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 time-scale for core collapse in spherical star clusters

The collapse time for a cluster of equal-mass stars is usually stated to be either 330 central relaxation times (t_rc) or 12-19 half-mass relaxation times (t_rh). But the first of these times applies only to the late stage of core collapse, and the second only to low-concentration clusters. To clarify how the time depends on the density profile, the Fokker-Planck equation is solved for the evolution of a variety of isotropic cluster models, including King models, models with power-law density cusps of ρ ∼ r^−γ, and models with nuclei. The collapse times for King models vary considerably with the cluster concentration when expressed in units of t_rc or t_rh, but vary much less when expressed in units of t_rc divided by a dimensionless measure of the temperature gradient in the core. Models with cusps have larger temperature gradients and evolve faster than King models, but not all of them collapse: those with 0 < γ < 2 expand because they start with a temperature inversion. Models with nuclei collapse or expand as the nuclei would in isolation if their central relaxation times are short; otherwise their evolution is more complicated. Suggestions are made for how the results can be applied to globular clusters, galaxies, and clusters of dark objects in the centers of galaxies.Scott D. Tremaine     

Bianchi type-I cosmological models for binary mixture of perfect fluid and dark energy

We consider a self consistent system of Bianchi Type-I cosmology and Binary Mixture of perfect fluid and dark energy. The perfect fluid is taken to be obeying equations of state p _PF =γρ _PF with γ∈[0,1]. The dark energy is considered to be obeying a quintessence-like equation of state where the dark energy obeys equation of state p _DE =ωρ _DE where ω∈[−1,0]. Exact solutions to the corresponding Einstein field equations are obtained. Some special cases are discussed and studied. Further more power law models and exponential models are investigated.     

Density fluctuations on super-Hubble scales

According to causality, the existence of density perturbations on scales larger than the present Hubble radiusy = 2c/H ₀ is crucial for discriminating between inflation and non-inflation models of the origin of inhomogeneity of the universe. Observations of the cosmic background radiation anisotropies favor a super-Hubble suppression on scales _max in the range 0.5–3.0y. Many of non-inflation models are consistent with such a suppression. Inflation models are certainly not in conflict with this suppression; however one important parameter, the duration of the epoch of inflation, may need to be fine-tuned.     

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.     

Viscous ΛCDM universe models

We explore flat ΛCDM models with bulk viscosity, and study the role of the bulk viscosity in the evolution of these universe models. The dynamical equations for these models are obtained and solved for some cases of bulk viscosity. We obtain differential equations for the Hubble parameter H and the energy density of dark matter ρ _m , for which we give analytical solutions for some cases and for the general case we give a numerical solution. Also we calculate the statefinder parameters for these models and display them in the s–r-plane.     

Three-component dust models for interstellar extinction

Interstellar extinction curves obtained from the ‘extinction without standard’ method were used to constrain the dust characteristics in the mean ISM (R _V = 3.1), along the lines of sight through a high latitude diffuse molecular cloud towards HD 210121 (R _V = 2.1) and in a dense interstellar environment towards the cluster NGC 1977 (R _V = 6.42). We have used three-component dust models comprising silicate, graphite and very small carbonaceous grains (polycyclic aromatic hydrocarbons) following the grain size distributions introduced by Li & Draine in 2001. It is shown that oxygen, carbon and silicon abundances derived from our models are closer with the available elemental abundances for the dust grains in the ISM if F & G type stars atmospheric abundances are taken for the ISM than the solar. The importance of very small grains in modelling the variation of interstellar extinction curves has been investigated. Grain size distributions and elemental abundances locked up in dust are studied and compared at different interstellar environments using these three extinction curves. We present the albedo and the scattering asymmetry parameter evaluated from optical to extreme-UV wavelengths for the proposed dust models.     

Evolution of temperature in granule and intergranular space

The temporal evolution of temperature in a dissolving granule and in an adjacent intergranular space is presented. The semi‐empirical evolutionary models have been calculated using an inversion method applied to 4‐min time series of Stokes I spectral line profiles. The models are presented in the form of the functional dependence of temperature T(log τ₅, t) on optical depth τ₅ at 500 nm and time t. The observed disappearance of the granule is accompanied with overall cooling of the granular photosphere. Temperature changes greater than 100 K have been found in deeper (log τ₅ ≥ 0) and upper layers (log τ₅ ≤ –2) whereas the intermediate layers are thermally stable. The intergranular space, which is 2 arcsec off the granule, keeps the temperature structure of the layers from log τ₅ = 0.5 to log τ₅ = –2 without global evolutionary changes except short‐term and spatially confined heating. Finally, the significant temperature changes in the upper layers (log τ₅ ≤ 2.5) observed during the time interval of 4 min are found to be typical for the granular and intergranular photosphere.     

