Expanding and superdense astrophysical systems in general relativistic hierarchical cosmology

A semi-continuous hierarchy, (i.e., one in which there are galaxies outside clusters, clusters outside superclusters etc.), is examined using an expression of the field equations of general relativity in a form due to Podurets, Misner and Sharp. It is shown (a) that for a sufficiently populous hierarchy, the thinning factor( _i+1/ _i [r _i /r _i+1] is approximately equal to the exponentN in a continuous density law (=aR ^–N) provided (r _i /r _i+1)^3-1; (b) that a hierarchical Universe will not look decidedly asymmetric to an observer like a human being because such salient observers live close to the densest elements of the hierarchy (viz stars), the probability of the Universe looking spherically symmetric (dipole anisotropy0.1 to such an observer being of order unity; (c) the existence of a semi-continuous or continuous hierarchy (Peebles) requires that 2 if galaxies, not presently bound to clusters were once members of such systems; (d) there are now in existence no less than ten arguments for believing 2, though recent number counts by Sandageet al. seem to be in contradiction to such a value; (e) Hubble's law, withH independent of distance, can be proved approximately in a relativistic hierarchy provided (i)N=2, (ii)2GM(R)/c ² R1; (iii)Rc (iv)M0 in a system of massM, sizeR (f) Hubble's law holds also in a hierarchy with density jumps; (g)H100 km s^–1 Mpc^–1; (h) objects forming the stellar level of the hierarchy (in a cosmology of the Wilson type) must once have had 2GM/c ² R1; (i) there is a finite pressurep=2Ga in all astrophysical systems (a=R ^N ,N2); (j) for the Galaxy, theory predictsp _G7×10^–12 dyn cm^–2, observation givesp _G5×10^–12 dyn cm^–2; (k) if the mass-defect (or excess binding energy) hypothesis is taken as a postulate, all non-collapsed astrophysical systems must be non-static, and any non-static, p0 systems must in any case be losing mass; (1) the predicted mass-loss rate from the Sun is 10¹² g s^–1, compared to 10¹¹ g s^–1 in the observed solar wind; (m) the mass-loss rates known by observation imply timescales of 5×10⁹ years for the Sun and 10¹⁰ years for other astrophysical systems; (n) degenerate superdense objects composed of fermions must haveN-2 if they were ever at their Schwarzschild radii and comprised a finite numberN _B of baryons; (o)N _B10⁵⁷N for degenerate fermion and boson systems; (p)285-4; (q) the metric coefficients for superdense bodies give equations of motion that imply equal maximum luminosities for all evolving superdense bodies (L _max10⁵⁹ erg s^–1); (r) larger bodies have longer time-scales of energy radiation atL _max (10^–5 s for stars,1 h for QSO's) (s) expansion velocities are c soon after the initial loss of equilibrium in a superdense object; (t) if the density parametera(t) in aR ^–N isa=a (non-atomic constants of physicsc, G, A), andA, thenN=2; (u) N2 is necessary to giveMM at the stellar level of the hierarchy;(v) systems larger than, and including, galaxies must have formed by clumping of smaller systems and not (as advocated by Wertz and others) in a multiple big bang.     

Unsteady hydromagnetic free convection flow past an accelerated infinite vertical porous plate

Unsteady laminar free convection flow of a viscous incompressible and electrically conducting fluid past an accelerated vertical infinite porous plate subjected to a suction velocity proportional to (time)^–1/2 is studied in presence of a uniform horizontal magnetic field. Results are discussed with the effects of the Grashof number Gr, and the magnetic field parameterM for Pr (the Prandtl number)=0.71 and 7.0 representing air and water respectively at 20 °C.Nomenclature a suction/injection parameter - C _p specific heat at constant pressure - B ₀ magnetic induction - g acceleration due to gravity - Gr Grashof number (vg(T'_w-T')/U ₀ ³ ) - K thermal conductivity - M magnetic field parameter (B ₀ ² _e ² /U ₀ ² ) - Pr Prandtl number (C _p/K) - T' temperature of the fluid near the plate - T' _w temperature of the plate - T' temperature of the fluid at infinity - t' time variable - t dimensionless time (t' U ₀ ² /v) - u non-dimensional velocity (u'/U ₀) - U' velocity of the plate - U dimensionless velocity of the plate (U'/U _o) - U ₀ reference velocity - v' ₀ suction velocity - v ₀ nondimensional suction velocity (v' ₀/U ₀)=at^–1/2 - v' normal velocity component - v dimensionless normal velocity - Ec Eckert number ((vU ₀)^2/3/C _p(T' _w -T' )) - T dimensionless temperature of the fluid near the plate ((T'-T' )/T' _w –T' )) - x',y' coordinates along and normal to the plate - y dimensionless ordinate (=y' U _o/v) - v kinematic viscosity - coefficient of volume expansion - electric conductivity of the fluid - similarity variable (y/2t) - _w density of the fluid at the plate - density of the fluid at infinity - ' skin-friction - dimensionless skin-friction - coefficient of viscosity - _e magnetic permeability     

