Static Equilibrium Configuration Research Articles

Entropy of self-gravitating radiation

We examine the entropy of self-gravitating radiation confined to a spherical box of radiusR in the context of general relativity. We expect that configurations (i.e., initial data) which extremize total entropy will be spherically symmetric, time symmetric distributions of radiation in local thermodynamic equilibrium. Assuming this is the case, we prove that extrema ofS coincide precisely with static equilibrium configurations of the radiation fluid. Furthermore, dynamically stable equilibrium configurations are shown to coincide with local maxima ofS. The equilibrium configurations and their entropies are calculated and their properties are discussed. However, it is shown that entropies higher than these local extrema can be achieved and, indeed, arbitrarily high entropies can be attained by configurations inside of or outside but arbitrarily near their own Schwarzschild radius. However, if we limit consideration to configurations which are outside their own Schwarzschild radius by at least one radiation wavelength, then the entropy is bounded and we find Smax ≲ MR, whereM is the total mass. This supports the validity for self-gravitating systems of the Bekenstein upper limit on the entropy to energy ratio of material bodies.

Nonlinear Static Analysis of Deep Ocean Mining Pipe—Part I: Modeling and Formulation

A static analysis procedure is formulated and implemented for the numerical determination of nonlinear static equilibrium configurations of deep ocean risers or mining pipes. This implementation involves selection of a finite element model, modeling of structure, surface and subsurface environment and external forces, and solution of nonlinear equilibrium equations. The riser is modeled by three-dimensional beam finite elements which include axial, bending, and torsional deformations. These deformations are coupled through geometrically nonlinear effects. The resulting tangent-stiffness matrix includes three contributions identified as linear, geometric (initial-stress) and initial-displacment stiffness matrices. For the solution, a combination of load-parameter incrementation, state updating of fluid properties, and corrective Newton-Raphson iteration is used. The resulting riser configuration reflects the realistic modeling of environments and external forces. The static equilibrium solution can be used as initial condition for vibration or transient analysis. Numerical studies are presented in Part II of this paper.

New method for extracting static equilibrium configurations in general relativity

We investigate the static equations of general relativity for spherical symmetry in Schwarzschild coordinates. A new formalism is discussed which allows integration of the equations for an isotropic fluid under a broad range of initial assumptions. We are able to derive many previous solutions using this technique, and find as well a number of new solutions to the equations. A process is discussed whereby already known solutions can be generalized to increase their flexibility and usefulness with regard to the mathematical modeling of ultradense objects such as neutron stars.

Energy principle and the interstellar medium. III - Plane-parallel configurations

view Abstract Citations (38) References (15) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Energy principle and the interstellar medium. III - Plane-parallel configurations Lachieze-Rey, M. ; Cesarsky, C. J. ; Asseo, E. ; Pellat, R. Abstract An instability criterion is derived which applies to any horizontal equilibrium state of the interstellar gas and fields system, i.e., such that the magnetic field lines remain parallel to the galactic plane, and that all the physical quantities depend only on the height above the plane; an analytic calculation of the growth rate of the instability is also given. Results are applied to the four models proposed by Badhwar and Stephens to fit radio-continuum observations, and find that they are all unstable in a time short compared to the galactic age. Following Zweibel and Kulsrud, some of the effects due to a microturbulent component in the interstellar magnetic field are introduced into calculations and it is confirmed that they tend to lead toward an overall stabilization of the system; in this case, the most unstable modes are not those with a very short wavelength in the direction across the field lines and parallel to the plane. Still, as long as the adiabatic index of the gas is less than about 1.5, it is noted that all the static equilibrium configurations proposed to date to describe the interstellar medium are unstable, and that the system is more likely to be in a state of permanent agitation, fed in part by the type of instabilities studied throughout this series. Publication: The Astrophysical Journal Pub Date: May 1980 DOI: 10.1086/157970 Bibcode: 1980ApJ...238..175L Keywords: Interstellar Magnetic Fields; Interstellar Matter; Magnetic Field Configurations; Magnetohydrodynamic Stability; Cosmic Rays; Lines Of Force; Milky Way Galaxy; Turbulence Effects; Astrophysics full text sources ADS |

