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.     

On the existence of cosmological evolutionary effects

Parameters of the distant galaxy clusters of 3C295 (z=0.46) and Cl 0024+1654 (z=0.39) are compared with the predictions made using galaxies of the local clusters Coma (z=0.023) and DC 0329–52 (z=0.057) taking theK-effect into account. The distributions of colour and morphological type, and the amplitudesF ⁺/F^– of the ₀ 4000 discontinuity are examined and no evidence for evolution of the galaxies and the clusters can be seen.     

Polytropic spheres with isothermal cores

The paper deals with massive fluid spheres with an isothermal core (having finite central density) and a polytropic envelope in terms of the General Relativity. The matching of the solutions in the core and envelope has been done using Bondi's condition,H=0 and also without it. The study reveals that since this condition does not correspond to any particular physical situation the maximum values of fractional core size, fractional core mass and the redshift do not occur atH=0, but that they occur at some other point. Within the permissible physical conditions (dP/dρ≤1) the maximum values ofM _core/M,R _core/R and the surface redshift, for an isothermal coreP=ρ/3, are respectively 0.473, 0.554, and 0.565. Using the conditionH=0, it has been shown that for isothermal cores corresponding to the equation of the stateP≥0.6ρ, the configurations are pulsationally unstable.     

Seven-digit tables of Lane-Emden functions

In Table I we present seven digit numerical solutions of the Lane-Emden equation for the plane-parallel, cylindrical, and spherical case for polytropic indices ofn=–10, –5, –4, –3, –2, –1.5, –1.01, –0.9, –0.5, 0, 0.5, 1, 1.5, 2, 3, 4, 5, 6, 10, 20, ±, supplemented byn=2.5, 3.5, 4.5, and 4.99 for the spherical case.In Table II some finite boundary values of polytropic slabs, cylinders, and spheres are summarized. For polytropic spheres (N=3) we have also quoted boundary values near the minimum of the dimensionless mass -² ₁₁ occurring atn4.823 (Seidov and Kuzakhmedov, 1978).     

Relativistic polytropic spheres in general relativity

Spherical-symmetric massive structures with a relativistic polytropic equation of state have been studied in detail. Thevarious static parameters have been calculated under all possible variations of ₀ (=P ₀/E ₀0 pertaining to the central value) and the polytropic indexn. In particular, the values of the static parameters have been identified under the extreme relativistic conditions and for the most bound structures. The structures have been used to model neutron stars and the various neutron parameters have been calculated. Radial pulsation and rotation of these configurations have also been discussed.     

Relativistic magnetic partially degenerate isothermal gas spheres

The effect of a poloidal magnetic field on the structure of isothermal gas spheres in hydrostatic equilibrium under the pressure of partially degenerate relativistic electrons and radiation has been considered. An equation of state involving Fermi-Dirac functionsF ₂() andF ₃() has been used. Modifications to the values of various structural parameters have been tabulated for the cases _c =0, 2, 3, 5, and 10, _c being the central degeneracy parameter.     

Intergalactic medium: X-ray background and compton parameter

A model of intergalactic medium heated by QSOs and cooled by the expansion of the universe and Compton cooling is studied in the framework of a Friedmann-Robertson-Walker universe. Cosmological evolution functions of the comoving density of QSO's as well as the case of no evolution are considered. The theoretical X-ray background spectrum (through thermal bremsstrahlung) and Comptony parameter are calculated including relativistic corrections in the electron-electron, electron-proton and electron-photon interactions. The observed X-ray background and the upper limit of the Compton parametery cobe given by the COBE satellite are used to adjust, for each value of reheating redshiftsz _c ranging from 0.1 to 5.0, the present values of the temperatureT ₀ and densityn ₀ of the intergalactic gas. Forz _c > 0.25, when the theoretical X-ray spectrum fits the observed one, the adjusted values ofT ₀ andn ₀ imply iny >y cobe. On the other hand, whenT ₀ andn ₀ are consistent withy cobe, the calculated X-ray spectrum is lower than the observed one. Unless 100% of the observed X-ray background is due to discrete sources and if the intergalactic medium contributes more than 2.5% to such background we come to the interesting result that the medium must have been heated atz _c < 1. In this case we shall have to explain the high energy rates necessary to heat the intergalactic medium. Forz _c 0.25, it is possible to find values ofT ₀ andn ₀ such that both the calculated X-ray background and the y parameter simultaneously reproduce the corresponding observed values. However, in this case, unless it could be shown to be otherwise by future observations or theoretical studies, it seems that the model of hot intergalactic medium is not plausible because of the high energies required to heat the intergalactic gas.     

