Self-gravitating Fluid Research Articles

On estimates for volume and surface potentials in domains with boundary of class W 2 l

We prove sharp estimates in the norms of the Sobolev-Slobodetskii space W l2 , l > 0, for volume and surface potentials in domains with boundary of Sobolev class. The estimates are used in the study of some problems in hydrodynamics for self-gravitating fluids. Bibliography: 4 titles.

The topographic limits of gravitationally bound, rotating sand piles

Rubble pile asteroids can attain shapes that are dramatically different from those of rotating, self-gravitating equilibrium fluids. A new numerical technique, called “seed growth,” is demonstrated for calculating three-dimensional bodies that are self-gravitating and rotating, and whose every surface is approximately at a constant angle, ϕ, with respect to the local horizontal. By altering the configuration of cusps, which are points along a constant longitude path where the surface angle changes sign but not magnitude, multiple solution shapes that satisfy the condition that all surface slopes are at a constant angle are possible. Five different cusp configurations are explored here, three of which yield solutions for 20 ° ⩽ ϕ ⩽ 30 ° . Rotational effects are explored, and it is found that for some solution shapes, the ratios of their shortest to longest dimensions, c / a , can fall outside the limits published in the literature for rotating, cohesionless, spheroidal bodies. Solution shapes show some similarities to observed small bodies, such as the saturnian satellite Atlas, the near-Earth Asteroid 1999 KW4, and some contact binary asteroids.

The Monge–Ampère–Kantorovich approach to reconstruction in cosmology

Motion of a continuous fluid can be decomposed into an “incompressible” rearrangement, which preserves the volume of each infinitesimal fluid element, and a gradient map that transfers fluid elements in a way unaffected by any pressure or elasticity (the polar decomposition of Y. Brenier). The Euler equation describes a system whose kinematics is dominated by incompressible rearrangement. The opposite limit, in which the incompressible component is negligible, corresponds to the Zel’dovich approximation, a model of motion of self-gravitating fluid in cosmology. We present a method of approximate reconstruction of the large-scale proper motions of matter in the Universe from the present-day mass density field. The method is based on recovering the corresponding gradient transfer map. We discuss its algorithmics, tests of the method against mock cosmological catalogues, and its application to observational data, which results in tight constraints on the mean mass density Ω m and age of the Universe.

R fluids

A theory of collisionless fluids is developed in a unified picture, where nonrotating (?f1 = ?f2 = ?f3 = 0) figures with some given random velocity component distributions, and rotating (?f1 = ?f2 = ?f3 ) figures with a different random velocity component distributions, make adjoint configurations to the same system. R fluids are defined as ideal, self-gravitating fluids satisfying the virial theorem assumptions, in presence of systematic rotation around each of the principal axes of inertia. To this aim, mean and rms angular velocities and mean and rms tangential velocity components are expressed, by weighting on the moment of inertia and the mass, respectively. The figure rotation is defined as the mean angular velocity, weighted on the moment of inertia, with respect to a selected axis. The generalized tensor virial equations (Caimmi and Marmo 2005) are formulated for R fluids and further attention is devoted to axisymmetric configurations where, for selected coordinate axes, a variation in figure rotation has to be counterbalanced by a variation in anisotropy excess and vice versa. A microscopical analysis of systematic and random motions is performed under a few general hypotheses, by reversing the sign of tangential or axial velocity components of an assigned fraction of particles, leaving the distribution function and other parameters unchanged (Meza 2002). The application of the reversion process to tangential velocity components is found to imply the conversion of random motion rotation kinetic energy into systematic motion rotation kinetic energy. The application of the reversion process to axial velocity components is found to imply the conversion of random motion translation kinetic energy into systematic motion translation kinetic energy, and the loss related to a change of reference frame is expressed in terms of systematic motion (imaginary) rotation kinetic energy. A number of special situations are investigated in greater detail. It is found that an R fluid always admits an adjoint configuration where figure rotation occurs around only one principal axis of inertia (R3 fluid), which implies that all the results related to R3 fluids (Caimmi 2007) may be ex- tended to R fluids. Finally, a procedure is sketched for deriving the spin parameter distribution (including imaginary rotation) from a sample of observed or simulated large-scale collisionless fluids i.e. galaxies and galaxy clusters.

