Case Of Thin Shell Research Articles

Constraining the Circumburst Medium of Gamma-Ray Bursts with X-Ray Afterglows

Abstract Long gamma-ray bursts (GRBs) are considered to originate from core collapse of massive stars. It is believed that the afterglow property is determined by the density of the material in the surrounding interstellar medium (ISM). Therefore, the circumburst density can be used to distinguish between an interstellar wind, n(R) ∝ R −k , and a constant-density medium (ISM), n ( R ) = const . Previous studies with different afterglow samples show that the circumburst medium of GRBs is neither simply supported by an interstellar wind nor completely favored by an ISM. In this work, our new sample consists of 39 GRBs with smoothly onset bump-like features in early X-ray afterglows, in which 20 GRBs have the redshift measurements. By using a smooth broken power-law function to fit the bumps of X-ray light curves, we derive the FWHM as the feature width (ω), as well as the rise and decay timescales of the bumps (T r and T d ). The correlations between the timescales of X-ray bumps are similar to those found previously in the optical afterglows. Based on the fireball forward shock model of the thin shell case, we obtain the distribution of the electron spectral index p and further constrain the medium density distribution index k. The new inferred k is found to be concentrated at 1.0, with a range from 0.2 to 1.8. This finding is consistent with previous studies. The conclusion of our detailed investigation for X-ray afterglows suggests that the ambient medium of the selected GRBs is not homogeneous, i.e., neither ISM nor the typical interstellar wind.

The Bright Reverse Shock Emission in the Optical Afterglows of Gamma-Ray Bursts in a Stratified Medium

Abstract The reverse shock (RS) model is generally introduced to interpret the optical afterglows with the rapid rising and decaying, such as the early optical afterglow of GRB 990123 (which is also called the optical flash). In this paper, we collected 11 gamma-ray burst (GRB) early optical afterglows, which have signatures of dominant RS emission. Since the temporal slopes of the optical flashes are determined by both the medium density distribution index k and the electron spectral index p, we apply the RS model of the thin shell case to the optical flashes and determine the ambient medium of the progenitors. We find that the k value is in the range of 0–1.5. The k value in this paper is consistent with the result in Yi et al., where the forward shock model was applied to some onset bumps. However, the method adopted in this paper is only applicable to GRB afterglows with significant sharp rising and decaying RS emission. Our results indicate that the RS model can also be applied to confirm the circumburst medium, further implying that GRBs may have diverse circumburst media.

Equilibrated stress space for nonlinear dimensionally reduced shell models in terms of first-order stress functions

Considering the power series expansion of the three-dimensional variables with respect to the shell thickness coordinate, an equilibrated stress space for the first Piola-Kirchhoff stress vectors is derived in convective curvilinear coordinate system. The infinite series of the two-dimensional translational equilibrium equations are satisfied by introducing two first-order stress function vectors expanded into power series. For the important case of thin shells, the infinite number of two-dimensional equilibrium equations is truncated to obtain a ‘first-order’ model, where the equilibrated stress-space requires three vectorial stress function coefficients only. The formulation presented for thin shells is compared to the nonlinear equilibrium equations of the classical shell theories, written in terms of the first Piola-Kirchhoff stress resultants and stress couples and satisfied by the introduction of three first-order stress function vectors.

Grasping Interface With Wet Adhesion and Patterned Morphology: Case of Thin Shell

In this letter, we present an analytical method for studying the role of the micropatterned morphology of soft fingers with wet adhesion in grasping a deformable thin shell. This letter originated from a project on a robotic platform for autonomous attachment and removal of a soft contact lens in a “human eye” under a wet environment. In this scenario, a contact lens (hemispherical thin shell) was gripped by a soft-fingered hand in three conditions: inside/outside a liquid environment and when the contact lens was in contact with a convex curved substrate (as an eye). Two fingertip surfaces were compared: a flat surface, and a micropatterned surface inspired by the adhesion mechanism of a tree-frog's toes. The pattern of the latter comprised 3600 square cells, each of size 85 $\mu$ m × 85 $\mu$ m interspaced by 15 $\mu$ m. A proposed analytical model was utilized to evaluate the grasp forces in both types of fingertip surface. The forces were subsequently verified by application in grasping a contact lens. The experimental results showed good agreement with the analysis as the micropattern pad decreased the preload and deformation of the shell 1.1–2× lower than that of the flat surface. This letter could be extended to model grasping interfaces with deformable curvature objects in wet and high moisture content environments.