Newtonian and relativistic periodic orbits around two fixed black holes

We compare families of simple periodic orbits of test particles in the Newtonian and relativistic problems of two fixed centers (black holes). The Newtonian problem is integrable, while the relativistic problem is highly non-integrable.The orbits are calculated on the meridian plane through the fixed centersM ₁ (atz=+1) andM ₂ (atz=–1) for energies smaller than the escape energyE=1. We use prolate spheroidal coordinates (, , =const) and also the variables =cosh and =–cos . The orbits are inside a curve of zero velocity (CZV). The Newtonian orbits are also limited by an ellipse and a hyperbola, or by two eillipses. There are 3 main types of periodic orbits (1) elliptic type (around both centers), (2) hyperbolic-type, and (3) resonant-type.The elliptic type orbits are stable in the Newtonian case and both stable and unstable in the relativistic case. From the stable orbits bifurcate double period orbits both symmetric and asymmetric with respect to thez-axis. There are also higher order bifurcations. The hyperbolic-type orbits are unstable. The Newtonian resonant orbits are defined by the ratiot _µ/t =n/m of oscillations along and during one period, and they are all marginally unstable. The corresponding relativistic orbits are stable, or unstable. The main families are figure eight orbits aroundM ₁, or aroundM ₂ (3/1 orbits); gamma, or inverse gamma orbits (4/2); higher resonant families 5/1,7/1,...,8/2,12/2,...;, more complicated orbits, like 5/3, and bifurcations from the above orbits. Satellite orbits aroundM ₁, orM ₂, and their bifurcations (e.g. double period) exist in the relativistic case but not in the Newtonian case. The characteristics of the various families are quite different in the Newtonian and the relativistic cases. The sizes of the orbits and their stabilities are also quite different in general. In the Appendix we study the various types of straight line orbits and prove that some subcases introduced by Charlier (1902) are impossible.     

Generalized polytropes: Stellar structure with a boltzmann factor

In order to study the structure of a chemically homogeneous star in equilibrium, a density profile of the form T ^N exp(–µm(–)/kT) is suggested. As for polytropes, qualitative aspects of the resulting stellar model can be discussed analytically. In particular it is shown that one reobtains forN=3 Eddington's standard model, whereas forN<3 nearly polytropic models result. WhenN>3, the effective polytropic index does vary appreciably over the star. Numerical results indicate that the proposed density profile is quite reasonable in view of the simplicity of the model. From a comparison of the degree of precision of a polytropic approximation with that of the newly proposed model it follows that the new approximation is definitely better than the polytropic one. It is suggested that the model may be useful to study the structure of stellar clouds, clusters and (spherical) galaxies.Now at Department of Applied Mathematics, Queen Mary College, University of London, England.     

Rapidly rotating supermassive stars in the first post-Newtonian approximation to general relativity