The effect of chemical abundance oscillations on the pulsational properties of A 5M ⊙ hydrogen-helium star

A model of a first generation intermediate star of 5M , with Z=0 has been considered. The model is at an advanced stage of its evolution and has a double shell burning. It burns helium in the inner shell, and hydrogen, via CNO cycle, in the outer shell. =(log/log) _T and _T =(log/logT) were computed allowing for the oscillations of the relative mass abundance of the reagents in nuclear reactions. Including =(log/log) _T and =(log/logT) of mean molecular weight and the effect of the oscillations of abundances due to nuclear reactions, stability was studied. Contrary to the results of the static calculations, we found that instability due to the excitation mechanism provided by the high temperature sensitivity of energy generation rate propagates up to the surface. Thus the model in question was found to be unstable against radial adiabatic pulsations, in its fundamental mode.     

Some applications of Jacobi dynamics to the stellar evolution

The stars in the Main Sequence are seen as a hierarchy of objects with different massesM and effective dynamical radiiR _eff=R/ given by the stellar radii and the coefficients for the inner structure of the stars.As seen in a previous work (Paper I), during the lifetime in the Main SequenceR _eff(t) remains a near invariant when compared to the variation in the time ofR(t) and (t).With such an effectiveR _eff one obtains the amounts of actionA _c(M), the effective densities _eff(M)=(M)³(M), the densities of action and of energy (or mean presures in the stellar interior)a _c(M),e _c(M), and the potential energiesE _p(M).The amounts of action areA _cM ^k withk1.87 for the M stars,k5/3 for the KGF stars, andk1.83 for the A and earlier stars, representing very simples conditions for the other dynamical parameters. For instancek5/3 means a near invariant effective density _eff for the KGF stars, while for such stars the mean densities and coefficients present the strongest variations with masses (M)M ^–1.81, (M)M^0.6.The cases for the M stars (e _c(M)M ^–1) and for the A and earlier stars (betweena _c(M)=constant and _eff(M)M ^–1) and also discussed. These conditions for the earlier stars also represent reasonable mean values for the whole stellar hierarchy in the range of masses 0.2M M25M .With all this, one can build dynamical HR diagrams withA _c(M), E_p(M), _eff M ^–p , etc., whose characteristics are analogous to these in the photometrical HR diagram. A comparison is made betweenA _c(M) from the models here and the HR diagram with the best known stars of luminosity classes IV, V, and white dwarfs.The comparison of the potential energiesE _p(M)M ^–p according to the stellar models used here and the observed frequency function (MM _–q (number of stars in a given interval of masses) from different authors suggests the possibility that the productE _p(M)(M) is a constant, but this must be confirmed with further studies of the function (M) and its fine structure.There are analogies between the formulation used here for the stellar hierarchy and other physical processes, for instance, in modified forms of the Kolmogorov law of turbulence and in the formulation used for the hierarchy of molecular clouds in gravitational equilibrium. Besides, the function of actionA _c(M) for the stars has analogous properties to the relations of angular momenta and massesJ(M) for different types of objects. The cosmological implications of all this are discussed.     

Theory of the Trojan asteroids,IV

In a previous publication (1977) the author has constructed a family () of long-periodic orbits in the Trojan case of the restricted problems of three bodies. Here he constructs the domain of the analytical solution of the problem of the motion, excluding the vicinity of thecritical divisor which vanishes at the exact commensurability of the natural frequencies ₁ and ₂. In terms of thecritical masses m_j(₂), or the associatedcritical energies _j ² (m), is the intersection of the intervals ofshallow resonance, of the form. Inasmuch as the intervals |₂– _j ² |<_j ofdeep resonance aredisjoint, it follows that (1) the disjointed family () embraces the tadpole branch, 0²1, lying in: and (2) despite the clustering of _j ² (m) atj=, the family () includes, for ²=1, an asymptoticseparatrix that terminates the branch in the vicinity of the Lagrangian pointL ₃.In a similar manner, the family () can be extended to the horseshoe branch 1<_{2 2} ² .     

