Elliptical Shell Research Articles

Mapping and spectroscopy of the planetary nebula NGC 7009 in the visual and infrared

NGC 7009 is a fascinating example of a high-excitation, elliptical planetary nebula (PN) containing circumnebular rings, and FLIERs and jets along the major axis. We present visual spectroscopy along multiple position angles through the nucleus, taken with the Observatorio Astronomico Nacional (Mexico); mid-infrared (MIR) spectroscopy and imaging acquired using the Infrared Space Observatory (ISO) and Spitzer Space Telescope (SST), and narrow-band imaging obtained using the Hubble Space Telescope (HST). The data show that the mid-infrared (MIR) continuum is dominated by a broad ≈100 K continuum, and a strong excess attributable to crystalline silicate emission. The primary peaks in this excess are similar to those observed in Forsterite and clino- and ortho-enstatite. The MIR images, by contrast, appear to be dominated by ionic transitions, with the 8.991 μm transition of [Ar iii] being important in the 8.0-μm band. The morphology and size of the envelope are found to vary with wavelength, with the largest dimensions occurring at 8.0 μm – a trend which is also reflected in an increase in the 8.0 μm/4.5 μm and 5.8 μm/4.5 μm ratios with distance from the nucleus. The visual spectroscopy permits us to map density and temperature throughout the shell, and confirm that the lowest values of ne are located close to the ansae, where densities appear to be of the order of 900–2600 cm−3. We provide mean line intensities for 116 transitions in six regions of the shell, and use mapping to confirm a systematic increase in excitation in the outer portions of the envelope. We finally use the ground-based spectroscopy, and ratioing of HST images to investigate the presence of shocks in the ansae and interior envelope. It is concluded that line ratios in the ansae may be partially consistent with shock excitation, although these features are primarily dominated by photoionization. We also note evidence for shock excitation at the limits of the interior elliptical shell, and for multiple bow-shock structures centred upon the ansae. The orientations of the easterly bow-shocks may have varied over time, indicating precession of the collimating engine at a rate of deg yr−1, whilst the outward splaying of the westerly ‘jet’ appears consistent with shock refraction modelling. We finally note that HST observations of the halo rings show them to have widths of the order of ∼1–3 arcsec, and steep changes in surface brightness consistent with local shock activity.

Vibration of elliptic cylindrical shells: Higher order shell theory

The equations for the free vibration of an elliptic cylindrical shell of constant thickness were derived using a Ritz approach. A higher order shell theory is employed that includes the effects of shear deformation, rotary inertia, and symmetric and antisymmetric thickness stretch deformations. The frequency-wavenumber spectrum has seven branches: flexural, extensional, torsional, two thickness shear, and two thickness stretch. The resulting seven coupled algebraic equations are symmetric and positive definite. The shell has a constant thickness, h, finite length, L, and is "simply supported" at its ends, (z=0,L), where z is the axial coordinate. The elliptic cross-section is defined by the shape parameter, a, and the half-length of the major axis, l. The modal solutions are expanded in a doubly infinite series of comparison functions in terms of circular functions in the angular and axial coordinates. Numerical results for the natural frequencies were obtained for two values of h/l and L/l, and various shape parameters, including the limiting case of a simply supported cylindrical shell (a approximately 100).

Forced vibrations of three-layer elliptic cylindrical shells under distributed loads

The dynamic behavior of a three-layer elliptic cylindrical shell is studied and associated problems are formulated. The refined Timoshenko model is used. The numerical method used to solve the equations of motion is based on the integro-interpolation differencing method. The dynamic behavior of a three-layer elliptic cylindrical shell under a distributed nonstationary load is studied.

