Flat Approximation Research Articles

Emissvity of metallic microcontoured surfaces

In this paper, we study the diffraction of electromagnetic radiation by a periodic micro-rough surface separating vacuum from a metal with a finite conductivity. We submit the integral method to the surface impedance boundary condition. Thus the numerical implementation is greatly reduced. We compare the numerical emissivities obtained by this approach to those we have calculated through the rigorous multilayer modal method. This enables us to show that the mentioned approximate method has two regions of validity: one corresponding to fairly flat surfaces and the other to very deep surfaces. It is well known that both the Kirchhoff approximation and the constant flat boundary impedance approximation are also valid for fairly flat surfaces. Our investigation aims also to establish whether these two approximate methods lead to the same results, and whether the integral method submitted to the surface impedance condition has a larger domain of validity. Concerning deep surfaces with a period smaller than the wavelength, we introduce the homogenization process in order to study its accuracy. Finally, this work permitted to identify three different regimes depending on the surface slopes: the simple scattering regime, the homogenization regime and the intermediate regime. For the latter, if the period is in the order of the wavelength, then we will show that the emissivity can be exalted.

Adhesive contact modeling for sub-5-nm ultralow flying magnetic storage head-disk interfaces including roughness effects

As the slider flying height decreases to sub-5-nm to obtain extremely high-density magnetic recordings of the order of 1Tbit∕in.2, problems of adhesion can cause catastrophic behavior at the magnetic recording head-disk interface (HDI). In the earlier part of the paper, a number of interfacial adhesive models were implemented for simplified HDI configurations (i.e., two flat parallel surfaces and a sphere on a flat surface). With the use of realistic HDI properties, individual adhesive force models, such as van der Waals and electrostatic forces, can provide initial approximations to the adhesive forces present during sub-5-nm flying. In the second part of the paper, realistic roughness conditions applicable to actual HDI’s were modeled using an improved Derjaguin–Muller–Toporov-based elastic-plastic rough surface adhesion model. Specifically, the proposed adhesion model accounts for roughness, the presence of molecularly thin lubricant, and includes electrostatic forces. Using experimentally measured roughness values from ultralow flying HDI’s (root-mean-square roughness of 0.65–1.62 nm), it was found that while the contact force is negligible for an interface with low roughness, the adhesive force dominates such interface. Moreover, the effect of roughness promotes adhesion at higher separations than if a two flat parallel surface configuration is considered. Prior to the onset of contact, the total adhesive force for an interface with low roughness is comparable to a two flat parallel surface approximation. However, the simple flat parallel surface approximation fails to predict the realistic onset of contact due to the exclusion of roughness.

Approximating Flats by Periodic Flats in CAT(0) Square Complexes

AbstractWe investigate the problem of whether every immersed flat plane in a nonpositively curved square complex is the limit of periodic flat planes. Using a branched cover, we reduce the problem to the case of VH-complexes. We solve the problem for malnormal and cyclonormal VH-complexes. We also solve the problem for complete square complexes using a different approach. We give an application towards deciding whether the elements of fundamental groups of the spaces we study have commuting powers. We note a connection between the flat approximation problem and subgroup separability.

The Collapse of Differentially Rotating Supermassive Stars: Conformally Flat Simulations

We investigate the gravitational collapse of rapidly rotating relativistic supermassive stars by means of a 3+1 hydrodynamical simulations in conformally flat spacetime of general relativity. We study the evolution of differentially rotating supermassive stars of $q \equiv J/M^{2} \sim 1$ ($J$ is the angular momentum and $M$ is the gravitational mass of the star) from the onset of radial instability at $R/M \sim 65$ ($R$ is the circumferential radius of the star) to the point where the conformally flat approximation breaks down. We find that the collapse of the star of $q \gtrsim 1$, a radially unstable differentially rotating star form a black hole of $q \lesssim 1$. The main reason to prevent formation of a black hole of $q \gtrsim 1$ is that quite a large amount of angular momentum stays at the surface. We also find that most of the mass density collapses coherently to form a supermassive black hole with no appreciable disk nor bar. In the absence of nonaxisymmetric deformation, the collapse of differentially rotating supermassive stars from the onset of radial instability are the promising sources of burst and quasinormal ringing waves in the Laser Interferometer Space Antenna.

