Symmetric Equilibrium Configurations Research Articles

Capillary surfaces in an exotic container: results from Space experiments

Experimental results from the Interface Configuration Experiment (ICE) performed aboard the Space Shuttle and the Mir Space Station are reported. The experiment concerns fluid interfaces in certain ‘exotic’ containers in a low-gravity environment. These containers are rotationally symmetric and have the property that for given contact angle and liquid volume, a continuum of distinct rotationally symmetric equilibrium configurations can appear, all of which have the same mechanical energy. These symmetric equilibrium configurations are unstable, in that deformations that are not rotationally symmetric can be shown mathematically to yield configurations with lower energy. It is found experimentally, in confirmation of mathematical results and of numerical computations, that distinct locally stable configurations can form that are not rotationally symmetric and have differing dynamic characteristics. It is found that this intriguing phenomenon of asymmetric local energy minimizers can occur even if conditions for an exotic container are not entirely met.

Boson-boson stars and their gravitational redshifts

We study the static spherically symmetric equilibrium configurations made from two kinds of bosons with different masses and their stabilities. The resulting configurations can have larger gravitational redshift than the ordinary boson stars. We find that the limiting value of the redshift is 1.08 for the mass ratio α≫1.

Nonlinear Evolution of Kink‐unstable Magnetic Flux Tubes and Solar δ‐Spot Active Regions

The motivation for the work described in this paper is to understand kink-unstable magnetic flux tubes and their role in the formation of δ-spot active regions. It has been proposed that, during their rise to the photosphere, a certain fraction of convection zone flux tubes become twisted to the point where they are unstable to the current driven kink instability. These kink-unstable flux tubes then evolve toward a new, kinked equilibrium as they continue to rise to the photosphere, appearing as δ-spot groups upon emergence. Because of their kinked nature, these flux tubes could be highly susceptible to flaring, explaining the very active nature of δ-spot groups. We study the kinking flux tube problem with a three-dimensional numerical model containing only the most basic features of a kink-unstable flux tube. We build on our earlier work describing the linear phase of the kink instability, and follow the evolution into the nonlinear regime: (1) We perform numerical simulations of constant-twist, kink-unstable flux tubes in an initially cylindrical equilibrium configuration in three dimensions, in a high-β pressure-confined environment. We consider many different initial configurations, including the Gold-Hoyle flux tube. (2) These numerical calculations confirm the growth-rate predictions of our earlier work, when viscous dissipation is included. They also confirm our velocity profile predictions. (3) The flux tubes evolve toward new helically symmetric equilibrium configurations. (4) The timescale for saturation to the kinked equilibrium configuration is τsat ~ 10/ω0, where ω0 is the linear growth rate calculated as in the earlier paper. (5) The cylindrically symmetric part of the kinked equilibrium is well described by the m = 0 Chandrasekhar-Kendall functions (i.e., the Lundquist field). The m = 1 helically symmetric part, however, is not well described by the m = 1 Chandrasekhar-Kendall functions. (6) The equilibrium kink amplitudes are not large, at less than one-third of the tube radius. (7) The peak kinetic energy of the instability can be predicted from the initial excess perpendicular magnetic energy. (8) The amplitudes of the kinked tubes are large enough to give a δ-spot region tilt angle of up to 30° away from that of an unkinked tube.

Numerical solutions of general-relativistic field equations for rapidly rotating neutron stars

Stationary axial symmetric equilibrium configurations rapidly rotating with uniform angular velocity in the framework of general relativity are considered. Sequences of models are numerically computed by means of a computer code that solves the full Einstein equations exactly. This code employs Neugebauer’s minimal surface formalism, where the field equations are equivalent to two-dimensional minimal surface equations for 4 metric potentials. The calculations are based upon 10 different equations of state. Results of various structures of neutron stars and the rotational effects on stellar structures and properties are reported. Finally some limits to equations of state of neutron stars and the stability for rapidly rotating relativistic neutron stars are discussed.