Analysis of nonlinear deformation and stability of reinforced elliptic cylindrical shells under bending with internal pressure

A finite element statement of solving problems on stability of reinforced elliptic cylindrical shells taking into account momentness and nonlinearity of their subcritical stress strain state is presented. The explicit expressions for displacements of noncircular cylindrical shell elements as rigid bodies are determined by integrating the equations obtained by equating the components of linear strains to zero. These expressions were used to construct the form functions for an efficient quadrilateral finite element of natural curvature. An efficient numerical algorithm of nonlinear deformation and stability of shells was developed. The stability of reinforced elliptic cylindrical shells under combined loading by bending moment, transverse boundary force and internal pressure is analyzed. We also examine how the critical loads are affected by the strain nonlinearity and ellipticity of shell deformation at the subcritical stage.

Nonlinear deformation and stability of reinforced elliptical cylindrical shells loaded by internal pressure under twisting and bending

The problem of stability of cylindrical shells with an elliptical cross-sectional contour reinforced by a set of stringers under combined loading by bending and twisting moments, transverse force, and internal pressure is studied with the use of the variational method of finite elements in displacements. The subcritical stress-strain state of the shells is assumed to be moment and nonlinear. The effect of nonlinearity of deformation of the shells and their ellipticity on the critical loads and buckling type is determined.

A HIDDEN POPULATION OF MASSIVE STARS WITH CIRCUMSTELLAR SHELLS DISCOVERED WITH THESPITZER SPACE TELESCOPE

We have discovered a large number of circular and elliptical shells at 24 microns around luminous central sources with the MIPS instrument on-board the Spitzer Space Telescope. Our archival follow-up effort has revealed 90% of these circumstellar shells to be previously unknown. The majority of the shells is only visible at 24 microns, but many of the central stars are detected at multiple wavelengths from the mid- to the near-IR regime. The general lack of optical counterparts, however, indicates that these sources represent a population of highly obscured objects. We obtained optical and near-IR spectroscopic observations of the central stars and find most of these objects to be massive stars. In particular, we identify a large population of sources that we argue represents a narrow evolutionary phase, closely related or identical to the LBV stage of massive stellar evolution.

Comparing different realizations of modified Newtonian dynamics: Virial theorem and elliptical shells

There exists several modified gravity theories designed to reproduce the empirical Milgrom's formula, modified Newtonian dynamics (MOND). Here we derive analytical results in the context of the static weak-field limit of two of them (bimetric MOND, leading for a given set of parameters to the quasilinear MOND, and Bekenstein's Tensor-Vector-Scalar). In this limit, these theories are constructed to give the same force field for spherical symmetry, but their predictions generally differ out of it. However, for certain realizations of these theories (characterized by specific choices for their free functions), the binding potential-energy of a system is increased, compared to its Newtonian counterpart, by a constant amount independent of the shape and size of the system. In that case, the virial theorem is exactly the same in these two theories, for the whole gravity regime and even outside of spherical symmetry, although the exact force fields are different. We explicitly show this for the force field generated by the two theories inside an elliptical shell. For more general free functions, the virial theorems are, however, not identical in these two theories. We finally explore the consequences of these analytical results for the two-body force.

Two-dimensional elliptical electromagnetic superscatterer and superabsorber

Using coordinate transformation stated earlier by Pendry et al. [Science 312, 1780 (2006)], we investigate the two-dimensional elliptical electromagnetic superscatterer and superabsorber, based on the concept of complementary media. Such an elliptical electromagnetic superscatterer (or superabsorber) is realized by coating an elliptical negative refractive material shell. The effectiveness of the elliptical electromagnetic superscatterer and superabsorber designs is verified by finite element simulations. The proposed design provides a more practical superscatterer (or superabsorber) geometry when compared to previous designs with axial and radial symmetries. Our results can be extended to an arbitrarily shaped electromagnetic superscatterer and superabsorber.

