Spectral Order Research Articles

Radial velocity variations in pulsating Ap stars

In this paper we present the first results from a high-precision radial velocity (RV) study of the rapidly oscillating Ap (roAp) star HR 1217. Data spanning a complete rotation period were acquired on 9 nights in late 1997 and early 1998 using the Harlan J. Smith 2.7 m telescope at McDonald Observatory. The RVs were measured using the wavelength range of each spectral order (≈100 A) of the 2dcoude echelle spectrograph. Most of the pulsational modes can be seen in all spectral regions but the amplitude varies by factors of ten or more between the different wavelength intervals. A detailed analysis of the spectral order centered on 5825 A, which had the largest amplitudes, showed the presence of all published photometric frequencies f1- f7. The multi-mode oscillations were also found in the RV variations of the high amplitude Nd III 5294 A feature. Our RV data reveal two new excited modes at frequencies f0 = 223.37 c d −1 (2585.3 µHz) and f−1 = 220.58 c d −1 (2553.0 µHz) that follow the odd and even alternative mode spacing with ∆ν0/2 � 34 µHz. An analysis of individual nights shows strong amplitude and phase modulation of all excited modes, including the two newly discovered ones. The amplitudes of the f2 = 229.21 c d −1 (2652.9 µHz) and f4 = 235.08 c d −1 (2720.85 µHz) modes are modulated with the published magnetic field variations and reach their maximum and minimum at magnetic extrema. However, the phase variability is in disagreement with that expected from the standard oblique pulsator model. The phase jumps for the f2 mode occur exactly at magnetic maximum and close to the phase of magnetic minimum, while the f3 and f4 modes have a continuous change. The phase variability is attributed to strong vertical phase changes in the line-forming layers of atmosphere. An echelle-diagram for all known excited modes in HR 1217 is constructed. We interpret the f7 = 242.41 c d −1 (2805.7 µHz) mode with the peculiar spacing as due to a mode of degree l = 4 that is the only observed member of another system of equally-spaced frequencies. We predict the existence of modes belonging to this system that should be equally spaced at 68 µHz with f7.

A Search for Discrete X‐Ray Spectral Features in a Sample of Bright γ‐Ray Burst Afterglows

We present uniform, detailed spectral analyses of γ-ray burst (GRB) X-ray afterglows observed with ASCA, BeppoSAX, Chandra, and XMM-Newton and critically evaluate the statistical significances of X-ray emission and absorption features in these spectra. The sample consists of 21 X-ray afterglow observations up to and including that of GRB 040106 with spectra of sufficient statistical quality to allow meaningful line searches, chosen here somewhat arbitrarily to be detections with more than 100 total (source plus background) counts. This sample includes all nine X-ray afterglows with published claims of line detections. Moderate resolution spectra are available for 16 of the 21 sources and for the remaining five the Chandra transmission grating spectrometers obtained high-resolution data. All of the data are available from the public archive. We test a simple hypothesis in which the observed spectra are produced by a power-law continuum model modified by photoelectric absorption by neutral material both in our Galaxy and possibly also local to the burst. As a sample, these afterglow spectra are consistent with this relatively simple model. However, since the χ2 statistic is not sensitive to weak and/or localized fluctuations, we have performed Monte Carlo simulations to search for discrete features and to estimate their significances. Our analysis shows that there are four afterglows (GRB 011211, GRB 030227, GRB 021004, and GRB 040106) with line-like features that are significant at the 3 σ level. We cautiously note that, in two cases, the features are associated with an unusual background feature; in the other two, the fractional magnitudes of the lines are small, and comparable to the expected level of systematic uncertainty in the spectral response. In addition, none of the statistically significant features are seen in more than one detector or spectral order where available. We conclude that, to date, no credible X-ray line feature has been detected in a GRB afterglow. Finally, in a majority of cases, we find no evidence for significant absorbing columns local to the GRB host galaxy, implying that there is little evidence from X-ray observations that GRB preferentially explode in high-density environments.

