Effects of Hall current on hydromagnetic free-convective flow through a porous medium

An analysis of the effects of Hall current on hydromagnetic free-convective flow through a porous medium bounded by a vertical plate is theoretically investigated when a strong magnetic field is imposed in a direction which is perpendicular to the free stream and makes an angle to the vertical direction. The influence of Hall currents on the flow is studied for various values of .Nomenclature c _p specific heat at constant pressure - e electrical charge - E Eckert number - E electrical field intensity - g acceleration due to gravity - G Grashof number - H ₀ applied magnetic field - H magnetic field intensity - (j _x , j _y , j _z ) components of current densityJ - J current density - K permeability of porous medium - M magnetic parameter - m Hall parameter - n _e electron number density - P Prandtl number - q velocity vector - (T, T _w , T ) temperature - t time - (u, v, w) components of the velocity vectorq - U ₀ uniform velocity - v ₀ suction velocity - (x, y, z) Cartesian coordinates Greek Symbols angle - coefficient of volume expansion - _e cyclotron frequency - frequency - dimensionless temperature - thermal conductivity - coefficient of viscosity - magnetic permeability - kinematic viscosity - mass density of fluid - _e charge density - electrical conductivity - _e electron collision time     

Search for the matter clumps in scalesZ ~ 1

The diagramV - log(1 +z _e ) as function of (, ) is considered for the quasars. HereV is the apparent visual magnitude,z _e is the emission line redshift, and are the equatorial coordinates. Two opposite extreme spots NE and SE are observed on the sky, where the inclination of the straight line fitting the dependenceV - log(1 +z _e ) is maximum and minimum. The coordinates of the centres of these extreme spots are ( _NE, _NE) = (282°, +42°) and ( _SE, _SE) = (70°, -38°) with errors 5°. A hypothesis of the Superattractor (SA) is proposed to explain such an effect. Two independent tests of this hypothesis are realized. First, the dependence or the frequency a of the absorbers in QSO spectra on (, ) is investigated. A region of the larger a is found. The coordinates of its centre are (, ) = (82°, - 10°) with error 5°. Second, the cases ofz _a >z _e are plotted in the Mercatorial projection (, ). The most of the casesz -z _e > 0.02 are concentrated within the circle with radiusR = 34° and centre (, ) = (50°, - 15°). The both anomalous regions overlap the Southern extreme spot around SE. The SA direction is (, ) = (67°, -21°) with errors about 12°. The redshift of SA isz _SA = 1.7 ± 0.3 that corresponds to the distancer _SA = (3100 ± 300)h ^–1 Mpc for the Hubble constantH ₀ = 75h kms^–1 Mpc^–1. The SA mass isM _SA ~ 10¹⁸-10²⁰ M . The orientation of the normal to the quasiperiodical large-scale sheet structure on the sky occurs near SA.     

The motion of a satellite in an axi-symmetric gravitational field

The equations for the variation of the osculating elements of a satellite moving in an axi-symmetric gravitational field are integrated to yield the complete first-order perturbations for the elements of the orbit. The expressions obtained include the effects produced by the second to eighth spherical harmonics. The orbital elements are presented in the most general form of summations by means of Hansen coefficients. Due to their general forms it is a simple matter to estimate the perturbations of any higher harmonic by simply increasing the index of summation. Finally, this paper gives the respective general expressions for the secular perturbations of the orbital elements. The formulae presented should be useful for the reductions of Earth-satellite observations and geopotential studies based on them.List of Symbols semi-major axis - C _jk ⁿ (, ) cosine functions of and - e eccentricity of the orbit - f acceleration vector of perturbing force - f sin²t - i inclination of the orbit - J _n coefficients in the potential expansion - M mean anomaly - n mean motion - p semi-latus rectum of the orbit - R, S, andW components of the perturbing acceleration - r radius-vector of satellite - r magnitude ofr - S _jk ⁿ (, ) sine functions of and - T time of perigee passage - u argument of latitude - U gravitational potential - true anomaly - V perturbing potential - G(M₊+m) (gravitational constant times the sum of the masses of Earth and satellite) - _n,k coefficients ofR component of disturbing acceleration (funtions off) - _n,k coefficients ofS andW components of disturbing acceleration (functions off) - mean anomaly at timet=0 - X ₀ ^n,m zero-order Hansen coefficients - argument of perigee - right ascension of the ascending node     