OBSERVATIONAL STUDY OF THE MULTISTRUCTURED PLANETARY NEBULA NGC 7354

We present an observational study of the planetary nebula (PN) NGC 7354 consisting of narrowband Halpha and [NII]6584 imaging as well as low- and high-dispersion long-slit spectroscopy and VLA-D radio continuum. According to our imaging and spectroscopic data, NGC 7354 has four main structures: a quite round outer shell and an elliptical inner shell, a collection of low-excitation bright knots roughly concentrated on the equatorial region of the nebula, and two symmetrical jet-like features, not aligned either with the shells' axes, or with each other. We have obtained physical parameters like electron temperature and electron density as well as ionic and elemental abundances for these different structures. Electron temperature and electron density slightly vary throughout the nebula. The local extinction coefficient c_Hbeta shows an increasing gradient from south to north and a decreasing gradient from east to west consistent with the number of equatorial bright knots present in each direction. Abundance values show slight internal variations but most of them are within the estimated uncertainties. In general, abundance values are in good agreement with the ones expected for PNe. Radio continuum data are consistent with optically thin thermal emission. We have used the interactive three-dimensional modeling tool SHAPE to reproduce the observed morphokinematic structures in NGC 7354 with different geometrical components. Our SHAPE model is in very good agreement with our imaging and spectroscopic observations. Finally, after modeling NGC 7354 with SHAPE, we suggest a possible scenario for the formation of the nebula.

KOITER ESTIMATE REVISITED

We prove a general adimensional energy estimate between the solution of the three-dimensional Lamé system on a thin clamped shell and a displacement reconstructed from the solution of the classical two-dimensional Koiter model. This estimate only involves the thickness parameter ε, constants attached to the mid-surface S, the two-dimensional energy of the solution of the Koiter model and "wavelengths" associated with this latter solution. This bound is in the same spirit as Koiter's heuristic estimate in Ref. 26 and can be viewed as an a posteriori estimation of the modeling error by means of the two-dimensional solution. It is general with respect to the geometry of the mid-surface S which is an arbitrary smooth manifold with boundary. Taking boundary layer terms into account, we prove that our estimates are sharp in the cases of plates and elliptic shells.

Prediction of breakout noise from an elliptical duct of finite length

This paper describes a predictive model for breakout noise from an elliptical duct or shell of finite length. The transmission mechanism is essentially that of ``mode coupling'', whereby higher structural modes in the duct walls get excited because of non-circularity of the wall. Effect of geometry has been taken care of by evaluating Fourier coefficients of the radius of curvature. The noise radiated from the duct walls is represented by that from a finite vibrating length of a semi infinite cylinder in a free field. Emphasis is on understanding the physics of the problem as well as analytical modeling. The analytical model is validated with 3-D FEM. Effects of the ovality, curvature, and axial terminations of the duct have been demonstrated. (C) 2010 Institute of Noise Control Engineering.

Two-Dimensional Scattering by a Conducting Elliptic Cylinder Coated With a Homogeneous Anisotropic Shell

A solution to the two-dimensional scattering properties of a conducting elliptic cylinder coated with a confocal homogeneous anisotropic elliptical shell is obtained. The transmitted field of the anisotropic shell is expressed as an integral equation based on waves with different wave numbers and different directions of propagation. The waves in all directions are represented as the eigenfunction expansion in elliptic coordinates in terms of Mathieu functions. In order to solve the nonorthogonality properties of Mathieu functions, Galerkin's method is applied and a matrix is required for the computation of unknown expansion coefficients of the scattered and transmitted fields. Only the transverse magnetic (TM) polarization is presented, while the transverse electric (TE) polarization can be obtained in the same way. Some numerical results are presented in graphical forms. The result is in agreement with that available as expected when a coated elliptic cylinder degenerates to the coated circular one.