Topological and physical effects of rotation and spin in the general relativistic theory of gravitation

Interrelations of the intrinsic momentum (spin), rotation of material distributions, and intrinsic momentum of the gravitational field are investigated in the context of the general relativistic theory of gravitation involving the general relativity theory (GRT) and the Einstein-Cartan theory. It is demonstrated that the spin density vector of the gravitational field s g i is equal to the rotor of the tetrad reference point ωi=ɛiklm e k (a) e(a)l,m/2 to within the factor 1/κ (s g i =ω/κc). It is demonstrated that the vector s g i is proportional to the spin density vector of the gravitating field si (ω)=jc(Ψγiγ5Ψ)/2 as well as the pseudovector of space-time torsion Qi in the Einstein-Cartan theory, which in both cases induces a cubic nonlinearity of the spinor field. An expression for the energy-momentum density tensor of the eddy gravitational field is derived. It is also demonstrated that the free eddy gravitational field with polarized spin can form “mole holes.” An ideal fast-rotating self-gravitating fluid can cause a similar effect. The corresponding exact solutions of joint systems of the Einstein and rotating ideal fluid equations are presented.

Asymptotic solutions of convection in rapidly rotating non-slip spheres

Asymptotic solutions describing the onset of convection in rotating, self-gravitating Boussinesq fluid spheres with no-slip boundary conditions, valid for asymptotically small Ekman numbers and for all values of the Prandtl number, are derived. Central to the asymptotic analysis is the assumption that the leading-order convection can be represented, dependent on the size of the Prandtl number, by either a single quasi-geostrophic-inertial-wave mode or by a combination of several quasi-geostrophic-inertial-wave modes, and is controlled or influenced by the effect of the oscillatory Ekman boundary layer. Comparisons between the asymptotic solutions and the corresponding fully numerical simulations show a satisfactory quantitative agreement.

An approximate global solution of Einstein’s equations for a rotating finite body

We obtain an approximate global stationary and axisymmetric solution of Einstein’s equations which can be considered as a simple star model: a self-gravitating perfect fluid ball with constant mass density rotating in rigid motion. Using the post-Minkowskian formalism (weak-field approximation) and considering rotation as a perturbation (slow-rotation approximation), we find second-order approximate interior and exterior (asymptotically flat) solutions to this problem in harmonic and quo-harmonic coordinates. In both cases, interior and exterior solutions are matched, in the sense of Lichnerowicz, on the surface of zero pressure to obtain a global solution. The resulting metric depends on three arbitrary constants: mass density, rotational velocity and the star radius at the non-rotation limit. The mass, angular momentum, quadrupole moment and other constants of the exterior metric are determined by these three parameters. It is easy to check that Kerr’s metric cannot be the exterior part of that metric.

Asymptotic estimates for a semigroup related to compressible viscous fluid flow

The paper is related to the question of stability for the motionless spherically symmetric equilibrium states of viscous, barotropic, self-gravitating fluids. It considers a perturbation of the linearization of the governing equations of this problem, taking a step in the derivation of estimates which will allow us to prove non-linear stability of the equilibria. The perturbed operator, like the linearization considered earlier, generates an analytic semigroup, which allows us to derive asymptotic estimates as

Wave mechanics and the adhesion approximation

The dynamical equations describing the evolution of a self-gravitating fluid of cold darkmatter (CDM) can be written in the form of a Schrödinger equation coupled to a Poissonequation describing Newtonian gravity. It has recently been shown that, in the quasi-linearregime, the Schrödinger equation can be reduced to the exactly solvable free-particleSchrödinger equation. The free-particle Schrödinger equation forms the basis of a newapproximation scheme—the free-particle approximation—that is capable of evolvingcosmological density perturbations into the quasi-linear regime. The free-particleapproximation is essentially an alternative to the adhesion model in which the artificialviscosity term in Burgers’ equation is replaced by a non-linear term known as thequantum pressure. Simple one-dimensional tests of the free-particle method haveyielded encouraging results. In this paper we comprehensively test the free-particleapproximation in a more cosmologically relevant scenario by appealing to anN-body simulation. We compare our results with those obtained from two establishedmethods: the linearized fluid approach and the Zeldovich approximation. Wefind that the free-particle approximation comprehensively out-performs both ofthese approximation schemes in all tests carried out and thus provides anotheruseful analytical tool for studying structure formation on cosmological scales.