The structure and stability of rapidly uniformly rotating supermassive stars is investigated using the full post-Newtonian equations of hydrodynamics. The standard model of a supermassive star, a polytrope of index three, is adopted. All rotation terms up to and including those of order ⁴, where is the angular velocity, are retained. The effects of rotation and post-Newtonian gravitation on the classical configuration are explicitly evaluated and shown to be very small. The dynamical stability of the model is treated by using the binding energy approach. The most massive objects are found to be dynamically unstable when =1/c ².p _c / _c 2.2 × 10^–3, wherep _c and _c are the central pressure and density, respectively. Hence, the higher-order terms considered in this analysis do not appreciably alter the previously known stability limits.The maximum mass that can be stabilized by uniform rotation in the hydrogen-burning phase is found to be 2.9×10⁶ M , whereM is the solar mass. The corresponding nuclear-generated luminosity of 6×10⁴⁴ erg/sec^–1 is too small for the model to be applicable to the quasi-stellar objects. The maximum kinetic energy of a uniformly rotating supermassive star is found to be 3×10^–5 Mc ², whereM is the mass of the star. Masses in excess of 10¹⁰ M are required if an adequate store of kinetic energy is to be made available to a pulsar like QSO. However such large masses have rotation periods in excess of 100 yr and thus could not account for any short term periodic variability. It is concluded then that the uniformly rotating supermassive star does not provide a suitable base for a model of a QSO.     

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.     

Secular evolution of the period and inclination of the magnetic to rotation axis and recycled pulsars

Statistical analysis has been carried out of the relations between period and the ageP–t _c, and the inclination of magnetic to rotation axis to the age –t _cof pulsars have been done.Up to characteristic agest _c=3×10⁷ years the period increases as expected for magneto-dipole radiation energy lossesP=P _m (1–exp(–t/ _B ))^1/n–1. Best-fitting parameters of this approximation are the time-scale of the magnetic moment decay _B =4×10⁶ years and breaking indexn=3.6. Fort _c>3×10⁷ years theP–t _cdependence is significantly different.The inclination of magnetic to rotation axis decreases versus age, showing a secular alignment of the axis. But this decrease continues also only up tot _c=3×10⁷ years. Thus bothP–t _cand –t _cdependencies indicate that most of the pulsars of agest _c>3×10⁷ years are not evolutionary continuations of more younger ones, but apparently represent another population of pulsars, which differ by their genetic history or physical processes. This population includes all known millisecond pulsars. We suggest, that this population is a so-called recycled pulsar. The list of candidates of recycled pulsars is presented.A new evaluation of the inclination of the magnetic to the rotation axis for 105 pulsars is presented.     

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.     

Rapidly rotating polytropes in the post-Newtonian approximation to general relativity

The study of uniformly polytropes with axial symmetry is extended to include all rotational terms of order ⁴, where is the angular velocity, consistently within the first post-Newtonian approximation to general relativity. The equilibrium structure is determined by treating the effects of rotation and post-Newtonian gravitation as independent perturbations on the classical polytropic structure. The perturbation effects are characterized by a rotation parameter = ²/2G _c and a relativity parameter, =p _c / _c C ² , wherep _c and _c are the central pressure and density respectively. The solution to the structural problem is obtained by following Chandrasekhar's series expansion technique and is complete to the post-Newtonian rotation terms of order ². The critical rotation parameterv _c , which characterizes the configuration with maximum uniform rotation, is accurately evaluated as a function of . Numerical values for all the structural parameters needed to determine the equilibrium configurations are presented for polytropes with indicesn=1, 1.5, 2, 2 5, 3, and 3.5.     

Gravitational instability of interstellar magnetic clouds

The gravitational instability of a nonrotating isothermal gaseous disk permeated by a uniform frozen-in magnetic field is investigated using a fourth-order perturbation technique. From the results it is found that the disk is stable whenn/B ₀ < (4/3³ G)^–1/2, wheren andB are the column density of the disk and unperturbed magnetic field, respectively, andG is the gravitational constant. The disk is gravitationally unstable only whenn/B ₀ > (4/3³ G)^–1/2.     

Variation in the Opacity of the Intergalactic Medium During its Reionization

It is now assumed that reionization of the intergalactic medium occurred under the action of ultraviolet radiation from hot stars contained in galaxies that formed early. Regions of ionized hydrogen (H II zones) were formed around such galaxies. The effects of hydrogen recombination and of the simultaneous cosmological expansion of such regions on their opacity are considered. It is shown that regions formed at z _i < 6 are opaque to L _c radiation, while for z _i 5 the intergalactic medium should be transparent at 0 < z < 5. This conclusion is in accord with recent observational data.     

