The equal area law of asymptotically AdS black holes in extended phase space

According to Maxwell’s equal area law we study the “real” phase diagram of the charged AdS black hole and the Kerr-AdS black hole. In the extended phase space constructed by treating the cosmological constant as pressure, the two kinds of AdS black hole display themselves like van der Waals system. Under the critical temperature T _c , there exists a regime where the condition of stable equilibrium will be violated. We employ the equal area law to find an isobar which is the real two phase coexistence line. As a byproduct we find a simple method to derive the critical value of the thermodynamic quantities.     

Temperature profile of accretion disk around a kerr black hole

We use a rigorous, general relativistic, viscous, fluiddynamical equation to calculate the temperature profile of the accretion disk around a Kerr black hole. Our result shows that there is indeed a maximum in the profile. If the radiation in the disk is black-body radiation, then the result is independent of the viscosity mechanism. Using a free-free radiation and the Lα viscosity law, our results are similar to our previous findings [1]. Our calculation also shows that within the radius of the temperature maximum, there exists a rapidly cooling ring-shaped region, possessing characteristics that are not found in the case with the usual temperature profile.     

Internal structure of massive terrestrial planets

Planetary formation models predict the existence of massive terrestrial planets and experiments are now being designed that should succeed in discovering them and measuring their masses and radii. We calculate internal structures of planets with one to ten times the mass of the Earth (Super-Earths) to obtain scaling laws for total radius, mantle thickness, core size and average density as a function of mass. We explore different compositions and obtain a scaling law of R∝M0.267-0.272 for Super-Earths. We also study a second family of planets, Super-Mercuries with masses ranging from one mercury-mass to ten mercury-masses with similar composition to the Earth's but with a larger core mass fraction. We explore the effect of surface temperature and core mass fraction on the scaling laws for these planets. The scaling law obtained for the Super-Mercuries is R∝M∼0.3.     

On the heat conduction in a high-temperature plasma in solar flares

We have developed three types of mathematical models to describe the mechanisms of plasma heating in the corona by intense heat fluxes from a super-hot (T _e ? 10⁸ K) reconnecting current layer in connection with the problem of energy transport in solar flares. We show that the heat fluxes calculated within the framework of self-similar solutions using Fourier’s classical law exceed considerably the real energy fluxes known from present-day multi-wavelength observations of flares. This is because the conditions for the applicability of ordinary heat conduction due to Coulomb collisions of thermal plasma electrons are violated. Introducing anomalous heat conduction due to the interaction of thermal runaway electrons with ion-acoustic turbulence does not give a simple solution of the problem, because it produces unstable temperature profiles. Themodels incorporating the effect of collisional heat flux relaxation describe better the heat transport in flares than Fourier’s law and anomalous heat conduction.     

The luminosity functions of the 1969 Perseid and Orionid meteor showers

Visual counts of the 1969 Perseid and Orionid meteor showers are presented, comprising 288 Perseids and 56 Orionids. On the basis of the maximum-likelihood method of determining the power law luminosity function index, we derive s ≡ 1 + 2.5 log(r) = 1.56 ± 0.06 for the Perseids with m_v = +1 to ?5, and s ≈ 1.85 ± 0.1 for the Orionids with m_v = +2 to ?3. These values are somewhat lower than those found by other observers, but we confirm the approximate power law nature of the luminosity functions. Under the assumption that the masses of visual meteors are proportional to a power law function of the luminosities, this implies power law mass functions. If mass is directly proportional to luminosity, we have power law mass functions with the indices s given above.     

