The gravitational field of a static infinite sheet of matter

The solution of the Einstein field equations in the case of an infinite, static, uniform distribution of matter lying parallel to thex–y plane is obtained. This metric is equivalent to the uniformly accelerated metric which causes the particles to move with uniform acceleration parallel to thez-axis.     

Convection in planetary interiors-the effects of rotation

The conservation equations of mass, energy and momentum are applied to the problem of thermal and non thermal convective motions inside a homogeneous, compressible fluid sphere of uniform viscosity which is rotating with a constant angular velocity about thez-axis. The resulting equations are manipulated into a form which should be suitable for solution.     

The flow of viscoelastic non-Newtonian fluid in the Ekman layer

In this paper the unsteady flow in the Ekman layer of a visco-elastic non-Newtonian fluid near a flat plate is discussed. Laplace transform technique has been employed to show the basic differential equations. Expressions for velocity profile, the skin friction have been calculated. It is shown that the time to attain the steady state increases with the elastic parameter. It is shown that normally the ultimate steady state is reached through a decay of inertial oscillations whose frequency decreases with increase in the elastic parameter. In the present study we examine the following unsteady problem in non-Newtonian fluid. Consider an infinite plate coinciding with the platez=0 and rotating in unison with elasticoviscous liquid occupying the regionz>0 with a uniform angular velocity about thez-axis for timet<-0. At timet>0, the plate starts moving with a uniform velocityU _o along thex-axis relative to the rotating frame of reference. The horizontal homogeneity of the problem demands that conditions depend onz andt only. The equation of continuity together with the no slip condition at the plate then shows that thez-component of the velocity vanishes everywhere.     

Modelling of the effect of current-carrying jets on the temperature of an astrophysical surrounding

Numerical solution of the effect of current-carrying jets on the temperature of an astrophysical surrounding is carried out using classical magnetohydrodynamic equations. Under the assumption of small hydrodynamic and magnetic Reynolds numbers and invoking a jet magnetic field intensityB , which confines high pressure jets along thez-axis, a non-linear equation is generated and solved by asymptotic approximation. It is found that when the field intensity is large, the temperature of the surrounding is small and vice-versa. The problem is of interest in the astrophysical studies of current-carrying jets or magnetised radio jets.     

Warped galactic disks and optical projection effects

Direct and indirect evidence for global warping in galactic disks is reviewed. The existing data suggest that inversion symmetric warping is common in spiral galaxies, that it can occur throughout the inner and outer disk regions, and that it can involve stellar, dust, and gas components. The amplitude of the warping tends to be proportional to the radial distance from the galactic center and varies from instances wherein the disk is essentially flat to cases wherein thez-axis displacement is at least 30% of the galactocentric distance. Projection effects that are inherent features of warped optical disks are discussed.     

Stability of motion in the Sitnikov 3-body problem

We study the stability of motion in the 3-body Sitnikov problem, with the two equal mass primaries (m ₁ = m ₂ = 0.5) rotating in the x, y plane and vary the mass of the third particle, 0 ≤ m ₃ < 10⁻³, placed initially on the z-axis. We begin by finding for the restricted problem (with m ₃ = 0) an apparently infinite sequence of stability intervals on the z-axis, whose width grows and tends to a fixed non-zero value, as we move away from z = 0. We then estimate the extent of “islands” of bounded motion in x, y, z space about these intervals and show that it also increases as |z| grows. Turning to the so-called extended Sitnikov problem, where the third particle moves only along the z-axis, we find that, as m ₃ increases, the domain of allowed motion grows significantly and chaotic regions in phase space appear through a series of saddle-node bifurcations. Finally, we concentrate on the general 3-body problem and demonstrate that, for very small masses, m ₃ ≈ 10⁻⁶, the “islands” of bounded motion about the z-axis stability intervals are larger than the ones for m ₃ = 0. Furthermore, as m ₃ increases, it is the regions of bounded motion closest to z = 0 that disappear first, while the ones further away “disperse” at larger m ₃ values, thus providing further evidence of an increasing stability of the motion away from the plane of the two primaries, as observed in the m ₃ = 0 case.     