Equilibrium configurations of fluids and their stability in higher dimensions

We study equilibrium shapes, stability and possible bifurcation diagrams of fluids in higher dimensions, held together by either surface tension or self-gravity. We consider the equilibrium shape and stability problem of self-gravitating spheroids, establishing the formalism to generalize the MacLaurin sequence to higher dimensions. We show that such simple models, of interest on their own, also provide accurate descriptions of their general relativistic relatives with event horizons. The examples worked out here hint at some model-independent dynamics, and thus at some universality: smooth objects seem always to be well described by both ‘replicas’ (either self-gravity or surface tension). As an example, we exhibit an instability afflicting self-gravitating (Newtonian) fluid cylinders. This instability is the exact analogue, within Newtonian gravity, of the Gregory–Laflamme instability in general relativity. Another example considered is a self-gravitating Newtonian torus made of a homogeneous incompressible fluid. We recover the features of the black ring in general relativity.

Unstable modes of non-axisymmetric gaseous discs

We present a perturbation theory for studying the instabilities of non-axisymmetric gaseous discs. We perturb the dynamical equations of self-gravitating fluids in the vicinity of a non-axisymmetric equilibrium, and expand the perturbed physical quantities in terms of a complete basis set and a small non-axisymmetry parameter ǫ. We then derive a linear eigenvalue problem in matrix form, and determine the pattern speed, growth rate and mode shapes of the first three unstable modes. In nonaxisymmetric discs the amplitude and phase angle of travelling waves are functions of both the radius R and the azimuthal angle φ. This is due to the interaction of different wave components in the response spectrum. We demonstrate that wave interaction in unstable discs with small initial asymmetries, can develop dense clumps during the phase of exponential growth. Local clumps, which occur on the major spiral arms, can constitute seeds of gas giant planets in accretion discs.

Local stability of self-gravitating fluid disks made of two components in relative motion

Context. We consider a simple self-gravitating disk, made of two fluid components characterized by different effective thermal speeds and interacting with one another only through gravity; two-component models of this type have often been considered in order to estimate the impact of the cold interstellar medium on gravitational instabilities in star-dominated galaxy disks. Aims. This simple model allows us to produce a unified description of instabilities in non-viscous self-gravitating disks, some originating from Jeans collapse, and others from the relative motion between the two components. In particular, the model suggests that the small streaming velocity between the two components associated with the so-called asymmetric drift may be the origin of instability for suitable non-axisymmetric perturbations. Methods. The result is obtained by examining the properties of a local, linear dispersion relation for tightly wound density waves in such two-component model. The parameters characterizing the equilibrium model and the related dispersion relation allow us to recover as natural limits the cases, known in the literature, in which the relative drift between the two components is ignored. Results. Dynamically, the instability is similar to (although gentler than) that known to affect counter-rotating disks. However, in contrast to the instability induced by counter-rotation, which is a relatively rare phenomenon, the mechanism discussed in this paper is likely to be rather common in nature. Conclusions. We briefly indicate some consequences of the instability on the evolution of galaxy disks and possible applications to other astrophysical systems, in particular to protostellar disks and accretion disks.

Steady states of self-gravitating incompressible fluid in two dimensions

In this paper we develop a simple model for the steady state of two-dimensional self-gravitating incompressible gas which is based on the hydrodynamic equations for stratified fluid. These equations are then reduced to a system of two equations for the mass density and the gravitational field. Analytical analysis and numerical solutions of these equations under different modeling assumptions (with special attention to the isothermal case) are then used to study the structure of the resulting steady state of the fluid.

