Uniform Distribution Of Energy Density Research Articles

Trapping of null geodesics in slowly rotating extremely compact Tolman VII spacetimes

Region of trapped null geodesics hidden inside of extremely compact objects is of astrophysical importance because of trapping of gravitational waves, or neutrinos. The trapping effect of null geodesics was extensively studied for spherically symmetric extremely compact objects. Recently, influence of rotation of the extremely compact objects on the trapping of null geodesics was treated in the simplest possible model of internal linearized Hartle–Thorne spacetime with uniform energy density distribution and uniform emissivity distribution of null geodesics. Here, we extend the study of the rotation influence on the trapping effect in the case of linearized Hartle–Thorne spacetimes based on the Tolman VII spherically symmetric solutions, where we assume the emissivity of the null geodesics proportional to the energy density of the Tolman VII object having quadratic radial profile. We demonstrate enhancement (suppression) of the trapping effect in the case of counter-rotating (co-rotating) null geodesics due to the behavior of the effective potentials and escape cones of the null geodesics in the linearized Hartle–Thorne–Tolman VII spacetimes. In dependence on the parameters of these spacetimes, we determine the “local” and “global” coefficients of efficiency of the trapping and compare the results to those related to the rotating spacetimes based on the internal Schwarzschild spacetimes. We demonstrate that in the Tolman VII spacetimes the trapping is more efficient, being allowed in objects with radii larger than those of the trapping internal Schwarzschild spacetimes, occurring even for \(R>3.3M\).

General relativistic polytropes with a repulsive cosmological constant

Spherically symmetric equilibrium configurations of perfect fluid obeying a polytropic equation of state are studied in spacetimes with a repulsive cosmological constant. The configurations are specified in terms of three parameters---the polytropic index $n$, the ratio of central pressure and central energy density of matter $\ensuremath{\sigma}$, and the ratio of energy density of vacuum and central density of matter $\ensuremath{\lambda}$. The static equilibrium configurations are determined by two coupled first-order nonlinear differential equations that are solved by numerical methods with the exception of polytropes with $n=0$ corresponding to the configurations with a uniform distribution of energy density, when the solution is given in terms of elementary functions. The geometry of the polytropes is conveniently represented by embedding diagrams of both the ordinary space geometry and the optical reference geometry reflecting some dynamical properties of the geodesic motion. The polytropes are represented by radial profiles of energy density, pressure, mass, and metric coefficients. For all tested values of $n>0$, the static equilibrium configurations with fixed parameters $n$, $\ensuremath{\sigma}$, are allowed only up to a critical value of the cosmological parameter ${\ensuremath{\lambda}}_{\mathrm{c}}={\ensuremath{\lambda}}_{\mathrm{c}}(n,\ensuremath{\sigma})$. In the case of $n>3$, the critical value ${\ensuremath{\lambda}}_{\mathrm{c}}$ tends to zero for special values of $\ensuremath{\sigma}$. The gravitational potential energy and the binding energy of the polytropes are determined and studied by numerical methods. We discuss in detail the polytropes with an extension comparable to those of the dark matter halos related to galaxies, i.e., with extension $\ensuremath{\ell}>100\text{ }\text{ }\mathrm{kpc}$ and mass $M>1{0}^{12}\text{ }{\mathrm{M}}_{\ensuremath{\bigodot}}$. For such largely extended polytropes, the cosmological parameter relating the vacuum energy to the central density has to be larger than $\ensuremath{\lambda}={\ensuremath{\rho}}_{\text{vac}}/{\ensuremath{\rho}}_{\mathrm{c}}\ensuremath{\sim}{10}^{\ensuremath{-}9}$. We demonstrate that the extension of the static general relativistic polytropic configurations cannot exceed the so-called static radius related to their external spacetime, supporting the idea that the static radius represents a natural limit on the extension of gravitationally bound configurations in an expanding universe dominated by the vacuum energy.