Free-convection flow of water at 4°C past an infinite vertical porous plate with constant heat flux

An analysis of the two-dimensional flow of water at 4°C past an infinite porous plate is presented, when the plate is subjected to a normal suction velocity and the heat flux at the plate is constant. Approximate solutions are derived for the velocity and temperature fields and the skin-friction. The effects ofG (Grashof number) andE (Eckert number) on the velocity and temperature fields are discussed.Nomenclature u, v velocity components of the fluid inx, y direction - g acceleration due to gravity - coefficient of thermal expansion of water at 4°C - v kinematic viscosity - density - T temperature inside thermal boundary layer - T free-stream temperature - k thermal conductivity - C _p specific heat at constant pressure     

Charged particle transport in turbulent magnetic fields: The perpendicular diffusion coefficient

The diffusion of charged particles in a static turbulent magnetic field, which is superimposed on a constant magnetic fieldB ₀ k, is considered. Previous calculations of the particle flux in a direction perpendicular tok have related the fluxS to the particle number densityf byS = – (f) where is found from the power spectrum of the turbulent magnetic field. It is shown that this formula is inconsistent with the notion, developed by Jokipii and Parker (1969), that the perpendicular particle flux primarily arises because of random-walking of magnetic field lines across the directionk. For a simple example of a turbulent magnetic field it is shown that the above expression forS is incorrect; the particle fluxS is recalculated and a new relationship betweenS andf is found. This new expression forS is shown to be consistent with particle diffusion across the directionk being due to random-walking of the magnetic field lines.     

On the reconstruction of the nonlinear force-free coronal magnetic field from boundary data

Using a simple model in which the corona is represented by the half-space domain = {z > 0} and the photosphere by the boundary plane = {z = 0}, we discuss some important aspects of the general problem of the reconstruction of the magnetic field B in a small isolated coronal region from the values of the vector B¦ measured by a magnetograph over its whole basis. Assuming B to be force-free in : (i) we derive a series of relations which must be necessarily satisfied by the boundary field B¦ , and then by the magnetograph data if the force-free assumption is actually correct; (ii) we show how to extract directly from the measured B¦ some useful informations about the energy of B in and the topological structure of its field lines; (iii) we present a critical discussion of the two methods which have been proposed so far for computing effectively B in from B¦ .     

Vertical motions in an intense magnetic flux tube

The recent discovery of localised intense magnetic fields in the solar photosphere is one of the major surprises of the past few years. Here we consider the theoretical nature of small amplitude motions in such an intense magnetic flux tube, within which the field strength may reach 2 kG. We give a systematic derivation of the governing expansion equations for a vertical, slender tube, taking into account the dependence upon height of the buoyancy, compressibility and magnetic forces. Several special cases (e.g., the isothermal atmosphere) are considered as well as a more realistic, non-isothermal, solar atmosphere. The expansion procedure is shown to give good results in the special case of a uniform basic-state (in which gravity is negligible) and for which a more exact treatment is possible.The form of both pressure and velocity perturbations within the tube is discussed. The nature of pressure perturbations depends upon a critical transition frequency, _p , which in turn is dependent upon depth, field strength, pressure and density in the basic (unperturbed) state of the tube. At a given depth in the tube pressure oscillations are possible only for frequencies greater than _p for frequencies below _p exponentially decaying (evanescent) pressure modes occur. In a similar fashion the nature of motions within the flux tube depends upon a transition frequency, _v . At a given depth within the tube vertically propagating waves are possible only for frequencies greater than _v ; for frequencies below _v exponentially decaying (evanscent) motions occur.The dependence of both _v and _p on depth is determined for each of the special cases, and for a realistic solar atmosphere. It is found that the use of an isothermal atmosphere, instead of a more realistic temperature profile, may well give misleading results.For the solar atmosphere it is found that _v is zero at about 12 km above optical depth ₅₀₀₀= 1, thereafter rising to a maximum of 0.04 s^–1 at some 600 km above ₅₀₀₀ = 1. Below ₅₀₀₀ = 1, in the convection zone, _v has a maximum of 0.013 s^–1. The transition frequency, _p , for the pressure perturbations, is peaked at 0.1 s^–1 just below ₅₀₀₀ = 1, falling to a minimum of 0.02 s^–1 at about one scale-height deeper in the tube     

О ФОрРИРОВНИИИ диаграммы излучения пульсаров

, v ₀c (c — ). , , . 1919, . ( ) 0532. , .     