On the stability of the singly and doubly charged beryllium trimers Be+3and Be++3

Extensive multireference configuration interaction calculations were carried out to generate adiabatic potential energy surfaces for the electronic ground state of Be+ 3 and Be++ 3. Resulting stability and other properties are investigated, showing that Be+ 3 is stable in a linear symmetric equilibrium configuration with a bond length R e = 2·28 Å and dissociation energy D e = 1·16 eV, the associated electronic state being 2Σ+ g. The doubly charged system Be++ 3 is found to be metastable due to relatively large and wide energy barriers. The corresponding local energy minimum occurs also in a linear symmetric geometry with R e = 2·17 Å, but is unbound by 1·44eV. A WKB estimate for the tunnelling dissociation shows that the lifetime of Be++ 3 is sufficiently long to allow its experimental observation.

An equilibrium analysis of helically symmetries plasmas using boundary element method

The authors present a computational scheme based on boundary-element methods for analyzing helically symmetric equilibrium configurations which are described by the MHD (magnetohydrodynamic) equilibrium equation. To solve the equation numerically, a boundary integral equation is introduced. The authors describe an effective scheme for calculating the principal solution to the equation which includes an infinite series of slow convergence. The vacuum fields enclosed by a helical perfectly conduction shell are calculated on the basis of this scheme. It is shown that the shifts of magnetic surfaces decrease with the helical pitch or the displacement of the helical shell.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Stability of boson stars.

Boson stars are gravitationally bound, spherically symmetric equilibrium configurations of cold, free, or interacting complex scalar fields \ensuremath{\varphi}. As these equilibrium configurations naturally present local anisotropy, it is sensible to expect departures from the well-known stability criteria for fluid stars. With this in mind, I investigate the dynamical instability of boson stars against charge-conserving, small radial perturbations. Following the method developed by Chandrasekhar, a variational base for determining the eigenfrequencies of the perturbations is found. This approach allows one to find numerically an upper bound for the central density where dynamical instability occurs. As applications of the formalism, I study the stability of equilibrium configurations obtained both for the free and for the self-interacting [with V(\ensuremath{\varphi})=(\ensuremath{\lambda}/4)\ensuremath{\Vert}\ensuremath{\varphi}${\ensuremath{\Vert}}^{4}$] massive scalar field \ensuremath{\varphi}. Instabilities are found to occur not for the critical central density as in fluid stars but for central densities considerably higher. The departure from the results for fluid stars is sensitive to the coupling \ensuremath{\lambda}; the higher the value of \ensuremath{\lambda}, the more the stability properties of boson stars approach those of a fluid star. These results are linked to the fractional anisotropy at the radius of the configuration.

Stability of rotating liquid drops: II. Homogeneously charged or self-gravitating drops

The oscillations and instabilities of a uniformly rotating and homogeneously charged or self-gravitating liquid spheroid with surface tension are investigated. The dispersion relation is obtained by means of an appropriate energy integral better suited to yield the higher modes than the virial method. Criteria are derived for the onset of secular instability that indicate bifurcation toward nonrotational symmetric equilibrium configurations, and of dynamical stability. In the oblate case the validity of the spheroidal approximation can be verified both quantitatively and qualitatively by comparison with two limiting cases: an uncharged or nongravitating rotating drop and a rotating drop without surface tension (astronomical limit). For the prolate spheroids there exists a clear difference between rotating and nonrotating configurations, for the equilibrium as well as for the stability.

Non-linear analysis of unbraced frames of variable geometry

A non-linear analysis of unbraced, rigid-jointed, and elastically restrained (against rotation) portal frames of variable geometry is presented. The loading system consists of eccentric concentrated loads and a transverse uniformly distributed load. Among the important conclusions one may list the following: (a) symmetric portal frames, loaded symmetrically, buckle through a stable bifurcation (sway-buckling) from a symmetric bent equilibrium configuration, and (b) nonsymmetric portal frames do not buckle; their behavior is similar to that of imperfect cantilever columns.

