Emden-Chandrasekhar axisymmetric,solid-body rotating polytropes

The basic theory on polytorpes is revisited and EC polytropes are defined. The first-order approximation theory of Chandrasekhar (1933a, b, c) and Chandrasekhar and Lebovitz (1962) is reviewed, refined and extended in such a way that better results are obtained without involving hard analytical or numerical techniques. A more precise equation is given in defining non-outer equipotential surfaces, and a new method is adopted in determining the explicit expression of the gravitational potential. This method essentially consists in equating the expression of the gravitational potential and its first radial derivative determined by accounting for the equilibrium condition, with the corresponding expression of the gravitational potential and its first radial derivative determined by accounting for mass distribution. Such expressions are to be calculated at convenient points — for instance, at the centre and at the pole of the system. In this way, an infinity of exact solutions is derived for the special casesn=0 andn=1, and we then have the problem: ‘Which of the infinite number of solutions available leads to the most stable configuration?’ The simplest of these solutions is taken into account in detail for bothn=0 andn=1; results relative to the latter case allow us to solve the Kopal (1937) problem. EC polytropes withn=5 are found to consist of an inner massive non-rotating component and an outer zero-density rotating atmosphere. It is seen that they are equivalent in some respects to Roche systems, and the corresponding exact solution is derived. Explicit expressions for characteristic physical parameters are also determened in the general case, relative to sequences of equilibrium states characterized by constant masses and angular momenta. Detailed results are given for the special casesn=0, 1 and 5. Finally, some properties of both EC polytropes and R polytropes withn=0 (i.e., generalized Roche systems) are reported and discussed. The conclusions of this paper make it highly desirable to have an extension of the method used here to general values ofn.     

Emden-chandrasekhar axisymmetric,rigidly rotating polytropes

We determine equilibrium configuration of Emden-Chandrasekhar axisymmetric, solid-body rotating polytropes, defined as EC polytropes, for polytropic indices ranging from 0 (homogeneous bodies) to 5 (Roche-type bodies). To this aim, we improve Chandrasekhar's method to determine equilibrium configurations on two respects: namely, (a) no distinction exists between undistorted and distorted terms in the expression of the potential, and (b) the comparison between the expressions of gravitational potential and its first derivatives inside and outside the body has to be made on the boundary of a sphere of radius Ξ_E, which does not necessarily coincide with the undistorted Emden's sphere of radius \(\bar \xi _0 \geqslant \Xi _{\text{E}} \) . We also allow different values of \(\bar \xi _0 \) for different physical parameters, and choose a special set which best fits more refined results (involving more complicated and more expensive computer codes) by James (1964). We find an increasing agreement with increasing values of polytropic indexn and vice-versa, while a large discrepancy arises for 0≤n<1, which makes the approximations used here too much rough tobe accepted in this range. A real slight non-monotonic trend is exhibited by axial rations and masses related to rotational equilibrium configurations — i.e., when gravity at the equator is balanced by centrifugal force-with extremum points for 4.8<n<4.85 in both cases. The same holds for masses related to spherical configurations, as already pointed out by Seidov and Kuzakhmedov (1978). Finally, it is shown that isotrophic, one-component models of this paper might provide the required correlation between the ratio of a typical rotation velocity to a typical peculiar velocity and the ellipticity, for about \(\tfrac{3}{4}\) of elliptical systems for which observations are available.     

Structure of polytropic models of stars containing poloidal magnetic fields

Polytropic models of axially-symmetric equilibrium stars of infinite conductivity with poloidal magnetic fields are constructed by numerical integration of the exact equations governing internal structure. The mathematical method used, a further generalization and improvement of Stoeckly's method, allows the construction of a sequence of equilibrium models starting with a spherically symmetric star (when no magnetic field is present) and terminating with a doughnut-shaped object (for a very strong magnetic field) — a fact already shown by Monaghan. Detailed results are given only for two polytropes with the indexn=1.5 and 3.0, although any other value ofn greater than or equal to one could have been selected. Contrary to Monaghan's results, it is found that along the sequence of configurations forn=3.0 the ratio of the magnetic and gravitational energy peaks out before a doughnut-shaped configuration is reached; but this effect does not characterize then=1.5 sequence. The calculations confirm, however, another result of Monaghan asserting that the magnetic field is a fairly insensitive function of the polytropic index.     

Can elliptical galaxies be equilibrium systems?