A model of the collapse and evaporation of charged black holes

In this paper, a natural generalization of KMY model is proposed for the evaporation of charged black holes. Within the proposed model, the back reaction of Hawking radiation is considered. More specifically, we consider the equation , in which the matter content is assumed spherically symmetric. We also assume the radiation to be massless and chargeless. With this equation of motion, the asymptotic behavior of the model is analyzed.Two kinds of matter contents are taken into consideration in this paper. In the first case (the thin-shell model), the infalling matter is simulated by a null-like charged sphere collapsing into its center. In the second case, we consider a continuous distribution of spherical symmetric infalling null-like charged matter. It is simulated by taking the continuous limit of many co-centric spheres collapsing into the center.We find that in the thin-shell case, an event horizon forms and the shell passes through the horizon before becoming extremal, provided that it is not initially near-extremal. In the case with continuous matter distribution, we consider explicitly an extremal center covered by neutral infalling matter, and find that the event horizon also forms. The black hole itself will become near-extremal eventually, leaving possibly a non-electromagnetic energy residue less than the order of .The details of the behavior of these models are explicitly worked out in this paper.

Photometric and Spectroscopic Observations of GRB 140629A

Abstract We present our optical photometric and spectroscopical observations of GRB 140629A. A redshift of z = 2.275 ±0.043 is measured through the metal absorption lines in our spectroscopic data. Using our photometric data and multiple observational data from other telescopes, we show that its optical light curve is well interpreted with the standard forward shock models in the thin shell case. Its optical–X-ray afterglow spectrum is jointly fitted with a single power-law function, yielding a photon index of −1.90 ± 0.05. The optical extinction and neutral hydrogen absorption of the gamma-ray burst (GRB) host galaxy are negligible. The fit to the light curve with the standard models shows that the ambient density is 60 ± 9 cm−3 and the GRB radiating efficiency is as low as ∼0.24%, likely indicating a baryonic-dominated ejecta of this GRB. This burst agrees well with the relation, but confidently violates those empirical relations involving geometric corrections (or jet break time). This gives rise to an issue of the possible selection effect on these relations since the jet opening angle of this GRB is extremely narrow (0.04 rad).

Dynamic snap-through buckling of CNT reinforced composite sandwich spherical caps

This paper investigates the nonlinear dynamic buckling response of axisymmetric carbon nanotubes reinforced composite sandwich spherical caps subjected to suddenly applied pressure. To solve this problem, a finite element approach is adopted using an axisymmetric shear flexible shell element, free from spurious constraints causing shear and membrane locking especially for the case of thin shells. The geometric nonlinearity is accounted for in the formulation using von Karman’s strain-displacement relations. Newmark’s integration technique along with the modified Newton-Raphson iteration scheme is employed to solve the nonlinear governing equations. The critical dynamic pressure value is considered as the pressure load beyond which the maximum average displacement response shows instant growth in the time history of the shell structure. The obtained results are validated against the available analytical solutions for the case of isotropic spherical caps. A detailed parametric study is carried out to bring out the effects of shell geometry parameter, volume fraction of the CNT, core-to-face sheet thickness, temperature, boundary conditions and different types of pulse on the dynamic snap-through characteristics of spherical shells.

Confined nematic liquid crystal between two spherical boundaries with planar anchoring

Nematic shells of liquid crystals have been provided in microscales. Defect structures in the shells are very essential in the electro-optical applications of such colloidal objects. We have numerically minimized the free energy of symmetric and asymmetric spherical shells of the nematic liquid crystal. Considering degenerate planar anchoring on the surfaces and isotropic nematic elasticity, a variety of defect structures are observed by controlling or varying the thicknesses of the shell and its degree of asymmetry. In symmetric shells, our calculations show that boojums (bipolar) defects appear in thick shells and tetrahedral (baseball) defects in thin shells. In asymmetric shells, while we are in the bipolar regime, the boojums defects transform to trigonal configurations. Free energy landscape shows that in this regime the inner droplet is not stable in the center and it is trapped in an off-center minimum energy position. For the case of thin shells, there are two degenerate director textures with similar tetrahedral configuration of the disclination lines. The levels are split in asymmetric shells. The stability of the inner droplet in the center position depends on director texture. It is stable for one texture and unstable for the other one. For an unstable pattern there is no minimum energy position for the inner droplet and it moves until it touches the outer boundary.

Extended rotation-free shell triangles with transverse shear deformation effects

The paper extends recent work of the authors to include transverse shear effects on rotation-free triangular element for plates (Onate and Zarate in Int J Numer Methods Eng 83(2):196---227, 2010). Two new shell triangular elements are presented, the EBST+ and the EBST+1. Transverse shear deformation effects are important for thick shells, as well when the shell is laminated or formed by composite material. The ingredients for the element formulation are: a Hu-Washizu type mixed functional and linear interpolation for the displacement field. In both elements presented a finite volume approach is used for computing the bending moments and the curvatures over a patch of elements. The nodal translational degrees of freedom of the original enhanced basic shell triangle (EBST) are extended with the two shear deformation angles via two different approaches. The first one uses a linear interpolation of the rotation angles inside the element (EBST+) and the second one assumes a constant field for the rotation angles (EBST+1). For the thin shell case the shear angles vanish and the new elements reproduce the good behaviour of the original thin EBST element. As a consequence the elements can reproduce the solutions for thick to thin shells situations without exhibiting shear locking. The numerical solution for the thick shell case can be found iteratively starting from the deflection values for the Kirchhoff theory using the original thin EBST element. Examples of the good performance of the new rotation-free shell triangles are given.

