The relationship betweenG andh in a closed Universe

In a closed expanding-contracting Universe, matter will be subject to an inward acceleration large enough to prevent perpetual expansion. A closed Universe must also perform a simple harmonic motion, which might consist either of one single cycle or of an infinite series of oscillations about a central point. It is the purpose of this study to find the rate ofa ₀, the cosmic acceleration, from which the gravitational constantG can be determined. It will be shown from Ampère's equation and Planck's radiation law that it is possible to derivea ₀=7.623×10^–12 ms^–2, a value which also conforms with the uncertainty principle. The relationship betweena ₀ and electromagnetic radiation is based on the concept that charges (such as electrons) must emit radiation while accelerating. The rate ofa ₀ yields a universal gravitational constant ofG=6.645×10^–11 N m² kg^–2.     

The origin of the Big-Bang

Within the framework of FLRW cosmology withk=+1 a singularity free model of the Universe is proposed which readily accounts for the origin of the Big-Bang and for the preponderance of matter over anti-matter. It is also free from the problems of accounting for the observed large-scale homogeneity and isotropy of the Universe as well as from the problems of horizon and flatness. It is pointed out that the collapsing universe might have acted as an ultra-high energy particle accelerator. In the collapsing phase of the Universe, when the interparticle distances10^–16 cm, the electromagnetic and weak interactions might have unified into electroweak interaction and as the collapse proceeded further the entire matter in the Universe might have been converted into quark-gluon plasma permeated by leptons. The gravitational energy released during the collapse might have been locked in this plasma. Ass approached 10^–28 cm, grand unification of electroweak and strong interactions might have occurred. It is also suggested that, with further collapse, whens<10^–33 cm super-symmetry (SUSY)—i.e., the unification of all the four interactions (viz., electromagnetic, weak, strong, and gravitational) might have occurred. During this process gravitinos, the supersymmetric partners of graviton might have been produced. As a result of the exchange of a pair of virtual gravitinos between two particles an ultra-strong repulsive force between them might have been generated. Due to this ultra-strong repulsive interaction between particles the motion of the Universe might have been reversed, i.e., the Universe might have started expanding. During expansion, whens10^–28 cm, SUSY might have broken down spontaneously toSU ₅ and gravity. Ass increased from 10^–28 to 10^–16 cm, the gravitational energy locked in the quark-gluon plasma might have been released with a gigantic explosion, the so-called Big-Bang. It is estimated here that during this Big-Bang more than 10⁸² GeV of energy might have been released. Whens10^–16 cm,SU ₅ might have broken down spontaneously toSU ₃ andU ₁. Expansion beyond this stage might have occurred in the manner described by the standard cosmology. It is further suggested that in due course of time expansion will be followed by contraction and the cycle of contraction-expansion-contraction will be repeated ad infinitum.     

Some requirements of a colliding comet source of gamma ray bursts

Colliding comets in the Solar System may be an important source of gamma ray bursts. The spherical gamma ray comet cloud required by the results of the Venera Satellites (Mazets and Golenetskii, 1987) and the BATSE detector on the Compton Satellite (Meeganet al., 1992a, b) is neither the Oort Cloud nor the Kuiper Belt. To satisfy observations ofN(>P _max) vsP _max for the maximum gamma ray fluxes,P _max > 10^–5 erg cm^–2 s^–1 (about 30 bursts yr^–1), the comet density,n, should increase asn a ¹ from about 40 to 100 AU wherea is the comet heliocentric distance. The turnover above 100 AU requiresn a ^–1/2 to 200 AU to fit the Venera results andn a ^1/4 to 400 AU to fit the BATSE data. Then the masses of comets in the 3 regions are from: 40–100 AU, about 9 earth masses,m _E; 100–200 AU about 25m _E; and 100–400 AU, about 900m _E. The flux of 10^–5 erg cm^–2 s^–1 corresponds to a luminosity at 100 AU of 3 × 10²⁶ erg s^–1. Two colliding spherical comets at a distance of 100 AU, each with nucleus of radiusR of 5 km, density of 0.5 g cm^–3 and Keplerian velocity 3 km s^–1 have a combined kinetic energy of 3 × 10²⁸ erg, a factor of about 100 greater than required by the burst maximum fluxes that last for one second. Betatron acceleration in the compressed magnetic fields between the colliding comets could accelerate electrons to energies sufficient to produce the observed high energy gamma rays. Many of the additional observed features of gamma ray bursts can be explained by the solar comet collision source.     