Temperature distribution in the transition region and inner corona

The energy balance equation for a general solar atmosphere without assuming the plane-parallel approximation is solved analytically. This leads to models for both the transition region and inner corona as well as for the outer corona. The form of the latter is very similar to that of the hydrostatic conduction model of Chapman (1957). However, in this paper we confine ourselves only to the former.Model I is electron-pressure dependent but model II depends particularly on the maximum coronal temperature T _m and its corresponding altitude h _m. Both the models are compared with recently constructed temperature models of Chiuderi and Riani (1974), McWhirter et al. (1975), and Gabriel (1976a). It is concluded that our model II reproduces these models within a factor of not more than 2.     

A search for a consistent model for the electromagnetic spectrum of the Crab nebula

An attempt is made to search for a consistent model to explain the electromagnetic spectrum of the Crab nebula (Tau A). It is assumed that there is a continuous injection of electrons at the centre of the nebula with an energy spectrumE ^–1.54 as evidenced by radio data. This spectrum must steepen to a slope larger than 2 at some energyE _i in order to ensure that the energy input into electrons remains finite. The spectrum must also steepen beyond an energyE _c depending on the magnetic field because of synchrotron energy losses. Two types of models are considered: Class I, in which the whole nebula is characterised by a uniform magnetic field, and Class II, in which besides the general fieldH ₀, small filamentary regions of strong fieldH _s are postulated.In models of Class I, the best fit to the observed data is obtained whenE _t >E _c andH ₀5×10^–4 gauss. However, this predicts a decrease in X-ray source size beyond 40 KeV. There are two possibilities of Class II model depending on the residence time of electrons in strong field regions being small or large. The former case explains the flattening in the optical spectrum.Experiments to distinguish between the various models are indicated.Presented at the International Conference on Cosmic Rays, Budapest, 1969.     

On Charged Analogues of Buchdahl’S Type Fluid Spheres

In this article the charged analogues of recently derived Buchdahl’s type fluid spheres have been obtained by considering a particular form of electric field intensity. In this process, Einstein–Maxwell field equations yield eight different classes of solutions, joining smoothly with the exterior Reissner–Nordstrom metric at the pressure free intersurface. Out of the eight solutions only seven could be utilized to represent superdense star models with ultrahigh surface density of the order 2×10¹⁴ gm cm⁻³. The maximum masses of the star models were found to be 8.223931M_Θ and 8.460857M_Θ subject to strong and weak energy conditions, respectively, which are much higher than the maximum masses 3.82M_Θ and 4.57M_Θ allowed in the neutral cases. The velocity of sound seen to be less than that of light throughout the star models.     

Theoretical models of planetary nebulae

Theoretical models are calculated for 15 planetary nebulae of medium-to-high excitation, following procedures previously described. Initial stellar energy distributions are adopted from Cassinelli (1971), but are subsequently modified to obtain the best representation of optical spectra for the selected objects. Other adjustable parameters include the stellar radius,R ^(*), the nebular density,N _H, the truncation radius,r _c, for the nebular shell, and the chemical composition. Excitationsensitive ratios are usually well-represented as are the actual observed intensities of spectral lines. Forbidden lines arising from 3p ³ configurations, e.g., those of [SII], [ArIV], and [ClIII] offer difficulties. For this sample of nebulae, the mean abundances seem to agree well with those found in an earlier study where the models were used as interpolation devices (Aller, 1978). Our objective is not to use the models to derive abundances explicitly, but rather to use them to find ionization correction factors. Some cautions and limitations of this procedure are described.     

The ratios I(K1)/I(H1) and I(K3)/I(H3) of ca II as diagnostics of the chromosphere above sunspots

The ratios I(K ₁)/I(H ₁) and I(K ₃)/I(H ₃) were calculated from four semi-empirical models of sunspot umbra. We determined the dependencies of both ratios of such parameters as temperature gradient and atmospheric opacity. A certain influence on the expected ratios I(K ₁)/I(H ₁) and I(K ₃)/I(H ₃) can also come from the FIP effect provided it exists in the chromosphere above sunspot umbra. Theoretical and observed values of I(K ₁)/I(H ₁) and I(K ₃)/I(H ₃) are compared. It is shown that for one of the sunspots we observed, the values obtained for the ratio I(K ₁)/I(H ₁) cannot be explained in terms of existing umbra models.     

Effects of variableC min on gamma-ray burstC max/C min distributions

Claims continue to be made that detector selection effects can explain the deviation of the gamma-ray burst brightness distributions from the -3/2 power law expected for homogeneous burst sources. However, these effects are insufficient to explain the BATSE observations. The BATSE sensitivity threshold does vary with time, independent of the burst brightness; however, a homogeneous distribution of standard candle sources would still produce a -3/2 power law. The variation in the threshold does affect inhomogeneous source models. As an example, the effect of a time-varyingC _min on theC _max/C _min distribution of an extended Galactic halo model is shown here. To fit the BATSEC _max/C _min distribution including a varyingC _min requires a larger observing distance (relative to the scale-height of the halo) than for a constantC _min; however, the observations can still be fit using the halo models.     