Mgix emission lines in an active region spectrum obtained with the Solar EUV Rocket Telescope and Spectrograph (SERTS)

Theoretical electron-temperature-sensitive Mgix emission line ratios are presented forR _I =I(443.96 )/I(368.06 ),R ₂ =I(439.17 )/I(368.06 ),R ₃ =I(443.37 )/I(368.06 ),R ₄ =I(441.22 )/I(368.06 ), andR ₅ =I(448.28 )/I(368.06 ). A comparison of these with observational data for a solar active region, obtained during a rocket flight by the Solar EUV Rocket Telescope and Spectrograph (SERTS), reveals excellent agreement between theory and observation forR ₁ throughR ₄, with discrepancies that average only 9%. This provides experimental support for the accuracy of the atomic data adopted in the line ratio calculations, and also resolves discrepancies found previously when the theoretical results were compared with solar data from the S082A instrument on boardSkylab. However in the case ofR ₅, the theoretical and observed ratios differ by almost a factor of 2. This may be due to the measured intensity of the 448.28 line being seriously affected by instrumental effects, as it lies very close to the long wavelength edge of the SERTS spectral coverage (235.46–448.76 ).     

Intensity fluctuations in Fraunhofer lines

An expression is derived for the fluctuation (t) in emergent intensity (observed at some wavelength in a Fraunhofer line or the continuum) caused by a perturbation in temperature (z, t) in the Sun's atmosphere. If the contribution function for the observed intensity is single-peaked near z and if (z) and p(z) are not too rapidly varying, then (t) m (z , t)+N p(z , t) where m and N depend on the structure of the atmosphere. We compute M, N, and contribution functions for several values of and in the inner wings of the K line (13933 Caii).Presently on leave of absence from the Institute for Astronomy, Honolulu, Hawaii.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Galactic4He and heavy elements enrichment by stellar nucleosynthesis

The contribution to the galactic abundance of He and heavy elements by stellar nucleosynthesis is calculated as a function of time, keeping account of present knowledge about stellar and galactic evolution. A model is used which distinguishes the phase of the contracting halo from the subsequent history of the disc. Various uncertainties involved both in stellar and in galactic evolutionary theory are discussed. The amount of⁴He produced by stars of different masses and ejected in interstellar medium is fairly well known from stellar theory, while we have assumed its primordial abundance as a free parameter, ranging from 0 up to 0.4. We find that stellar activity provides a significant contribution to the cosmic⁴He, though not sufficient to explain the observed abundance. The best agreement with observational data (Y 0.26 andY _now0.28) is obtained starting with a primordial abundanceY =(0.20–0.23), which is consisten with the Big-Bang theory predictions and with recent observational estimates. The contribution to the abundance of heavy elements depends on the last stellar stages and on the final explosion mechanism, which are only now beginning to be understood. Nevertheless, in the framework of present theories, we individuate a stellar evolutionary scheme reproducing the observedZ abundances for Populationi and Populationii stars, with the correctly estimated Y/Z value. In this scheme, only stars belonging to two narrow mass ranges (10m/m 15 andm/m 80) are allowed to eject metal-enriched matter, possibly with the solar (C+O)/(Si+Fe) ratio.     

Central holes in disks of spiral galaxies

On the basis of the solutions obtained in the previous paper, the changes in the scenario of the standard model of the Big Bang are found. The chaos degree (constrainst on fluctuation spectra) is obtained, which could be still preserved by the initially completely chaotic Universe at the time of light elements nucleosynthesist _es. The time boundaries of hadron and lepton eras and the time the electron neutrinos and neutrons become frozen in reactions of weak interaction may be shifted up to 1.4 times. The corresponding temperatures may shift off from the standard ones 0.88 times if the mean-square level of fluctuations is close to unity. If the density of the energy of fluctuations concentrated in the short-wave region of the spectrum is less than 1.5 ^– , the nucleosynthesis leads to a helium abundance coinciding with the observe one. If at the timet _es the maximum of the spectral density of the energy is in the long-wave region, that is _max ct _es the level of the chaos during the period of nucleosynthesis is restricted to 1.76 (where |C _K |² d³ K,C _K is Fourier component of the amplitude of metric fluctuations). In particular, the protogalactic vortical disturbances with a wide spectrum 4 × 10^{3 -1}( = K/K, = /_crit) are compatible with the observed helium abundance.     

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.     

Stellar structure with a Boltzmann factor: An extension to chemically heterogeneous H−He-stars

We present an attempt for an extension of the modified Boltzmann model, which was introduced by Callebautet al. (1982) as an improvement of the polytropic models, to the case of chemically-heterogeneous stars in equilibrium, containing H and He, by proposing a density profile of the formp=C ₁ T ^N exp (–_H m(–_*)/kT) +C ₂ T ^N exp (–_He m(–_*)/kT. Analytical properties are derived and numerical as well as analytical arguments are presented for the conclusion that this hypothesis for a density profile imposes an almost constant chemical profile to the model as a whole, thereby making it in this form unsuited for the study of heterogeneous stars. A comparison is made with the former Boltzmann model in the homogeneous limit.     