R3 fluids

The current paper is aimed at getting more insight on three main points concerning large-scale astrophysical systems, namely: (i) formulation of tensor virial equations from the standpoint of analytical mechanics; (ii) investigation on the role of systematic and random motions with respect to virial equilibrium configurations; (iii) determination of extent to which systematic and random motions are equivalent in flattening or elongating the shape of a mass distribution. The tensor virial equations are formulated regardless of the nature of the system and its constituents, by generalizing and extending a procedure used for the scalar virial equations in presence of discrete subunits (Landau and Lifchitz 1966). In particular, the self potential-energy tensor is shown to be symmetric with respect to the exchange of the indices, (Epot)pq = (Epot)qp. Then the results are extended to continuous mass distributions. The role of systematic and random motions in collisionless, ideal, self-gravitating fluids is analysed in detail including radial and tangential velocity dispersion on the equatorial plane, and the related mean angular velocity, ?, is conceived as a figure rotation. R3 fluids are defined as ideal, self-gravitating fluids in virial equilibrium, with systematic rotation around a principal axis of inertia, taken to be x3. The related virial equations are written in terms of the moment of inertia tensor, Ipq, the self potential-energy tensor, (Epot)pq, and the generalized anisotropy tensor, ?pq (Caimmi and Marmo 2005, Caimmi 2006a). Additional effort is devoted to the investigation of the properties of axisymmetric and triaxial configurations. A unified theory of systematic and random motions is developed for R3 fluids, taking into consideration imaginary rotation (Caimmi 1996b, 2006a), and a number of theorems previously stated for homeoidally striated Jacobi ellipsoids (Caimmi 2006a) are extended to the more general case of R3 fluids. The effect of random motion excess is shown to be equivalent to an additional real or imaginary rotation, respectively, inducing flattening (along the equatorial plane) or elongation (along the rotation axis). Then it is realized that a R3 fluid always admits an adjoint configuration with isotropic random velocity distribution. In addition, further constraints are established on the amount of random velocity anisotropy along the principal axes, for triaxial configurations. A necessary condition is formulated for the occurrence of bifurcation points from axisymmetric to triaxial configurations in virial equilibrium, which is independent of the anisotropy parameters. A particularization of general relations is made to the special case of homeoidally striated Jacobi ellipsoid, and some previously known results (Caimmi 2006a) are reproduced. .

A Geometric Approach to the Classification of the Equilibrium Shapes of Self-Gravitating Fluids

The classification of the equilibrium shapes that a self-gravitating fluid can take in a Riemannian manifold is a classical problem in Mathematical Physics. In this paper it is proved that the equilibrium shapes are isoparametric submanifolds. Some geometric properties of them are also obtained, e.g. classification and existence for some Riemannian spaces and relationship with the isoperimetric problem and the group of isometries of the manifold. Our approach to the problem is geometrical and allows to study the equilibrium shapes on general Riemannian spaces.

Linear perturbative theory of the discrete cosmologicalN-body problem

We present a perturbative treatment of the evolution under their mutual self-gravity of particles displaced off an infinite perfect lattice, both for a static space and for a homogeneously expanding space as in cosmological $N$-body simulations. The treatment, analogous to that of perturbations to a crystal in solid state physics, can be seen as a discrete (i.e. particle) generalization of the perturbative solution in the Lagrangian formalism of a self-gravitating fluid. Working to linear order, we show explicitly that this fluid evolution is recovered in the limit that the initial perturbations are restricted to modes of wavelength much larger than the lattice spacing. The full spectrum of eigenvalues of the simxple cubic lattice contains both oscillatory modes and unstable modes which grow slightly faster than in the fluid limit. A detailed comparison of our perturbative treatment, at linear order, with full numerical simulations is presented, for two very different classes of initial perturbation spectra. We find that the range of validity is similar to that of the perturbative fluid approximation (i.e. up to close to ``shell-crossing''), but that the accuracy in tracing the evolution is superior. The formalism provides a powerful tool to systematically calculate discreteness effects at early times in cosmological $N$-body simulations.

Static configurations and nonlinear waves in rotating nonuniform self-gravitating fluids

The equilibrium states and low-frequency waves in rotating nonuniform self-gravitating fluids are studied. The effect of a central object is included. Two-dimensional static configurations accounting for self-gravity, external gravity, and nonuniform rotation are considered for three models connecting the pressure with the mass density: thermodynamic equilibrium, polytropic pressure, and constant mass density. Explicit analytical solutions for equilibrium have been found in some cases. The low-frequency waves arising due to the vertical and horizontal fluid inhomogeneities are considered in the linear and nonlinear regimes. The relationship between the background pressure and mass density is supposed to be arbitrary in the wave analysis. It is shown that the waves considered can be unstable in the cases of polytropic pressure and constant mass density. The additional nonlinear term proportional to the product of the pressure and mass density perturbations, which is usually omitted, is kept in our nonlinear equations. There have been found conditions for this term to be important. Stationary nonlinear wave equations having solutions in the form of coherent vortex structures are obtained in a general form. The importance of involving real static configurations in the consideration of wave perturbations is emphasized.