On the adjustment of outer solar layer models

Using the electron density n _e as an independent variable agreement between the models of the convective zone, photosphere, chromosphere, corona and solar wind is obtained. As a base the known data about the mean models of the individual layers of the quiet Sun are taken (i.e. without taking account of inhomogeneities and deviations from spherical symmetry). The chromospheric region is the exclusion. Here the run of T _e (n _e ) is revised anew to provide a smooth transition from temperature minimum to the corona and to satisfy the observed intensity distribution in the shortwave radio emission spectrum.A plot of the gas density versus n _e permits to get a clear representation about the rate of change of the degree of ionization x and to evaluate quickly the numerical values of x.     

Higher-dimensional vacuum bianchi-mixmaster cosmologies

We derive some new exact 7-dimensional cosmological solutions |R⊗ I ⊗N, whereN = I, II, VI₀, VII₀, VIII and IX are the various 3-dimensional Bianchi models. The solutions given are higher-dimensional generalizations of the mixmaster cosmologies. There is a strong influence of the extra spacesN, which results in a fundamental change of the 3-dimensional cosmology.     

Neutron star sequences and the starquake glitch model for the Crab and the Vela pulsars

We construct for the first time, the sequences of stable neutron star (NS) models capable of explaining simultaneously, the glitch healing parameters, Q, of both the pulsars, the Crab (Q≥0.7) and the Vela (Q≤0.2), on the basis of starquake mechanism of glitch generation, whereas the conventional NS models cannot give such consistent explanation. Furthermore, our models also yield an upper bound on NS masses similar to those obtained in the literature for a variety of modern equations of state (EOSs) compatible with causality and dynamical stability. If the lower limit of the observational constraint of (i) Q≥0.7 for the Crab pulsar and (ii) the recent value of the moment of inertia for the Crab pulsar (evaluated on the basis of time-dependent acceleration model of the Crab Nebula), I _Crab,45≥1.93 (where I ₄₅=I/10⁴⁵ g cm²), both are imposed together on our models, the models yield the value of matching density, E _b =9.584×10¹⁴ g cm⁻³ at the core-envelope boundary. This value of matching density yields a model-independent upper bound on neutron star masses, M _max≤2.22M _⊙, and the strong lower bounds on surface redshift z _R ≃0.6232 and mass M≃2.11M _⊙ for the Crab (Q≃0.7) and the strong upper bound on surface redshift z _R ≃0.2016, mass M≃0.982M _⊙ and the moment of inertia I _Vela,45≃0.587 for the Vela (Q≃0.2) pulsar. However, for the observational constraint of the ‘central’ weighted mean value Q≈0.72, and I _Crab,45>1.93, for the Crab pulsar, the minimum surface redshift and mass of the Crab pulsar are slightly increased to the values z _R ≃0.655 and M≃2.149M _⊙ respectively, whereas corresponding to the ‘central’ weighted mean value Q≈0.12 for the Vela pulsar, the maximum surface redshift, mass and the moment of inertia for the Vela pulsar are slightly decreased to the values z _R ≃0.1645, M≃0.828M _⊙ and I _Vela,45≃0.459 respectively. These results set an upper and lower bound on the energy of a gravitationally redshifted electron-positron annihilation line in the range of about 0.309–0.315 MeV from the Crab and in the range of about 0.425–0.439 MeV from the Vela pulsar.     

Some magnetized bulk viscous string cosmological models in?General Relativity

Some Bianchi type-I viscous fluid string cosmological models with magnetic field are investigated. The viscosity coefficient of bulk viscous fluid is assumed to be a power function of mass density ξ(t)=ξ ₀ ρ ^m , where ξ ₀ and m are constants. To get a determinate model, we assume conditions ρ=(1+ω)λ, where ρ is rest energy density, ω a positive constant and λ the string tension density and expansion θ is proportional to eigen value σ ₁¹ of the shear tensor σ _j ⁱ . The behaviour of the models from physical and geometrical aspects in presence and absence of magnetic field is discussed.      

Collinear Equilibrium Solutions of Four-body Problem

We discuss the equilibrium solutions of four different types of collinear four-body problems having two pairs of equal masses. Two of these four-body models are symmetric about the center-of-mass while the other two are non-symmetric. We define two mass ratios as μ ₁ = m ₁/M _T and μ ₂ = m ₂/M _T, where m ₁ and m ₂ are the two unequal masses and M _T is the total mass of the system. We discuss the existence of continuous family of equilibrium solutions for all the four types of four-body problems.     

A model atmosphere analysis of the binary 112 Herculis

Double-lined spectra have been obtained at 2 Å/mm for the Mn star 112 Herculis AB. The components are shown to be a B6.5V star and A2V star. Hydrogen-line-blanketed models are used to calculate atmospheric abundances. The best models appear to beT _e=13500 K, logg=4.0, _T =2.5 km/sec for the primary; andT _e=10000K, logg=4.2, _T =6 km/sec for the secondary. Both stars appear to have the same abundances. Helium, carbon, magnesium, aluminum, silicon, calcium, vanadium, and nickel appear to be considerably underabundant; phosphorus, scandium, titanium, manganese, iron, gallium, and mercury considerably overabundant; and chromium, strontium, and zirconium normal in abundance.