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     

Compton scattering in sources with synchrotron reabsorption

By considering the consecutive effects of synchrotron reabsorption, Compton scattering and other kinds of energy losses of relativistic electrons, it may be possibile to form a universal distribution of electrons in the region of reabsorption (synchrotron reactor). This will be either a power law with a power index of the energy spectrumn _r=3–5, or a relativistic Maxwell distribution with an electron temperatureT _e=4T _b(1+), where is the ratio of Compton (or other losses) to synchrotron losses, andT _bis the brightness temperature of the radiation. Since the total energy losses of electrons in the reactor is equal to zero, this ensures the continuous existence and accumulation of relativistic electrons in the region of reabsorption and their associated hard scattered radiation. Multiple Compton scattering produces a specific stepped power distribution of scattered radiation by which we can identify the reactor. In the nuclei of quasars W _Hand, therefore,n _r=3; hence the spectral index of scattered radiation in the corresponding ranges (optical, UV, X- and -ray) is .Consideration of other kinds of losses and gains of energy by electrons can lead to the dependencen =3–5(E) — where (E) may have either positive or negative values—which, in turn, leads to the frequency dependence of the spectral index of scattered radiation = 1 – (), |()| < 1, |(E)| < 1.Within the framework of the model being considered, the physical parameters of the nucleus of quasar 3C 273 are calculated.     

Cyclotron waves in the solar wind

Cyclotron waves in the solar wind near 1 AU with frequencies well below the electron cyclotron frequency and wavelengths much larger than the electron cyclotron radius but less than the proton cyclotron radius are considered. The cyclotron radii are defined from parallel thermal velocity of electron component and proton component with respect to the interplanetary magnetic field. No LH cyclotron waves are found to propagate for _p < 0, where _p 1 –T _p/T _p is the temperature anisotropy of the proton component with respect to the interplanetary magnetic field. The damping or growth of RH cyclotron waves is found to depend on the frequency range and the temperature anisotropy of the proton component. The RH cyclotron waves are damped in the frequency range _r | _p | _p for _p < 0, where _p is the proton cyclotron frequency. RH cyclotron instabilities occur in the frequency range | _p | _p > _r > | _p | _p /(1– _r ) for _p < 0. The marginal state is at _r =| _p | _p .Abstract presented at theInternational Symposium on Solar-Terrestrial, São Paulo, Brazil, 17–22 June, 1974     

Рассеяние света на анизотропных частицах

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Unsteady free-convective flow and mass transfer in a rotating porous medium with Hall current,I

The Hall effect on the unsteady hydromagnetic free-convection resulting from the combined effects of thermal and mass diffusion of an electrical-conducting liquid through a porous medium past an infinite vertical porous plate in a rotating system have been analysed. The expressions for the mean velocity, mean skin friction, and mean rate of heat transfer on the plate are derived. The effects of magnetic parameterM, Hall parameterm, Ekman numberE, and permeability parameterK ^* on the flow field are discussed with the help of graphs and tables.Nomenclature C _p specific heat at constant pressure - C the species concentration inside the boundary layer - C _w the species concentration at porous plate - C the species concentration of the fluid at infinite - C dimensionless species concentration - D chemical molecular diffusivity - E Ekman number - Ec Eckert number - g acceleration due to gravity - Gr Grashof number - Gm modified Grashof number - H ₀ applied magnetic field - (J _x, J_y, J_z) components of current density - M magnetic parameter - m Hall parameter - P Prandtl number - q _m mean rate of heat transfer - Sc Schmidt number - t time - t dimensionless time - T temperature of fluid - T _w temperature of the plate - T temperature of fluid at infinite - T dimensionless temperature - (u, v, w) components of the velocityq - w ₀ suction velocity - (x, y, z) Cartesian coordinates - z dimensionless coordinate normal to the plate Greek symbols coefficient of volume expansion - * coefficient of thermal expansion with concentration - frequency - dimensionless frequency - k thermal conductivity - K ^* permeability parameter - dinematic viscosity - density of the fluid in the boundary layer - coefficient of viscosity - _e magnetic permeability - angular velocity - electrical conductivity of the fluid - _m mean skin friction - _mn mean skin friction in the direction ofx - _mv mean skin friction in the direction ofy     

