A continuous system such as a galactic disc is shown to be well approximated by an N-ring differentially rotating self-gravitating system. Lowest order (m=1) non-axisymmetric features such as lopsidedness and warps are global in nature and quite common in the discs of spiral galaxies. Apparently these two features of the galactic discs have been treated like two completely disjoint phenomena. The present analysis based on an eigenvalue approach brings out clearly that these two features are fundamentally similar in nature and they are shown to be very Low frequency Normal Mode (LNM) oscillations manifested in different symmetry planes of the galactic disc. Our analysis also show that these features are actually long-lived oscillating pattern of the N-ring self-gravitating system.
We investigate the structure of dynamics of large self-gravitating astrophysical systems using a self-interacting two-component model. We consider two cases, galaxy clusters and cosmic walls, for illustrations. In both cases stability analyses are conducted using perturbative expansion. We have found that waves and solitons are easily generated in these systems. Our analysis shows that dark matter can be Jeans unstable in the very inner regions of galaxy clusters if it has a large internal degree of freedom. The dark matter core may collapse under external perturbations. We also discuss dark-m…
We summarize the general formalism describing surface flows in three-dimensional space in a form which is suitable for various astrophysical applications. We then apply the formalism to the analysis of non-radial perturbations of self-gravitating spherical fluid shells. Spherically symmetric gravitating shells (or bubbles) have been used in numerous model problems especially in general relativity and cosmology. A radially oscillating shell was recently suggested as a model for a variable cosmic object. Within Newtonian gravity we show that self-gravitating static fluid shells are unstable with…
This review starts with a discussion of the hierarchy of scales, relevant to the description of superfluids in neutron stars, which motivates a subsequent elementary exposition of the Newtonian superfluid hydrodynamics. Starting from the Euler equations for a superfluid and a normal fluid we apply the tensor virial method to obtain the virial equations of the first, second, and third order and to compute their Eulerian perturbations. Special emphasis is put on the computation of perturbations of the new terms due to mutual gravitational attraction and mutual friction between the two fluids. Th…
We investigate the orbital evolution of planetesimals in a self-gravitating circumstellar disc in the size regime ($\sim 1-5000$ km) where the planetesimals behave approximately as test particles in the disc's non-axisymmetric potential. We find that the particles respond to the stochastic, regenerative spiral features in the disc by executing large random excursions (up to a factor of two in radius in $\sim 1000$ years), although typical random orbital velocities are of order one tenth of the Keplerian speed. The limited time frame and small number of planetesimals modeled does not permit us …
We study the probability of close encounters between stars from a nuclear cluster and a massive black hole. The gravitational field of the system is dominated by the black hole in its sphere of influence. It is further modified by the cluster mean field (a spherical term) and a gaseous disc/torus (an axially symmetric term) causing a secular evolution of stellar orbits via Kozai oscillations. Intermittent phases of large eccentricity increase the chance that stars become damaged inside the tidal radius of the central hole. Such events can produce debris and lead to recurring episodes of enhanc…
Flow nonnormality induced linear transient phenomena in thin self-gravitating astrophysical discs are studied in the shearing sheet approximation. The considered system includes two modes of perturbations: vortex and (spiral density) wave. It is shown that self-gravity considerably alters the vortex mode dynamics -- its transient (swing) growth may be several orders of magnitude stronger than in the non-self-gravitating case and 2-3 times larger than the transient growth of the wave mode. Based on this finding, we comment on the role of vortex mode perturbations in a gravitoturbulent state. Al…
The issue of fragmentation in self-gravitating gaseous accretion discs has implications both for the formation of stars in discs in the nuclei of active galaxies, and for the formation of gaseous planets or brown dwarfs in circumstellar discs. It is now well established that fragmentation occurs if the disc is cooled on a timescale smaller than the local dynamical timescale, while for longer cooling times the disc reaches a quasi-steady state in thermal equilibrium, with the cooling rate balanced by the heating due to gravitational stresses. We investigate here how the fragmentation boundary d…
(Abridged) We develop an analytical spectral method to solve the equations of equilibrium for a self-gravitating, magnetized fluid body, under the only hypotheses that (a) the equation of state is isothermal, (b) the configuration is scale-free, and (c) the body is electrically neutral. All physical variables are represented as series of scalar and vector spherical harmonics of degree l and order m, and the equilibrium equations are reduced to a set of coupled quadratic algebraic equations for the expansion coefficients of the density and the magnetic vector potential. The method is general, a…
The steady-state structure of a disc with a corona is analyzed when the vertical component of the gravitational force due to the self-gravity of the disc is considered. For the energy exchange between the disc and the corona, we assume a fraction f of the dissipated energy inside the accretion disc is transported to the corona via the magnetic tubes. Analytical solutions corresponding to a prescription for f (in which this parameter directly depends on the ratio of the gas pressure to the total pressure) or free f are presented and their physical properties are studied in detail. We show that …