This paper deals with the question of whether elliptical galaxies can be considered as equilibrium systems (i.e., the gravitational+centrifugal potential is constant on the external surface). We find that equilibrium models such as Emden-Chandrasekhar polytropes and Roche polytropes withn=0 can account for the main part of observations relative to the ratio of maximum rotational velocity to central velocity dispersion in, elliptical systems. More complex models involving, for example, massive halos could lead to a more complete agreement. Models that are a good fit to the observed data are characterized by an inner component (where most of the mass is concentrated) and a low-density outer component. A comparison is performed between some theoretical density distributions and the density distribution observed by Younget al. (1978) in NGC 4473, but a number of limitations must be adopted. Alternative models, such as triaxial oblate non-equilibrium configurations with coaxial shells, involve a number of problems which are briefly discussed. We conclude that spheroidal oblate models describing elliptical glaxies cannot be ruled out until new analyses relative to more refined theoretical equilibrium models (involving, for example, massive halos) and more detailed observations are performed.     

The equilibrium and stability of magnetopolytropes

The theory of the oscillations of axisymmetric gaseous configurations with a prevalent magnetic field is presented. The virial tensor method is used to obtain the nine second harmonic modes of oscillations of the system. It is found that out of the nine modes, three are neutral, four are non-radial, and two are coupled. For the Prendergast spherical model it is found that one of the coupled modes is radial and the other non-radial. Both the radial and the non-radial modes obtained in this case agree with the corresponding formulae obtained byChandrasekhar andLimber (1954) andWoltjer (1962).The equilibrium structure of gaseous polytropes with toroidal magnetic fields is also investigated in detail for values of the polytropic indexn=1, 1.5, 2, 3 and 3.5. For this model the components of the moment of intertia and potential energy tensors together with the non-zero components of the supermatrix potential are obtained. The final results in terms of the effect of weak toroidal magnetic fields on the characteristic frequencies of distorted polytropes are presented in the form of tables.     

Gravitational Radiation of Relativistic <Emphasis Type="Italic">n</Emphasis> = 1 Polytropes

The gravitational radiation of n = 1 polytropes undergoing quasiradial pulsations is examined. The intensity of the gravitational radiation and the gravitational wave amplitudes are calculated for polytropic models of white dwarfs and neutron stars when the energy of rotation of the object serves as the source of the radiated energy. Calculations of h₀ show that objects with a polytropic equation of state can describe the expected gravitational radiation from white dwarfs and neutron stars. The gravitational radiation of polytropic models of galactic nuclei and quasars is also examined. These objects can create a high enough background of gravitational radiation at frequencies of 10^-8–10^-11 Hz for gravitational wave detectors operating in this frequency range. __________ Translated from Astrofizika, Vol. 48, No. 4, pp. 603–612 (November 2005).     

The structure of the tidally and rotationally distorted polytropes

A study of the structure of tidally and rotationally distorted polytropes has been carried out using Monaghan and Roxburgh (1965) method for rapidly rotating polytropes, at the new interfacial points (Singh and Singh, 1982), ξ_f = 3.025, 3.470, and 4.950 for the polytropic models with the indicesn=1.5, 2.0, and 3.0, respectively. The structure has been studied for various mass ratios too.     

Potential energy of gravitationally interacting spherical galaxies

A theory has been developed for obtaining the potential energy of two interpenetrating spherically symmetric galaxies of unequal dimensions due to their mutual gravitational interaction. The mass distribution in both the galaxies is assumed to be that of a polytrope of integral index. A basic function that occurs in the theory has been tabulated for the cases of polytropes of indicesn=0 and 4 for four ratios of the radii.     

Stellar models with generalized polytropic equation of state

We find new classes of exact solutions to the Einstein field equations where the matter distribution satisfies a generalized polytropic equation of state. The matter distribution is uncharged with anisotropic pressures. Equations of state for polytropes and quark matter are contained as special cases. The matter variables and metric potentials can be obtained explicitly. Known solutions, for the choice of the gravitational potential made in this analysis, arise as special cases for particular choice of the equation of state parameters. A detailed physical analysis indicates that the model is well behaved.     

Physical characteristics ofN-dimensional,radially-symmetric polytropes

The physical characteristics radius, mass, mean density, gravitational potential and acceleration, gravitational and internal energy are presented with the aid of the gamma function forN-dimensional, radially-symmetric polytropes. The virial theorem with external pressure is derived in the relativistic limit, with Newtonian gravitation still valid. The gravitational energy of polytropes obeying the generalized Schuster—Emden integral is shown to be finite. Finiteness of mass and radius is discussed for the cases of practical interestN=1 (slab),N=2 (cylinder), andN=3 (sphere). Uniform contraction or expansion ofN-dimensional polytropes is considered in the last section.     