AKARIINFRARED OBSERVATIONS OF THE SUPERNOVA REMNANT G292.0+1.8: UNVEILING CIRCUMSTELLAR MEDIUM AND SUPERNOVA EJECTA

We present the results of AKARI observations of the O-rich supernova remnant (SNR) G292.0+1.8 using six Infrared Camera (IRC) and four Far-Infrared Surveyor bands covering 2.7–26.5 μm and 50–180 μm, respectively. The AKARI images show two prominent structures; a bright equatorial ring (ER) structure along the east–west direction and an outer elliptical shell structure surrounding the remnant. The ER structure is clumpy and incomplete with its western end opened. The outer shell is almost complete and slightly squeezed along the north–south direction. The central position of the outer shell is ∼1' northwest from the embedded pulsar and coincides with the center of the ER structure. In the northern and southwestern regions, there is also faint emission with a sharp boundary beyond the bright shell structure. The ER and the elliptical shell structures were partly visible in optical and/or X-rays, but they are much more clearly revealed in our AKARI images. There is no evident difference in infrared colors of the two prominent structures, which is consistent with the previous proposition that both structures are of circumstellar origin. However, we have detected faint infrared emission of a considerably high 15/24 μm ratio associated with the supernova (SN) ejecta in the southeastern and northwestern areas. Our IRC spectra show that the high ratio is at least partly due to the emission lines from Ne ions in the SN ejecta material. In addition, we detect a narrow, elongated feature outside the SNR shell. We derive the physical parameters of the infrared-emitting dust grains in the shocked circumstellar medium (CSM) and compare the result with model calculations of dust destruction by an SN shock. The AKARI results suggest that the progenitor was at the center of the infrared circumstellar shell in the red supergiant stage and that the observed asymmetry in the SN ejecta could be a result of either a dense CSM in the equatorial plane and/or an asymmetric explosion.

Studies of nonlinear deformation and stability of oval and elliptic cylindrical shells under axial compression

The stability of noncircular shells, in contrast to that of circular ones, has not been studied sufficiently well yet. The publications about circular shells are counted by thousands, but there are only several dozens of papers dealing with noncircular shells. This can be explained on the one hand by the fact that such shells are less used in practice and on the other hand by the difficulties encountered when solving problems involving a nonconstant curvature radius, which results in the appearance of variable coefficients in the stability equations. The well-known solutions of stability problems were obtained by analytic methods and, as a rule, in the linear approximation without taking into account the moments and nonlinearity of the shell precritical state, i.e., in the classical approximation. Here we use the finite element method in displacements to solve the problem of geometrically nonlinear deformation and stability of cylindrical shells with noncircular contour of the transverse cross-section. We use quadrilateral finite elements of shells of natural curvature. In the approximations to the element displacements, we explicitly distinguish the displacements of elements as rigid bodies. We use the Lagrange variational principle to obtain a nonlinear system of algebraic equations for determining the unknown nodal finite elements. We solve the system by a step method with respect to the load using the Newton-Kantorovich linearization at each step. The linear systems are solved by the Kraut method. The critical loads are determined with the use of the Silvester stability criterion when solving the nonlinear problem. We develop an algorithm for solving the problem numerically on personal computers. We also study the nonlinear deformation and stability of shells with oval and elliptic transverse cross-section in a wide range of variations in the ovalization and ellipticity parameters. We find the critical loads and the shell buckling modes. We also examine how the critical loads are affected by the strain nonlinearity and the ovalization and ellipticity of shells.

Sensitivity of applanation tonometry readings to the geometrical parameters of the sclera and the shape of the cornea

