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.     

Pulsars: Observational parameters and a discussion on dispersion measures

The relevant data for the known 147 pulsars are presented in graphical and tabular forms. Various data correlations are discussed, and a detailed analysis of pulsar dispersion measures and distances is given. The range of the electron densities in the diffuse interstellar medium is found to be 0.01 cm^–3n _e0.1 cm^–3, and n _e0.03 cm^–3. The dispersion scale height for pulsars is found to be 5.9±0.7 pc cm^–3 implying a linear scale height of 200 pc, which is much smaller than the electron scale height of our Galaxy.Astrophysics and Space Science Review Paper.     

The peculiarities of diffusion of the galactic cosmic rays at low energies

I give an interpretation of a result of Simpsonet al. (1988) on the variation with kinetic energyT _i of the mean pathlengthX _m (T _i ) of the galactic cosmic rays (CGRs) in the range 0.1T _i 10.0 GeV nucl^–1. I argue that the data onX _m (T _i ) may be interpreted in terms of a model of GCR diffusion on the one-dimensional Alfvén-wave turbulence, having a cutoff in the spectrum at frequencies _h , where _h is the proton gyrofrequency. The cutoff results in changing of the character of variation of the GCR diffusion coefficientD(T _i )T ^a in the rangeT _i 1 GeV nucl^–1 towards some more complicated variation at 0.1T _i 1.0 GeV nucl^–1 due to the peculiarities of the pitch-angle scattering at 90⁰.     

Effects of mass transfer on the unsteady free convection flow past an accelerated vertical porous plate with variable suction

An exact analysis of the effects of mass transfer on the flow of a viscous incompressible fluid past an uniformly accelerated vertical porous and non-porous plate has been presented on taking into account the free convection currents. The results are discussed with the effects of the Grashof number Gr, the modified Grashof number Sc, the Schmidt number Sc, and the suction parametera for Pr (the Prandtl number)=0.71 representating air at 20°C.Nomenclature a suction parameter - C species concentration - C species concentration at the free stream - g acceleration due gravity - Gc modified Grashof number (vg^*(C –C )/U ₀ ³ ) - Pr Prandtl number (C _p/K) - T temperature of the fluid near the plate - T dimensionless temperature near the plate ((T-T )/(T -T )) - U(t) dimensionless velocity of the plate (U/U ₀) - v normal velocity component - v ₀ suction/injection velocity - x, y coordinate along and normal to the plate - v kinematic viscosity (/gr) - C _p specific heat at constant pressure - C _w species concentration at the plate - C non-dimensional species concentration ((C-C )/(C _w -C )) - Gr Grashof number (g(T _w -T )/U ₀ ³ ) - D chemical molecular diffusivity - K thermal conductivity - Sc Schmidt number (/D) - T _w temperature of the plate - T free stream temperature - t time variable - t dimensionless time (tU ₀ ² /) - U ₀ reference velocity - u velocity of the fluid near the plate - u non-dimensional velocity (u/U ₀) - v dimensionless velocity (v/U ₀) - v ₀ non-dimensionalv ₀ (v ₀ /U₀)=–at^–1/2 - y dimensionless ordinate (yU ₀/) - density of the fluid - coefficient of viscosity     

Small dense molecular clouds which envelope groups of T-Tauri stars

Dense molecular clouds within the Taurus and NGC 2264 regions have undergone gravitational collapse and fragmentation to form groups of low mass (1M ) T-Tauri stars which are still embedded within the clouds and which are kinematically associated with them. Molecular column densities on the order of 10¹⁴ cm^–2 are inferred from the emission lines of OH and NH₃. Emission line widths are 2 km s^–1 and the antenna beamwidths include linear extents of order 0.1 pc. The OH emission appears to be in a condition of local thermodynamic equilibrium, and it cannot arise from circumstellar sheils similar to those surrounding the masing infrared stars. The OH and NH₃ emission occurs in clouds of 1 pc in extent with optical depths of 0.1 to 1.0 and excitation temperatures of the order of 10 K. The molecular clouds have radii of 0.5 pc, molecular hydrogen densities of 4000 cm^–3, masses of 100 solar masses, and kinetic temperatures of 20 K. The observed data are not inconsistent with the molecular clouds being in a state of hydrostatic equilibrium.Paper presented at the Conference on Protostars and Planets, held at the Planetary Science Institute, University of Arizona, Tucson, Arizona, between January 3 and 7, 1978.     