A theory of the onset of solar eruptive processes

The present theory assumes that solar eruptive processes - such as flares or eruptive prominences - occur at a critical stage when a configuration evolves in a quasi-static way. The onset criterion is not based on standard linear stability considerations but on the fact that under suitable conditions static equilibrium configurations cease to exist. This starting point leads us to present a general discussion of existence properties of the corresponding set of equations. It turns out that the existing mathematical literature provides very useful pertinent information. In fact, some important questions of the existence can be answered without even solving the equations. Nevertheless, several explicit examples are discussed for reasons of illustration. For the major part of the paper the configurations are characterized by a single parameter. In a particularly simple (one-dimensional) case we also discuss solutions depending on two parameters. The results can be discussed in terms of catastrophe theory. The theory is valid for two space dimensions and contains topological changes of the magnetic field, although the latter feature is not necessary for the theory to apply. The theory of two-dimensional force-free fields is contained as a special case.

Static and dynamic interaction of superconducting vortices with a periodic pinning potential

A model is developed which describes the static and dynamic interaction of the vortex lattice with the pinning potential due to harmonic one-dimensional variations of the thickness of superconducting films. Using a London-Pearl approach, an expression is derived for the free energy of the mixed state in thin modulated films from which static equilibrium configurations of the vortex lattice and the corresponding deformations due to the harmonic pininng force are deduced. Lattice configurations of lowest energy are found when the magnetic induction B corresponds to matching of the (undistorted) triangular vortex lattice to the periodic pinning structure. Particular attention is devoted to lattice configurations slightly deviating from a matching situation. The flux-flow regime is also investigated when steady-state vortex motion results from the equilibrium of the Lorentz driving force, the viscous damping force, the harmonic pinning force and the pinning-induced lattice restoring force. A highly coherent flux-flow regime with rf properties similar to those of series arrays of resistively shunted Josephson junctions acting in phase and frequency coherence (super-radiant state) is found in the dynamic matching state. For nonmatching configurations we study the influence of dynamically excited lattice deformation modes on the current-voltage characteristics when vortex motion is driven bymore » dc or by superimposed dc and rf transport currents. In particular, for nearly matching configurations the width of the rf-induced interference transitions is related to the shear modulus of the vortex lattice and to the flux-flow resistivity.« less

The dynamic behaviour of stiffened strings

The small oscillation of a stiffened string about its static equilibrium configuration is studied. The solution requires both the static and dynamic shape of the rod to be determined and since the governing equations do not lend themselves to an analytical solution a Runge-Kutta integration technique was used. Experimental results for the natural frequencies are compared to the numerical solution for a particular stiffened string.

Solar activity and the general absence of hydrostatic equilibrium in an azimuthal magnetic field

It is shown that any temperature variation along the axis of symmetry of an azimuthal field causes continual convective activity. There is no static equilibrium configuration except in very special, and hence improbable, cases. It is suggested that this dynamic effect contributes to the activity associated with the flux tubes extending through the solar photosphere. The X-ray bright spots, evidently caused by the emergence and expansion of small bipolar regions, may be produced in part by the convective effect of an azimuthal field.

The Orowan mechanism in anisotropic crystals

Abstract The static equilibrium configurations of a dislocation bypassing a periodic row of impenetrable circular obstacles were obtained for various crystals by means of a self-stress method wherein the elastic anisotropy of the crystal and the dislocation self-interaction could be taken into account. A correlation was developed from the results which related the Orowan stress values to the obstacle size and spacing. This correlation demands two parameters to account for the anisotropy, and these appear as a suitably defined anisotropic shear modulus and Poisson's ratio. The self-interactions are taken into account by a single logarithmic parameter which is the harmonic mean of the obstacle size and spacing, and is independent of anisotropy. The shapes of the bowing dislocation loops are strongly influenced by the anisotropy and interactions. The combined effects on shape can bo simply described in the line-tension framework, using the de Wit—Koehler solutions. In particular, the swept areas and overall ...