The velocity sensitivity of resonant scattering spectrometers employing a piezoelastic modulator

The resonant scattering spectrometers of the IRIS ground-based network for measuring whole-disc solar velocity oscillations make use of a piezoelastic modulator. The velocity noise generated by this optical component is analysed with particular emphasis on the required stability of the amplitude of oscillation, a. The product of the absolute stability ¦ a – a _m ¦/a _m and the relative stability a _r.m.s./a _m may not be larger than 10 ^–4 to 10 ^–5 (depending on specific wishes), where a _m is the optimum amplitude. The velocity noise due to photon statistics is slightly enhanced, but other instrumental sources of velocity noise remain unaffected.     

Boltzmann constant in the model of expansive non-decelerative universe

From the observed present parameters of the Universe and the model properties of an expansive non-decelerative universe it results that the value of Boltzmann's constant (coefficient)k does not change only before the end of radiation era, but also in the matter era; with the increase of gauge factora, it decreases as (a ^–1)^1/4.     

Models of the universe compatible with present observations

It is shown that cosmic radiation almost follows a Planck distribution, because just as matter is formed, its density of energy is negligible in comparison with that of radiation, and that the present age of the Universe does not depend on the particular manner in which the matter is formed.Thus, if the results of the latest observations (which imply a deceleration parameterq=1.6) are combined with the assumption that the present age of the Universe is at least 12×10⁹ yr, they lead to a hyperbolic oscillating universe with a negative cosmological constant (<–1.53×10^–56 cm^–2) and a present mass-density _m of less than 1.2×10^–30 g cm^–3. If the cosmological constant is taken to be zero, a solution is only possible if we are prepared to admit a rate of evolution of galaxies with a deceleration parameterq<0.52. Three types of oscillating universe are then possible, but the heperbolic type is the most probable. If Hubble's constant is greater than 63.4 km s^–1 Mpc^–1, the solutions are only hyperbolic universes with <+0.45×10^–56 cm^–2 and _m <4.8×10^-30g cm^-3.

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Sommaire On montre que le rayonnement cosmique général suit pratiquement une loi de Planck parce que la densité d'énergie de la matière au moment de sa formation est négligeable à côté de celle du rayonnement et que l'âge actuel de l'Univers ne dépend pas du mode de formation de la matière.Dans ces conditions, si l'on combine les derniers résultats d'observations (qui impliquent un paramètre de décélérationq=1.6) avec l'hypothèse que l'âge actuel de l'Univers est au moins de 12×10⁹ années on est conduit à un Univers hyperbolique oscillant à constante cosmologique négative (<–1.53×10^–56 cm^–2) et où l'actuelle densité de matière _m est moindre que 1.2×10^–30 g cm^–3. Si la constante cosmologique est supposée nulle, une solution ne peut être obtenue que si l'on admet un certain taux d'évolution des Galaxies et un paramètre de décélérationq<0.52. Alors, les trois types d'Univers oscillants sont possibles, mains les Univers hyperboliques paraissent plus probables. Enfin, si la constante de Hubble est plus grande que 63.4 km s^–1 Mpc^–1 les solutions ne peuvent être que des Univers hyperboliques avec <+0.45×10^–56 cm^–2 et _m <4.8×10^-30g cm^-3.

     

Investigation of galaxy and quasar luminosity evolution in the model of standard cosmology

It is shown that the Hubble curvem(z) for galaxies and quasars averaged over a large volume of data forms in the first approximation a single continuous curve in the interval of red shifts 10^–2.5z4.5, which is satisfactorily described by the dependence .A large deviation of the observed mean dependence from the theoretical one predicted by the standard cosmology is explained by the evolution of the galaxy and especially quasar luminosity. The corresponding mid-statistical function of the absolute luminosity variation for the last 4/5 times of existence in the Universe is equal toM(z)–M(z ₀)=logz/z ₀+2z–0.4z ².The luminosity of the most far distant from the observed quasars on the average by 5–6 stellar magnitudes high than the luminosity of near galaxies and quasars. It is obtained that even the most far distant quasars atz5 are in the maximum of luminosity, or their extinction has just began, thus the quasar formation should be expected forz>(5–6). The relative rate of the luminosity decrease of galaxies and near quasars is rather accurately amounts in the recent epoch 7% per 10⁹ years. The obtained average Hubble curve of galaxies and quasars is evidently the main cause of their evolution in the Universe.     