The spectrum of titan in the far-infrared and microwave regions

The pressure-induced absorptions of gaseous nitrogen (N₂) and methane (CH₄) are computed on the basis of the collisional lineshape theory of G. Birnhaum and E.R. Cohen [Canad. J. Phys.54, 593–602 (1976)]. Laboratory data at 300 and 124°K for N₂ and at 296 and 195°K for CH₄ are used to determine the collisional time constant and their temperature dependence. The spectrum of Titan from the microwave to the far-infrared region (0.1–600 cm^?1) is then modeled using these opacities and a temperature profile of Titan's atmosphere derived from the Voyager 1 radio occultation experiment. The model atmosphere is composed of N₂ and CH₄, their relative proportions being determined by the vapor pressure law of CH₄. A model with gaseous opacity alone is ruled out by the far-infrared observations. An additional opacity, thought to be associated with a methane cloud, is confirmed. The effective temperature of Titan is estimated at T_e = 83.2 ± 1.4°K.     

The heating of the earth during its formation

The thermal state of the Earth accumulating from solid bodies is investigated. The conductivity equation is deduced for a growing spherically symmetrical planet which takes into account heating by impacts of bodies, by radioactivity, and by compression of its material. The cooling is produced mainly by impact mixing, which is approximated by extrapolating the parameters from known impact craters to larger sizes. The solution of a more simple conductivity equation for a uniformly heated plane parallel layer with moving boundaries is found. It can be considered as an approximate quasi-stationary solution for the temperature distribution in the outer parts of the growing Earth. The result depends substantially on the sizes of impacting bodies but almost not at all on the time scale of the accumulation. The latter only weakly affects the surface temperature and does not affect the temperature distribution in the layer. For bodies of small radii, r′ < r₁, where the size of the crater is not affected appreciably by gravitation (for the present mass of the Earth r₁ ≈ 1 km), the heating is small. For bodies with r′ > r₁, the heating of the layer is roughly proportional to the ratio $\text{r′}\text{r}{}_1$. Toward the end of the Earth's accumulation the melting point can be reached in the outer layer at r′_M ? 60 km, where r′_M is the radius of the largest body in the power law size spectrum of falling bodies. The estimates of the initial temperature of the Earth can vary within wide limits depending on the mass distribution of large protoplanetary bodies, which at the present time is not known correctly. The initial melting of an upper layer of the Earth a few hundred kilometers thick seems to be possible.     

Self-modulational formation of pulsar microstructures

In this paper, we used the more rigorous general relativistic viscous hydrodynamical equation to discuss and calculate the temperature profile in the accreting disk around a Kerr black hole. It is found that for an accreting disk around a Kerr black hole, the occurrence of extremum values in the temperature profile is indeed possible. Furthermore, the temperature profile is always the same—no matter what kind of viscous mechanism is adopted—if the radiative mechanism of the materials in the disk is black-body radiation. For free-free radiation, we adopted the so-called α-viscosity law model to get a solution analogous to that obtained earlier (Fanget al. 1980). The calculations show that there exists a fast cooling ring region lying just inside the temperature maximum. In this ring region, there are probably some pecularities which do not exist in the usual temperature profile calculation (Fang, 1981).     

Extinction law at a distance up to 25 kpc toward the Galactic poles

Photometry from the Tycho-2, 2MASS, andWISE catalogues for clump and branch giants at a distance up to 25 kpc toward the Galactic poles has allowed the variations of various characteristics of the infrared interstellar extinction law with distance to be analyzed. The results obtained by the extinction law extrapolation method are consistent for different classes of stars and different characteristics as well as with previous studies. The conventional extinction law with a low infrared extinction is characteristic of only a thin layer no farther than 100 pc from the Galactic plane and of two thin layers near Z = ?600 and +500 pc. Far from the Galactic plane, in the Galactic halo, the infrared extinction law is different: the extinction in the Ks, W1, W2, W3, and W4 bands is, respectively, 0.17, 0.16, 0.16, 0.07, and 0.03 of the extinction in the V band. The accuracy of these coefficients is 0.03. If the extinction law reflects primarily the grain size distribution, then the fraction of large dust grains far from the Galactic plane is greater than that in the circumsolar interstellar medium.     