A halo approach to the evaluation of the cross-correlation between the SZ sky and galaxy surveys

Using a purely analytic approach to gaseous and dark matter halos, we study the cross-correlation between the Sunyaev-Zel'dovich (SZ) sky and galaxy survey under flat sky approximation, in an attempt to acquire the redshift information of the SZ map. The problem can be greatly simplified when it is noticed that the signals of the SZ-galaxy correlation arise only from hot gas and galaxies inside the same massive halos (i.e. clusters), and field galaxies make almost no contribution to the cross-correlation. Under the assumption that both the hot gas and galaxies trace the common gravitational potential of dark halos, we calculate the expected cross SZ-galaxy power spectra for the WMAP/Planck SZ maps and the SDSS galaxy sample at small scales $100<l<1000$. It turns out, however, that it is not presently feasible to measure such small angular cross power spectra because of the high noise levels at $l>400$ with the WMAP/Planck experiments. Future SZ observations with better angular resolutions and sufficiently wide sky coverages will be needed if this technique is applied for the statistical measurement of redshift distribution of the SZ sources.

Recent Advances in Spectral Nodal Methods for Numerically Solving Neutron‐Diffusion Eigenvalue Problems

In this paper we present the recent advances in spectral nodal methods for numerically solving neutron-diffusion eigenvalue problems in Cartesian geometry. For one‐dimensional two‐energy group diffusion criticality problems, we describe the use of nonconventional albedo boundary conditions that substitute approximately the reflective properties of the nonmultiplying media around the core. We also present the Chebyshev acceleration scheme for the power method used in the outer iterations. For three‐dimensional one‐speed diffusion eigenvalue problems, we present a spectral nodal method that is based on transverse‐integrating the diffusion equation separately in X, Y, and Z directions, and then considering flat approximations for the transverse leakage terms. Numerical results for typical model problems are given to illustrate the accuracy.

Relativistic calculations of coalescing binary neutron stars

We have designed and tested a new relativistic Lagrangian hydrodynamics code, which treats gravity in the conformally flat approximation to general relativity. We have tested the resulting code extensively, finding that it performs well for calculations of equilibrium single-star models, collapsing relativistic dust clouds, and quasi-circular orbits of equilibrium solutions. By adding a radiation reaction treatment, we compute the full evolution of a coalescing binary neutron star system. We find that the amount of mass ejected from the system, much less than a per cent, is greatly reduced by the inclusion of relativistic gravitation. The gravity wave energy spectrum shows a clear divergence away from the Newtonian point-mass form, consistent with the form derived from relativistic quasi-equilibrium fluid sequences.

Abstractive dissociation of oxygen over Al(111): a nonadiabatic quantum model.

The dissociation of oxygen on a clean aluminum surface is studied theoretically. A nonadiabatic quantum dynamical model is used, based on four electronically distinct potential energy surfaces characterized by the extent of charge transfer from the metal to the adsorbate. A flat surface approximation is used to reduce the computation complexity. The conservation of the helicopter angular momentum allows Boltzmann averaging of the outcome of the propagation of a three degrees of freedom wave function. The dissociation event is simulated by solving the time-dependent Schrödinger equation for a period of 30 femtoseconds. As a function of incident kinetic energy, the dissociation yield follows the experimental trend. An attempt at simulation employing only the lowest adiabatic surface failed, qualitatively disagreeing with both experiment and nonadiabatic calculations. The final products, adsorptive dissociation and abstractive dissociation, are obtained by carrying out a semiclassical molecular dynamics simulation with surface hopping which describes the back charge transfer from an oxygen atom negative ion to the surface. The final adsorbed oxygen pair distribution compares well with experiment. By running the dynamical events backward in time, a correlation is established between the products and the initial conditions which lead to their production. Qualitative agreement is thus obtained with recent experiments that show suppression of abstraction by rotational excitation.