A COMPARISON OF HIGHER-ORDER COMPACT FINITE DIFFERENCE SCHEMES THROUGH BURGERS' EQUATION

Two higher-order compact finite difference approaches, the Hermitian and the Lax-Wendroff are examined by applying them to the viscous Burgers' equation. The difference equations obtained by the two methods were integrated in time through the third-order Strong-Stability-Preserving Runge-Kutta scheme. Absolute errors are computed by using an exact solution. The results are also compared with a second order central difference scheme. The Hermitian approach is far easier to implement. On uniform grids the Lax-Wendroff scheme produces smaller errors during the initial stages, but both methods are equally good for larger durations of integration. The convergence rate of the Hermitian scheme is slightly higher than the Lax-Wendroff scheme although both are of fourth order. Both schemes are unstable beyond a certain step size in time and space. When numerical boundary conditions are imposed, second-order conditions produce the best results whereas linear extrapolation proved to be the worst. It was also observed that large domains were required to implement the numerical boundary conditions properly. There was no detrimental effect on the accuracy of the results obtained through either of the two schemes when the size of the domain was greatly increased. Both schemes showed remarkable improvement in accuracy when clustered grids were employed. However much smaller time steps are required for stable solutions.

Spectral shorted matrices

Given a n× n positive semidefinite matrix A and a subspace S of C n , Σ( S,A) denotes the shorted matrix of A to S . We consider the notion of spectral shorted matrix ρ( S,A)= lim m→∞Σ( S,A m) 1/m. Wecompletely characterize this martix in terms of S and the spectrum and the eigenspaces of A. We show the relation of this notion with the spectral order of matrices and the Kolmogorov’s complexity of A to a vector ξ∈ C n .

Three-dimensional instability of standing waves

We investigate the three-dimensional instability of finite-amplitude standing surface waves under the influence of gravity. The analysis employs the transition matrix (TM) approach and uses a new high-order spectral element (HOSE) method for computation of the nonlinear wave dynamics. HOSE is an extension of the original high-order spectral method (HOS) wherein nonlinear wave–wave and wave–body interactions are retained up to high order in wave steepness. Instead of global basis functions in HOS, however, HOSE employs spectral elements to allow for complex free-surface geometries and surface-piercing bodies. Exponential convergence of HOS with respect to the total number of spectral modes (for a fixed number of elements) and interaction order is retained in HOSE. In this study, we use TM-HOSE to obtain the stability of general three-dimensional perturbations (on a two-dimensional surface) on two classes of standing waves: plane standing waves in a rectangular tank; and radial/azimuthal standing waves in a circular basin. For plane standing waves, we confirm the known result of two-dimensional side-bandlike instability. In addition, we find a novel three-dimensional instability for base flow of any amplitude. The dominant component of the unstable disturbance is an oblique (standing) wave oriented at an arbitrary angle whose frequency is close to the (nonlinear) frequency of the original standing wave. This finding is confirmed by direct long-time simulations using HOSE which show that the nonlinear evolution leads to classical Fermi–Pasta–Ulam recurrence. For the circular basin, we find that, beyond a threshold wave steepness, a standing wave (of nonlinear frequency . From the cases we consider, the critical wave steepness is found to generally decrease/increase with increasing radial/azimuthal mode number of the base standing wave. Finally, we show that the instability we find for both two- and three-dimensional standing waves is a result of third-order (quartet) resonance.

Hyperbolic polynomials and spectral order

The spectral order on R induces a partial ordering on the manifold H n of monic hyperbolic polynomials of degree n. We show that the semigroup S generated by differential operators of the form (1−λ d dx ) e λ d/ dx , λ∈ R , acts on the poset H n in an order-preserving fashion. We also show that polynomials in H n are global minima of their respective S -orbits and we conjecture that a similar result holds even for complex polynomials. Finally, we show that only those pencils of polynomials in H n which are of logarithmic derivative type satisfy a certain local minimum property for the spectral order. To cite this article: J. Borcea, B. Shapiro, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Relative Flux Calibration of Keck HIRES Echelle Spectra11Based on data obtained with the Kast spectrograph on the Lick Observatory 3 m Shane telescope and with the HIRES and ESI spectrographs at the W. M. Keck Observatory, which is a joint facility of the University of California, the California Institute of Technology, and NASA.

