Effects of variations of parallel angular velocity and vorticity on the oscillations of compressible homogeneous rotating ellipsoids

Earlier work on the oscillations of an ellipsoid is extended to investigate the behaviour of a nonequilibrium compressible homogeneous rotating gaseous ellipsoid, with the components of the velocity field as linear functions of the coordinates, and with parallel angular velocity and uniform vorticity. The dynamical behaviour of the ellipsoid is obtained by numerically integrating the relevant differential equations for different values of the initial angular velocity and vorticity. This behaviour is displayed by the (a ₁,a ₂) and (a ₁,a ₃) phase plots, where thea _i's (i = 1, 2, 3) are the semi-diameters, and by the graphs ofa ₁,a ₂,a ₃, the volume, and the angular velocity as functions of time.The dynamical behaviour of the nonequilibrium ellipsoid depends on the deviation of the angular momentum from its equilibrium value; for larger deviations, the oscillations are more nonperiodic with larger amplitudes.An initially ellipsoidal configuration always remains ellipsoidal, but it cannot become spheroidal about its rotation axis, though it may become spheroidal instantaneously about either one of the other two principal axes.For an ellipsoid approaching axisymmetry about its axis of rotation, the angular velocity can suddenly increase by a large amount. Thus if an astrophysical object can be modelled by a nonequilibrium ellipsoid, it may occasionally undergo sudden large increases of angular velocity.     

Rapidly rotating stellar models with radiation pressure

A modified perturbation technique, developed earlier by the authors, has been used to study the effect of rapid uniform and differential rotation on the equilibrium structure of early type stars. The models are in convective equilibrium with significant radiation pressure. The effect of both uniform and differential rotation on various structural parameters like mass, radius, central condensation etc., are calculated for different values ofy _c (the ratio of central radiation pressure to central gas pressure).     

Hydrogen ionization and n=2 population for model spicules and prominences

Using slab model atmospheres that are irradiated from both sides by photospheric, chromospheric, and coronal radiation fields we have determined the ionization and excitation equilibrium for hydrogen.The model atom consists of two bound levels (n = 1 and n = 2) and a continuum. Ly- was assumed to be optically thick with the transition in detailed radiative balance. The Balmer continuum was assumed to be optically thin with the associated radiative ionization dominated by the photospheric radiation field (T _rad = 5940 K). The ionization equilibrium was determined from an exact treatment of the radiative transfer problem for the internally generated Ly-c field and the impressed chromospheric and coronal field (characterized by T _rad = 6500K).Our calculations corroborate the hypothesis that N₂, the n = 2 population density, is uniquely determined by the electron density N _e. We also present ionization curves for 6000K, 7500K, and 10000K models ranging in total hydrogen density from 1 × 10¹⁰/cm³ to 3 × 10¹²/cm³. Using these curves it is possible to obtain the total hydrogen density from the n = 2 population density in prominences and spicules.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Empirical Models for Estimating Net Radiative Flux: A Case Study for Three Mid-Latitude Sites with Orographic Variability

Based on a 4-year REKLIP data-set of global solar radiation G, shortwavesurface albedo a, air temperature T _a and net radiative flux R _n,four types of regression models for the estimation of net radiative fluxfor three sites at different altitudes, located in the southern UpperRhine valley have been proposed. In order to make for the limitationassociated with the basic regression model (BRM) which relates netradiative flux over a surface to only incoming shortwave radiation, alongwave exchange coefficient has been introduced thus givingrise to the modified regression model (MRM). During daytime, the longwaveexchange coefficient is observed to be negative for all three sitesaveraging about –0.20. The suitability of MRM over BRM becomesparticularly obvious with respect to the mountainous site of Feldbergwhere the mean absolute error between measured and simulated R _n usingMRM amounts to just half of that observed using BRM. Furthermore the roleof clearness index and air temperature in the estimation of the netradiative flux have each been examined. The incorporation of the former isto make up for the effect of cloudiness on the net radiative flux budget,while the latter is an independent variable arising from the effectiveterrestrial radiation which thus allow for the estimation of the netradiative flux during all hours of the day. The regression models beenproposed here have each been validated and their efficiency in reproducingactual measurements have been reported.     