Thermal emission from a multiple scattering model of Saturn's rings

Models of Saturn's B ring have been investigated which include the shadowing mechanism, realistic phase functions for the ring particles, and the effects of multiple scattering and a particle size dispersion. These models are based on the assumption that the rings form a layer many particles thick. A power law relation dn ∝ ?^?s is used for the size dispersion law of the ring particles, where dn is the number of particles with radii between ? and ? + d?. In the calculation of the infrared brightness temperature of the rings, the effect of mutual heating among the ring particles is considered quantitatively for the first time. The parameters of the polydisperse s = 2 model can be chosen to satisfy both optical (λ ? 1.1 μ) and infrared data, but the situation could be much clarified if a good phase curve for the rings were available in the red, if the ring brightness were known accurately for λ > 1 μ, and if it could be established whether the ring particles are rotating synchronously.     

A postulate leading to the Titius-Bode law

The revised Titius-Bode law (Balsano and Hughes, 1979) giving distances of planets from the Sun shows integers that recall the Bohr law in the early quantum theory of hydrogen atom. The author searchs for a formalism, similar to the Sommerfeld's one accounting for the planetary distribution of the solar system. It is shown that such a formalism might be started with the assumed relation $$\int_{r_0 }^{r_n } {U(r)dr = nk,} $$ whereU(r) stands for the gravitation potential created by the Sun at distancer,k=cte,r _o=cte,r _n=distance of then ^th planet andn=1, 2, 3... Although the inspiration of the present note came from the early quantum theory, it is emphasized that there is no connection between the above assumed equation and the Sommerfeld quantum rule, but a pure formal similarity. the true significance of that equation is still unknown. It is either a fortuitous coincidence leading to the T-B law or any possible unsuspected property of gravitation.     

Ion-ion recombination rates in the earth's atmosphere

The temperature dependence of the binary recombination coefficient, α₂, for the reaction NO⁺+NO₂^? → products has been obtained over the range 185–530 K. It is found that the corresponding mean cross section $\text{σ}$ is described by the power law $\text{σ}\text{? A · T}{}^{\mathrm{?0.9}}$, and that α₂ ? B · T^?0.4. Data has also been obtained for two cluster ion recombination reactions which indicate that their recombination cross sections are only about 40% larger than for the parent ions at a given temperature, the cross sections for these reactions also apparently increasing with decreasing temperature. In the light of this data and by considering the most probable positive and negative ions existing at various altitudes up to 90km in the atmosphere, the most appropriate ionic recombination coefficients in various altitude ranges are deduced. Thus, between 30 and 90 km, where the recombination process is two-body, the coefficient varies over the narrow range 5–9 × 10^?8 cm³s^?1, while below 30 km the process is predominantly three-body with an effective two-body rate increasing rapidly to a maximum value ≈3 × 10^?6 cm³s^?1 in the troposphere, these deductions being based on published laboratory determinations of three-body recombination coefficients.     

Volume Filling Factors of the DIG in M 31

The volume filling factor f _v of the diffuse ionized gas in the bright emission ring of M 31 is derived from radio continuumobservations. The dependence of f _v on the local mean electron density n _e is a power law, f _v(n _e) = a n _e ^-bwith a = 0.011± 0.003 and b = -1.2± 0.3, where n _e is in cm^-3. The same power law was recently found for the DIG in the solar neighbourhood from pulsar data.     

