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.     

Radiation pressure effects in the oscillations of compressible rotating homogeneous spheroids

Earlier models of compressible, rotating, and homogeneous ellipsoids with gas pressure are generalized to include the presence of radiation pressure. Under the assumptions of a linear velocity field of the fluid and a bounded ellipsoidal surface, the dynamical behaviour of these models can be described by ordinary differential equations. These equations are used to study the finite oscillations of massive radiative models with masses 10M and 30M in which the effects of radiation pressure are expected to be important.Models with two different degrees of equilibrium are chosen: an equilibrium (i.e., dynamically stable) model with an initial asymmetric inward velocity, and a nonequilibrium model with a nonequilibrium central temperature and which falls inwards from rest. For each of these two degrees of equilibrium, two initial configurations are considered: rotating spheroidal and nonrotating spherical models.From the numerical integration of the differential equations for these models, we obtain the time evolution of their principal semi-diametersa ₁ anda ₃, and of their central temperatures, which are graphically displayed by making plots of the trajectories in the (a ₁,a ₃) phase space, and of botha ₁ and the total central pressureP _c against time.It is found that in all the equilibrium radiative models (in which radiation pressure is taken into account), the periods of the oscillations of botha ₁ andP _c are longer than those of the corresponding nonradiative models, while the reverse is true for the nonequilibrium radiative models. The envelopes of thea ₁ oscillations of the equilibrium radiative models also have much longer periods; this result also holds for the nonequilibrium models whenever the envelope is well defined. Further, as compared to the nonradiative models, almost all the radiative models collapse to smaller volumes before rebouncing, with the more massive model undergoing a larger collapse and attaining a correspondingly larger peakP _c.When the mass is increased, the dynamical behavior of the radiative model generally becomes more nonperiodic. The ratio of the central radiation pressure to the central gas pressure, which is small for low mass models, increases with mass, and at the center of the more massive model, the radiation pressure can be comparable in magnitude to the gas pressure. In all the radiative models, the average periods as well as the average amplitudes of both thea ₁ andP _c oscillations also increase with mass.When either rotation or radiation pressure effects or both are included in the equilibrium nonradiative model, the period of the envelope of thea ₁ oscillations is increased. The presence of rotation in the equilibrium radiative model, however, decreases this period.Some astrophysical implications of this work are briefly discussed.     

The seven components of Hα and the 9873 MHz line

Under conditions pertaining to the middle chromosphere, the nlj-substates of the n = 2 and n = 3 states of hydrogen are found to be populated proportionally to their degree of degeneracy. Thus, the non-LTE formation of the H line is not expected to be influenced by some exotic excitation conditions in one of its seven components. The overpopulation of the 2S _1/2 substate relative to the 2P _3/2 substate does not depend upon details of the radiative transfer in H, so that it rests wholly on the transfer in L (Milkey and Mihalas, 1973). One-component plane parallel models of the chromosphere thus further predict that the 2S _1/2-2P _3/2, 9873 MHz (3.04 cm) line will not be observable in the radio-spectrum of the quiet Sun.Astronomische Mitteilungen der Eidgenössischen Sternwarte Zürich, Nr. 367.     

Relativistic isentropic stellar models

Relativistic, isentropic, homogeneous models are constructed by a method that automatically detects instabilities, and evolutionary tracks of central conditions are shown on a (T, ) diagram. Models heavier than 20M become unstable because of pair creation. Iron photodisintegration causes instability in the mass range between 1.5M and 20M . General relativistic effects bring about the onset of instability in models of 1.2–1.5M when the central density is about 10¹⁰ g/cm³. Lighter models become white dwarfs. It is pointed out that general relativistic instability will prevent the formation of neutron stars through hydrostatic evolution and may be relevant in setting off low-mass supernovae.     