Linear waves in a radiating and scattering grey medium

The problem on linear waves in a radiating and scattering grey medium is studied using Whitham's method. Analysis of the basic equations distinguishes two limiting cases: the one is theequilibrium case in which the energy exchange between the gas and radiation plays an essential role, and the other is theScattering case in which the effect of energy exchange is negligible. A new type ofradiation acoustic wave with the speed is found in the scattering case. The governing equations for linearized one-dimensional flow are reduced to one equation of radiative acoustics valid to order 1/c, and the criterion for the two limiting cases is derived from studying this equation. The harmonic solution is analytically studied to show that theeffective optical depth corresponding to the wavelength of perturbation gives the measure of the interaction between the gas and radiation. When _eff1, the sound speeda _g ² =P _g / and the propagating speed of radiative disturbancea _f ² =fc ² appear as the modified classical and radiation-induced modes respectively, wheref is the Eddington factor. When _eff1, the isentropic sound speeda _s ² =(P _g +P _r / appears in the equilibrium case, and the radiation acoustic speeda _A ² appears in the scattering case. The dispersion relation of the harmonic solution is numerically calculated. The result shows that the wave pattern depends critically on the ratio=P _g /(P _g +P _r ). When , the speeda _S anda _A arise from the modified classical mode, and when , they originate from the radiation-induced mode.     

Three-dimensional periodic oscillations generating from plane periodic ones around the collinear Lagrangian points

The three families of three-dimensional periodic oscillations which include the infinitesimal periodic oscillations about the Lagrangian equilibrium pointsL ₁,L ₂ andL ₃ are computed for the value =0.00095 (Sun-Jupiter case) of the mass parameter. From the first two vertically critical (|a _v |=1) members of the familiesa, b andc, six families of periodic orbits in three dimensions are found to bifurcate. These families are presented here together with their stability characteristics. The orbits of the nine families computed are of all types of symmetryA, B andC. Finally, examples of bifurcations between families of three-dimensional periodic solutions of different type of symmetry are given.     

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.     

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.     

Formation of resonance doublet profiles in rapidly expanding envelopes

Generalization of the escape probability method introduced by Sobolev allows us to study the transfer of spectral line radiation for a resonance doublet in rapidly expanding envelopes.For the cases of outward-accelerating (or equivalently inward-decelerating) and outward-decelerating (or equivalently inward-accelerating) envelopes we derive, in the frame of a three-level atom model, the expressions for the spectral radiation fieldsJ ₁₂ andJ ₁₃, for the resulting radiative force F^RE exerted per atom and for the resonance doublet profileE(X)/C _c; we take into account the complex radiative coupling, in both resonance transitions 12 and 13, between distant parts of the atmosphere.For various physical and geometrical conditions prevailing in the expanding media, we illustrate and discuss the behaviours of those quantities as well as their dependence on the parameters of the model. Namely, we deduce criteria under which resonance doublet profiles formed in outward-accelerating and/or outward-decelerating envelopes would appear to be resolved into double P Cygni profiles.We also stress the importance of treating a resonance doublet as being formed by two distinct resonance transitions when evaluating the resulting radiative force F^RE acting on an atom. It is indeed shown that if we use a two-level atom model to represent a resonance doublet-i.e., assigning to it an oscillator strength equal to the sum of the oscillator strengths of both resonance transitions-the amplitude of the resulting radiative force can be underestimated by factors reaching 100% and more in the regions of the expanding envelope which are optically thick to the spectral line radiation. In this context, it would be essential to revise the previous models of radiation-driven winds developed for early-type stars in which the lines belonging to any multiplet were treated as a single line.Also, Aspirant au Fonds National de la Recherche Scientifique (Belgium).     

Exact MHD solutions for rotating,magnetic stars

Simple exact solutions of the magnetohydrodynamic equations are found for rotating, magnetic stars. The velocity and magnetic field are axisymmetric and purely toroidal, and the magnetic energy density equals the kinetic energy density. For constant mass density, the solution reduces to that of Chandrasekhar (1956), which is stable even against non-axisymmetric perturbations. For an ideal gas equation of state, the condition for radiative thermal equilibrium is solved to lowest order in the non-spherical perturbation. The velocity, magnetic field and non-spherical pressure and temperature perturbations all vanish within cones centered around the rotation axis, |cos |>x _i a zero of a Legendre polynomial. Low-order, long-period stellar oscillations may be excited by MHD instabilities near the equatorial region which become damped near the axis.     

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.     

Exact solution of the transport equation for radiative transfer with scattering albedo ωO < 1 using the Laplace transform and the Wiener-Hopf technique and an expression ofH-function

We have considered the transport equation for radiative transfer to a problem in semi-infinite non-conservative atmosphere with no incident radiation and scattering albedo ₀ < 1. Usint the Laplace transform and the Wiener-Hopf technique, we have determined the emergent intensity and the intensity at any optical depth. We have obtained theH-function of Dasgupta (1977) by equating the emergent intensity with the intensity at zero optical depth.     

Structure of sunspots

The sunspot models published so far do not reproduce the observed run of the umbral continuum intensities over the entire spectral range 0.5 < < 4 m. Moreover, in several previous models is the temperature gradient smaller than both the adiabatic and the radiative equilibrium gradient.Agreement between intensities computed from acceptable models and measured intensities can be obtained by introducing an additional opacity for 0.8 m, which is probably caused by the crowding of atomic and molecular lines. We present a new umbral model atmosphere with a wavelength dependent opacity enhancement factor which explains the continuum intensities and also reproduces plausible center-to-limb variations and line profiles. This model is in radiative equilibrium down to _0.5 = 1.5, with an effective temperature of 4000 ± 100K. For the deeper superadiabatic layers a small but probably significant departure from radiative equilibrium is indicated by the intensities in the range 1.5 < < 2.4 m.The uncertainties in the present model and the effect of the additional opacity on line profiles are briefly discussed.     