Equilibrium of self-gravitating polytropic cylinders with a magnetic field

The effect of a prevalent magnetic field on static and uniformly rotating self-gravitating cylinders of infinite length is examined. The magnetic field is assumed to consist ofH andH _z components, which are taken to be functions of the radial coordinate alone. A variety of magnetic-field configurations are shown to be admissible solutions of equations of motion, from which some feasible cases are presented. A particular magnetic-field configuration having bothH andH _z components is studied in detail. The configuration is such that the assumption of a polytropic equation of state reduces the equation governing the density function to a non-homogeneous cylindrical analogue of the Lane-Emden equation for spherical polytropes. The homogeneous case is also studied and shows interesting magnetic-field patterns.     

Equilibrium structure of polytropes with toroidal magnetic fields

Structure equations for the equilibrium of the gaseous polytropes with toroidal magnetic fields are solved numerically for two values of polytropic indexn-1.5 and 3, using a variant of Stoeckly's method. In addition to the structure parameters frequencies of the characteristic modes of oscillation are calculated. The results are considerably different from the values obtained by Anand for weak toroidal fields.     

Rosette Central Configurations, Degenerate Central Configurations and Bifurcations

In this paper we find a class of new degenerate central configurations and bifurcations in the Newtonian n-body problem. In particular we analyze the Rosette central configurations, namely a coplanar configuration where n particles of mass m₁ lie at the vertices of a regular n-gon, n particles of mass m₂ lie at the vertices of another n-gon concentric with the first, but rotated of an angle π /n, and an additional particle of mass m₀ lies at the center of mass of the system. This system admits two mass parameters μ = m₀/m₁ and ε = m₂/m₁. We show that, as μ varies, if n > 3, there is a degenerate central configuration and a bifurcation for every ε > 0, while if n = 3 there is a bifurcation only for some values of ε.     

Similarity between the structure and stability of isothermal and polytropic gas spheres

As shown by Chiéze, polytropic gas spheres imbedded in an ambient medium with a given pressure exhibit a relation between the mass and radius similar to isothermal gas spheres, providedn(polytropic index)>3 or <–1. Here, further investigation is made of the polytropic spheres. Not only the relation between radius and external pressure, but the relation between the central density ( _c ) and the mass (M) is similar to isothermal spheres and neutron star models. Dynamical calculation shows that the stability of polytropic spheres is lost at the configuration which corresponds to the first peak in the _c —M relation, as is the case for neutron star models in general relativity.     

Rotational distortion of polytropes by analytic approximation

This paper modifies the first-order perturbation theory of Chandrasekhar, for rotational distortion of polytropes. Comparison with numerical integrations by other authors demonstrates that the present analytic theory is as accurate as other published first-order theories. The present theory is in a form permitting rapid calculation of boundary shapes as a function of the rotation parameter,v, and the polytrope index,n. Results are presented for a critically rotating polytrope, for the casen=3.     

Bifurcation of central configuration in the 2N+1 body problem

The bifurcation of central configuration in the Newtonian N-body problem for any odd number N ≥ 7 is shown. We study a special case where 2n particles of mass m on the vertices of two different coplanar and concentric regular n-gons (rosette configuration) and an additional particle of mass m₀ at the center are governed by the gravitational law he 2n+1 body problem. This system is of two degrees of freedom and permits only one mass parameter μ =m ₀/m. This parameter μ controls the bifurcation. If n≥ 3, namely any odd N ≥ 7, then the number of central configurations is three when μ ≥ μ _c , and one when μ ≥ μ _c . By combining the results of the preceding studies and our main theorem, explicit examples of bifurcating central configuration are obtained for N ≤ 13, for any odd N ∈ [15,943], and for any N ≥ 945.     

Emden-chandrasekhar axisymmetric,rigidly rotating polytropes

An approximate analytical method of solving the polytropic equilibrium equations, first developed by Seidov and Kuzakhmedov (1978), has been extended and generalized to equilibrium configurations of axisymmetric systems in rigid rotation, with polytropic index,n =n _p + _n , nearn _p =0, 1, and 5. Though the details of the method depend on the value ofn _p , acceptable results are obtained for | _n | 0.5 to describe slowly rotating configurations in the range 0n1.5, 4.5n5. In the limit of rotational equilibrium configurations, when the distorsion may be large enough, a satisfactory approximation holds only in the range 0n, 1n1.5, 4.5n5.     

Evolution of Polytropic Reimann Disks in Halos

The equilibrium of elliptical Riemann disks with a polytropic equation of state and their evolution under the influence of viscosity and gravitational radiation inside spheroidal halos with a relative surface mass density k. The evolutionary trajectory of a disk inside a halo with k<0.5, which is analogous to the evolution of an isolated disk, differs from that of a disk inside a denser halo.