A possible model of supernovae: Detonation of12C

Stars of intermediate mass (4M M9M ) may ignite the¹²C+¹²C reaction explosively because of the high degree of electron degeneracy in their central regions. After the exhaustion of helium burning in the core of such stars, a helium-burning shell develops which is thermally unstable. Approximating this shell by suitable boundary conditions, the subsequent evolution of the core is examined quantitatively by standard techniques. An explosive instability due to ignition and detonation of¹²C+¹²C develops at a central density _c 2 × 10⁹. Subsequent hydrodynamic expansion is computed; final velocities of expansion up tov20 000 km/sec are found. The star is totally disrupted; no condensed remnant is left. Such an explosion may be a plausible model for a significant fraction of supernovae. Investigation of the relevant nuclear reaction network shows that the entire core (M _core1.37M ) is processed through¹²C burning,¹⁶O burning and silicon burning. Significant amounts of⁵⁶Ni are produced. This nucleosynthesis is critically sensitive to the exact central density at which the¹²C+¹²C reaction ignites; several factors which affect this critical density are discussed. A brief summary of other supernovae thories which have been expounded in detail is presented for comparison.Supported in part by the National Science Foundation [GP-9433, formerly GP-7976], [GP-9114], and the Office of Naval Research [Nonr-220(47)].     

The pulsation of Delta-Scuti stars

Evolution, linear pulsation, and nonlinear pulsation codes were combined to produce nonlinear models of Delta-Scuti stars in an instability region extending over 3.8T _e<3.95 and 0.6L/L <2.0. The linear analysis upheld the consensus that they are normal Population I stars of about 2M , in stages of evolution corresponding to central hydrogen burning and shell hydrogen burning. The growth rates were very slow; driving was due to an opacity mechanism in the second helium ionization region; periods and period ratios of the lowest modes of the models were in the same range as those observed. A wide range of nonlinear models was investigated. When eigenfunctions from the linear analysis were used as initial velocity profiles, it was found that the dominant peak in the periodogram of the light curve corresponded to the mode initiated. For a small subset of models, limiting amplitudes were identified, and were found to be in close agreement with observed light amplitudes.     

Neutrino luminosity by the ordinary URCA process in an intense magnetic field

The neutrino luminosity by the ordinary URCA process in a strongly magnetized electron gas is computed. General formulae are presented for the URCA energy loss rates for an arbitrary degree of degeneracy. Analytic expressions are derived for a completely degenerate, relativistic electron plasma in the special case of neutron-proton conversion. Numerical results are given for more general cases.The main results are as follows: the URCA energy loss rates are drastically reduced for the regime of great degeneracy by a factor up to 10^–3 for 1, andT ₉10, where =H/H _q ,H _q =m ² c ³/eh=4.414×10¹³ G. In the non-degenerate regime the neutrino luminosity is enhanced approximately linearly with for the temperature range 1T ₉10. Possible applications to white dwarfs and neutron stars are briefly discussed.We have been recently informed that in Gamow home-dialect (Odessa dialect) URCA means thief — (Private communication from Prof. G. Wataghin).     

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.     

Radial oscillations of finite-temperature white dwarfs

Models of finite temperature completely degenerate stellar configurations are considered. The frequencies of fundamental radial modes of oscillations for these white dwarf models have been computed for different values of the central degeneracy parameter 1/y ₀ ² and for uniform temperatures of 20×10⁶ K and 10⁸ K. A variational formulation as well as a direct approach is employed to calculate the temperature induced modifications in the frequencies of oscillation of these white-dwarf models.     

Study of the X-ray selected BL Lac object 1727+502

In this paper, accretion disc and synchrotron emission models have been used to analyse simultaneous IR-optical-UV data of the BL Lac object 1727+502. In the following, some of its properties have been discussed. It is shown that the temperature of the disc is about 19, 000°k, the mass of the central black hole isM ₈ 5.4, and the accretion ratio is 10^–3 M /yr.     

The parameters of planetary nebulae and their central stars derived from observations

On the basis of data on planetary nebula (PN) central star temperatures obtained by measurements in the ultraviolet (UV) range, the empirical calibration dependence between the number of Lyman photons emitted by a central starS and PN diameterD, is constructed. The temperatures of 118 PN central stars are estimated with this dependence. It is shown that the central star masses are distributed in a wide interval from 0.5 to 1.2M . About 60% of all stars have masses <0.6M , about 25% have masses >0.6M and the remainder have masses 0.6M . The averaged empirical tracks of evolution of low-mass (<0.6M ) and massive (>0.6M ) central stars differing considerably from each other are constructed. It is shown that the majority of central stars may possess hot chromospheres (T>2×10⁵ K) which spread for several tens of radii of the central star. The PN originates as a result of ionization of the matter ejected by a red giant at the superwind stage. The cause for this ionization is the UV radiation of the PN central star.     