On the Collective Magnetic Field Strength and Vector Structure of Dark Umbral Cores Measured by the Hinode Spectropolarimeter

We study 7530 sunspot umbrae and pores measured by the Hinode Spectropolarimeter (SP) between November 2006 and November 2012. We primarily seek confirmation of the long term secular decrease in the mean magnetic field strength of sunspot umbrae found by Penn and Livingston (IAU Symp. 273, 126, 2011) between 1998 and 2011. The excellent SP photometric properties and full vector magnetic field determinations from full-Stokes Milne–Eddington inversions are used to address the interrelated properties of the magnetic field strength and brightness temperature for all umbral cores. We find non-linear relationships between magnetic field strength and umbral temperature (and continuum contrast), as well as between umbral radius and magnetic field strength. Using disambiguated vector data, we find that the azimuths measured in the umbral cores reflect an organization weakly influenced by Joy’s law. The large selection of umbrae displays a log-normal size spectrum similar to earlier solar cycles. Influenced by the amplitude of the solar cycle and the non-linear relationship between umbral size and core magnetic field strength, the distribution of core magnetic field strengths, fit most effectively with a skew-normal distribution, shows a weak solar cycle dependence. Yet, the mean magnetic field strength does not show a significant long term trend.     

Reconstruction of f(R), f(T) and f(\mathcal{G}) models inspired by variable deceleration parameter

We study an special law for the deceleration parameter, recently proposed by Akarsu and Dereli, in the context of f(R), f(T) and $f(\mathcal{G})$ theories of modified gravity. This law covers the law of Berman for obtaining exact cosmological models to account for the current acceleration of the universe, and also gives the opportunity to generalize many of the dark energy models having better consistency with the cosmological observations. Our aim is to reconstruct the f(R), f(T) and $f(\mathcal{G})$ models inspired by this law of variable deceleration parameter. Such models may then exhibit better consistency with the cosmological observations.     

Gamma-ray bursts in the Swift era

GRBs are the most energetic combination of jets and disks in the Universe. Observations made using Swift reveal a complex temporal and spectral behaviour. We propose that this behaviour can be used to refine the GRB classification scheme and align it better with progenitor types. The early (prompt) X-ray light curve can be well described by an exponential which relaxes into a power law. The transition time between the exponential and the power law gives a well-defined timescale, T _p , for the burst duration which we use with the spectral index of the prompt emission, β _p , and the prompt power law decay index, α _p to define four classes of burst: short, slow, fast and soft. Short bursts tend to decline more gradually than long bursts. Most GRBs display a second “afterglow” component which can be fitted in a similar way to the early emission. During the decay of this second component, few GRBs show jet breaks in accord with pre-Swift predictions. However, the start time of the final afterglow decay, T _a , correlates with the peak of the prompt γ-ray emission spectrum, E _peak, in an analogous way to the Ghirlanda relation found between optical “jet-break” times, t _j , and E _peak. These data are inconsistent with simple achromatic jet-break models casting doubt on the reliability of using late temporal breaks to determine the jet collimation.     

On the restricted circular three-charged-body problem

Two-charged bodiesM ₁ andM ₂ revolve round their centre of mass in circular orbits under Newton's inverse-square law and the so similar Coulomb's law. A third-charged-bodyM, without mass and charge (i.e., such that it is attracted or repulsed byM ₁ andM ₂, but does not influence their motion), moves in a field with a force function, namely $$U = {\text{ }}\frac{{q - \mu }}{{r_1 }}{\text{ }} + {\text{ }}\frac{{\mu - q}}{{r_2 }}$$ , which is created byM ₁ andM ₂. In what follows, the existence and location of the collinear and equilateral Lagrangian points or solutions with be discussed and the interpretation of them will be given. This work is a generalization of the classical restricted circular three-body problem.     

Martian glaciation and the flow of solid CO2

The flow law determined experimentally for solid CO₂ establishes that a hypothesis of glacial flow of CO₂ at the Martian poles is not physically unrealistic. Compression experiments carried out under 1 atm pressure and constant strain rate demonstrate that the strength of CO₂ near its sublimation point is considerably less than the strength of water ice near its melting point. The data fit a power law “creep” equation of the form

$\text{?}\text{?}\text{= (4 × 10}{}^6\text{) σ}{}^{3.9}\text{exp}\text{(}\text{?12 200}\text{RT}\text{)}$