The Schoenberg-Chandrasekhar limit: A polytropic approximation

The existence of a maximum isothermal core mass fraction (q _max), the Schoenberg-Chandrasekhar limit, is one of the classic results from the theory of stellar structure. This limit can be demonstrated through a simplified composite polytrope model in which an isothermal core is surrounded by ann=1 polytrope envelope. While this model underestimatesq _mas by 25% in the homogeneous case, it is accurate to within 5% in the more realistic inhomogeneous situation.     

Period distributions in pulsating and binary stars

We analyze the hypothesis of quantization in bands for the angular momenta of binary systems and for the maount of actionA _c in stable and pulsating stars. This parameter isA _c=Mv _eff R _eff, where the effective velocity corresponds to the kinetic energy in the stellar interior and the effective radius corresponds to the potential energyGM ²/R _eff. Analogous parameters can be defined for a pulsating star withm=M where is the rate of the massm participating in the oscillation to the total massM andv _osc,R _osc the effective velocity and oscillation radius.From an elementary dimensional analysis one has thetA _c (energy x time) (period)^1/3 independently ifA _c corresponds to the angular momentum in a binary system, or to the oscillation in a pulsating star or the inner energy and its time-scaleP _eff in a stable star.From evolving stellar models one has that P _effP _eff(solar)1.22 hr a near-invariant for the Main Sequence and for the range of masses 0.6M <M<1.6M .With this one can give scalesn _k=kn ₁ withk integers andn ₁=(P/P ₁)^1/3 withP ₁=P _eff1.22 hr. In these scales proportional toA _c, one sees that the periods in binary and pulsating stars are clustered in discrete unitsn ₁,n ₂,n ₃, etc.This can be seen in pulsating Scuti, Cephei, RR Lyrae, W Virginis, Cephei, semi-regular variables, and Miras and in binary stars as cataclysmic binaries, W Ursa Majoris, Algols, and Lyrae with the corresponding subgroups in all these materials. Phase functions (n _k) in RR Lyrae and Cephei are also associated with discrete levelsn _k.the suggested scenario is that the potential energies and the amounts of actionE _p(t), A_c(t) are indeed time-dependent, but the stars remain more time in determinated most proble states. The Main Sequence itself is an example of this. These most probable states in binary systems, or pulsating or stable stars, must be associated with velocities sub-multiplesc/ _F , given by the velocity of light and the fine structure constant.Additional tests for such a hypothesis are suggested when the sufficient amount of observational data are available. They can made with oscillation velocities in pulsating stars and velocity differences of pairs of galaxies.     

The primeval hadron: Origin of stars,galaxies and astronomical universe

The relationship between mass and angular momentum for the known cosmic objects has been examined, and it is shown that they are described by the generalized Regge-like dependence of the formJ=(m/m _p)^1+1/n wheren=2 for galaxies, their clusters and superclusters, andn=3 for asteroids, planets and stars. It offers the possibility, that Ambartsumian's superdense proto-matter has hadronic nature. This allows us to give a realistic and quantitave explanation with a minimum number of arbitrary assumptions for the origin of cosmic angular momenta, cosmic magnetic fields and offers the framework for other cosmogonic implications. This approach incorporates in a natural way the fundamental quantum-mechanical parameters andm _p, besides the classical parametersG andc, and allows us to derive simple expressions for masses and spins of cosmic objects through fundamental constants, some of which coincide with Eddington-Dirac's Large Number relations.     

The effect of a toroidal magnetic field on the non-radial oscillations of polytropes

Frequencies of non-radial oscillation of polytropic models of stars, belonging to spherical harmonics of ordersl=1, 2 and 3, are evaluated, in a second approximation, by a variational method. Equilibrium configurations in the presence of toroidal magnetic fields are obtained numerically without any restriction on the field strength. The value of the ratio of the specific heats, , is assumed to be equal to 5/3 and only two polytropic indeces,n=1.5 and 3.0, are considered. It is found that a polytropic star stays stable for magnetic fields considerably stronger than expected from the results obtained by the weak field perturbation methods.