Exploring the relativistic regime with Newtonian hydrodynamics: an improved effective gravitational potential for supernova simulations

We investigate the possibility approximating relativistic effects in hydrodynamical simulations of stellar core collapse and post-bounce evolution by using a modified gravitational potential in an otherwise standard Newtonian hydrodynamic code. Different modifications of a previously introduced effective relativistic potential are discussed. Corresponding hydrostatic solutions are compared with solutions of the TOV equations, and hydrodynamic simulations with two different codes are compared with fully relativistic results. One code is applied for one- and two-dimensional calculations with a simple equation of state and employs either the modified effective relativistic potential in a Newtonian framework or solves the general relativistic field equations under the assumption of the conformal flatness condition (CFC) for the three-metric. The second code allows for full-scale supernova runs including a microphysical equation of state and neutrino transport based on the solution of the Boltzmann equation and its moments equations. We present prescriptions for the effective relativistic potential for self-gravitating fluids to he used in Newtonian codes, which produce excellent agreement with fully relativistic solutions in spherical symmetry, leading to significant improvements compared to previously published approximations. Moreover, they also approximate qualitatively well relativistic solutions for models with rotation.

Envelope expansion with core collapse — II. Quasi-spherical self-similar solutions for an isothermal magnetofluid

We investigate self-similar magnetohydrodynamic (MHD) processes in an isothermal self-gravitating fluid with a quasi-spherical symmetry and extend the envelope expansion with core collapse (EECC) solutions of Lou & Shen by incorporating a random magnetic field. Magnetized expansion-wave collapse solutions (mEWCS) can be readily constructed as a special case. This inside-out MHD collapse occurs at the magnetosonic speed and magnetized EECC solutions are obtained systematically. Stagnation surfaces of EECC solutions that separate core collapse and envelope expansion propagate at constant speeds either sub-magnetosonically or super-magnetosonically. These similarity MHD solutions show various behaviours, such as radial inflow or contraction, core collapse, oscillations and outflow or wind as well as shocks. For solutions to go across the magnetosonic line smoothly, we carefully analyse topological properties of magnetosonic critical points. Without crossing the magnetosonic critical line, continuous spectra of magnetized EECC and envelope contraction with core collapse (ECCC) solutions are readily obtained. Crossing the magnetosonic line twice analytically, there exists an infinite number of discrete magnetized EECC and ECCC solutions. One or more sub-magnetosonic stagnation surfaces associated with these discrete solutions travel outwards at constant yet different speeds in a self-similar manner. In addition to the EECC shock solution, which could change the central accretion rate, the magnetic field can also affect the core accretion rate. As the magnetic parameter lambda increases, the core accretion rate appropriate for the MHD EWCS becomes larger. Under the frozen-in approximation, magnetic fields in the envelope expansion portion would scale as B proportional to r(-1), while in the core collapse portion they would scale as B proportional to r(-1/2). We discuss several astrophysical applications of EECC similarity solutions to the formation process of protoplanetary nebulae connecting the asymptotic giant branch phase and the planetary nebula, to supernova remnants, to collapse of magnetized molecular clouds, to H II clouds surrounding massive OB stars and to a certain evolution phase of galaxy clusters.

Spherical isothermal self-similar shock flows

We explore self-similar dynamical processes in a spherical isothermal self-gravitational fluid with an emphasis on shocks and outline astrophysical applications of such shock solutions. The previous similarity shock solutions of Tsai & Hsu and of Shu et al. may be classified into two types: Class I solutions with downstream being free-fall collapses; and Class II solutions with downstream being Larson‐Penston (LP) type solutions. By the analyses of Lou & Shen and Shen & Lou, we further construct similarity shock solutions in the ‘semicomplete space’. These general shock solutions can accommodate and model dynamical processes of radial outflows (wind), inflows (accretion or contraction), subsonic oscillations and free-fall core collapses all with shocks in various settings such as star-forming molecular clouds, ‘champagne flows’ in H II regions around luminous massive OB stars or surrounding distant quasars, dynamical connections between the asymptotic giant branch phase to the protoplanetary nebula phase with a central hot white dwarf as well as accretion shocks around compact objects such as white dwarfs, neutron stars and supermassive black holes. By a systematic exploration, we are able to construct families of infinitely many discrete Class I and Class II solutions matching asymptotically with a static outer envelope of a singular isothermal sphere; the shock solutions of Tsai & Hsu form special subsets. These similarity shocks travel at either subsonic or supersonic constant speeds. We also construct twin shocks as well as an ‘isothermal shock’ separating two fluid regions of two different yet constant temperatures. Ke yw ords: hydrodynamics ‐ stars: AGB and post AGB ‐ stars: formation ‐ H II regions ‐ planetary nebulae: general ‐ quasars: general.