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.     

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.     

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.     

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.     

Emden-Chandrasekhar axisymmetric,rigid-body rotating polytropes

In connection with the basic theory reported in a previous paper (Paper I) for EC1 (rigidly rotating) polytropes, we define exact configurations as configurations for which the equilibrium equation has solutions which are infinitely close to some analytical function and the related gravitational potential coincides, in fact, with the gravitational potential due to mass distribution, at any point not outside the system. Then we restrict to the special casen=5 and divide the related polytropes into two components, a massive body where each mass element has a finite (polytropic) distance from the centre, and a massless atmosphere where each mass element has an infinite (polytropic) distance from te centre. It is found a single exact configuration exists, which under some assumptions may be related to Roche systems. In the special casen=0 it is shown a particular configuration, the spheroidal one, is an exact configuration and evidence is given that spheroidal configurations are the stablest among all the allowed (axisymmetric) configurations. It is also pointed out that EC1 polytropes withn=0 and incompressible MacLaurin spheroids belong to different sequences, even if they exhibit some common features. In the special casen=1 it is shown each allowed configuration is expressible by a convenient series development, which reduces to the relatedn=0 configuration by maintaining only the first two or the first one terms of the sum. It is also deduced, by analogy with the casen=0, that pseudospheroidal configurations are exact and the stablest among all the allowed (axisymmetric) configurations.     

Equilibrium structure of differentially rotating polytropic models of the stars

In this paper we propose a method for computing the equilibrium structure of differentially rotating polytropic models of the stars. A general law of differential rotation of the type ²=b ₀+b ₁ s ²+b ₂ s ⁴, which can account for a reasonably large variety of possible differential rotations in the stars has been used. The distortional effects have been incorporated in the structure equations up to second order of smallness in distortion parametersb ₀,b ₁, andb ₂ using Kippenhahn and Thomas' averaging approach in conjunction with Kopal's results on Roche equipotentials in manner similar to the one earlier used by Mohan and Saxena for computing the equilibrium structure of polytropes having solid body rotation. Numerical results have been obtained for various types of differentially rotating polytropic models of stars of polytropic indices 1.5, 3, and 4. Certain differentially rotating models of the Sun which are possible with such a type of law of differential rotation, have also been computed.     

The structure of tidally distorted polytropes

A numerical study of the structure of tidally-distorted polytropes has been carried out by using the Monaghan and Roxburgh (1965) method.     

A note on critically rotating polytropes

The structure of critically rotating polytropes is calculated using two (reated) Roche-type approximation schemes, one of which has been developed for stellar models. Comparisons with other results are made as a validation of these methods in stellar structure calculations.     

Equilibrium structure of stars obeying a generalized differential rotation law

In the present paper we have considered the problem of determining the equilibrium structure of differentially rotating stars in which the angular velocity of rotation varies both along the axis of rotation and in directions perpendicular to it. For this purpose, a generalized law of differential rotation of the type ² =b ₀+b ₁ s ²+b ₂ s ⁴+b ₃ z ²+b ₄ z ⁴+b ₅ z ² s ² (here is a nondimensional measure of the angular velocity of a fluid element distants from the axis of rotation andz from the plane through the centre of the star perpendicular to the axis of rotation, andb's are suitably chosen parameters) has been used. Whereas Kippenhahn and Thomas averaging approach has been used to incorporate the rotational effects in the stellar structure equations, Kopal's results on Roche equipotentials have been used to obtain the explicit form of the stellar structure equations, which incorporate the rotational effects up to second order of smallness in the distortion parameters. The method has been used to compute the equilibrium structure of certain differentially rotating polytropes. Certain differentially rotating polytropes. Certain differentially rotating models of the Sun have also been computed by using this approach.     

Comment on “Non-vacuum conformally flat space-times: dark energy”

Ibohal, Ishwarchandra and Singh (Ibohal et al., Astrophys. Space Sci. 335, 581, 2011) proposed a class of exact, non-vacuum and conformally flat solutions of Einstein’s equations whose stress tensor T _ab has negative pressure. We show that T _ab corresponds to an anisotropic fluid and the equation of state parameter seems not to be ω=?1/2. We consider the authors’ constant cannot be the mass of a test particle but is related to a Rindler acceleration of a spherical distribution of uniformly accelerating observers.     

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.     