The geometry of the Roche coordinates

The exact geometry of the Roche curvilinear coordinates (, , ) in which corresponds to the zero-velocity surfaces is investigated numerically in the plane, as well as in the spatial, case for various values of the mass-ratio between the two point-masses (m ₁,m ₂) constituting a binary system.The geometry of zero-velocity surfaces specified by -values at the Lagrangian points are first discussed by taking their intersections with various planes parallel to thexy-, xz- andyz-planes. The intersection of the zero-velocity surface specified by the -value at the Lagrangian equilateral-triangle pointsL _4,5 with the planex=1/2 discloses two invariable curves passing through the pointsL _4,5 and situated symmetrically with respect to thexy-plane whose form is independent of the mass-ratio.The geometry of the remaining two coordinates (, ) orthogonal to the zero-velocity surfaces is investigated in thexy- andxz-planes from extensive numerical integrations of differential equations generated from the orthogonality relations among the coordinates. The curves (x, y)=constant in thexy-plane are found to be separated into three families by definite envelopes acting as boundaries whose forms depend upon the mass-ratio only: the inner -constant curves associated with the masspointm ₁, the inner -constant curves associated with the mass-pointm ₂ and the outer -constant curves. All the -constant curves in thexy-plane coalesce at either of the Lagrangian equilateraltriangle pointsL _4,5, except for a limiting case coincident with thex-axis. The curves (x, z)=constant in thexz-plane are also separated by definite envelopes depending upon the mass-ratio into different families: the inner -constant curves associated with the mass-pointm ₁, the inner -constant curves associated with the mass-pointm ₂ and the outer -constant curves on both sides out of the envelopes. For larger values ofz, the curves =constant tend asymptotically to the line perpendicular to thex-axis and passing through the centre of mass of the system, except for a limiting case coincident with thex-axis. The geometrical aspects of the envelopes for the curves (x, y)=constant in thexy-plane and the curves (x, z)=constant in thexz-plane are also discussed independently.In the three-dimensional space, the Roche coordinates can be conveniently defined in such a way as to correspond to the polar coordinates in the immediate neighbourhood of the origin, and to the cylindrical coordinates at great distances. From numerical integrations of simultaneous differential equations generating spatial curves orthogonal to the zero-velocity surfaces, the surfaces (x, y, z)=constant and the surfaces (x, y, z)=constant are constructed as groups of such spatial curves with common values of some parameters specifying the respective surfaces.On leave of absence from the University of Tokyo as an Honorary Fellow of the Victoria University of Manchester.     

Inhomogeneous perfect fluid universe with electromagnetic field

Cylindrically symmetric inhomogeneous cosmological model for perfect fluid distribution with electromagnetic field is obtained. The source of the magnetic field is due to an electric current produced along the z-axis. F ₁₂ is the non-vanishing component of electromagnetic field tensor. To get the deterministic solution, it has been assumed that the expansion θ in the model is proportional to the shear σ. Physical and geometric aspects of the models are also discussed in presence and absence of magnetic field.      

The galactic surface density derived from K giants

We compare theW velocity dispersions of Brosche, Schwan & Schwarz (2001) with more recent results. The increase with the distance |z | from the galactic plane is confirmed, although perhaps with reduced amplitude. This could be interpreted either as one homogenous population or as a superposition of (at least) two populations with two constant dispersions and two scale heights. For each of the possibilities we propose a simple model. Combined with two observational variants for the velocity variation, we obtain surface densities up to |z | = 250 pc ranging from 5 to 44 M_⊙ pc^–2. Thus the case for considerable dark matter in the neighbourhood of the galactic plane is not supported     

Integrated colors in the solar neighborhood

The bivariate spectral type-luminosity class distribution combined with thez-distribution and broad-band photometric data have been used in order to derive integrated colors in Johnson's UBVRIJKL system for the solar neighborhood.The frequency distribution of white dwarfs is also taken into account for theU-B, B-V colors.     

Magnetized cosmological model in Lyra manifold

A spatially-homogeneous and anisotropic magnetized cosmological model in Lyra's manifold is obtained when the source of the gravitational field is a perfect fluid distribution. The magnetic field is due to an electric current produced along thex-axis. The physical behaviour of the model is discussed.     