Radio source orientation and the cosmological interpretation of the angular size-redshift relation for double-lobed quasars

The angular size-redshift test for quasars was compared with various cosmological models including non standard models. The possible effects of radio source orientation and relativistic beaming were taken into account in the analysis.It was found that orientation effects alone were not sufficient to explain the observed-z relation in terms of Friedmann models. In addition, linear size evolution of the formD ~ (1 +z)^–n , with 0.75 n 1.2 would be required for 0 1.0, or possibly an inverse correlation between luminosity and linear size. The non-standard cosmological models all gave better fits to the deprojected data than the Friedmann models in the absence of evolutionary effects, with the tired light effect providing the best fit.     

The proton-proton reaction and related reactions in an intense magnetic field

The reaction rates for the proton-proton reaction and the related electron capture reaction in a strongly magnetized relativistic electron gas of arbitrary degree of degeneracy are computed. The proton-proton reaction rates are unaffected by the presence of the magnetic field for field strengths up to the critical valueH _q =m ² c ³/e=4.414×10¹³G. For fields greater thanH _q , the proton-proton reaction rates are enhanced linearly with (=H/H _q ).The PeP reaction is investigated in detail for a wide range of temperatures, densities and magnetic field strengths that are of interest. The main results are as follows: In the non-degenerate regime the reaction rates are significantly reduced for high temperatures (T ₉5) and low fields (1). For instance, _pep(H)=0.04 _pep(O) at =10^–3,T ₉=10. For relatively high fields (>1) and low temperatures (T ₉2), the reaction rates are enhanced approximately linearly with . In the complete degenerate regime the reaction rates are reduced up to one-third of the field-free value for moderate densities (₆/_e10). At high densities (₆/_e10) the reaction rates are unaffected by the magnetic field.     

Redshift cutoff and evolution of QSOs

In this paper I present a new evolution model of QSOs luminosity. The model is based on edges distribution of apparent magnitude-redshift of QSOs. After the quasars were formed, the luminosities were increasing until they attained their maximum value atz=2+a, where –0.1a0.6, then the luminosities were decreasing. If the QSOs originate from superconducting cosmic string of same initial massM _i 10¹² M , the formation epochs are different, most of the quasars start atz _cutoff5.6. The most luminous QSOs start at later epochz _cutoff5.15. The present sky survey echniques may give us the possibility to see the formation of QSOs at apparent magnitudem _V 22.5 by chance of 0.3%.     

Abundances of heavy elements in the sun

Solar abundances have been derived for Pb, Bi, Th and U. For the three first elements, the abundances are determined from the profiles of their lines obtained with the high resolution solar spectrometer installed at the Jungfraujoch Scientific Station (Switzerland) and from new oscillator strengths based on life-time measurements (Pb and Bi). The possible presence of U in the solar spectrum is also examined.Our results are (in the log N _H = 12.00 scale): logN _Pb = 1.83, logN _Bi 0.80, logN _Th = 0.82, and logN _U 0.60.The meaning of these new solar abundances is discussed from the point of view of the nucleosynthesis theories.This work has been sponsored in part by the Air Force Cambridge Research Laboratories, OAR, through the European Office of Aerospace Research, U.S.A.F., under Contract AF 61 (052)-955 and by the Comité National Belge de la Coopération Géophysique Internationale (C.N.B.C.G.I.).     