The evolution of nonlinear hydrodynamical density fluctuations of the photon-plasma gas during the recombination era of the Universe

On the basis of the hydrodynamical equations of a two-component gas (photons and hydrogen with coupling via Thomson scattering) in the recombination era of the Universe (standard model), the evolution of the density perturbations up to second order are calculated. It is shown, that the generated second-order amplitudes of the density fluctuations of the matter reach values of the same order as the first-order amplitudes within only one tenth of the expansion time for fluctuations with wavelengths corresponding to 10⁷ M . Upper limits in the density fluctuations (for the gravitationally instable modes) up to which first-order calculations are valid, are given. This calculation indicates that the linear perturbation analysis is very restricted, especially at wavelengths near the lower limit of the Jeans length.The linear analysis would be a good approximation only for density fluctuations of the matter with the density contrast less than 10^–5–10^–4 at the recombination era. Therefore, a nonlinear analysis which is not based on a perturbation series is required for studying the evolution of the density perturbations, because for this we need a density contrast of 10^–2–10^–3 at the end of the recombination era.     

Theorie litterale du neuvieme satellite de Saturne,Phoebe

We study a theory for the ninth satellite of Saturn, Phoebe, based on the literal solution we have obtained in the main problem of the lunar theory.These series were computed by solving, by successive approximations, the Lagrange's equations expressed in variables, functions of the elliptic elements.We may consider the case of Phoebe simpler than a lunar case because we seek less precision (1/10 geocentric) than in the Lunar case, although the eccentricity of Phoebe is stronger.Main problem: our series are computed to the complete seventh order and a great part of the perturbations of the eighth and ninth order, where we have attributed to the small lunar parameters the order 1 tom ₀=n/n ₀,e ₀,e, sin (i ₀/2), the order 2 to ₀=(a ₀/a)((M ₁–)/(M ₁+M)) and the order 4 toµ ₀(a ₀/a)M ₁ M/M ₁ ² –M ².In the case of Phoebe,µ ₀ equal zero and ±₀ is the ratioa ₀/a.We study the further development of these series by using, instead of parameterm ₀, the quantity m ₀=n/n ₀–m ₁ wherem ₁ is an approached value ofm ₀, in order to accelerate the convergence of the series with respect tom ₀.Comparison with a numerical integration we are adjusting a numerical integration to the observations. We have already more than 100 observations, for the period 1900–1957.At first, we compare the series of the main problem to a numerical integration of the Keplerian problem.

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Proceedings of the Conference on Analytical Methods and Ephemerides: Theory and Observations of the Moon and Planets. Facultés universitaires Notre Dame de la Paix. Namur, Belgium, 28–31 July, 1980.     

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.     

Ionization feedback mechanism in the recombination era of the Universe

Within the cosmological standard models of the Universe, we inquire the 2-fluid radiation hydrodynamics during and before the recombination era. If we neglect all nonlinearities with exception of those contained in the coupling term between the two fluids via the degree of ionization, there exists a strong feedback mechanism on the instable modes which prevents them from growing if their relative amplitudes of the density contrast have values in the order of 10^–1–10^–2. The non-linear equations are numerically solved by adiabatic elimination technique used usually in self-organization problems. The effect depends very on the redshift z and become maximum atz1350. It depends also on the masses involved for the rangeM>10¹¹ M . The lowest limits in the amplitudes (10^–2) are imposed on the largest masses (10¹⁷ M ). In the range 10⁶M/M 10¹¹, the effect is independent of the mass.     

Relativistic hierarchical cosmology

While Euclidean models with uniform matter density have a number of radio sources of flux density greater thanF at frequencyv that varies asN(>F, v)¹ F ^–3/2, hierarchical models with = ₀ r ^–2 haveN(>F,)F ^–1/2 (Section 1). Since the observed dependency isN(>F,)F ^–1.8, severe density and/or luminosity evolution must be present in a workable hierarchical cosmology (Section 2). The same argument applies (Section 3) to the number of sources of apparent luminosity greater thanl,N(>l) and (Section 4) to the number of sources within redshift distancez from the local origin. To give agreement with empirical data demandsq _o=+1 and large first and second derivatives with respect to time of the number source density (Section 5). The adoption ofq _o=+1 allows one to show (Section 6) that a Lemaitre-type hierarchical Universe with a long coasting or waiting time can give agreement with observations of the numbers of QSO's etc. if the age of the Universe is more than 10¹³ yr. The dependence of the effective Hubble parameter onk(t), (t) andR (Section 7) leads one to suggest that ak=0, =0 hierarchy with 0 might be the simplest acceptable form of model Universe. Section 8 (Conclusion) points out that further data on source count anisotropies should allow the component levels of the hierarchy to be delineated.     