A new analytic approach to the Sitnikov problem

A new analytic approach to the solution of the Sitnikov Problem is introduced. It is valid for bounded small amplitude solutions (z _max = 0.20) (in dimensionless variables) and eccentricities of the primary bodies in the interval (–0.4 < e < 0.4). First solutions are searched for the limiting case of very small amplitudes for which it is possible to linearize the problem. The solution for this linear equation with a time dependent periodic coefficient is written up to the third order in the primaries eccentricity. After that the lowest order nonlinear amplitude contribution (being of order z ³) is dealt with as perturbation to the linear solution. We first introduce a transformation which reduces the linear part to a harmonic oscillator type equation. Then two near integrals for the nonlinear problem are derived in action angle notation and an analytic expression for the solution z(t) is derived from them. The so found analytic solution is compared to results obtained from numeric integration of the exact equation of motion and is found to be in very good agreement. CERN SL/AP     

The color-redshift relation for giant elliptical galaxies

The colors of giant elliptical (gE) galaxies in clusters out to redshiftz=0.2, observed by Oke and Sandage (1968), are studied for systematic color-redshift effects. To reveal any intrinsic changes, theK term is subtracted from each color, after correction for galactic reddening and the resulting (B-V) _c -K _B-V versus z relation analysed. If the Oke and SandageK terms (relevant to nuclear colors) are used, the best fitting linear relation shows negligible change withz. But if the Whitford (1970)K terms (relevant to integrated colors) are used, there is a trend to bluer colors, by 0.07 mag.atz=0.2 if higher weight is given the better observed clusters. An upper limit, of ±0.08 mag. atz=0.2, is set to possible systematic aperture effects, by the total change between nuclear and integrated B-V and U-B of nearbygE galaxies.The color-redshift trends, interpreted as evolutionary changes, are related to evolution in the magnitude-redshift relation by means of models of stellar evolution in agE galaxy. If one uses the linear fit to the color-redshift relation obtained with WhitfordK terms and neglect of any aperture effects (which is the appropriate case if Oke and Sandage used large enough apertures), and if one adopts plausible limits to the ratio between color and magnitude evolution, the effect of evolution in the magnitude-redshift relation results in a negative value ofq ₀. There are still substantial observational and theoretical uncertainties affecting this conclusion.     

Simulations of jet formation from a magnetized accretion disk

Axisymmetric magnetohydrodynamic (MHD) simulations have been made of the formation of jets from a Keplerian disk threaded by a magnetic field. The disk is treated as a boundary condition, where matter with high specific entropy is ejected with a Keplerian azimuthal speed and a poloidal speed less than the slow magnetosonic velocity, and where boundary conditions on the magnetic fields correspond to a highly conducting disk. Initially, the space above the disk, the corona, is filled with high specific entropy plasma in the thermal equilibrium in the gravitational field of the central object. The initial magnetic field is poloidal and is represented by the superposition of the fields of monopoles located below the plane of the disk.The rotation of the disk twists the initial poloidal magnetic field lines, and this twist propagates into the corona pushing matter into jet-like outflow in a cylindrical region. After the first switch-on wave, which originates during the first rotation period of the inner radius of the disk, the matter outflowing from the disk starts to flow and accelerate in thez-direction owing to both the magnetic and pressure gradient forces. The flow accelerates through the slow magnetosonic and Alfvén surfaces and at larger distances through the fast magnetosonic surface. The flow velocity of the jet is approximately parallel to thez-axis, with the collimation mainly a result of the pinching force of the toroidal magnetic field. The energy flux of the flow increases with increasing magnetic field strength on the disk. Some of the cases studied have been run for long times, 60 rotation periods of the inner radius of the disk, and show indications of approaching a stationary state.     