Gas,dust and molecules in the Galaxy

The relative abundances of cool neutral hydrogen, carbon monoxide and formaldehyde are studied using all the available observational data in the literature. The obtained mean valuesN _{H 1}/ ,N _{H 1}/N _CO,N _CO/ are approximately constant in the dark clouds of the solar neighbourhood and in the distant molecular clouds.The observed correlationsN _CO,A _v and ,A _v show that formaldehyde can also be used as an indicator of molecular hydrogen. The ratioN H₁/A _v depends on densities and decays considerably in the ranges of visual absorptions in which the molecules become detectable (A _v 2 mg); an average of /N _{H 1}10 is calculated for the dense dark clouds.Indications of systematic temperature gradiens T/A _v are found for formaldehyde and neutral hydrogen inside the dark clouds, and qualitative comparisons are made with theoretical quantum mechanics calculations.The observed carbon monoxide and formaldehyde abundances, the free electron layer in the Galaxy, the distribution of neutral hydrogen in different states are only compatible if an ionization rate of 10^–16 is accepted, provided presumably by 2 MeV protons of cosmic radiation.Three main states for neutral hydrogen and dust are identified from different kinds of observational data (21 cm line in emission, absorption in galactic radio sources and self-absorption in the hot gas background): (1) a homogeneous intercloud stratum of tenuous gas and dust with a galactic halfwidth of 350 pc and mean parametersn _H=0.2 atom cm^–3, spin temperatureT _s 10000 K andn _d 0.3 mg kpc^–1; (2) cool gas and dust concentrated in spiral features with a galactic half-width of less than 100 pc, probably forming clouds with diffuse and indefinite limits, with mean parametersn _H2 atom cm^–3,T _s <1100 K (probable average,T _s =135 K) andn _d 3 mg kpc^–1; (3) dense gas and dust clouds with a mean diameter of 7 pc and mean parametersn _H700 atom cm^–3 (90% in a molecular state),T _s 63 K andn _d 1 mg pc^–1 on which molecules as CO and H₂CO are formed.The application of the Jeans criteria for gravitational instability shows that the dense clouds are gravitationally bound while the gas in the intermediate state (2) can be protected against collapse by the total internal energy in the medium increasing due to cosmic rays and the magnetic field in the Galaxy.The observed velocity halfwidths and galacticZ-halfwidths in states (1) and (2) are compatible with a total mass density in the galactic layer of 90M pc^–2 (gas plus stars) according to the barometric equation.The relative abundancesN _{H 1}/N _CO, calculated from C¹²O and C¹³O data and comparisons with studies in the 21 cm emission line, show that the antenna temperatureT _A ⁺ in the 2.6 mm line of C¹²O is a good indicator of the cool gas densities in the Galaxy. The possible application of this for studies in galactic structure is discussed and hypothetical distributions of carbon monoxide in the zones outside the galactic planeB=0° are presented.From a synthesis based on the results obtained, a cycle is postulated for the neutral hydrogen in the Galaxy: condensation and cooling of gas molecular formation gravitational collapse and star formation gas dissipation and heating by cosmic rays and UV radiation.     

The abundance of chlorine in the Sun

The Cli line 8375.943 (4s ⁴P_5/2 – 4p ⁴D7/2) is identified in the solar spectrum. This is the first identification of a chlorine line in solar spectrum. The measured equivalent width (W = 0.8 mÅ) corresponds to an abundance log N(Cl) = 5.65 on the scale log N(H) = 12.00.     

Stability criteria in many-body systems

An expansion of the force function ofn-body dynamical systems, where the equations of motion are expressed in the Jacobian coordinate system, is shown to give rise naturally to a set of (n–1) (n–2) dimensionless parameters ^ki _li {i = 2,...,n;k = 2,...,i – 1 (i 3);l =i + 1,...,n (i n – 1)}, representative of the size of the disturbances on the Keplerian orbits of the various bodies. The expansion is particularized to the casen=3 which involves the consideration of only two parameters ²³ and ₃₂. Further, the work of Szebehely and Zare (1977) is reviewed briefly with reference to a sufficient condition for the stability of corotational coplanar three-body systems, in which two of the bodies form a binary system. This condition is sufficient in the sense that it precludes any possibility of an exchange of bodies, i.e. Hill type stability, however, it is not a necessary condition. These two approaches are then combined to yield regions of stability or instability in terms of the parameters ²³ and ₃₂ for any system of given masses and orbital characteristics (neglecting eccentricities and inclinations) with the following result: that there is a readily applicable rule to assess the likelihood of stability or instability of any given triple system in terms of ²³ and ₃₂.Treating a system ofn bodies as a set of disturbed three-body systems we use existing data from the solar system, known triple systems and numerical experiments in the many-body problem to plot a large number of triple systems in the ²³, ₃₂ plane and show the results agree well with the ²³, ₃₂ analysis above (eccentricities and inclinations as appropriate to most real systems being negligible). We further deal briefly with the extension of the criteria to many-body systems wheren>4, and discuss several interesting cases of dynamical systems.     

General integral transform of the exponential integrals

General integral transform of the exponential integralsE _n is considered and will be denoted asB _(k) ⁿ (). Different expressions and the equations satisfied byB _(k) ⁿ are developed. Two-term recurrence formula forB _(k) ⁿ (0) and three-term recurrence formula forB _(k) ⁿ (); 0 will be established for a givenk1 andn=2,3, ...,N. The computational algorithms based on these formulae are also constructed for the casesk=1,2,3, andn2. Finally the numerical results fork=2,3 andn=2(1)25 are presented to 15-digit accuracy