Slowly-rotating partially-degenerate standard model

Slowly-rotating partially-degenerate standard models have been computed for different values of central degeneracy parameter _c . For increased central degeneracy the models get more distorted, since the central condensation increases and this results in more severe rotational effects in the outer layers of the configuration.     

Probabilistic approach to radiative transfer in supersonic flows

The relationship J=(1 – )S + _c I _c common to most supersonic radiative transfer theories, is given a detailed interpretation in terms of atomic rates and probabilities. It is also generalized to cases whereI _c depends on angle. The nature of the escape process and the expressions for the escape probability are clarified.On leave from Michigan State University.     

On the possibility of constructing a radiative sunspot model in magnetohydrostatic equilibrium

It is currently believed that it is impossible to construct a radiative sunspot model in magnetohydrostatic equilibrium unless magnetic fields below the surface are excessively large (> 100 kG). This belief is based on results obtained using the mixing length theory of convection. We wish to point out that by using a different theory of convection, due to Öpik (1950), it is possible to compute a radiative sunspot model in which the field becomes no greater than 9000 G. By applying two boundary conditions, (i) depth of spot equals depth of convection zone, (ii) magnetic field has zero gradient at the base of the spot, we show that a radiative spot has a unique effective temperature for a given Wilson depression, . For = 650 km, we find T _e = 3800K ; for = 150 km, T _e = 3950K. According to our model, spots having T _e cooler than these values should not exist.     

On the directional dependence of the emission of acoustic noise by convective turbulence in a gravitational atmosphere

The directional dependence of the emission of sound waves in the solar atmosphere is studied. It is shown that quadrupole acoustic radiation generated in convective turbulence is strongly enhanced in the direction of the mean convective flow. The intensity in a direction with the convective velocity is proportional to (1 – M _c cos)^–17/2, where M _c is the Mach number of the convective motion. The influence of the atmospheric cut off frequency on the transmitted acoustic spectrum is taken into account. It is suggested that low frequency atmospheric oscillations may modulate the flux of high frequency sound waves.     

The effect of mechanical waves on empirical solar models

Empirical solar models contain the effect of heating due to radiative energy loss from acoustic waves. We estimate here the temperature difference between the radiative equilibrium model and the empirical model. At optical depth ₅₀₀₀ = 0.1 this difference is small, but near the temperature minimum (₅₀₀₀ = 10^–4) it reaches between 53 and 83 K. The temperature difference between the equator and the poles caused by a hypothetical difference in the heating is estimated.     

Nonradial oscillations of a neutron star in an inhomogeneous hydrodynamic model

An inhomogeneous model neutron star with a variable density profile of the type ₀(r)=_c[1–(2/3)r²/R²]exp(–r²/R²) is considered, where _c is the central density, R is the star's radius, and is the inhomogeneity parameter in the radial mass distribution. This parameterization adequately reproduces the results of numerical evolutionary calculations of the density profile and enables one to obtain in analytical form the parameters of hydrostatic equilibrium and the eigenmodes of nonradial oscillations of a nonrotating neutron star, modeled by a spherical mass of incompressible, inviscid liquid. It is shown that a characteristic manifestation of the star's inhomogeneity is the presence of a stable dipole f-mode, the lowest one in the spectrum of natural oscillations. The presence of this mode serves as a general and primary criterion that evidently distinguishes all inhomogeneous hydrodynamic models from the homogeneous Kelvin model, in which the quadrupole mode is the lowest stable mode. Estimates obtained for the periods of nonradial pulsations coincide with the periods of micropulses observed in the average pulse profiles of c-pulsars. This suggests that the detected variations in emission intensity in the range of micropulse duration (on the order of 10^–4 sec) are associated with nonradial stellar oscillations.Translated from Astrofizika, Vol. 39, No. 3, pp. 475–488, July–September, 1996.     

The maximum limit of the density variation parameter λ for a stable neutron star

A static spherically-symmetric model, based on an exact solution of Einstein's equation, gives the permissible matter density 2×10¹⁴ g cm^–3. If we use the change in the ratio of central density to the radiusr=a (i.e., central density per unit radius (₀/a), we call it radius density) minimum, we have estimated the upper limit of the density variation parameter () and minimum mass limit of a superdense star like a neutron star. This limit gives an idea of the domain where the neutron abundance with negligible number of electrons and protons (may be treated as pure neutrons) and equilibrium in neutrons begins.