In this paper we examine the issue of characterising the transport associated with gravitational instabilities in relatively cold discs, discussing in particular the conditions under which it can be described within a local, viscous framework. We present the results of global, three-dimensional, SPH simulations of self-gravitating accretion discs, in which the disc is cooled using a simple parametrisation for the cooling function. Our simulations show that the disc settles in a ``self-regulated'' state, where the axisymmetric stability parameter $Q\approx 1$ and where transport and energy diss…
An unsolved issue in the standard core accretion model for gaseous planet formation is how kilometre-sized planetesimals form from, initially, micron-sized dust grains. Solid growth beyond metre sizes can be difficult both because the sticking efficiency becomes very small, and because these particles should rapidly migrate into the central star. We consider here how metre-sized particles evolve in self-gravitating accretion discs using simulations in which the gravitational influence of the solid particles is also included. Metre-sized particles become strongly concentrated in the spiral stru…
We study effects of winds on the time evolution of isothermal, self-gravitating accretion discs by adopting a radius dependent mass loss rate because of the existence of the wind. Our similarity and semi-analytical solution describes time evolution of the system in the slow accretion limit. The disc structure is distinct in the inner and outer parts, irrespective of the existence of the wind. We show that existence of wind will lead to a reduction of the surface density in the inner and outer parts of the disc in comparison to a no-wind solution. Also, the radial velocity significantly increas…
For a few nearby active nuclei, the disc orbiting the black hole is traced by water maser emission. By combining a simple model together with observed velocity profiles, we show that it is possible to put constraints on the black hole mass and on the distribution of matter (shape, density, size) in the outer disc. We then report possible parameters for the non-keplerian disc and for the black hole in NGC 1068, and mention an uncertainty of at least 25% on the central mass in NGC 4258.
We discuss the nature of phase transitions in the self-gravitating Fermi gas at non-zero temperature. This study can be relevant for massive neutrinos in Dark Matter models and for collisionless self-gravitating systems experiencing a ``violent relaxation''. Below a critical energy (in the microcanonical ensemble) or below a critical temperature (in the canonical ensemble), the system undergoes a gravitational collapse leading to a compact object with a massive degenerate nucleus (fermion ball).
In this paper we examine the issue of characterizing the transport associated with gravitational instabilities in relatively cold discs, discussing in particular under which condition it can be described within a local, viscous framework. We present the results of global, three-dimensional, SPH simulations of self-gravitating accretion discs, in which the disc is cooled using a simple parameterization for the cooling function. Our simulations show that the disc settles in a ``self-regulated'' state, where the axisymmetric stability parameter $Q\approx 1$ and where transport and energy dissipat…
We study the perturbative dynamics of an infinite gravitating Nambu-Goto string within the general-relativistic perturbation framework. We develop the gauge invariant metric perturbation on a spacetime containing a self-gravitating straight string with a finite thickness and solve the linearized Einstein equation. In the thin string case, we show that the string does not emit gravitational waves by its free oscillation in the first order with respect to its oscillation amplitude, nevertheless the string actually bends when the incidental gravitational waves go through it.
In this paper we consider the evolution of small planetesimals in marginally stable, self-gravitating protoplanetary discs. The drag force between the disc gas and the embedded planetesimals generally causes the planetesimals to drift inwards through the disc at a rate that depends on the particle size. In a marginally stable, self-gravitating disc, however, the planetesimals are significantly influenced by the non-axisymmetric spiral structures resulting from the growth of the gravitational instability. For small particles, that are strongly coupled to the disc gas, and for large particles, t…
Self-gravitating stellar disks with random motion support both exponentially growing and, in some cases, purely oscillatory axisymmetric bending modes, unlike their cold disk counterparts. A razor-thin disk with even a very small degree of random motion in the plane is both unstable and possesses a discrete spectrum of neutral modes, irrespective of the sharpness of the edge. Random motion normal to the disk plane has a stabilizing effect but at the same time allows bending waves to couple to the internal vibrations of the particles, which causes the formerly neutral modes to decay through Lan…
I review recent progresses in the dynamics and the evolution of self-gravitating accretion discs. Accretion discs are a fundamental component of several astrophysical systems on very diverse scales, and can be found around supermassive black holes in Active Galactic Nuclei (AGN), and also in our Galaxy around stellar mass compact objects and around young stars. Notwithstanding the specific differences arising from such diversity in physical extent, all these systems share a common feature where a central object is fed from the accretion disc, due to the effect of turbulence and disc instabilit…
We revisit the modeling of ion-neutral (or ambipolar) diffusion with two fluid smoothed particle hydrodynamics, as discussed by Hosking & Whitworth. Some parts of the technique are optimized to testify the pioneer works on behavior of the ambipolar diffusion in an isothermal self-gravitating layer. The frictional heating by ambipolar diffusion is examined, and its effect on fragmentation of the layer is studied. The results are compared to the thermal phases of instability as obtained by Nejad-Asghar.