Emden-chandrasekhar axisymmetric,solid-body rotating polytropes

According to the general results of a previous work (Caimmi, 1980; hereafter referred to as Paper I), solutions to EC equation, which expresses a necessary and sufficient condition for equilibrium of Emden-Chandresekhar axisymmetric, solid-body rotating polytropes (EC polytropes), are taken into consideration, of the type $$\vartheta (\xi ,\mu ) = A_0 \vartheta _0 (\upsilon ,\xi ) + \sum\limits_l^\infty {_l {\rm A}_{2l} (\upsilon )\vartheta _{2l} (\xi )P_{2l} (\mu ),} $$ with ? _2l later defined as the EC associated function of degree 2l. Thus the EC equation, involving (?, μ), is found to be equivalent to the infinite set of EC associated equations, involving ? _2l (μ). We approximate g (?, μ) by neglecting all terms of degree higher than 2 which appear in the above expression, and then search power series solutions to EC associated equations of degree 0 and 2, corresponding to any choice ofn (polytropic index, related to density distribution) andv (related to rotational distorsion). To this aim, we extend the methods used by Seidov and Kuzakhmedov (1977), and Mohan and Al-Bayaty (1980), to construct power series of the type outlined above, related to solid-body rotating configurations and originating both inside and outside the radial boundary (defined as the first zero of ?₀(μ)=0). The corresponding expressions of ?₀ and ?₂ may serve to derive an approximate expression of, and future work becomes possible concerning the determination of some physical parameters (such as volume, mass, potential energy, angular momentum) related to any choice ofn andv. Computations have been performed forn=k/4 (0≤k≤20, i.e. 0≤n≤5) andv=0,v≈v _R/2,v≈v _R, withv _R lowest value ofv leading to balance between gravitation and centrifugal force at the equator of the system. An upper limit to the error, ε^*(μ), done in computing ? _2l , ?? _2l , and ?? _2l at any point ? for a given choice ofn andv, is estimated, ranging from large values (ε^*=1E-2) forn close enough to 0 and ? close enough or outside the radial boundary, to low values (ε^*=1E-10) forn far enough from 0 and no constraint on ?. Comparison between results of this paper and the accurate results by Linnell (1977, 1981) obtained using a different approach and available forn=2,v=0, andn=3,v=0, lead to a fair agreement (up to (1E?5?1E?6). It is apparent that the method followed here continues to hold when the first EC associated functions up to degree 2l are taken into account, leading — at least in way of principle — to a more refined approximation to the EC function; this would only make the related calculations much more complicated.     

Dust ion-acoustic shock waves in nonextensive dusty plasma

A parametric survey on the propagation characteristics of the dust ion-acoustic (DIA) shock waves showing the effect of nonextesivity with nonextensive electrons in a dissipative dusty plasma system has been carried out using the reductive perturbation technique. We have considered continuity and momentum equations for inertial ions, q-distributed nonextensive electrons, and stationary charged dust grains, to derive the Burgers equation. It has been found that the basic features of DIA shock waves are significantly modified by the effects of electron nonextensivity and ion kinematic viscosity. Depending on the degree of nonextensivity of electrons, the dust ion-acoustic shock structures exhibit compression and rarefaction. The implications of our results would be useful to understand some astrophysical and cosmological scenarios like stellar polytropes, hadronic matter and quark-gluon plasma, protoneutron stars, dark-matter halos, etc., where effects of nonextensivity can play the significant roles.     

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.     

Small Fe bearing ring molecules of possible astrophysical interest: molecular properties and rotational spectra

A number of small, cyclic molecules containing several carbon atoms in their ring structure has been identified in different astrophysical environments. It is the aim of this work to study important molecular properties of such heterocyclic species bearing an iron atom, which is one of the most abundant cosmic elements. Quantum theoretical calculations based on a density functional approach have been employed to investigate physical properties of six small cyclic carbon and hydrocarbon molecules containing iron as a hetero atom, viz. FeC₂H _n and FeC₃H _n (n=0,2,4). The full geometry optimisation at the chosen level of electronic structure theory (B3LYP/6-31G(d)) including vibrational anharmonicities and non-rigidity, has furnished values for the rotational constants of these species to an expected accuracy of about one per cent. We present structural, electronic, vibrational, and rotational molecular properties including line frequencies, line strengths, and transition probabilities. These results may be helpful for identifying these molecules in future laboratory experiments in view of tentative astronomical observations.     

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.     

The equilibrium structure of loaded rotating polytropes

The equilibrium structure of rotating polytropes with a compact core has been studied by means of Chandrasekhar's first-order perturbation theory. Several numerical solutions are given. The results show that the larger the core mass, the smaller the critical central angular velocity will be, and for the same angular velocity, the larger the core mass, the more oblate the rotation ellipsoid will be.