Remarks on the one-photon-pair annihilation in strong magnetic fields

We report specifically on a quantum electrodynamic feature of the one-photon-pair annihilation. Most of the calculation related with this process (not excluding other ones) have been carried out usually utilizing the wave functions obtained from the Dirac equation in the Landau gauge A = (0,Bx, 0), where B is oriented along the z-axis (Johnson and Lippman, 1949). Although, the eigenstates of the Dirac Hamiltonian as it was introduced by Johnson and Lippman, do not consider the coupling to the radiation field and consequently they reflect only forp _z = 0 the same linear combination of the two degenerate polarization states.We report the transition rate function for the one-photon-pair annihilation in a strong magnetic field by using the Sokolov and Ternov eigenstate |± as far asp _z = 0 andN > 0 is concerned. The difference between the expression for the transition rate by using the Sokolov and Ternov eigenstates and the one calculated by (Wunneret al., 1986), is just the functionI _s,s which corresponds to the degeneracy of the orbit center of the electrons characterized by the quantum numberss ands. Presented at the 2nd UN/ESA Workshop, held in Bogotá, Colombia, 9-13 November, 1992.     

Some features of the solar microwave emission and their connection with geomagnetic activity

A statistical investigation of the slowly varying component of the 9.1-cm solar radio emission is based upon the Stanford radioheliograms covering the years 1962–66. On the average the peak value of the brightness temperature T _b is proportional to the area covered by the corresponding spot group. However, in individual cases the observed T _b is definitely lower or higher than is to be expected from the size of the spot group. We introduce the concept microwave importance I _m of the spot group, which is the T _b to be expected from the Zürich class and spot number; and the concept relative brightness B _r, which is the ratio of the observed T _b to I _m. This leads to the distinction of faint, normal and bright sources with B _r 0.8, 0.8 < B _r < 1.2 and B _r 1.2 respectively. B _r is correlated with the maximum magnetic-field strength H observed in the spot group and with the flux-density spectrum of the source. The yearly average of B _r and the average flux-density spectrum vary with the phase of the solar cycle.An analysis of the results is based upon the electron-density distribution in the condensation, which was visible at the solar limb during the eclipse of February 5, 1962, and on an adopted temperature distribution with a central value of 4 × 10⁶ K. The computed T _b, including gyro-resonance absorption, agrees with the value derived from the microwave importance of the spot group and B _r = 0.5, which shows that in the current gyro-resonance models the electron density is underestimated. The variation of T _b with the size of the spot group can be explained by varying the dimensions of the condensation in area and in height, if the central density and temperature remain constant. The statistical relationships between B _r and H and between B _r and the flux-density spectrum yield a model for the differences between faint and bright sources: B _r increases with the contribution from the gyro-resonance absorption and with the central electron density.For paper I see Solar Phys. 7, 448. For paper III see Solar Phys. 8, 450.     

Theoretical models of planetary nebulae

Sets of theoretical models of spherically symmetrical gaseous nebulae are calculated for a fixed chemical composition and with central star, energy distributions given by Cassinelli (1971) and by Kunasz, Mihalas and Hummer (1975). A central region of constant densityN _H=100 cm^–3 is surrounded by a broken shell consisting of zones of different density and optical thickness. It is found that for a fixed chemical composition and central star flux, a considerable range can be found in the emitted fluxes of certain lines such as 4686 which are often used to fix the level of excitation of a nebula. Although it is possible to produce models yielding a variety of nebular line ratios in this way, these truncated inhomogeneous shell models have not reproduced the integrated spectra of nebulae that simultaneously show extreme features of high and low excitation with great strength-e.g. [Nii], [Oii] and 4686 Heii.     