Wetting of fine wires and fibers by a liquid film

When a fine wire or fiber is coated with a liquid film, the latter breaks up spontaneously into beads. Such unusual wetting behavior is a particular manifestation of the general phenomenon of liquid column instability first studied by Plateau and Rayleigh. In this work various physical factors that influence the instability and the subsequent beading are examined in detail. In the first part hydrodynamic equations are solved to obtain the rate of growth of disturbances and the wavelength of the most rapidly growing disturbance. These quantities are governed primarily by the viscosity of the liquid and the thickness of the liquid film as compared to the wire radius. In the second part the static equilibrium configuration of the beads attained at the completion of the dewetting process is derived. The bead shape is determined by the contact angle of the liquid on wire and the volume of liquid available per bead. It is shown that even when the contact angle is zero, not more than three quarters of the wire surface is covered by the liquid beads.

Magnetohydrodynamic aspects of the Earth's bow shock 2. Motions induced by directional discontinuities

From measurements made by the Heos 2 satellite a spread of bow shock locations is observed, oblique shocks (in the hydromagnetic sense) lying more earthward than perpendicular ones. This result is interpreted as a dynamic effect rather than as being due to different field-determined static equilibrium configurations. A simple hydromagnetic model for the impulsive interaction of the bow shock with a discontinuity in magnetic field direction is stated and discussed quantitatively. It is found that velocities of the order of 10 km/s can be induced thereby. The observed spread in bow shock positions as well as the correlation with magnetic field orientation can be explained by our model.

Computer Evaluation of Overhead Equipment for Electric Railroad Traction

The mathematical models and associated computer programs for analyzing a number of designs of overhead equipment such as simple catenary, compound catenary, stitched catenary, tunel equipment, and single-wire system, together with many possible variations are described. The programs are able to deal with up to four pantographs, and the initial conditions assumed are that the wires have their static equilibrium configurations and that the front pantograph is at the beginning of the first span touching the contact wire (or trolley wire) and moving tangentially to it. Any subsequent pantographs reach the beginning of the equipment after the start of the run and are then dealt with appropriately. The output of the programs list the input data, the total percentage contact loss between the panhead and contact wire in each span and the pantograph heights and contact forces at appropriate intervals of time. Graph plots of the pantograph heights and contact forces are also produced. The results of these programs are shown to give satisfactory agreement with previously published results obtained on a scaled model.

Two axisymmetric black holes cannot be in static equilibrium

No static equilibrium configuration of two black holes can exist in an axisymmetric asymptotically flat vacuum space-time.

Librations of Gravity-Oriented Satellites in Elliptic Orbits through Atmosphere

The coupled librational dynamics of gravity-orien ted, axisymmetric satellites in elliptic orbits is investigated. An approximate closed form solution is obtained using variation of parameter approach to study the effect of eccentricity and inertia on the response. The plots for stability in the large, generated numerically, show the transverse motion to be relatively more stable. Since in the presence of atmosphere the static equilibrium configuration varies continuously with the satellite's orbital position, the response becomes relatively complex and the stability region diminishes rapidly. However, with a suitable choice of a velocity-sensitive, semipassive controller, the normally destabilizing aerodynamic moment can be used to advantage in damping the librational motion.