An efficient algorithm for solving Poisson's equation in two and three spatial dimensions is discussed. The algorithm, which is described in detail, is based on the integral form of Poisson's equation and utilizes spherical coordinates and an expansion into spherical harmonics. The solver can be applied to and works well for all problems for which the use of spherical coordinates is appropriate. We also briefly discuss the implementation of the algorithm into hydrodynamic codes which are based on a conservative finite--difference scheme.
The equations governing the vertical structure of a stationary keplerian accretion disc are presented. The model is based on the alpha-viscosity, includes self-gravity, convective transport and turbulent pressure. A few properties of the model are discussed for circumstellar and AGN discs. We show the strong sensitivity of the disc structure to the viscous energy deposition towards the vertical axis, specially when entering inside the self-gravitating part of the disc. The local version of the alpha-prescription leads to a "singular" behavior which is also predicted by the vertically averaged …
It has already been shown, using a local model, that accretion discs with cooling times t_cool <= 3 Omega^-1 fragment into gravitationally bound objects, while those with cooling times t_cool > 3 Omega^-1 evolve into a quasi-steady state. We present results of three-dimensional simulations that test if the local result still holds globally. We find that the fragmentation boundary is close to that determined using the local model, but that fragmentation may occur for longer cooling times when the disc is more massive or when the mass is distributed in such a way as to make a particular region o…
Accretion flows with pressure gradients permit the existence of standing waves which may be responsible for observed quasi-periodic oscillations (QPO's) in X-ray binaries. We present a comprehensive treatment of the linear modes of a hydrodynamic, non-self-gravitating, polytropic slender torus, with arbitrary specific angular momentum distribution, orbiting in an arbitrary axisymmetric spacetime with reflection symmetry. We discuss the physical nature of the modes, present general analytic expressions and illustrations for those which are low order, and show that they can be excited in numeric…
We discuss the virial balance of all members of a cloud ensemble in numerical simulations of self-gravitating MHD turbulence. We first discuss the choice of reference frame for evaluating the terms entering the virial theorem (VT), concluding that the balance of each cloud should be measured in its own reference frame. We then report preliminary results suggesting that a) the clouds are far from virial equilibrium, with the ``geometric'' (time derivative) terms dominating the VT. b) The surface terms in the VT are as important as the volume ones, and tend to decrease the action of the latter. …
A modern approach to the evolution of a mixed (stars and gas) self-gravitating system is the fully Lagrangian particle approach. The gaseous (particle) phase differs from the compact because the mutual force is given by the sum of gravity and pressure gradient. In this note we report of some characteristics, advantages and limitations of this approach for what regards the evaluation of forces in the system. In particular, a comparison between classic tree-code and fast multipole algorithm to evaluate gravitational forces is discussed.
Using a local model Gammie (2001) has shown that accretion discs with cooling times t_cool <= 3/Omega fragment into gravitationally bound objects, while those with cooling times t_cool > 3/Omega evolve into a quasi-steady state. We use three-dimensional smoothed particle hydrodynamic simulations of protoplanetary accretion discs to test if the local results hold globally. We find that for disc masses appropriate for T Tauri discs, the fragmentation boundary still occurs at a cooling time close to t_cool = 3/Omega. For more massive discs, which are likely to be present at an earlier stage of th…
An improved linear stability theory of small-amplitude oscillations of a self-gravitating, infinitesimally thin gaseous disk of spiral galaxies has been developed. It was shown that in the differentially rotating disks for nonaxisymmetric perturbations Toomre's modified hydrodynamical simulations to test the validity of the modified local criterion.
A new phase of temporal evolution of the one-dimensional self-gravitating system is numerically discovered. Fractal structure is dynamically created from non-fractal initial conditions. Implication to astrophysics and mathematical physics is discussed.