ABSTRACTWe describe a method to calibrate the relative flux levels in spectra from the High‐Resolution Echelle Spectrograph (HIRES) on the Keck I telescope. Standard data reduction techniques that transfer the instrument response between HIRES integrations leave errors in the flux of 5%–10% because the effective response varies during the night. The flux errors are most severe near the ends of each spectral order, where there can be discontinuous jumps. The source of these errors is uncertain but may include changes in the vignetting connected to the optical alignment. Our flux calibration method uses a calibrated reference spectrum of each target to calibrate individual HIRES integrations. We determine the instrument response independently for each integration, and hence we avoid the need to transfer the instrument response between HIRES integrations. The procedure can be applied to any HIRES spectrum or any other spectrum. While the accuracy of the method depends upon many factors, we have been able to flux‐calibrate a HIRES spectrum to 1% over scales of 200 Å that include order joins. We illustrate the method with spectra of QSO 1243+3047, toward which we have measured the deuterium‐to‐hydrogen abundance ratio.

Osmium atomic-oxygen protection by an iridium overcoat for increased extreme-ultraviolet grating efficiency.

The deposition of a 30-A-thick layer of iridium upon a 250-A-thick osmium reflective layer for use as a diffraction grating in the Cosmic Hot Interstellar Plasma Spectrometer (CHIPS) satellite observatory has provided sufficient protection from an expected maximum orbital atomic-oxygen fluence of 1 x 10(16) atoms/cm2. The grating parameters of groove constant and depth, efficiencies of zeroth-order, first and second inside orders, and first inside-order efficiency positional uniformity as well as stray light near the first inside order of the Ir-Os-coated grating were measured within a CHIPS spectral bandpass of 90-260 A. Stray-light measurements were also made near the first inside spectral order at 304, 584, and 1216 A. The results make the Ir-Os coat an acceptable grating reflectivity layer for CHIPS and other spaceborne extreme-ultraviolet spectrometers that employ grazing-incidence reflection optics.

Echelle gratings acting as one.

A new device for producing order-sortable echelle spectra by use of consecutive diffactions in several echelle grating surfaces is described. This echelle emulating device (EED) comprises a prism placed in the path between a pair of echelle grating surfaces. The refractive angle of the prism is fixed through a simple relation. The device reproduces all the main properties of a single virtual echelle, obtained from a simple grating equation describing the combined action of the gratings and the prism. The spectral order of the EED is the sum of the spectral orders of the individual gratings. A spectrograph that utilizes the device is described, and several applications are discussed.

Computation of multiphase systems with phase field models

Phase field models offer a systematic physical approach for investigating complex multiphase systems behaviors such as near-critical interfacial phenomena, phase separation under shear, and microstructure evolution during solidification. However, because interfaces are replaced by thin transition regions (diffuse interfaces), phase field simulations require resolution of very thin layers to capture the physics of the problems studied. This demands robust numerical methods that can efficiently achieve high resolution and accuracy, especially in three dimensions. We present here an accurate and efficient numerical method to solve the coupled Cahn–Hilliard/Navier–Stokes system, known as Model H, that constitutes a phase field model for density-matched binary fluids with variable mobility and viscosity. The numerical method is a time-split scheme that combines a novel semi-implicit discretization for the convective Cahn–Hilliard equation with an innovative application of high-resolution schemes employed for direct numerical simulations of turbulence. This new semi-implicit discretization is simple but effective since it removes the stability constraint due to the nonlinearity of the Cahn–Hilliard equation at the same cost as that of an explicit scheme. It is derived from a discretization used for diffusive problems that we further enhance to efficiently solve flow problems with variable mobility and viscosity. Moreover, we solve the Navier–Stokes equations with a robust time-discretization of the projection method that guarantees better stability properties than those for Crank–Nicolson-based projection methods. For channel geometries, the method uses a spectral discretization in the streamwise and spanwise directions and a combination of spectral and high order compact finite difference discretizations in the wall normal direction. The capabilities of the method are demonstrated with several examples including phase separation with, and without, shear in two and three dimensions. The method effectively resolves interfacial layers of as few as three mesh points. The numerical examples show agreement with analytical solutions and scaling laws, where available, and the 3D simulations, in the presence of shear, reveal rich and complex structures, including strings.

