of a large number of gravitating bodies are discussed. If in the initial stage the dynamics of the system is determined by large-scale perturbations of the gravitational potential associated with excitations of a few collective degrees of freedom, then one can assume, by analogy with chaos in the several-body problem (Poincarè chaos), that randomization will occur in the system over several average crossing times. In the next stage of evolution, the energy of collective modes should be transferred by the cascade mechanism to ever smaller scales, down to invididual particles. Numerical experiments and gross-dynamical considerations that could verify this picture and bring out details are discussed.     

Regular and chaotic motion in softened gravitational systems

The stability of the dynamical trajectories of softened spherical gravitational systems is examined, both in the case of the full N -body problem and that of trajectories moving in the gravitational field of non-interacting background particles. In the latter case, for   N 10 000  , some trajectories, even if unstable, had exceedingly long diffusion times, which correlated with the characteristic e-folding time-scale of the instability. For trajectories of   N ≈100 000  systems this time-scale could be arbitrarily large – and thus appear to correspond to regular orbits. For centrally concentrated systems, low angular momentum trajectories were found to be systematically more unstable. This phenomenon is analogous to the well-known case of trajectories in generic centrally concentrated non-spherical smooth systems, where eccentric trajectories are found to be chaotic. The exponentiation times also correlate with the conservation of the angular momenta along the trajectories. For N up to a few hundred, the instability time-scales of N -body systems and their variation with particle number are similar to those of the most chaotic trajectories in inhomogeneous non-interacting systems. For larger N (up to a few thousand) the values of the these time-scales were found to saturate, increasing significantly more slowly with N . We attribute this to collective effects in the fully self-gravitating problem, which are apparent in the time variations of the time-dependent Liapunov exponents. The results presented here go some way towards resolving the long-standing apparent paradoxes concerning the local instability of trajectories. This now appears to be a manifestation of mechanisms driving evolution in gravitational systems and their interactions – and may thus be a useful diagnostic of such processes.     

Riemann Ellipsoids with Spherical Halos

Possible ellipsoidal figures of equilibrium are obtained for a rotating, gravitating fluid mass with internal mass flows of constant vorticity, embedded inside a homogeneous gravitating sphere. The classical ellipsoidal figures of equilibrium are generalized and new S-ellipsoids and ellipsoids with oblique rotation are obtained. The stability of embedded S-ellipsoids is investigated and the criterion for their stability is obtained. The existence of an ellipsoid with oblique rotation of type II inside a relatively dense halo becomes impossible.     

Interaction between a high-velocity star and a hot dense cloud

In order to investigate the fate of a high-velocity star confined to a massive gas cloud, the hydrodynamic behavior of a radiation-dominated flow past a finite-size gravitating object is calculated. Such a study is important in the context of quasars and active galactic nuclei, since large gas clouds have been suggested to be associated with the phenomena occurring in these systems. In particular, we study the processes of mass and energy exchange between the ambient gas and the star. A significant result is that the effective accretion cross-section is found to be a substantial fraction of the stellar geometrical cross-section. Consequently, the buildup of massive stars via accretion of the surrouding gas may be important in determining the evolution of the system of stars confined to the cloud.     

Analytical solutions for the Newtonian gravitational field induced by matter within axisymmetric boundaries

An analytical method originally applied to the problem of the actuator disc in fluid mechanics has been applied to the closely analogous problem of constructing the classical Newtonian potential and attractions. The method can treat axisymmetric problems and also non-axisymmetric cases where matter is confined within axisymmetric boundaries. The potential and attractions for the generalized thin finite disc can be given in closed form in terms of elliptic integrals and elementary functions. For the general case of matter within an axisymmetric boundary, the potentials and attractions can be evaluated as one-dimensional integrals of albeit complex analytical expressions. These expressions represent the fields induced by matter in an extended region as a distribution of gravitating discs. For certain special cases, such as matter bounded by a circular cylinder and also for matter distributed in a spherical region, closed-form solutions can be given that appear to be new. Some non-axisymmetric results are also given for the thin disc of infinite radial extent.     

On the possibility of studying the dynamics of astrophysical disks in a two-dimensional formulation

A closed system of two-dimensional equations describing the dynamics of rotating, gravitating gas disks is derived. It is an integrodifferential system for barotropic disks and a differential system for polytropic disks. For both barotropic and polytropic disks, these equations differ both from the dynamical equations used in the literature for astrophysical disks and from the traditional equations of two-dimensional hydrodynamics. The sufficient conditions under which the dynamics of a disk can be described in a two-dimensional formulation are obtained. The first condition reflects the thin-disk approximation. The second condition imposes a limit on the characteristic times of processes studied in a two-dimensional formulation. In most cases this condition limits the characteristic frequency of a process to the disk's rotational frequency.Translated from Astrofizika, Vol. 39, No. 3, pp. 441–466, July–September, 1996.     

Instability of spherical accretion — I. Shock-free Bondi accretion

We examine the spatial stability of spherical adiabatic Bondi accretion on to a point gravitating mass against external perturbations. Both transonic critical and subsonic subcritical accretion are shown to be stable against purely radial acoustic, vortex or entropy perturbations. In the case of non-radial perturbations the amplitude of the perturbations grows without limit with smaller radii. Instability manifests itself only if the size of the accreting body is much less than the Bondi radius so that the inflow is highly supersonic or highly subsonic at the surface of the accretor in the case of critical or subcritical accretion respectively. These asymptotics hold and consequently the instability may develop for adiabatic index of accreting gas γ < 5/3. We suggest that this instability may lead to an essential thermalization of accreting flow thus, particularly, solving the problem of otherwise inefficient energy release in spherical accretion on to a black hole.     

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.     

Mutual gravitational potential,force, and torque of a homogeneous polyhedron and an extended body: an application to binary asteroids

Binary systems are quite common within the populations of near-Earth asteroids, main-belt asteroids, and Kuiper belt asteroids. The dynamics of binary systems, which can be modeled as the full two-body problem, is a fundamental problem for their evolution and the design of relevant space missions. This paper proposes a new shape-based model for the mutual gravitational potential of binary asteroids, differing from prior approaches such as inertia integrals, spherical harmonics, or symmetric trace-free tensors. One asteroid is modeled as a homogeneous polyhedron, while the other is modeled as an extended rigid body with arbitrary mass distribution. Since the potential of the polyhedron is precisely described in a closed form, the mutual gravitational potential can be formulated as a volume integral over the extended body. By using Taylor expansion, the mutual potential is then derived in terms of inertia integrals of the extended body, derivatives of the polyhedron’s potential, and the relative location and orientation between the two bodies. The gravitational forces and torques acting on the two bodies described in the body-fixed frame of the polyhedron are derived in the form of a second-order expansion. The gravitational model is then used to simulate the evolution of the binary asteroid (66391) 1999 KW4, and compared with previous results in the literature.