Hydroelastic behaviour of a conical shell impacting on a quiescent-free surface of an incompressible liquid

Hydroelastic effects are simulated by coupling a hydrodynamic Wagner model to a linear model of elasticity for thin shells. Applications are done for a cone falling on a flat-free surface of an incompressible liquid. Both hydrodynamic and structural models are linearized on the basis of a flat disk approximation. This is justified when the deadrise angle is small. In the hydrodynamic Wagner model, the main task is to evaluate the time-varying expansion of the wetted surface. The coupling with the linear model of elasticity is achieved via a modal-based method. This means that the hydrodynamic variables must project onto the family of eigenfunctions; this is the second main difficulty of the present problem. The coupled problem is solved mainly analytically. Special attention is paid to the energy conservation law. In particular, it is shown that kinetic energy evacuated in the jet plays as significant a role in the distribution of energy as the kinetic energy transmitted to the fluid or the kinetic and potential energies of the elastic shell. The importance of elasticity is discussed by comparing the rigid and elastic behaviours for free drop tests. A parametric study shows the influence of each parameter: thickness, deadrise angle and drop height. Comparisons with available experimental pressure data show a reasonable agreement. On the basis of this work, lines of future research are outlined.

The influence of quark matter at high densities on binary neutron star mergers

We consider the influence of potential quark matter existing at high densities in neutron star interiors on gravitational waves (GW) emitted in a binary neutron star merger event. Two types of equations of state (EoS) at zero temperatures are used, one describing pure nuclear matter, the other nuclear matter with a phase transition to quark matter at very high densities. Binary equilibrium sequences close to the innermost stable circular orbit (ISCO) are calculated to determine the GW frequencies just before merger. It is found that EoS effects begin to play a role for gravitational masses larger than $M_\infty\simeq1.5M_\odot$. The difference in the gravitational wave frequency at the ISCO grows to up to $\simeq 10%$ for the maximal allowed mass given by the EoSs used. Then, we perform 3D hydrodynamic simulations for each EoS varying the initial mass and determine the characteristic GW frequencies of the merger remnants. The implications of quark matter show up mainly in a different collapse behaviour of the merger remnant. If the collapse does not take place immediately after merger, we find a phase difference between two EoS's in the post-merger GW signal. We also compare the GW frequencies emitted by the merger remnant to values from simulations using a polytropic EoS and find an imprint of the non-constant adiabatic index of our EoSs. All calculations are based on the conformally flat (CF) approximation to general relativity and the GW signal from the merger simulation is extracted up to quadrupole order.

Cosmic microwave background lensing reconstruction on the full sky

Gravitational lensing of the microwave background by the intervening dark matter mainly arises from large-angle fluctuations in the projected gravitational potential and hence offers a unique opportunity to study the physics of the dark sector at large scales. Studies with surveys that cover greater than a percent of the sky will require techniques that incorporate the curvature of the sky. We lay the groundwork for these studies by deriving the full sky minimum variance quadratic estimators of the lensing potential from the CMB temperature and polarization fields. We also present a general technique for constructing these estimators, with harmonic space convolutions replaced by real space products, that is appropriate for both the full sky limit and the flat sky approximation. This also extends previous treatments to include estimators involving the temperature-polarization cross-correlation and should be useful for next generation experiments in which most of the additional information from polarization comes from this channel due to sensitivity limitations.

Nonsymmetric unified theory of gravitation, electromagnetism and Yang–Mills field

The recently studied new variation of Einstein's metric nonsymmetric unified field theory of gravitation and electromagnetism is enlarged to include the Yang–Mills field theory. It is shown that the antisymmetric part of the metric tensor, now a 2 × 2 matrix, can be made to describe both a field obeying Maxwell's equations and a field obeying Yang–Mills's field equations in the flat space linear approximation, thereby making its identification with the sum of a generalized electromagnetic and isotopic field strength tensors a possibly consistent procedure. The theory is shown to be free of unphysical ghost-negative energy radiative modes even when expanded on a curved Riemannian background. The Einstein–Maxwell–Yang–Mills theory is contained in the first approximation of the field equations on a curved general relativity background.