Abstract Purpose To study the effect of the geometrical parameters of a cornea (the thickness and difference of corneal shape from spherical segment) on applanation tonometry readings. Methods The mathematical model for Goldmann and Maklakov tonometers is developed. The corneoscleral shell is modeled as two conjugated transversal isotropic elliptic (with non‐zero eccentricity) shells (cornea and sclera) with different mechanical properties. The baseline (prior to loading) two‐segment shell is assumed to be filled with incompressible liquid under pressure. Simulated IOP measurements are obtained for Goldman and 5‐g and 10‐g Maklakov tonometers. To analyze the effect of the eyeball geometry on IOP readings the elastic constants of cornea and sclera are varied and geometrical parameters (such as central corneal thickness, cornea and sclera radii of curvature, scleral thickness, ratio of anterior‐posterior eye axis lengths vs. the equator diameter) are varied. Simulated calculations are compared with experimental data. Results The results are obtained over a wide range of parameters of the sclera and cornea. Conclusion If the shape of a cornea differs from spherical it could cause the measurement error of tonometry IOP readings ranged between 3% and 25%. Significant sensitivity of the Goldmann tonometer to corneal thickness compared to Maklakov's method may be explained with smaller contact zone and, hence, the larger influence of the flexural (bending) deformation, which depends on the thickness of a shell. Contrariwise the change of the radius of curvature of cornea affects greatly on the results obtained with Maklakov tonometer.

Hybrid/mixed assumed stress element for anisotropic laminated elliptical and parabolic shells

A variety of elliptical/parabolic dome type structures are used in important aerospace and civil structural systems such as underwater vehicles, stadium covers, exhibition halls, auditoriums and museum halls. For the analysis of such structures, a shear deformable four-noded finite element based on a hybrid/mixed assumed stress is presented in this paper. The element called iHES (improved Hybrid and Enhanced Shell element) is developed assuming the most general arbitrary orthogonal coordinate system. The element is based on a first-order shear flexible formulation and essentially consists of a combination of drilling degrees of freedom with assumed stress and enhanced strain techniques. Using the element developed here, a detailed parametric study of anisotropic elliptical and parabolic shells of various configurations is carried out to investigate the effects of aspect and height ratios as well as layer lay-up schemes.

Influence of geometrical parameters on the stress state of longitudinally corrugated elliptic cylindrical shells

An approach developed to solve static problems for longitudinally corrugated elliptic cylindrical shells is used to analyze the influence of their geometric parameters and thickness on the stress–strain state. The circumferential distribution of stresses and displacements is analyzed for different values of the aspect ratio and number of corrugations

Wave Equations and Solutions of In Vacuo and Fluid-Filled Elliptical Cylindrical Shells

A perturbation-based approach is used to formulate the governing equations for wave propagation at high frequencies in infinite in vacuo and fluid-filled elliptical cylindrical shells. The in vacuo equations thus formulated have the form of a perturbation over the corresponding equations for the circular cylindrical shell. Here, eccentricity of the cross-section is assumed to be small and used as the perturbation parameter. Next, the coupled equations for the fluid-filled elliptical shell are obtained as a perturbation over the in vacuo shell equations (by using a single fluid-loading parameter μ). Asymptotic arguments are used to neglect various terms in the derivation, and a compact form of the non-dimensional governing equations is found. Presenting the governing equations in this form is the main contribution of this work. Due to the eccentricity, all spatial quantities need to be represented in terms of a harmonic series instead of a single harmonic term. Using symmetry and asymptotic arguments, the nature of the harmonic series is obtained. Using the harmonic series expansion, the dispersion relation is formulated for both the in vacuo and the fluid-filled cases. The in vacuo dispersion equation is solved using the regular perturbation method, while the transcendental coupled dispersion equation is solved numerically.

Material parameter equation for rotating elliptical spherical cloaks

By using the coordinate transformation method, we have deduced the material parameter equation for rotating elliptical spherical cloaks and carried out simulation as well. The results indicate that the rotating elliptical spherical cloaking shell, which is made of meta-materials whose permittivity and permeability are governed by the equation deduced in this paper, can achieve perfect invisibility by excluding electromagnetic fields from the internal region without disturbing any external field.

Refined analysis of the stress state of orthotropic elliptic cylindrical shells with variable geometrical parameters

An approach developed earlier to solve boundary-value problems is used to analyze the behavior of the stress-strain state of orthotropic elliptic cylindrical shells with variation in the geometric parameters of their cross section at constant volume (weight)