Структура и эволюция Сверхмассивных вращающихся магнитных политроп

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The structure of rotating magnetic polytropes is considered in Roche approximation. Investigation of the influence of poloidal as well as toroidal magnetic fields on the conditions of the beginning of matter outflow due to rotational instability is carried out. The influence of the turbulent convection and twisting of magnetic force-lines on the time of smoothing of differential rotation is considered. The estimate of the magneto-turbulence energy generated by differential rotation is presented. Both maximum possible energy output and duration of the quasi-statical evolution phase up to the appearance of hydrodynamic instability due to the effects of general relativity are calculated for supermassive magnetic polytropes of index three with uniform or differential rotation. The radius-mass relation is obtained for supermassive differentially-rotating magnetic polytropes referring to the longest part of the quasi-statistical evolution stage; some consequences are pointed out, including the period-luminosity relation.The evolution of the considered models of supermassive rotating magnetic polytropes with different character of rotation and different geometry of a magnetic field is discussed.The results obtained are summarized in the last section.

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English translation will appear in the next issue ofAstrophys. Space Sci.

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Receipt delayed by postal strike in Great Britain     

Индуцированное комптоновское рассеяние изотропного излучения релятивистскими электронами

, kT _b mc ² . . , , . E ^–5, - E ², mc ²(kT /mc ²)^1/7. , (>2.10⁵ cm ^–3) . , -, , .     

A discussion of Hill's method of secular perturbations and its application to the determination of the zero-rank effects in non-singular vectorial elements of a planetary motion

Knowledge of the perturbations of zero-rank is essential for the understanding of the behavior of a planetary or cometary orbit over a long interval of time. Recent investigations show that these zero-rank perturbations can cause large oscillations in both the shape and position of the orbit. At present we lack a complete analytical theory of these perturbations that can be applied to cases where either the eccentricity or inclination is large or has large oscillations. For this reason we here develop formulas for the numerical integration of the zero-rank effects, using a modified Hill's theory and suitable vectorial elements. The scalar elements of our theory are the two components of Hamilton's vector in a moving ideal reference frame and the three components of Gibb's rotation vector in an inertial system. The integration step can be taken to be several hundred years in the planetary or cometary case, and a few days in the case of a near-Earth space probe. We re-discuss Hill's method in modern symbolism and by applying the vectorial analysis in a pseudo-euclidean spaceM ₃, we obtain a symmetrical computational scheme in terms of traces of dyadics inM ₃. The method is inapplicable for two orbits too close together. In Hill's method the numerical difficulty caused by such proximity appears in the form of a small divisor, whereas in Halphen's method it appears as a slow convergence of a hypergeometric series. Thus, in Hill's method the difficulty can be watched more directly than in Halphen's method. The methods of numerical averaging have, at the present time, certain advantages over purely analytical methods. They can treat a large range of eccentricities and orbital inclinations. They can also treat the free secular oscillations as well as the forced ones, and together with their mutual cross-effects. At the present time, no analytical theory can do this to the full extent.Basic Notations m the mass of the disturbed body - M the mass of the Sun - f the gravitational constant - f(M+m) - r the heliocentric position vector of the disturbed body - r |r| - r ⁰ the unit vector alongr - n ⁰ the unit vector normal tor and lying in the orbital plane of the disturbed body - a the semi-major axis of the orbit of the disturbed body - e the eccentricity of the orbit of the disturbed body - g the mean anomaly of the disturbed body - the eccentric anomaly of the disturbed body - p a(1–e ²) - P ₁ the unit vector directed from the Sun toward the perihelion of the disturbed body - P ₂ the unit vector normal toP ₁ and lying in the orbital plane of the disturbed body - s - the true orbital longitude of the disturbed body, reckoned from the departure point of the ideal system of coordinates - X the true orbital longitude of the perihelion of the disturbed body in the ideal system of coordinates reckoned from the departure point - the angular distance of the ascending node from the departure point - R ₁,R ₂,R ₃ the unit vectors along the axes of the ideal system of coordinates,R ₁ andR ₂ are in the osculating orbital plane of the disturbed body,R ₃ is normal to this plane. The intersection ofR ₁ with the celestial sphere is the departure point - R ₃ P ₁×P ₂ - S ₁,S ₂,S ₃ the initial values ofR ₁,R ₂,R ₃, respectively - q the Gibb's vector. This vector defines the rotation of the orbital plane of the disturbed body from its initial position to the position at the given timet - m the mass of the disturbing body - r the heliocentric position vector of the disturbing body - a the semi-major axis of the orbit of the disturbing body - e the eccentricity of the orbit of the disturbing body - g the mean anomaly of the disturbing body - the eccentric anomaly of the disturbing body - P₁ the unit vector directed from the Sun toward the perihelion of the disturbing body - P₂ the unit vector normal toP₁ and lying in the orbital plane of the disturbing body - A₁ a P₁ - A₂ - |r–r|     