Relativistic simulations of rotational core collapse II. Collapse dynamics and gravitational radiation

We have performed hydrodynamic simulations of relativistic rotational supernova core collapse in axisymmetry and have computed the gravitational radiation emitted by such an event. The Einstein equations are formulated using the conformally flat metric approximation, and the corresponding hydrodynamic equations are written as a first-order flux-conservative hyperbolic system. Details of the methodology and of the numerical code have been given in an accompanying paper. We have simulated the evolution of 26 models in both Newtonian and relativistic gravity. The initial configurations are differentially rotating relativistic -polytropes in equilibrium which have a central density of . Collapse is initiated by decreasing the adiabatic index to some prescribed fixed value. The equation of state consists of a polytropic and a thermal part for a more realistic treatment of shock waves. Any microphysics like electron capture and neutrino transport is neglected. Our simulations show that the three different types of rotational supernova core collapse and gravitational waveforms identified in previous Newtonian simulations (regular collapse, multiple bounce collapse, and rapid collapse) are also present in relativistic gravity. However, rotational core collapse with multiple bounces is only possible in a much narrower parameter range in relativistic gravity. The relativistic models cover almost the same range of gravitational wave amplitudes ( for a source at a distance of 10 kpc) and frequencies () as the corresponding Newtonian ones. Averaged over all models, the total energy radiated in the form of gravitational waves is in the relativistic case, and in the Newtonian case. For all collapse models that are of the same type in both Newtonian and relativistic gravity, the gravitational wave signal is of lower amplitude. If the collapse type changes, either weaker or stronger signals are found in the relativistic case. For a given model, relativistic gravity can cause a large increase of the characteristic signal frequency of up to a factor of five, which may have important consequences for the signal detection. Our study implies that the prospects for detection of gravitational wave signals from axisymmetric supernova rotational core collapse do not improve when taking into account relativistic gravity. The gravitational wave signals obtained in our study are within the sensitivity range of the first generation laser interferometer detectors if the source is located within the Local Group. An online catalogue containing the gravitational wave signal amplitudes and spectra of all our models is available at the URL http://www.mpa-garching.mpg.de/Hydro/hydro.html.

Estimate of the cosmological bispectrum from the MAXIMA-1 cosmic microwave background map.

We use the measurement of the cosmic microwave background taken during the MAXIMA-1 flight to estimate the bispectrum of cosmological perturbations. We propose an estimator for the bispectrum that is appropriate in the flat sky approximation, apply it to the MAXIMA-1 data, and evaluate errors using bootstrap methods. We compare the estimated value with what would be expected if the sky signal were Gaussian and find that it is indeed consistent, with a chi(2) per degree of freedom of approximately unity. This measurement places constraints on models of inflation.

Relativistic simulations of rotational core collapse I. Methods, initial models, and code tests

We describe an axisymmetric general relativistic code for rotational core collapse. The code evolves the coupled system of metric and fluid equations using the ADM 3 + 1 formalism and a conformally flat metric approximation of the Einstein equations. Within this approximation the ADM 3 + 1 equations reduce to a set of five coupled non-linear elliptic equations for the metric components. The equations are discretized on a 2D grid in spherical polar coordinates and are solved by means of a Newton-Raphson iteration using a block elimination scheme to solve the diagonally dominant, sparse linear system arising within each iteration step. The relativistic hydrodynamics equations are formulated as a first-order flux-conservative hyperbolic system and are integrated using high-resolution shock-capturing schemes based on Riemann solvers. We assess the quality of the conformally flat metric approximation for relativistic core collapse and present a comprehensive set of tests that the code successfully passed. The tests include relativistic shock tubes, the preservation of the rotation profile and of the equilibrium of rapidly and differentially rotating neutron stars (approximated as rotating polytropes), spherical relativistic core collapse, and the conservation of rest-mass and angular momentum in dynamic spacetimes. The application of the code to relativistic rotational core collapse, with emphasis on the gravitational waveform signature, is presented in an accompanying paper.

Conformally flat smoothed particle hydrodynamics application to neutron star mergers

We present a new 3D smoothed particle hydrodynamics code that solves the general relativistic $\mathrm{field}+\mathrm{hydrodynamic}$ equations in the conformally flat approximation. Several test cases are considered to test different aspects of the code. We then apply the code to the coalescence of a neutron star binary system. The neutron stars are modeled by a polytropic equation of state with adiabatic indices $\ensuremath{\Gamma}=2.0, \ensuremath{\Gamma}=2.6,$ and $\ensuremath{\Gamma}=3.0.$ We calculate the gravitational wave signals, luminosities, and frequency spectra by employing the quadrupole approximation for emission and back reaction in the slow motion limit. In addition, we consider the amount of ejected mass.