The distinctive feature of relativistic stellar systems

In this paper we adopt the method of relativistic fluid dynamics to examine the number density distribution of stars around a massive black hole in the core of stellar clusters. We obtain extensive results,n(r) r ^–a, 3/2a9/2, which include, respectively, then(r) r ^–7/4 power law obtained by Bahcall and Wolf and then(r) r ^–9/4 power law by Peebles. Sincen(r) is not an observable quantity for star clusters, we also consider general relativity effects, i.e., the consequence of the bending of light, in calculating the projected density of stars in such a system. As an example we employ a massive black hole 10³ M inlaid in the center of a globular cluster and calculate various projected densities of stars. The results show that cusp construction occurs in all cases unless the central black hole massM=0, and the polytropic index does not affect at all the position of the capture radiusr _a. The obvious differences in the surface density is only embodied in the interior of the capture radius. At the outer regions of the core, the surface density of stars declines rapidly with ar ^–5 power law in all cases. These results can be applied to cases of unequal-mass and non-steady state.     

Coronal environment of quiescent prominences

With thespectro-coronagraph and themultichannel subtractive double pass spectrograph (MSDP) at the Pic du Midi Observatory two quiescent prominences were observed simultaneously. From the spectro-coronagraph observations 2D maps of Hei 10830 , Fexiii 10798 and 10747 line intensities were obtained. In addition, we obtained 2D maps of the ratioR of the two iron lines. This ratio is used as a diagnostic for determining the density of the hot coronal plasma surrounding prominences. We found that the electron density is higher at the location of the prominences than in the corona, whereas small regions (40) of lower electron density are unevenly distributed around the prominences indicating that the surrounding corona is highly inhomogeneous. The density of the cavity is reduced by a factor 1.5 compared to the density of the prominence environment (5 × 10⁸ cm^–3). We discuss the existence of cavities around these prominences according to the orientation of their axes relative to the line of sight and according to the velocity field inside the prominences. Constraints on models for prominence formation are derived.     

Absolute and relative amplitudes of variations in radius of classical cepheids

On the basis of observational data for the absolute R and relative R/R amplitudes of variations in radius of galactic classical cepheids (55 stars from Balona and Stobie (1979) and 30 stars from Sollazzoet al. (1981)), four kinds of empirical linear relations are obtained: log(P V)–logR, logP–logR, log(P V)–log(R/R), and logP–log(R/R);P, R, and V are the pulsation periods, the mean stellar radii, and the amplitudes of light variations, respectively. Three groups of stars are considered: short-period cepheids (SPC)-with logP1.1; long-period cepheids (LPC)-with logP>1.1; and s-cepheids (sC). Both the R values and the R/R values increase withP andP V, for a given group of variables. A comparison is performed with our results obtained from data in other sources (Kurochkin, 1966; Gieren, 1982; etc.). The investigated relations can be applied for determining R and R/R of galactic classical cepheids, by using their observedP and V. All studied galactic classical cepheids have R/R<0.35, R<10R for SPC and 10R <R60R for LPC. The sC have smaller R and R/R values than other classical cepheids, at the same periods (the difference is about 2 times for R and 1.4–2.8 times for R/R); the studied sC have R/R in the range 0.025–0.075 and R in the range 1–3R (only Y Oph has R8R ).     

Hidden mass in the solar neighborhood

It is suggested that the minimum mass of a star at the time of its formation is approximately 0.01M . Making use of this fact and the stellar mass functionF(M) M ^–, it is found that the hidden mass (or the missing mass) in the solar neighborhood may be explained by the presence of a large number of invisible stars of very low mass (0.01M M<0.07M ).     

The Energy Equation in the Lower Solar Convection Zone

As a consequence of the Taylor–Proudman balance, a balance between the pressure, Coriolis and buoyancy forces in the radial and latitudinal momentum equations (that is expected to be amply satisfied in the lower solar convection zone), the superadiabatic gradient is determined by the rotation law and by an unspecified function of r, say, S(r), where r is the radial coordinate. If the rotation law and S(r) are known, then the solution of the energy equation, performed in this paper in the framework of the ML formalism, leads to a knowledge of the Reynolds stresses, convective fluxes, and meridional motions. The ML-formalism is an extension of the mixing length theory to rotating convection zones, and the calculations also involve the azimuthal momentum equation, from which an expression for the meridional motions in terms of the Reynolds stresses can be derived. The meridional motions are expanded as U _r(r,)=P ₂(cos)₂(r)/r ²+P ₄(cos)₄(r)/r ² +..., and a corresponding equation for U (r,). Here is the polar angle, is the density, and P ₂(cos), P ₄(cos) are Legendre polynomials. A good approximation to the meridional motion is obtained by setting ₄(r)=–H₂(r) with H–1.6, a constant. The value of ₂(r) is negative, i.e., the P ₂ flow rises at the equator and sinks at the poles. For the value of H obtained in the numerical calculations, the meridional motions have a narrow countercell at the poles, and the convective flux has a relative maximum at the poles, a minimum at mid latitudes and a larger maximum at the equator. Both results are in agreement with the observations.     