Study on Applicability of Large-Area EB Irradiation to Micro-Deburring

In a large-area electron beam (EB) irradiation method developed recently, high energy EB can be obtained without focusing the beam, and large-area EB with almost uniform energy density distribution of 60mm in diameter can be used for instantly melting and evaporating metal surface. Then, the surface smoothing and surface modification of metal molds made of steel, cemented carbide and ceramics can be performed efficiently. When the large-area EB is irradiated to a sharp edge, the material removal remarkably progresses at the edge due to the heat accumulation and EB concentration there, which leads to the edge rounding. Our previous study clarified that micro-deburring was also possible by using this phenomenon. Moreover, this method would be effective for removal of discontinuous micro burrs generated by thermal machining processes, such as EDM and laser cutting. In this study, the possibility to completely remove micro burrs generated in EDM was experimentally investigated. The difference in micro-deburring characteristics with the type of workpiece material was also investigated. As a result, the EDM micro burrs with a height of about up to 50μm could be completely removed by this method. Furthermore, it was made clear that the magnetic property of workpiece materials greatly influenced the deburring characteristics, and the thermal property such as melting point and thermal conductivity also did.

Thermo-mechanical solution of film/substrate systems under local thermal load and application to laser lift-off of GaN/sapphire structures

Film/substrate structures may undergo a localized thermal load, which can induce stresses, deformation and defects. In this paper, we present the solutions of temperature and stresses in a film/substrate structure under a local thermal load on the film surface. Then, the generalized Stoney formula, which connects the curvature of deformation and the stress field is obtained. The present solution takes into account the non-uniformity of the temperature field both in the width and thickness directions of the film. The thermo-mechanical solution is applied to the analysis of the temperature distribution, stresses, and damage of a GaN/sapphire system during the laser lift-off (LLO) process. It is shown that the laser with the Gaussian distribution of energy density causes much smaller tensile stresses at the edge of the heated area in the film than the laser with the uniform distribution of energy density, and thus can avoid damage to the GaN films separated from the substrate.

Neutrino trapping in extremely compact objects described by the internal Schwarzschild-(anti-)de Sitter spacetimes

Extremely compact stars (ECS) (having radius R < 3GM/c2) contain captured null geodesics. Certain part of neutrinos produced in their interior will be trapped, influencing thus their neutrino luminosity and thermal evolution. The trapping effect has been previously investigated for the internal Schwarzschild spacetimes with the uniform distribution of energy density. Here, we extend our earlier study considering the influence of the cosmological constant Λ on the trapping phenomena. Our model for the interior of ECS is based on the internal Schwarzschild-(anti-)de Sitter (S(a)dS) spacetimes with uniform distribution of energy density matched to the external vacuum S(a)dS spacetime with the same cosmological constant. Assuming uniform and isotropic distribution of local neutrino emissivity we determine behavior of the trapping coefficients, i.e., “global” one representing influence on the neutrino luminosity and “local” one representing influence on the cooling process. We demonstrate that the repulsive (attractive) cosmological constant has tendency to enhance (damp) the trapping phenomena.

Neutrino trapping in braneworld extremely compact stars

Extremely Compact Stars (ECS) contain trapped null geodesics. When such objects enter the evolution period admitting geodetical motion of neutrinos, certain part of neutrinos produced in their interior will be trapped influencing their neutrino luminosity and thermal evolution. We study neutrino trapping in the braneworld ECS, assuming uniform distribution of neutrino emissivity and massless neutrinos. We give the efficiency of the neutrino trapping effects in the framework of the simple model of the internal spacetime with uniform distribution of energy density, and external spacetime described by the Reissner-Nordström geometry characterized by the braneworld “tidal” parameter b. For b < 0 the external spacetime is of the black-hole type, while for b > 0 the external spacetime can be of both black-hole and naked-singularity type. Then the ECS surface radius R can be located also above the unstable (outer) photon circular orbit. Such basically new types of the spacetimes strongly alter the trapping phenomena as compared to the standard case of b = 0. It is shown that the neutrino trapping effects are slightly lowered by the presence of physically more plausible case of b < 0, as compared to the standard internal Schwarzschild spacetime, while they can be magnified by positive tidal charges if b < 1 and lowered for b > 1. However, potential astrophysical relevance of the trapping phenomena is strongly enhanced for negative tidal charges enabling a significant enlargement of the ECS surface radius to values coherent with recent observations.