Binary black holes in circular orbits. II. Numerical methods and first results

We present the first results from a new method for computing spacetimes representing corotating binary black holes in circular orbits. The method is based on the assumption of exact equilibrium. It uses the standard 3+1 decomposition of Einstein equations and conformal flatness approximation for the 3-metric. Contrary to previous numerical approaches to this problem, we do not solve only the constraint equations but rather a set of five equations for the lapse function, the conformal factor and the shift vector. The orbital velocity is unambiguously determined by imposing that, at infinity, the metric behaves like the Schwarzschild one, a requirement which is equivalent to the virial theorem. The numerical scheme has been implemented using multi-domain spectral methods and passed numerous tests. A sequence of corotating black holes of equal mass is calculated. Defining the sequence by requiring that the ADM mass decrease is equal to the angular momentum decrease multiplied by the orbital angular velocity, it is found that the area of the apparent horizons is constant along the sequence. We also find a turning point in the ADM mass and angular momentum curves, which may be interpreted as an innermost stable circular orbit (ISCO). The values of the global quantities at the ISCO, especially the orbital velocity, are in much better agreement with those from third post-Newtonian calculations than with those resulting from previous numerical approaches.

Binary black holes in circular orbits. I. A global spacetime approach

We present a new approach to the problem of binary black holes in the pre-coalescence stage, i.e. when the notion of orbit has still some meaning. Contrary to previous numerical treatments which are based on the initial value formulation of general relativity on a (3-dimensional) spacelike hypersurface, our approach deals with the full (4-dimensional) spacetime. This permits a rigorous definition of the orbital angular velocity. Neglecting the gravitational radiation reaction, we assume that the black holes move on closed circular orbits, which amounts to endowing the spacetime with a helical Killing vector. We discuss the choice of the spacetime manifold, the desired properties of the spacetime metric, as well as the choice of the rotation state for the black holes. As a simplifying assumption, the space 3-metric is approximated by a conformally flat one. The problem is then reduced to solving five of the ten Einstein equations, which are derived here, as well as the boundary conditions on the black hole surfaces and at spatial infinity. We exhibit the remaining five Einstein equations and propose to use them to evaluate the error induced by the conformal flatness approximation. The orbital angular velocity of the system is computed through a requirement which reduces to the classical virial theorem at the Newtonian limit.

Computation of gravitational waves from inspiraling binary neutron stars in quasiequilibrium circular orbits: Formulation and calibration

Gravitational waves from binary neutron stars in quasiequilibrium circular orbits are computed using an approximate method which we propose in this paper. In the first step of this method, we prepare general relativistic irrotational binary neutron stars in a quasiequilibrium circular orbit, neglecting gravitational waves. We adopt the so-called conformal flatness approximation for a three-metric to obtain the quasiequilibrium states in this paper. In the second step, we compute gravitational waves, solving linear perturbation equations in the background spacetime of the quasiequilibrium states. Comparing numerical results with post Newtonian waveforms and luminosity of gravitational waves from two point masses in circular orbits, we demonstrate that this method can produce accurate waveforms and luminosity of gravitational waves. It is shown that the effects of tidal deformation of neutron stars and strong general relativistic gravity modify the post Newtonian results for compact binary neutron stars in close orbits. We indicate that the magnitude of a systematic error in quasiequilibrium states associated with the conformal flatness approximation is fairly large for close and compact binary neutron stars. Several formulations for improving the accuracy of quasiequilibrium states are proposed.

Quantum wave packet approach to the Eley–Rideal reactive scattering between a gas phase atom and an adsorbate

A quantum wave packet code for studying Eley–Rideal scattering reactions of gas phase atoms with atoms adsorbed onto a rigid substrate is described. The flat surface approximation reduces the full treatment to three-dimensional (3D) numerical resolutions. The time evolution relies on the split operator propagator in a discrete, cylindrical coordinate representation. A pseudospectral strategy is used to evaluate efficiently the Hamiltonian operation by means of sequential 1D transformations between coordinate and momentum spaces. Fast Fourier transforms are performed for the two Cartesian coordinates whereas a discrete Bessel transform is applied for the cylindrical radius. The potential energy representation is based on the so-called LEPS model. Flux analysis in the product molecular region yields energy-resolved reaction cross sections and rovibrational distributions.