Self-gravitating Polytropes Research Articles

The equilibrium of overpressurized polytropes

We investigate the impact of an external pressure on the structure of self-gravitating polytropes for axially symmetric ellipsoids and rings. The confinement of the fluid by photons is accounted for through a boundary condition on the enthalpy $H$. Equilibrium configurations are determined numerically from a generalised "Self-Consistent-Field"-method. The new algorithm incorporates an intra-loop re-scaling operator ${\cal R}(H)$, which is essential for both convergence and getting self-normalised solutions. The main control parameter is the external-to-core enthalpy ratio. In the case of uniform rotation rate and uniform surrounding pressure, we compute the mass, the volume, the rotation rate and the maximum enthalpy. This is repeated for a few polytropic indices $n$. For a given axis ratio, over-pressurization globally increases all output quantities, and this is more pronounced for large $n$. Density profiles are flatter than in the absence of an external pressure. When the control parameter asymptotically tends to unity, the fluid converges toward the incompressible solution, whatever the index, but becomes geometrically singular. Equilibrium sequences, obtained by varying the axis ratio, are built. States of critical rotation are greatly exceeded or even disappear. The same trends are observed with differential rotation. Finally, the typical response to a photon point source is presented. Strong irradiation favours sharp edges. Applications concern star forming regions and matter orbiting young stars and black holes.

The shape of a rapidly rotating polytrope with index unity

We show that the solutions obtained in the paper `An exact solution for arbitrarily rotating gaseous polytropes with index unity' by Kong, Zhang, and Schubert represent only approximate solutions of the free-boundary Euler-Poisson system of equations describing uniformly rotating, self-gravitating polytropes with index unity. We discuss the quality of such solutions as approximations to the rigidly rotating equilibrium polytropic configurations.

Perturbative solution to the Lane–Emden equation: an eigenvalue approach

Under suitable scaling, the structure of self-gravitating polytropes is described by the standard Lane-Emden equation (LEE), which is characterised by the polytropic index $n$. Here we use the known exact solutions of the LEE at $n=0$ and $1$ to solve the equation perturbatively. We first introduce a scaled LEE (SLEE) where polytropes with different polytropic indices all share a common scaled radius. The SLEE is then solved perturbatively as an eigenvalue problem. Analytical approximants of the polytrope function, the radius and the mass of polytropes as a function of $n$ are derived. The approximant of the polytrope function is well-defined and uniformly accurate from the origin down to the surface of a polytrope. The percentage errors of the radius and the mass are bounded by $8.1 \times 10^{-7}$ per cent and $8.5 \times 10^{-5}$ per cent, respectively, for $n\in[0,1]$. Even for $n\in[1,5)$, both percentage errors are still less than $2$ per cent.

Effective geometries in self-gravitating polytropes

Perturbations of a perfect barotropic and irrotational Newtonian self-gravitating fluid are studied using a generalization of the so-called ``effective geometry'' formalism. The case of polytropic spherical stars, as described by the Lane-Emden equation, is studied in detail in the known cases of existing explicit solutions. The present formulation gives a natural scenario in which the acoustic analogy has relevance for both stellar and galactic dynamics.

Anomalous diffusion and collapse of self-gravitating Langevin particles in D dimensions.

We address the generalized thermodynamics and the collapse of a system of self-gravitating Langevin particles exhibiting anomalous diffusion in a space of dimension D. This is a basic model of stochastic particles in interaction. The equilibrium states correspond to polytropic configurations similar to stellar polytropes and polytropic stars. The index n of the polytrope is related to the exponent of anomalous diffusion. We consider a high-friction limit and reduce the problem to the study of the nonlinear Smoluchowacute;ski-Poisson system. We show that the associated Lyapunov functional is the Tsallis free energy. We discuss in detail the equilibrium phase diagram of self-gravitating polytropes as a function of D and n, and determine their stability by using turning point arguments and analytical methods. When no equilibrium state exists, we investigate self-similar solutions of the nonlinear Smoluchowski-Poisson system describing the collapse. Our stability analysis of polytropic spheres can be used to settle the generalized thermodynamical stability of self-gravitating Langevin particles as well as the nonlinear dynamical stability of stellar polytropes, polytropic stars and polytropic vortices. Our study also has applications concerning the chemotactic aggregation of bacterial populations.

Instability of Rapidly Rotating Polytropes

The linear stability theory of rapidly rotating, self-gravitating polytropes is developed by an asymptotic, shallow-layer method. This reduces the general three-dimensional stability problem to an integro-differential eigenvalue problem (a Fredholm integral equation of the second kind) for normal modes. At leading order, the asymptotic analysis produces familiar, zero-thickness disk equations. In subsequent orders, stabilizing effects due to compressibility enter. We solve the stability equations numerically and construct approximate solutions using short-wavelength arguments. The eigenspectrum of a disk can have various forms; instabilities of a pressure-less configuration form a continuous piece of the spectrum, but polytropic disks can have discrete, unstable eigenvalues. A further example is provided by the rigidly rotating disk, which, at leading order, can be solved exactly.

Equilibria of rapidly rotating polytropes

The equilibrium structures of self-gravitating polytropes are found in the limit of rapid rotation by the method of matched asymptotic expansions. For physical values of the polytropic index, the discs are thin throughout most of their radii, but they flare out at their edges where the local Jeans length of the configuration becomes large. This signals a breakdown in the asymptotic expansion that we correct by including a boundary layer at the rim of the disc. A variety of results are illustrated for differing rotation curves and polytropic indices

Entropy of nonsingular self-gravitating polytropes and their TOV equation

This paper has discussed the behaviour of total entropy of self-gravitating polytropes in the nonsingular region and proved that the self-gravitating polytropes in the nonsingular region satisfy the equation of general relativistic hydrostatic equilibrium (TOV equation). We obtain that the magnitude order of total entropy of the polytropes with massM⊙ is (1040÷1041) erg·K−1 by calculating the nonsingular entropy of self-gravitating polytropes with state equationp=Kπγ, whereK≠−1, −3−2√2 0, Γ is a constant.