Unseen matter and angular momentum in the Milky Way and other galaxies: Simple two-component models

Simple two-component (dark+bright) models are built up for the Milky Way, where both the density distribution and the rotation curve are deduced from known observations. The derived dark to bright mass ratio turns out to be in the range 10, in close agreement with the results of more refined approaches, with a weak dependence on the geometry of the model. The related angular momentum appears to be well in agreement with theoretical predictions, if proto-galaxies gain angular momentum via either gravitational interactions or peculiar velocities of their own sub-units, according to a logarithmic distribution of the squared fractional angular momenta close to a Maxwellian one. The rougher assumption that the whole system is represented by a rigidly rotating polytrope leads to dark components rounder than _D 0.7 if proto-galaxies gain angular momentum via gravitational interactions, and to much more flattened dark components if proto-galaxies gain angular momentum via peculiar velocities of their own sub-units and few (4) sub-units are present at the beginning. To fit the observed positions of several galaxies on the (Oê _B q _B ) plane-ê _B representing the ellipticity andq _B close to the ratio of maximum rotational to central peculiar velocity, averaged for all the inclinations to the line of sight — galaxies are modelled by two-component (dark+bright) rigidly rotating, concentric, co-polar, homogeneous spheroids and the Galaxy is assumed to be a typical system. An acceptable fit is produced only under the assumption that protogalaxies gain their angular momentum in late stages of evolution, i.e., after having decoupled from the Hubble flow.     

Central Configurations of the Symmetric Restricted 4-Body Problem

We consider the symmetric planar (3 + 1)-body problem with finite masses m ₁ = m ₂ = 1, m ₃ = µ and one small mass m ₄ = . We count the number of central configurations of the restricted case = 0, where the finite masses remain in an equilateral triangle configuration, by means of the bifurcation diagram with as the parameter. The diagram shows a folding bifurcation at a value consistent with that found numerically by Meyer [9] and it is shown that for small > 0 the bifurcation diagram persists, thus leading to an exact count of central configurations and a folding bifurcation for small m ₄ > 0.     

Genericity of the non-periodic solutions of the central force problem

We study the classical problem of two-dimensional motion of a particle in the field of a central force proportional to a real power of the distancer. for negative energy and (0, 2), each energy levelI _h is foliated by the invariant toriI _hc of constant angular momentumc and, by Liouville-Arnold's theorem, the flow on eachI _hc is conjugated to a linear flow of rotation number _h (c).A well-known result asserts that if we require _h (c) to be rational for every value ofh andc, the, must be equal to one (Kepler's problem). In this paper we prove that for almost every (0, 2) _h (c) is a non-constant continuous function ofc, for everyh<0. In particular, we deduce that motion under central potentials is generically non-periodic.Partially supported by CIRIT under grant No. EE88/2.     

Equation of state of dense nuclear matter and an upper bound on neutron star masses

We have discussed, in general, the important physical parameters, likemaximum mass, radius, and the minimum rotation period of self-bound,causally consistent, and pulsationally stable neutron stars (Q-starmodels) by using a realistic stiff EOS (such that, the speed of sound,v P ^N, or nP=K(E-E _a ) ⁿ, where K 1 and n =1/(1-2N);where P and E represent respectively, the pressure and theenergy-density, and E _a is the value of E at the surface (r = a) of the configuration) within the two constraints imposed by: (i) The minimumrotation period, P _rot, for the pulsar known to date corresponds to1.558 ms, and (ii) The maximum number density anywhere inside thestructure for the models described as Q-stars cannot exceed 1nucleon/fm³. By using the empirical formula given by Koranda,Stergioulas and Friedman (1997) (KSF-formula), and imposing constraint(i), we have obtained an upper bound of M _max 7.76 M radius a 32.5 km, and the central energy-density around 2.17 ×10¹⁴ g cm^-3 (for n =1.01). Constraint (ii) provides the minimumrotation period, P _rot 0.489 ms for the maximum mass M _max 2.4 M, and the central energy-density around 2.20 ×10¹⁵ g cm^-3 (for n =1.01). The speed of sound at the centre ofthese models approaches 99% of the speed of light `c' (in thevacuum) and vanishes at the surface of the configuration together withpressure. If we relax the maximum Kepler frequency imposed by the fastestrotating pulsar known to date (constraint (i)), in view of certainobservational effects and theoretical evidences, and allow the present EOSto produce larger rotation rates than the 1.558 ms pulsar, the maximummass of the non-rotating model drops down to a value 7.2 M .The higher values of masses ( 7 M ) and radii (31-32 km) obtained in this study imply that these models may representthe massive compact objects like Cyg X-1, Cyg XR-1, LMC X-3, and otherswhich are known as black hole candidates (BHCs). This study also suggestthat the strongest contender for black hole at present might be recurrentnova V404 Cyg (mass estimate 8 -12 M ).     