Formation of stars by implosion of a gas cloud

In the standard Friedmann cosmology the black-body radiation spectrum is usually taken (without explicit proof as far as we know) to have the same familiarT ⁴-form that it has in a flat space. With explicit use of the equation of motion of a quantized massless field propagating in a curved background Robertson-Walker metric we show (for the readily tractable scalar field case) that the assumption is in fact true for an open Universe. For a closed Universe, we find that there is an in principle modification to theT ⁴-law. Unfortunately, the correction turns out to be too small to be experimentally detectable. In passing, we also obtain a simple derivation for the cosmological red shift of frequencies.     

Coronal Mass Ejection of 15 May 2001: I. Evolution of Morphological Features of the Eruption

We study the initiation and development of the limb coronal mass ejection (CME) of 15 May 2001, utilizing observations from Mauna Loa Solar Observatory (MLSO), the Solar and Heliospheric Observatory (SOHO), and Yohkoh. The pre-eruption images in various spectral channels show a quiescent prominence imbedded in the coronal void, being overlaid by the coronal arch. After the onset of rapid acceleration, this three-element structure preserved its integrity and appeared in the MLSO MK-IV coronagraph field of view as the three-part CME structure (the frontal rim, the cavity, and the prominence) and continued its motion through the field of view of the SOHO/LASCO coronagraphs up to 30 solar radii. Such observational coverage allows us to measure the relative kinematics of the three-part structure from the very beginning up to the late phases of the eruption. The leading edge and the prominence accelerated simultaneously: the rapid acceleration of the frontal rim and the prominence started at approximately the same time, the prominence perhaps being slightly delayed (4 – 6 min). The leading edge achieved the maximum acceleration a_max 600 ± 150 m s^–2 at a heliocentric distance 2.4 –2.5 solar radii, whereas the prominence reached a_max 380± 50 m s^–2, almost simultaneously with the leading edge. Such a distinct synchronization of different parts of the CME provides clear evidence that the entire magnetic arcade, including the prominence, erupts as an entity, showing a kind of self-similar expansion. The CME attained a maximum velocity of v_max 1200 km s^–1 at approximately the same time as the peak of the associated soft X-ray flare. Beyond about 10 solar radii, the leading edge of the CME started to decelerate at a–20 m s^–2, most likely due to the aerodynamic drag. The deceleration of the prominence was delayed for 10 –30 min, which is attributed to its larger inertia.     

A complete model of the propagation of solar-flare cosmic rays

A two-stage model of the propagation of 1–50 MeV solar-flare cosmic rays is presented. The first stage consists of a thin spherical shell of radius r _a near the Sun which feeds particles into interplanetary space (the second stage) where they propagate along the Archimedean mean interplanetary magnetic field under the influences of anisotropic diffusion, convection, and energy changes. To calculate the time dependence at a fixed point in space, account is taken of the corotation of flux tubes past the observer.It is shown that the well-known east-west effect of the time-to-maximum cannot be obtained if the injection from the first stage is impulsive and thus a time and longitude dependent release for the second stage is essential. This is achieved by treating the first stage as a thin, spherical, diffusing shell of radius r _a with diffusion coefficient _s, from which particles leak into interplanetary space at a rate determined by the leakage coefficient .With this model we are able to reproduce simultaneously four principal features of solar events observed at r = 1 AU: (i) the east-west effect, i.e. the time-to-maximum as a function of flare longitude; (ii) the three phases of the anisotropy vector variation; (iii) the time-to-convective-phase as a function of flare longitude; and (iv) the longitudinal distribution of the differential intensity. Our best estimates of the parameters of the near-Sun propagation are that 0.01 hr^–1 _s/r _a ² 0.02 hr^–1 and 1/15 hr^–1 1/10 hr^–1. For the interplanetary propagation we estimate /V - 1.2AU with , the effective cosmic-ray diffusion coefficient and V, the solar-wind speed.     