Structural and dynamic characterization of a neuron-specific protein kinase C substrate, neurogranin.

Neurogranin/RC3 is a neuron-specific, Ca(2+)-sensitive calmodulin binding protein and a specific protein kinase C substrate. Neurogranin may function to regulate calmodulin levels at specific sites in neurons through phosphorylation at serine residue within its IQ motif, oxidation outside the IQ motif, or changes in local cellular Ca(2+) concentration. To gain insight into the functional role of neurogranin in the regulation of calmodulin-dependent activities, we investigated the structure and dynamics of a full-length rat neurogranin protein with 78 amino acids using triple resonance NMR techniques. In the absence of calmodulin or PKC, neurogranin exists in an unfolded form as evidenced by high backbone mobility and the absence of long-range nuclear Overhauser effect (NOE). Analyses of the chemical shifts (13)C(alpha), (13)C(beta), and (1)H(alpha) reveal the presence of a local alpha-helical structure for the region between residues G25-A42. Three-bond (1)H(N)-(1)H(alpha) coupling constants support the finding that the sequence between residues G25 and A42 populates a non-native helical structure in the unfolded neurogranin. Homonuclear NOE results are consistent with the conclusions drawn from chemical shifts and coupling constants. (15)N relaxation data indicate motional restrictions on a nanosecond time scale in the region from D15 to S48. Spectral densities and order parameters data further confirm that the unfolded neurogranin exists in conformation with residual secondary structures. The medium mobility of the nascent helical region may help to reduce the entropy loss when neurogranin binds to its targets, but the complex between neurogranin and calmodulin is not stable enough for structural determination by NMR. Calmodulin titration of neurogranin indicates that residues D15-G52 of neurogranin undergo significant structural changes upon binding to calmodulin.

Automated determination of stellar parameters from simulated dispersed images for DIVA

We have assessed how well stellar parameters (T e f f , log g and [M/H]) can be retrieved from low-resolution dispersed images to be obtained by the DIVA satellite. Although DIVA is primarily an all-sky astrometric mission, it will also obtain spectrophotometric information for about 13 million stars (operational limiting magnitude V ≃ 13.5 mag). Constructional studies foresee a grating system yielding a dispersion of ≃ 200 nm/mm on the focal plane (first spectral order). For astrometric reasons there will be no cross dispersion which results in the overlapping of the first to third diffraction orders. The one-dimensional, position related intensity function is called a DISPI (DISPersed Intensity). We simulated DISPIS from synthetic spectra taken from Lejeune et al. (1997) and Lejeune et al. (1998) but for a limited range of metallicites, i.e. our results are for [M/H] in the range -0.3 to 1 dex. We show that there is no need to deconvolve these low resolution signals in order to obtain basic stellar parameters. Using neural network methods and by including simulated data of DIVA's UV telescope, we can determine T e f f . to an average accuracy of about 2% for DISPIS from stars with 2000 K ≤ T e f f < 20000 K and visual magnitudes of V = 13 mag (end of mission data). log g can be determined for all temperatures with an accuracy better than 0.25 dex for magnitudes brighter than V = 12 mag. For low temperature stars with 2000 K ≤ T e f f < 5000 K and for metallicities in the range -0.3 to +1 dex a determination of [M/H] is possible (to better than 0.2 dex) for these magnitudes. For higher temperatures, the metallicity signatures are exceedingly weak at DISPI resolutions so that the determination of [M/H] is there not possible. Additionally we examined the effects of extinction E(B - V) on DISPIS and found that it can be determined to better than 0.07 mag for magnitudes brighter than V = 14 mag if the UV information is included.