A simplified model solar atmosphere

A simplified representation of the temperature distribution in the solar photosphere is proposed: ( ₀) = ₀ - ₁ log ₀. An expression is derived for the emergent continuous spectrum from the simple model. The limitations and applications of the simple model are discussed.     

Asymptotic properties of disk dynamo

The asymptotic properties of a turbulent disk dynamo at large dimensionless numbersR andR characterizing the helicity and the differential rotation are analysed. Three types of generations in the dependence of the relations betweenR andR are found: ²-dynamo and two types of -dynamo. For each of these types the rates of growth are obtained and the forms of solution are pointed out. Boundaries of the disk dynamo approximation are given.     

Rapidly rotating polytropes in the post-Newtonian approximation to general relativity

The study of uniformly polytropes with axial symmetry is extended to include all rotational terms of order ⁴, where is the angular velocity, consistently within the first post-Newtonian approximation to general relativity. The equilibrium structure is determined by treating the effects of rotation and post-Newtonian gravitation as independent perturbations on the classical polytropic structure. The perturbation effects are characterized by a rotation parameter = ²/2G _c and a relativity parameter, =p _c / _c C ² , wherep _c and _c are the central pressure and density respectively. The solution to the structural problem is obtained by following Chandrasekhar's series expansion technique and is complete to the post-Newtonian rotation terms of order ². The critical rotation parameterv _c , which characterizes the configuration with maximum uniform rotation, is accurately evaluated as a function of . Numerical values for all the structural parameters needed to determine the equilibrium configurations are presented for polytropes with indicesn=1, 1.5, 2, 2 5, 3, and 3.5.     

Late-time emission of the type II Supernovae powered by the radioactive decay56Ni−56Co−56Fe

A one-zone model for the late time SN II energized by the radioactive decay⁵⁶Ni–⁵⁶Co–⁵⁶Fe is presented. The model succeeds in reproducing for the late time evolution of H and [Oi] 6300 emission lines in SN1970g for the reasonable set of parameters: mass of ejecta 4M , boundary velocityv ₀=4000 km s^–1 and amount of⁵⁶NiM _Ni=0.02M . However, a one-zone model does not account for the late time continuum. In the case of SN1980k the radioactive model fits H and [Oi] 6300 emissions att250 day satisfactory but fails at very late time, e.g.,t=670 day when the predicted value of the ratioL(H)/L(6300) is two orders of magnitude smaller than the observed one. We suggest that the strong H emission in SN1980k on the 670th day is due to the interaction of the supernova envelope with the pre-SN wind. The radioactive model for the late time SN II predicts strong Mgii 2800 line and detectable Hei 10830 line in emission and absorption.