The product of the globular cluster collapse

In this paper we calculate the number of close binaries formed during the evolution process of a globular cluster core. The globular cluster core is assumed to contain a massive black hole at its center. We show that the central black hole can drive binaries formation in the core and the rate of binaries formation depends on the mass of the black hole at its center. When the massM of the black hole is between 10² M and 3×10³ M , there will be a few binaries formed. When the mass of the black hole is 4×10³ M M6×10³ M , the number of binary star formation will suddenly increase with a jump to the maximum value 58. When the mass of the black hole is 7×10³ M M9×10³ M , the number of binary star will immediately decrease. Whether cluster X-ray is produced mainly by the central black hole or by binaries in the core depends on the mass of the central black hole. Therefore, two cases arise: namely, black hole accretion domination and binaries radiation domination. We do think that we cannot exclude the possibility of the existence of a central black hole even when binary radiation characteristics have been observed in globular cluster X-ray sources.     

Chemical evolution of two-component galaxies

In this paper, we try to insert into a single evolutionary scheme — in dealing with chemical evolution of galaxies — two different viewpoints that (at least in not too much complicated models) have been treated separately: namely, theS models, allowing mass conservation; andI models, allowing initial zero masses and no mass conservation due to gas inflow. The true evolution of a real proto-galaxy (after reaching the state of maximum expansion) is simulated as follows: A spheroidal gas mass continued to collapse and form stars until a flat configuration is reached after a timeT _c has elapsed, while a given amount of gas flows in on a time-scale . According to this scheme, the basic equations of chemical evolution are derived and models which simulate the history of solar neighborhood, other regions and Galactic spheroid component are built up, in the whole range between theS-limit (mass conservation) and theI-limit (zero initial mass and subsequent accretion due to inflowing gas). Concerning the solar neighbourhood, we find that neither the occurrence of gas inflow nor inflow on time-scales 2–3 10⁹ yr are necessary in order to reproduce the temporal behaviour and the empirical distribution of metal content, as pointed out by some authors. On the contrary, the constraint on the lower mass limit for stars formed,m _mf0.01, allows only models with T _c (i.e. inflow time-scale of the order of the contraction time), while the constraint on the disk mass fraction,R _D(T _a)0.75, rules out the cases near theI-limit forT _c0.55 but permits all cases forT _c2.75. Concerning other regions, models are built up which roughly simulate elliptical, spiral and irregular galaxies, and all less extended regions resembling such systems.If the stellar birthrate function is assumed to be an universal law, the chemical evolution of the Galactic disk may be understood in terms of different zones (that might be thought as concentric and coaxial rings) the total density of which decreases monotonically, owing to a corresponding decrease in total mass and/or increase in volume, when passing from the center to the border of the disk. The constraintsm _mf0.01 andR _D(T _a)0.75 for different regions of the Galactic disk would also rule out all models well beyond theS-limit, but further results are required in order to confirm this conclusion. Finally, concerning the Galactic spheroid component, it is found that onlyS models with massive halos (R _D(T _a)0.01) are able to reproduce in an acceptable way the empirical metal abundance distribution. In order to obtain a complete fit, a spheroid component has to be assumed, with a steeper mass spectrum exponent in the stellar birthrate function, and a lower yield of metallicity, in respect to the disk component. According to this last model, a mean value of disk metal content (with respect to spatial distribution) of the order of the solar value also results.