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.     

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,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.     

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.     

Magnetically distorted polytropes

The fundamental frequencies of the non-radial mode of oscillation belonging to the second harmonic (l=2) of magnetically distorted polytropic gas spheres are evaluated in the second approximation by a variational method. The magnetic field is assumed to have both the toroidal and the poloidal components. We find that the frequencies of oscillation are increased due to the presence of the magnetic field and that these depend only slightly on the value of , the ratio of the specific heats. We have also determined the value of <1+1/n for the mode of oscillation which exhibits convective instability. This value is lower than the one which is obtained in the absence of a magnetic field.     

Magnetically distorted polytropes

The oscillations of a gaseous polytrope with a magnetic field having both a toroidal and a poloidal component are examined using the second-order tensor virial equations on the assumption that the magnetic energy is small compared with the gravitational energy. The frequencies of oscillation of the transverse shear, the toroidal and the coupled pulsation modes are tabulated for polytropic indicesn=1, 1.5, 2, 3 and 3.5. It is found that the magnetic field decreases the frequency of oscillation of (i) the transverse shear mode and (ii) the mode which starts as a radial pulsation in the absence of a magnetic field while it increases the frequency of oscillation of (i) the toroidal mode and (ii) the Kelvin mode. In all cases the shift in frequency decreases with increasingn.     

Rotational distortion of polytropes by analytic approximation,III

Paper II of this series was less rigorous than desirable, particularly in the fitting of internal and external potentials. A method is developed in this paper properly to fit the external and internal potentials on the distorted configuration boundary. Displacement of the fiting radius from the radius of a nonrotating configuration in general changes the distorted boundary location by an insignificant amount.A comparison of this theory with the numerical integrations of James (1964) shows excellent agreement to at least an angular velocity equalling 70% of critical rotation. The comparison is for polytropic indices of 1.0, 1.5, 2.0 and 3.0.     

Constant-mass sequences of differentially-rotating polytropes

In this paper, differentially-rotating polytropic constant-mass sequences are compated by implementation of the so-called constant units technique. Then, numerical results concerning such constant-mass sequences are compared with their respective values obtained when corresponding constant-centraldensity sequences are computed.     

Rotational distortion of polytropes by analytic approximation,II

Chandrasekhar's (1933) paper on rotational distortion of polytropes contained a perturbation term in the potential which was linear inv, the rotation parameter. The same paper, and subsequent papers by various authors, developed an analytic expression for the boundary also linear inv. The latter expression is equivalent to a two term Taylor series about the unperturbed boundary, and is in error by 12% near critical rotation, for a polytropic index 3.0. The boundary can be located directly from the functions representing density, potential, and the potential gradient. The boundary error by this procedure is 0.2% near critical rotation.     

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.     

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.     

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.     

Post-newtonian equilibrium configurations of rotating polytropes

Post-Newtonian equations are solved numerically for stellar models with a polytropic pressure-density relation for the case of uniform rotation, no meridional currents, and axial symmetry. The solution is obtained by following Stoeckly's numerical technique. Parameters characterizing the critical configuration are determined and compared with the values obtained recently by Fahlman and Anand, who followed Chandrasekhar's series expansion method.     

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.     

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.     

A boundary equation for uniformly rotating polytropes

A linear approximating equation exists for the boundary of a uniformly rotating polytrope. The equation in η=(ξ₁−ξ)/ξ₁ permits rapid calculation of the polytrope radius for any latitude, and is accurate for angular velocities of rotation nearly to critical rotation. Data in this paper apply to a polytrope indexn=3.0.