Vacuum energy in cosmic dynamics

The analysis of the Th/U ratio in meteorites and the evolutionary ages of globular clusters favour values of the cosmic age of (19±5)×10⁹ yr. This evidence together with a Hubble parameterH ₀>70 km s^–1 Mpc^–1=(14×10⁹ yr)^–1 cannot be reconciled in a Friedmann model with =0. It requires a cosmological constant in the order of 10^–56 cm^–2, equivalent to a vacuum density _v =10^–29 g cm^–3 The Friedmann-Lemaître models (>0) with a hot big-bang have been calculated. They are based on a present value of the baryonic matter density of ₀=0.5×10^–30 g cm^–3 as derived from the primordial⁴He and²H abundances.For a Hubble parameter ofH ₀=75 km s^–1 Mpc^–1, our analysis favours a set of models which can be represented by a model with Euclidean metric (density parameter ₀=1.0, deceleration parameterq ₀=–0.93, aget ₀=19.7×10⁹ yr) and by a closed model with perpetual expansion (₀=1.072,q ₀=–1.0, aget ₀=21.4×10⁹ yr). A present density parameter close to one can indeed be expected if the conjecture of an exponential inflation of the very early universe is correct.The possible behaviour of the vacuum density is demonstrated with the help of Streeruwitz' formula in the context of the closed model with an inflationary phase at very early times.     

The masers of E-type methanol in orion KL and SGR B2

Using a simplified model the statistical equilibrium and radiative transfer equations of E-type-CH₃OH are solved for Orion KL and SgrB2. According to our calculation results and the observation data taken by Matsakiset al. (1980) and Morimotoet al. (1985a, b), the physical conditions of both sources are estimated. In theJ ₂-J ₁ E methanol maser region of Orion KL, the density, kinetic temperature, dust temperature, and the fractional abundance are 0.8–2×10⁶ cm^–3, 150, 30–90 K, 0.8–8×10^–6. In the 4_–1-3₀ E and 5_–1-4₀ E methanol maser region of Sgr B2 the correspondance physical conditions above are 10⁴ cm³, 45, 23 K, and 7×10^–7, respectively.     

The Hypothesis of Additional Acceleration in the Motion of Comet Harrington–Abell from Jupiter

By processing 494 observations of Comet Harrington–Abell, we obtained a unified system of elements that includes its turn around the Sun during which it closely approached Jupiter to a minimum distance of 0.037 AU in 1974. A study of the cometary orbit before and after the approach showed that, probably, at the approach of the comet to Jupiter, apart from the well-known gravitational perturbations, its motion was affected by an additional force. An improvement of the cometary orbit by assuming that an additional acceleration inversely proportional to the square of the distance to Jupiter exists in its motion yielded the following values: (4.57 ± 0.42) × 10^–10 and (–7.20 ± 0.42) × 10^–10 AU day^–2 for the radial and transversal acceleration components, respectively. As a plausible explanation of the changes in the cometary orbit, we additionally considered a model based on the hypothesis of partial disintegration of the cometary nucleus. The parameter that characterizes the instant displacement of the center of inertia along the jovicentric radius vector was estimated to be –1.83 ± 0.75 km. Based on a unified numerical theory of cometary motion, we determined the nongravitational parameters using Marsden's model for two periods: A ₁ = (11.68 ± 1.74) × 10^–10 AU day^–2, A ₂ = (0.53 ± 0.0357) × 10^–10 AU day^–2 for 1975–1999 and A ₁ = (5.92 ± 5.86) × 10^–10 AU day^–2, A ₂ = (0.08 ± 0.028) × 10^–10 AU day^–2 for 1955–1969, under the assumption that the nongravitational acceleration changed at the approach of the comet to Jupiter.     

A self-consistent world model

The field equations of the generalized field theory constructed by Mikhail and Wanas have been applied to a well-established geometrical structure given earlier by H. P. Robertson in connection with the cosmological problem. A unique solution, representing a specified expanding Universe (withq ₀=0, ₀=0.75,k=–1) has been obtained. The model obtained has been compared with cosmological observations and with FRW-models of relativistic cosmology. It has been shown that the suggested model is free of particle horizons. The existence of singularities has been discussed.The two cases, when the associated Riemannian-space has a definite or indefinite metric are considered. The case of indefinite metric with signature (+ – – –) is found to be characterized byk=–1, while the case of +ve definite metric is characterized byk=+1. Apart from that difference, the two cases give rise to the same cosmological parameters. It has been shown that energy conditions are satisfied by the material contents in both cases.     

Intergalactic suprathermal grains

The physical conditions under which suprathermal grains may loose energy and the processes involving the grains (a3×10^–6 cm) destruction are investigated. It is found that the dust grain once attaining the velocityU (10⁵ cm s^–1) may attain suprathermal energy (v _g3×10⁸ cm s^–1) and subsequently may also attain relativistic energy are almost indestructible in the accelerating phase.