In this paper and a companion work, we report on the first global numerical simulations of self-gravitating magnetized tori, subject in particular to the influence of the magnetorotational instability (MRI). In this work, paper I, we restrict our calculations to the study of the axisymmetric evolution of such tori. Our goals are twofold: (1) to investigate how self-gravity influences the global structure and evolution of the disks; and (2) to determine whether turbulent density inhomogeneities can be enhanced by self-gravity in this regime. As in non self-gravitating models, the linear growth …
Ordinary differential equations describing the self-similar collapse of a rotating, magnetized, self-gravitating and isothermal filament are derived. Explicit homologous solutions are studied with special emphasis on the bifurcation that occurs at the magnetosonic critical point. It is shown that there is a critical value for the toroidal magnetic field slope at the origin above which no bifurcation occurs, the solution remains homologous, and below which the density and the poloidal magnetic field tend to zero at large radius (envelope) whereas the toroidal magnetic field and azimuthal veloci…
Discreteness effects are a source of uncontrolled systematic errors of N-body simulations, which are used to compute the evolution of a self-gravitating fluid. We have already developed the so-called "Particle Linear Theory" (PLT), which describes the evolution of the position of self-gravitating particles located on a perturbed simple cubic lattice. It is the discrete analogue of the well-known (Lagrangian) linear theory of a self-gravitating fluid. Comparing both theories permits to quantify precisely discreteness effects in the linear regime. It is useful to develop the PLT also for other p…
It is shown that the diluted thermodynamical limit of a self-gravitating system proposed by de Vega and Sanchez suffers from the same problems as the usual thermodynamical limit and leads to divergent thermodynamical functions. This question is also discussed from the point of view of mean field theory.
After introducing the fundamental properties of self-gravitating systems, we present an application of Tsallis' generalized entropy to the analysis of their thermodynamic nature. By extremizing the Tsallis entropy, we obtain an equation of state known as the stellar polytrope. For a self-gravitating stellar system confined within a perfectly reflecting wall, we discuss the thermodynamic instability caused by its negative specific heat. The role of the extremum as a quasi-equilibrium is also demonstrated from the results of N-body simulations.
The shear free condition is studied for dissipative relativistic self-gravitating fluids in the quasi-static approximation. It is shown that, in the Newtonian limit, such condition implies the linear homology law for the velocity of a fluid element, only if homology conditions are further imposed on the temperature and the emission rate. It is also shown that the shear-free plus the homogeneous expansion rate conditions are equivalent (in the Newtonian limit) to the homology conditions. Deviations from homology and their prospective applications to some astrophysical scenarios are discussed, a…
We discuss the nature of phase transitions in self-gravitating systems both in the microcanonical and in the canonical ensemble. We avoid the divergence of the gravitational potential at short distances by considering the case of self-gravitating fermions and hard spheres models. Three kinds of phase transitions (of zeroth, first and second order) are evidenced. They separate a ``gaseous'' phase with a smoothly varying distribution of matter from a ``condensed'' phase with a core-halo structure. We propose a simple analytical model to describe these phase transitions. We determine the value of…
We compute statistical equilibrium states of rotating self-gravitating systems enclosed within a box by maximizing the Fermi-Dirac entropy at fixed mass, energy and angular momentum. We increase the rotation up to the Keplerian limit and describe the flattening of the configuration until mass shedding occurs. At the maximum rotation, the system develops a cusp at the equator. We draw the equilibrium phase diagram of the rotating self-gravitating Fermi gas and discuss the structure of the caloric curve as a function of degeneracy parameter and angular velocity. We argue that systems described b…
This work assembles some basic theoretical elements on thermal equilibrium, stability conditions, and fluctuation theory in self-gravitating systems illustrated with a few examples. Thermodynamics deals with states that have settled down after sufficient time has gone by. Time dependent phenomena are beyond the scope of this paper. While thermodynamics is firmly rooted in statistical physics, equilibrium configurations, stability criteria and the destabilizing effect of fluctuations are all expressed in terms of thermodynamic functions. The work is not a review paper but a pedagogical introduc…
A class of five-dimensional warped solutions is presented. The geometry is everywhere regular and tends to five-dimensional anti-de Sitter space for large absolute values of the bulk coordinate. The physical features of the solutions change depending on the value of an integer parameter. In particular, a set of solutions describes generalized gravitating kinks where the scalar field interpolates between two different minima of the potential. The other category of solutions describes instead gravitating defects where the scalar profile is always finite and reaches the same constant asymptote bo…
The perturbative modes propagating along an infinite string are investigated within the framework of the gauge invariant perturbation formalism on a spacetime containing a self-gravitating straight string with a finite thickness. These modes are not included in our previous analysis. We reconstruct the perturbation formalism to discuss these modes and solve the linearized Einstein equation within the first order with respect to the string oscillation amplitude. In the thin string case, we show that the oscillations of an infinite string must involve the propagation of cosmic string traveling w…
It is natural important question for us to ask what the nonextensive parameter stands for when Tsallis statistics is applied to the self-gravitating systems. In this paper, some properties of the nonextensive parameter and Tsallis equilibrium distribution for the self-gravitating system are discussed in the framework of nonextensive kinetic theory. On the basis of the solid mathematical foundation, the nonextensive parameter can be expressed by a formula with temperature gradient and the gravitational potential and can be presented the physical meaning with regard to the non-isothermal (nonequ…