Early afterglows from radially structured outflows and the application to X-ray shallow decays

In the fireball model, it is more physically realistic that gamma-ray burst (GRB) ejecta have a range of bulk Lorentz factors (assuming M α −S). The low Lorentz factor part of the ejecta will catch up with the high Lorentz factor part when the latter is decelerated by the surrounding medium to a comparable Lorentz factor. Such a process will develop a long-lasting weak reverse shock until the whole ejecta are shocked. Meanwhile, the forward shocked materials are gradually supplied with energy from the ejecta that are catching-up, and thus the temporal decay of the forward shock emission will be slower than that without an energy supply. However, the reverse shock may be strong. Here, we extend the standard reverse-forward shock model to the case of radially nonuniform ejecta. We show that this process can be classified into two cases: the thick shell case and the thin shell case. In the thin shell case, the reverse shock is weak and the temporal scaling law of the afterglow is the same as that in Sari & Mészáros (2000). However, in the thick shell case, the reverse shock is strong and thus its emission dominates the afterglow in the high energy band. Our results also show slower decaying behavior of the afterglow due to the energy supply by low Lorentz factor materials, which may help the understanding of the plateau observed in the early optical and X-ray afterglows.

The initial Lorentz factors of fireballs inferred from the early X-ray data of SWIFT GRBs

We intend to determine the type of circumburst medium and measure directly the initial Lorentz factor $\Gamma_0$ of GRB outflows. If the early X-ray afterglow lightcurve has a peak and the whole profile across the peak is consistent with the standard external shock model, the early rise profile of light curves can be used to differentiate whether the burst was born in interstellar medium (ISM) or in stellar wind. In the thin shell case, related to a sub-relativistic reverse shock, the peak time occurring after the end of the prompt emission, can be used to derive an accurate $\Gamma_0$, especially for the ISM case. The afterglow lightcurves for a flat electron spectrum $1<p<2$ have been derived analytically. In our GRB sample, we obtain $\Gamma_0 \sim 300$ for the bursts born in ISM. We did not find any good case for bursts born in stellar wind and behaving as a thin shell that can be used to constrain $\Gamma_0$ reliably.

Revisiting chameleon gravity: Thin-shell and no-shell fields with appropriate boundary conditions

We derive analytic solutions of a chameleon scalar field $\ensuremath{\phi}$ that couples to a nonrelativistic matter in the weak gravitational background of a spherically symmetric body, paying particular attention to a field mass ${m}_{A}$ inside of the body. The standard thin-shell field profile is recovered by taking the limit ${m}_{A}{r}_{c}\ensuremath{\rightarrow}\ensuremath{\infty}$, where ${r}_{c}$ is a radius of the body. We show the existence of ``no-shell'' solutions where the field is nearly frozen in the whole interior of the body, which does not necessarily correspond to the ``zero-shell'' limit of thin-shell solutions. In the no-shell case, under the condition ${m}_{A}{r}_{c}\ensuremath{\gg}1$, the effective coupling of $\ensuremath{\phi}$ with matter takes the same asymptotic form as that in the thin-shell case. We study experimental bounds coming from the violation of equivalence principle as well as solar-system tests for a number of models including $f(R)$ gravity and find that the field is in either the thin-shell or the no-shell regime under such constraints, depending on the shape of scalar-field potentials. We also show that, for the consistency with local gravity constraints, the field at the center of the body needs to be extremely close to the value ${\ensuremath{\phi}}_{A}$ at the extremum of an effective potential induced by the matter coupling.

The Onset of Gamma‐Ray Burst Afterglow

We discuss the reference time t0 of afterglow light curves in the context of the standard internal-external shock model. The decay index of early afterglow is very sensitive to the reference time one chooses. In order to understand the nature of early afterglow, it is essential to take a correct reference time. Our simple analytic model provides a framework for understanding special relativistic effects involved in early afterglow phase. We evaluate light curves of reverse shock emission as well as those of forward shock emission, based on full hydrodynamic calculations. We show that the reference time does not shift significantly even in the thick-shell case. For external shock emission components, measuring times from the beginning of the prompt emission is a good approximation and it does not cause an early steep decay. In the thin-shell case, the energy transfer time from fireball ejecta to ambient medium typically extends to thousands of seconds. This might be related to the shallow decay phases observed in early X-ray afterglow at least for some bursts.

Asymptotic integration of the dynamic equations of the theory of elasticity for the case of thin shells

Asymptotic integration of the three-dimensional dynamic equations of the theory of elasticity is carried out for the case of thin shells. Specific features of the asymptotic properties of the stress-strain state (SSS) of the shell in dynamic problems are discussed. Limiting two-dimensional systems of equations are derived.