Subsurface temperature—depth profiles, anomalies due to climatic ground surface temperature changes or groundwater flow effects

Climatic temperature changes at the ground surface propagate downward to the subsurface creating transient disturbances to the temperature—depth (T(z)) profile. Due to the poor thermal diffusivity of rocks the disturbances are preserved long times in the bedrock, and in a conductive regime it is possible to reveal the ground surface temperature (GST) history from borehole temperature data with inversion techniques. Geothermal temperature measurements thus provide a source of palaeoclimatic information which so far has not been utilized extensively. Inversion of GST history is, however, not straightforward and any disturbing effects should be excluded before the data can be utilized in inversion. Groundwater flow is of special importance in this respect because it is a common phenomenon in bedrock and convection often produces temperature—depth profiles resembling those affected by palaeoclimatic GST changes. In interpreting temperature—depth (T(z)) logs it is therefore not always clear whether the recorded vertical gradient variations should be attributed to the effects of palaeoclimatic ground surface temperature (GST) changes or to groundwater circulation. Using several synthetic T(z) profiles and applying general least squares inversion techniques we simulate a situation of “misinterpreting” the curvature of the T(z) profile in terms of palaeoclimatic GST changes, although it is actually produced by convective heat transfer due to groundwater flow. For comparison the opposite case is also studied, namely, genuine palaeoclimatic effects are misinterpreted as being due to disturbances caused by groundwater flow. A homogeneous half-space model is used to model T(z) profiles disturbed conductively by GST changes during the time interval 10–10000 yr B.P. and a one-dimensional porous layer model is applied for convective heat transfer calculations. The results indicate that a given T(z) profile can be attributed to either of these effects with reasonable parameter values. In addition to the synthetic T(z) profiles, a case history from a 958 m deep drill hole at Lavia, southwestern Finland, is presented. Special care is needed in analyzing T(z) data. A knowledge of geothermal data, such as temperature, thermal conductivity and diffusivity is not necessarily adequate for determining which of the phenomena (or whether a combination of them) provides the most probable interpretation of a T(z) profile. Additional information on the hydrogeological properties of the drilled strata is essential.     

The existence of three-dimensional symmetric orbits in the magnetic-binary problem

We study the existence of three-dimensional symmetric orbits in a magnetic-binary system. We point out that only two kinds of such orbits exist, depending on the orientation of both magnetic momentsM _i,i=1, 2; one with respect to the plane,y=0 and one with respect to thex-axis of the rotating-coordinate system.     

Magnetostatic stiff-fluid model of cylindrical symmetry in general relativity

We study the static stiff-fluid model for perfect fluid distributions in the presence of incident magnetic field. The magnetic field is surrounded by static stiff fluid of infinite electric conductivity and it is due to the electric current flowing along theZ-axis. The various physical and geometrical properties together with the state of model in absence of magnetic field are also discussed.     

On the existence of cosmological evolutionary effects

Parameters of the distant galaxy clusters of 3C295 (z=0.46) and Cl 0024+1654 (z=0.39) are compared with the predictions made using galaxies of the local clusters Coma (z=0.023) and DC 0329–52 (z=0.057) taking theK-effect into account. The distributions of colour and morphological type, and the amplitudesF ⁺/F^– of the ₀ 4000 discontinuity are examined and no evidence for evolution of the galaxies and the clusters can be seen.     

Dynamical problems of thermoelastic solids

The dynamical problem for thermal stresses in an infinite isotropic elastic cylinder of radius a with its axis along the z-axis, subject to fixed boundary conditions is studied. The Fourier heat conduction equation has been solved applying the Fourier transform and the theory of complex variable. The thermoelastic equation of motion has been separated into two wave equations which can be solved separately. The temperature, the displacement and the stress components have been obtained in analytical form as series involving Bessel function of first kind and of order zero.     

An equation for the characteristic curve of a family of symmetric periodic orbits

Szebehely's equation for the inverse problem of Dynamics is used to obtain the equation of the characteristic curve of a familyf(x,y)=c of planar periodic orbits (crossing perpendicularly thex-axis) created by a certain potentialV(x,y). Analytic expressions for the characteristic curves are found both in sideral and synodic systems. Examples are offered for both cases. It is shown also that from a given characteristic curve, associated with a given potential, one can obtain an analytic expression for the slope of the orbit at any point.