Photon and neutrino redshift in the field of braneworld compact stars

We study gravitational redshift of photons and neutrinos radiated by the braneworld neutron or quark stars that are considered in the framework of the simple model of the internal spacetime with uniform distribution of energy density, and the external spacetime described by the Reissner-Nordström geometry characterized by the braneworld ``tidal'' charge b. For negative tidal charges, the external spacetime is of the black-hole type, while for positive tidal charges, the external spacetime can be of both black-hole and naked-singularity type. We consider also extremely compact stars allowing existence of trapped null geodesics in their interior. We assume radiation of photons from the surface at radius R, neutrinos from the whole compact star interior, and their motion along radial null geodesics of the spacetime. In dependency on the compact stars parameters b and R, the photon surface redshift is related to the range of the neutrino internal redshift and the signatures of the tidal charge and possible existence of extremely compact stars are discussed. When both surface (photon) and internal (neutrino) redshift are given by observations, both compact star parameters R and b can be determined in the framework of our simple model.

Neutrino trapping in extremely compact objects: I. Efficiency of trapping in the internal Schwarzschild spacetimes

Extremely compact objects (9GM/4c2 < R < 3GM/c2) contain trapped null geodesics. When such objects enter the evolution period corresponding to the geodetical motion of neutrinos, a certain part of neutrinos produced in their interior will be trapped influencing thus neutrino luminosity of the objects and consequently their thermal evolution. The existence of trapped neutrinos indicates possibility of ‘two-temperature’ cooling regime of the extremely compact objects. We present upper estimates on the efficiency of the neutrino trapping effects obtained in the framework of the simplest model of the internal Schwarzschild spacetime with uniform distribution of energy density, assuming uniform distribution of neutrino emissivity. We introduce a ‘global’ luminosity trapping coefficient representing influence of the trapping effect on the total neutrino luminosity of the extremely compact objects and cooling trapping coefficients of both ‘local’ and ‘global’ kinds characterizing influence of the trapping on the cooling process. It is shown that the trapping of neutrinos can be relevant to moderately or even slightly extremely compact objects.

Formulation of the multiple non-isotropic scattering process in 3-D space on the basis of energy transport theory

Summary High-frequency seismograms of local earthquakes are considered to consist of incoherent body waves scattered by random inhomogeneities in the earth medium. For the study of seismogram envelopes, we can describe the scattering process on the basis of energy transport theory by representing the random inhomogeneities as distributed point-like scatterers. By introducing the concept of directional distribution of energy density, we propose a formulation of the multiple non-isotropic scattering process in 3-D space. We suppose that the axially symmetric non-isotropic scattering is described by a directional scattering coefficient, the medium is characterized by one wave velocity, and the source radiation is spherical. In addition to the Fourier transformation in space and the Laplace transformation in time used to solve for energy density in isotropic scattering media, we use a spherical harmonic series expansion in solid angle. Then, the energy transport equation given as an integral can be written as simultaneous linear equations, where coefficients are given by the Wigner 3-j symbols and the spherical harmonic expansion coefficients of the directional scattering coefficient. The lowest term of the spherical harmonic series corresponds to isotropic scattering. When the directional scattering coefficient is written as a finite-length spherical harmonic series, the simultaneous linear equations can be solved for a finite number of unknowns and we can obtain the spatio-temporal distribution of energy density. The numerical calculation for a case with strong forward scattering shows a uniform distribution of energy density around the hypocentre at large lapse times.

Laser surface refinement of YBa2Cu3Ox superconductor

A novel laser-processing technique that produces bulk YBa2Cu3Ox (1 2 3) plates has been developed. Through the application of a square CO2 laser beam with uniform energy density distribution to the surface of 1 2 3 powder compact, a single piece of ribbon-like plate is produced. This plate may be separated from the powder compact after laser scanning. The width of the plate is ≈ 6 mm, while its thickness is 0.1–0.2 mm. Powder X-ray diffraction indicates that laser-treated samples contain both orthorhombic and tetragonal 1 2 3 phases, as well as Y2O3 (2 0 0), Y2BaCuO5 (2 1 1), BaCuO2 (0 1 1), and CuO (0 0 1) phases. Scanning electron microscopy reveals a pattern of phase segregation along the transverse cross-section after solidification of the plate. After oxygen annealing of a single ribbon piece, Tc is found to be 90 K. This technique may be applicable to the mass production of 1 2 3 bulk superconductor by continuous melting of 1 2 3 powders. In addition to its potential for practical applications, the laser technique also helps to explain the complex phases and microstructure formation during melting and solidification of laser-melted 1 2 3 liquid. A model relating the microstructure to the thermal history inside the laser-affected region and to the phase diagram of incongruently melting 1 2 3 material has been developed to analyse phase formation during laser melting and solidification processes. Reasonable correspondence between theoretical analysis and experimental results was obtained.