A Rotating Disk Model for a Quasar

It has often been suggested that the energy supply of a quasar is derived from a central body with a mass of 109−1010M⊙. However, it is known that such bodies have no stable and static equilibrium configurations beyond a relatively low degree of condensation. Only a very limited amount of energy can be extracted before instability sets in. Basically the instability is due to the dominance of radiation pressure in supporting the mass against its self-gravitation.But if a massive object is in rapid rotation, then it is partly supported by centrifugal force, and is much less liable to an instability of this kind. It is therefore interesting to investigate the properties of a rotating disk-like object with a mass of the order of 109−1010M⊙, and to see whether its overall properties match those of any quasar. The most important parameters are 2σ0, the mass per unit area of the disk, and Z, its half-thickness. In the case of the quasar 3C 273 B, assumed to be at a distance of 450 Mpc, the most suitable pair of values are σ0 = 107 gm/cm2 and Z = 1016 cm.

Dynamics of a Semi-Infinite Crystal Lattice in a Quasiharmonic Approximation. I. The Static Equilibrium Configuration of a Semi-Infinite Lattice

This is the first of two papers dealing with the dynamics of semi-infinite crystals. The static equilibrium structure of a semi-infinite crystal at zero temperature is considered in the first paper. The discussion of small-amplitude vibrations of the atoms about their static equilibrium positions is taken up in the second paper. The static equilibrium configuration of a semi-infinite, classical, crystal lattice at zero temperature, having a free boundary, is described in terms of a set of displacements relating the actual positions to those which the lattice particles would assume if the semi-infinite lattice were embedded in an infinite lattice. These displacements are calculated from a set of difference equations which specify the conditions of static equilibrium for the semi-infinite lattice in the absence of external forces. Several unexpected results concerning the structure of the physical boundary region were obtained. These will be shown, in a forthcoming paper, to provide a consistent interpretation of some features of low-energy electron diffraction data, which, until now, could not be interpreted. Of particular interest is the conclusion that the two-dimensional period of the semi-infinite lattice in the planes parallel to the boundary may be larger than that deduced from the bulk lattice structure. The diffraction of low-energy electrons from the surface of single crystals of Ge and Si provides a clear illustration of this effect. It is also shown that the change in the spacing of atomic planes (in the direction normal to the boundary) varies in a nonmonotonic fashion with the distance from the boundary. The effect of appreciable distortions in the surface region is considered in some detail.

Static frenkel dislocation model

A graphical method is used to determine the static equilibrium configurations of atoms which correspond to a one-dimensional model of a crystal disturbed by a dislocation. The question of the number of equilibrium configurations and their stability with respect to a homogeneous stress is solved and cases of an anomally low Peierls stress are studied.

Theory of the Thermal Breakaway of a Pinned Dislocation Line with Application to Damping Phenomena

Although the mechanical theory of Granato and Lücke for the strain-amplitude-dependent internal friction and modulus changes of solids containing dislocations gives a fair account of many of the observed effects, simple theoretical considerations show that the effect of thermal fluctuations should be very important. To extend the theory to finite temperatures, a detailed study of the possible static equilibrium configurations of a pinned dislocation line as a function of external stress is required. This is done here for the two specific examples of a dislocation with a single pinning point and a dislocation with a continuous pinning agent. It is shown in a general way how the damping and modulus change can be computed from these results, and a qualitative discussion of expected behavior is given. For high concentrations of pinning points, the theory is also applicable in the field of yield point phenomena, and the results found here are compared with previous calculations.

Stability of a rotating pinch

The problem of pinch stabilisation is of interest in connection with plasma confinement and has been the subject matter of various investigations in recent years, (Kruskal andSchwarzschild, Taylor, Shafranov). The theoretical investigations in the past have been restricted to static equilibrium configurations. It is of some interest to explore whether an initial motion of plasma tends to make the pinch more, or less stable than when the fluid velocity is zero. Keeping this in mind let us investigate theaxisymmetric stability of an infinitely long cylindrical plasma column of radiusR rotating uniformly with angular velocityΩ about theZ-axis, the fluid being assumed to the incompressible, inviscid, and of infinite electrical conductivity. The plasma carries a uniform axial current of densityj0 and a uniform axial magnetic fieldH1. The field outside is taken to beH2 inZ-direction and a toroidal component which is continuous across the boundary.