Statistical survey of planetary nebulae: Distances,masses, and distribution in the galaxy

On the basis of empirical (D)-dependency at the frequency of 5 GHz constructed using 15 planetary nebulae with the independently measured distances (10^–171×10^–20 W m^–2 Hz^–1 ster^–1), we evaluated distances of 335 objects. Independent evidence of the correctness of the accepted scale are given. Then(D)-dependency is constructed and it is shown that atD<0.08 pc the mean electron density is higher than the one determined by the Seaton method. We showed that the filling factor diminishes with the increase of the PN diameter (1 atD0.08 pc and 0.2 atD0.4 pc). the ionized mass of 33 PNs is determined. With the diameter increase the ionized mass grows and atD0.4 pc reaches the valueM0.07M . We used the new distance scale when investigating the space distribution of PNs. The mean scale height =130±15 pc and the mean gradient of the change of surface densitym=0.37, which allowed us to estimate the total number of nebulae in the GalaxyN4×10⁴. We divided the PNs according to their velocities (withV _LSR>35 km s^–1 andV _LSR<35 km s^–1) and permitted us to confirm that the PN belong to different sub-systems of the Galaxy. The estimated local formation rate of PNs [=(4.6±2.2)×10^–12 pc^–3 yr^–1] is a little higher than the one of the white dwarfs. That can be explained by a large number of PNs having binary cores, which used in our sample. The statistical estimation of PN expansion velocity showed thatV _ex increases from 5–7 km s^–1 (atD0.03 pc) to 40–50 km s^–1 (atD0.8 pc).     

Almost stable regions for an area-preserving mapping

The area preserving mapping x = x + a(y – y ³), y = y – a(x – x³), for 0.3 a 2.0 has been studied to locate approximately the x-axis points bounding almost stable regions. For each value of a, these are fixed points with variational trace just greater than 2.0. Transition to chaos can occur rapidly as a increases (with n/k fixed).     

Einstein-like field equations with conserved source and decreasing ∧ term; Their cosmological consequences

The consequences of a cosmological term varying asS ^–2 in a spatially isotropic universe with scale factorS and conserved matter tensor are investigated. One finds a perpetually expanding universe with positive and gravitational constantG that increases with time. The hard equation of state 3P>U (U mass-energy density,P scalar pressure) applied to the early universe leads to the expansion lawSt (t cosmic time) which solves the horizon problem with no need of inflation. Also the flatness problem is resolved without inflation. The model does not affect the well known predictions on the cosmic light elements abundance which come from standard big bang cosmology.In the present, matter dominated universe one findsdG/dt=2H/U (H is the Hubble parameter) which is consistent with observations provided <10^–57 cm^–2. Asymptotically (S) the term equalsGU/2, in agreement with other studies.     

Kelvin-Helmholtz instability of composite plasma in an oblique magnetic field with resistivity

Analytic structure of high-density steady isothermal spheres is discussed using the TOV equation of hydrostatic equilibrium which satisfies an equation of state of the kind:P = K _g , = _g c ².Approximate analytical solutions to the Tolman-Oppenheimer-Volkoff (TOV) equations of hydrostatic equilibrium in (, ), (,U) and (u, v) phase planes in concise and simple form useful for short computer programmes or on small calculator, have been given. In Figures 1, 2, and 3, respectively, we display the qualitative behaviours of the ratio of gas density _g to the central density _gc , _g / _gc ; pressureP to the _gc ,P/ _gc ; and the metric componente ^–, for three representative general relativistic (GR) isothermal configurations =0.1, 0.2, and 0.3. Figure 4 shows the solution curve (, ) for =0.1, 0.2, and 0.3 (=0 represents the classical (Newtonian) curve). Numerical values of physical quantitiesv (=4r ² P ^*(r)), in steps ofu (=M(r)/r)=0.03, and the mass functionU, in steps of =0.2 (dimensionless radial distance), are given, respectively, in Tables I and II. Other interesting features of the configurations, such as ratio of gravitational radius 2GM/c ² to the coordinate radiusR, mass distributionM(r)/M, pressure (or density) distributionP/P _c , binding energy (B.E.), etc., have also been incorporated in the text. It has further been shown that velocity of sound inside the configurations is always less than the velocity of light.Part of the work done at Azerbaijan State University, Baku, U.S.S.R., and Mosul University, Mosul, Iraq, 1985-1986     