, where ? is compressive strain rate (sec^?1), σ is compressive stress (bars), R is the gas constant in calories per mole, and T is absolute temperature. The exponent of σ of 3.9 contrasts with a value near 3.1 for water ice, and indicates that the strain rate is somewhat more sensitive to stress for CO₂ than for water. Likewise, the low activation energy for creep, 12 200 cal mole^?1, illustrates that CO₂ is not highly sensitive to temperature and is thus likely to flow over a broad range of temperatures below its melting point. Strength values for CO₂ are of the order of one-tenth to one-third the strength of ice under equivalent conditions.A plausible glacial model for the Martian polar caps can be constructed and is helpful in explaining the unique character of the polar regions. CO₂-rich layers deposited near the pole would have flowed outward laterally to relieve high internal shear stresses. The topography of the polar caps, the uniform layering of the layered deposits, and the general extent of the polar “sediments” could all be explained using this model. Flow of CO₂ rather than water ice greatly reduces the problems with Martian glaciation. Nevertheless, problems do remain, in particular the large amounts of CO₂ necessary, the need to increase vapor pressure and temperature with depth in the polar deposits, and the lack of good observational evidence of flor features. Within the limits of the present knowledge of surface conditions of Mars, CO₂ glaciation appears to be a realistic alternate working hypothesis for the origin of the polar features.     

Vaidya black hole in non-stationary de Sitter space: Hawking’s temperature

In this paper we present a class of non-stationary solutions of Einstein’s field equations describing embedded Vaidya-de Sitter black holes with a cosmological variable function Λ(u). The Vaidya-de Sitter black hole is interpreted as the radiating Vaidya black hole is embedded into the non-stationary de Sitter space with variable Λ(u). The energy-momentum tensor of the Vaidya-de Sitter black hole is expressed as the sum of the energy-momentum tensors of the Vaidya null fluid and that of the non-stationary de Sitter field, and satisfies the energy conservation law. We study the energy conditions (like weak, strong and dominant conditions) for the energy-momentum tensor. We find the violation of the strong energy condition due to the negative pressure and leading to a repulsive gravitational force of the matter field associated with Λ(u) in the space-time. We also find that the time-like vector field for an observer in the Vaidya-de Sitter space is expanding, accelerating, shearing and non-rotating. It is also found that the space-time geometry of non-stationary Vaidya-de Sitter solution with variable Λ(u) is Petrov type D in the classification of space-times. We also find the Vaidya-de Sitter black hole radiating with a thermal temperature proportional to the surface gravity and entropy also proportional to the area of the cosmological black hole horizon.     

Fragmentation of a Molecular Cloud Core versus Fragmentation of the Massive Protoplanetary Disk in the Main Accretion Phase

We present three-dimensional numerical simulations on binary formation through fragmentation. The simulations follow gravitational collapse of a molecular cloud core up to growth of the first core by accretion. At the initial stage, the gravity is only slightly dominant over the gas pressure. We made various models by changing initial velocity distribution (rotation speed, rotation law, and bar-mode perturbation). The cloud fragments whenever the cloud rotates sufficiently slowly to allow collapse but faster enough to form a disk before first-core formation. The latter condition is equivalent to Ω₀ t _ff ? 0.05, where Ω₀ and t _ff f denote the initial central angular velocity and the freefall time measured from the central density, and the condition is independent of the initial rotation law and bar-mode perturbation. Fragmentation is classified into six types. When the initial cloud rotates rigidly the cloud collapses to form a adiabatic disk supported by rotation. When the bar-mode perturbation is very minor, the disk deforms to a rotating bar, and the bar fragments. Otherwise, the adiabatic disk evolves into a central core surrounded by a circumstellar disk, and the the circumstellar disk fragments. When the initial cloud rotates differentially, the cloud deforms to a ring or bar in the isothermal collapse phase. The ring fragments into free or more cores, while the bar fragments into only two cores. In the latter case, the core merges due to low orbital angular momentum and new satellite cores form in the later stages.