Turbulence in Interstellar Clouds

Above masses of the order of lOO M⊙, molecular clouds have masses and sizes which scale like those of self-gravitating polytropes bounded by an external pervading pressure. It is unlikely that this scaling is due to mere observational bias. But the physics underlying this behaviour is far from being understood. In particular, the possible contribution of turbulence to both the ambient pressure and the internal pressure (whose dependence with the density would mimic a polytropic behavior) is a difficult and much debated issue. The clouds mass, size and internal velocity dispersion are such that they are observed to be in approximate virial balance between their self-gravity, the surface energy term due to the ambient pressure and their internal energy. The latter is dominated by the kinetic energy of disordered internal motions. However, there has been little evidence so far that these motions are actually turbulent rather than simply disordered. The transition to turbulence in a flow occurs when the non linear advection term in the momentum equation, v.Δv, considerably exceeds the viscous dissipation term, vΔv (wherevis the kinematic viscosity). Non linearities therefore dominate the physics of a turbulent flow and the velocities are not randomly distributed. Most of the previous attempts to determine a well-defined correlation length in the velocity field (Kleiner and Dickman 1985,aandb;Scalo 1984), which is predicted to be close to the scale at which the energy is injected, or to characterize the expected hierarchical structure (Pérault et al. 1986) have been plagued by the lack of dynamical range in the data set and the range of scales over which the correlation functions have been computed. The most plausible determination, that of Kleiner and Dickman (1987) who claim to have found a correlation length of0.2 pcin the Taurus cloud, gives a result which is so close to the angular resolution of the observations that it is doubtful.

Circumstellar ring formation in rapidly rotating protostars

view Abstract Citations (65) References (32) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Circumstellar Ring Formation in Rapidly Rotating Protostars Williams, Harold A. ; Tohline, Joel E. Abstract A three-dimensional hydrodynamic computer program is used to study rapidly rotating, self-gravitating polytropes with polytropic indices n = 0.8 and n = 1.8. The initial ratio of the rotational to gravitational potential energy is βi = 0.310 in both models. Both models, initially axisymmetric, are dynamically unstable toward the development of a nonaxisymmetric, two-armed spiral structure. Here the models are shown as they evolve to extremely nonlinear amplitudes: the end result in both cases is a type of fission. The models shed a fraction of their mass and angular momentum, producing a ring which surrounds a more centrally condensed object with βd= 0.27. The central object is a triaxial figure that is rotating about its shortest axis. The low-mass, roughly axisymmetric ring probably is capable of condensing into planets. Publication: The Astrophysical Journal Pub Date: November 1988 DOI: 10.1086/166849 Bibcode: 1988ApJ...334..449W Keywords: Protostars; Star Formation; Stellar Envelopes; Stellar Rotation; Angular Momentum; Computerized Simulation; Hydrodynamics; Molecular Clouds; Potential Energy; Astrophysics; HYDRODYNAMICS; PLANETS: GENERAL; STARS: CIRCUMSTELLAR SHELLS; STARS: ROTATION full text sources ADS |

Linear and nonlinear dynamic instability of rotating polytropes

A three-dimensional hydrodynamic computer program is used to study the growth of nonaxisymmetric structures in rapidly rotating, self-gravitating polytropes. Models with polytropic index n = 0.8, 1.0, 1.3, 1.5, and 1.8 are studied. The initially axisymmetric equilibria are constructed by the Ostriker-Mark self-consistent-field method. The nonaxisymmetric pattern that develops out of low-amplitude random noise is a two-armed spiral with a well-defined pattern speed and growth rate which closely match properties of the toroidal mode predicted from the linear, second-order tensor-virial equation. A Fourier analysis of each polytrope's azimuthal density distribution shows that, even in the linear amplitude regime, higher-order angular patterns also develop exponentially in time. The higher-order patterns ultimately move in synchronization with the broad two-armed spiral, creating a narrow two-armed spiral. As the polytropic index is decreased, a more open and centrally more barlike pattern develops.

QUIPS - Time-dependent properties of quasi-invariant self-gravitating polytropes

view Abstract Citations (18) References (9) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS QUIPS - Time-dependent properties of quasi-invariant self-gravitating polytropes Munier, A. ; Feix, M. R. Abstract A time parameter is introduced into Emden's static solution in a quasi-invariant time-dependent analysis of the expansion and/or contraction of a self-gravitating sphere of perfect gas with a predetermined polytropic index. Oscillations of the gas sphere are ignored, and transformations are defined in a generalized canonical form, i.e., to ensure a linear time transformation. Additionally, the Hamiltonian is formulated to be invariant, and a dual space is defined which includes friction, a sink term, and a centripetal/centrifugal force. A characteristic frequency is obtained that is related to Jean's frequency, and is significant for the quasi-invariant polytrope. The internal structure of the gas sphere is found to be steady whether in contraction or expansion, whereas an expansion is stable while a contraction is not. It is noted that the numerical model developed is not periodic. Publication: The Astrophysical Journal Pub Date: April 1983 DOI: 10.1086/160874 Bibcode: 1983ApJ...267..344M Keywords: Hydrodynamics; Polytropic Processes; Stellar Gravitation; Stellar Oscillations; Stellar Structure; Time Dependence; Canonical Forms; Gas Expansion; Gravitational Effects; Hamiltonian Functions; Ideal Gas; Linear Transformations; Spheres; Stellar Models; Astrophysics full text sources ADS |