Electron densities above a polar coronal hole based on improved Si IX density diagnostics

Using new close-coupling excitation rates for the C-like ion Siix, density-diagnostic ratios based on Siix lines have been re-evaluated and applied to a sequence of CDS observations taken above a polar coronal hole. The derived electron densities are in excellent agreement with previous values of N_eestimated from the N-like ion Siviii for another coronal hole. The confirmed trend is for a fall-off of one order of magnitude within the first 0.3 Rabove the limb. These densities are well fitted with an analytic formula for the density profile out to at least 8 R, by which stage the electron density has fallen to 4×10³ cm^–3, from 1.5×10⁸ cm^–3at 1.0 R.     

Spatial correlation of Hα filaments and photospheric velocity

The location of H filaments is compared with maps of the photospheric line of sight velocity V and the magnetic field H . It is found that (1) H filaments are associated with regions of ¦V ¦ 300m s^–1, (2) always both positive as well as negative velocities are present under H structures, (3) stable (long lasting) portion of filaments frequently occur at the position of H = 0 as well as V = 0 lines, (4) this association remains valid for the longitudes less than 50° from central meridian.     

The north-south distribution of major solar flare events,sunspot magnetic classes and sunspot areas (1955–1974)

The north-south incidence has been studied of 31 white-light flares observed since 1859 and of 1669 events meeting the criteria for major flares of Dodson and Hedeman (1971) for the period 1955–1974. The asymmetry in favor of the northern hemisphere increases strikingly with the importance of the events. Similarly, magnetically complex sunspot groups (Mt. Wilson classes, and) display a more pronounced asymmetry in favor of the north than non-complex groups for 1962–1970. Contrary to the flare asymmetry, the spottedness asymmetry is independent of the size of sunspots.     

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.     

Lunar properties from transient and steady magnetic field measurements

The electrical conductivity of the lunar interior has been determined from magnetic field step transients measured on the lunar dark side. The simplest model which best fits the data is a spherically symmetric three layer model having a nonconducting outer crust of radial thickness 0.03R _moon; an intermediate layer of thicknessR0.37R _moon, with electrical conductivity ₁ 3.5 × 10^–4 mhos/m; and an inner core of radiusR ₂ 0.6R _m with conductivity ₂ 10^–2 mhos/m. Temperatures calculated from these conductivities in the three regions for an example of an olivine Moon are as follows: crust, < 440 K; intermediate layer, 890 K; and core, 1240 K. The whole-moon relative permeability has been calculated from the measurements to be/ ₀ = 1.03 ± 0.13. Remanent magnetic fields at the landing sites are 38 ± 3 at Apollo 12, 43 ± 6 and 103 ± 5 at two Apollo 14 sites separated by 1.1 km, and 6 ± 4 at the Apollo 15 site. Measurements show that the 38 remanent field at the Apollo 12 site is compressed to 54 by a solar wind pressure increase of 7 × 10^–8 dynes/cm².National Research Council Postdoctoral Associate.     

Comparison of large-scale structures and velocities in the local universe

Comparison of the large-scale density and velocity fields in the local universe shows detailed agreement, strengthening the standard paradigm of the gravitational origin of these structures. Quantitative analysis can determine the cosmological density parameter, , and biasing factor,b; there is virtually no sensitivity in any local analyses to the cosmological constant,. Comparison of the dipole anisotropy of the cosmic microwave background with the acceleration due to theIRAS galaxies puts the linear growth factor in the range ^0.6 /b = 0.6 _–0.3 ^+0.7 (95% confidence). A direct comparison of the density and velocity fields of nearby galaxies gives = 1.3 _–0.6 ^+0.7 , and from nonlinear analysis the weaker limit > 0.45 forb > 0.5 (again 95% confidence). A tighter limit, > 0.3 (4–6), is obtained by a reconstruction of the probability distribution function of the initial fluctuations from which the structures observed today arose. The last two methods depend critically on the smooth velocity field determined from the observed velocities of nearby galaxies by thePOTENT method. A new analysis of these velocities, with more than three times the data used to obtain the above quoted results, is now underway and promises to tighten the uncertainties considerably, as well as reduce systematic bias.