Gravitational Field Equations with Cross‐terms and Static Equilibrium Configurations

AbstractConsidering Einstein's equations modified by additional cross‐terms we calculate spherically symmetric equilibrium configurations of cold degenerated matter. The effects of such modifications proposed by TREDER [5] to describe properties of a unitary field theory depend on the sign of the coupling and on some critical energy density. If this density is higher than the nuclear density, we do not find significantly larger equilibrium masses than in general relativity. However the modifications of the gravitional dynamics resulting from the cross‐terms become physically significant.

On the maximization of the central gravitational red-shift of a schwarzschild sphere with a given surface gravitational red-shift

Following Bondi static, spherically symmetric equilibrium configurations with a core and an envelope have been considered. It has been shown that for any configurations with nonnegative pressure and density and with a surface red-shiftzs ⩽ 4.77 arbitrarily large central red-shiftszc are possible in the limiting case of arbitrarily large radius. The effects of imposition of further constraints in the form of a real speed of sound not exceeding the speed of light are also examined. It is seen that for a given limiting sound-to-light-speed ratio √λ. (i) There exists a limiting surface red-shiftzs(λ) ⩽ 1.71. (ii) A configuration withzs >zs(λ) is not possible, (iii) A configuration withzs=zs(λ) has a unique and finitezc=zc(λ). (iv) Forzs<zs(λ) arbitrarily large central red-shifts can be obtained for configurations with arbitrarily large radii.

Zwischengitteratome in kubisch-flächenzentrierten kristallen, insbesondere in kupfer

We give a comprehensive discussion of the geometry of all the possible equilibrium or non-equilibrium configurations of interstitial atoms in f.c.c. crystals. It is shown that these occur within a polyhedron which occupies only a small fraction of the atomic volume. The interstitial in the stable equilibrium configuration, the so-called 〈100〉 dumb-bell, has two elementary modes of movement (migration and rotation). We characterize them by specifying the side of the polyhedron through which the interstitial atom has to move. The reorientation process of the dumb-bells under external strains gives rise to a relaxation effect, which is discussed in some detail. This relaxation process has been experimentally verified for cold-worked and irradiated nickel. The quantitative calculations of energies and displacements in seven symmetric equilibrium configurations is carried out for copper, employing an improved version of the model due to Seeger and Mann. The improvements allow, in particular, for the anisotropy of the elastic displacement field and for the large relative displacements of neighbouring atoms. The influence of the choice of the Born-Mayer potential and of various cut-off and correction procedures are discussed and allowed for in some detail. The more important numerical results are: energy of migration of an interstitial atom in copper, 0·5 ev; energy of formation, 2·8 ev; increase of the volume of a finite crystal due to one interstitial atom, 0·4 atomic volumes. The expansion of the crystal in the environment of the interstitial atoms was found to be slightly less than one atomic volume. Contrary to previous views, the region of the crystal containing the interstitial atom appears to be more densely packed than the ideal crystal. The relative change in the elastic constants due to the relaxation process of an atomic concentration c of interstitial atoms in copper is calculated to be of the order 20 c. As far as they could be checked, these results are in fair agreement with the experimental data. By introducing additional data, we find for the electrical resistivity in copper 2·5 μΩ cm/per cent Frenkel pairs and 0·9 μΩ cm/per cent interstitials.

Axialsymmetrische Lösungen der magnetohydrostatischen Gleichung mit Oberflächenströmen II

Axially symmetric magneto-hydrodynamic equilibrium configurations are considered where the plasma is contained in a torus of arbitrary cross-section and electric currents flow in the surface of the plasma only. It is shown that the magnetic field is uniquely determined by the values of the total current in azimuthal and in meridional direction respectively and by the gas pressure. A method is given to compute the location of discontinuities of the magnetic field outside of the torus. The singularities are found explicitly in the case of elliptic or circular cross-section. In the circular case only the magnetic field is regular everywhere outside except on the axis of symmetry.