A uniformly stable family of mixed hp-finite elements with continuous pressures for incompressible flow

A new family of mixed hp-finite elements is presented for the discretization of planar Stokes flow on meshes of curvilinear, quadrilateral elements. The elements involve continuous pressures and are shown to be stable with an inf-sup constant bounded below independently of the mesh-size h and the spectral order p. The spaces have balanced approximation properties - the orders of approximation in h and p are equal for both the velocity and the pressure. This is the first example of a uniformly stable method with continuous pressures for spectral element discretization of Stokes equations, valid for geometrically refined meshes and curvilinear elements.

On a question of Furuta on chaotic order

The chaotic order A≫ B among positive invertible operators A, B>0 on a Hilbert space is introduced by log A⩾log B. Uchiyama's method brings us the Furuta inequality for the chaotic order from the Furuta inequality. Related to this, Furuta posed the following question: For A, B>0, A≫ B if and only if (Q) A r−t⩾ A r/2 A −t/2B pA −t/2 sA r/2 (r−t)/((p−t)s+r) holds for all p⩾1, r⩾ t, s⩾1 and t∈[0,1]? Recently he gave a counterexample to the “only if” part. In this note, we point out that condition (Q) characterizes the operator order A⩾ B. Moreover, (Q) characterizes the spectral order by extending the bounds of t.

Stabilised hp -Finite Element Approximation of Partial Differential Equations with Nonnegative Characteristic Form

This paper is devoted to the a priori error analysis of the hp-version of a streamline-diffusion finite element method for partial differential equations with nonnegative characteristic form. This class of equations includes second-order elliptic and parabolic problems, first-order hyperbolic problems and second-order problems of mixed elliptic-parabolic-hyperbolic type. We derive error bounds which are simultaneously optimal in both the mesh size h and the spectral order p. Numerical examples are presented to confirm the theoretical results.

Spectral Galerkin Discretization for Hydrodynamic Stability Problems

A spectral Galerkin discretization for calculating the eigenvalues of the Orr-Sommerfeld equation is presented. The matrices of the resulting generalized eigenvalue problem are sparse. A convergence analysis of the method is presented which indicates that a) no spurious eigenvalues occur and b) reliable results can only be expected under the assumption of scale resolution, i.e., that Re/p 2 is small; here Re is the Reynolds number and p is the spectral order. Numerical experiments support that the assumption of scale resolution is necessary in order to obtain reliable results. Exponential convergence of the method is shown theoretically and observed numerically.

Computational aspects of the spectral Galerkin FEM for the Orr-Sommerfeld equation

Since Orszag's paper [‘Accurate solution of the Orr–Sommerfeld stability equation’, J. Fluid Mech., 50, 689–703 (1971)], most of the subsequent spectral techniques for solving the Orr–Sommerfeld equation (OSE) employed the Tau discretization and Chebyshev polynomials. The use of the Tau discretization appears to be accompanied by so-called spurious eigenvalues not related to the OSE and a singular matrix B in the generalized eigenvalue problem. Starting from a variational formulation of the OSE, a spectral discretization is performed using a Galerkin method. By adopting integrated Legendre polynomials as basis functions, the boundary conditions can be satisfied exactly for any spectral order and the non-singular matrices A and B are obtained in Ax=λBx. For plane Poiseuille flow, the stiffness and the mass matrices are sparse with bandwidths 7 and 5 respectively, and the entries can be calculated explicitly (thus avoiding quadrature errors) for any polynomial flow profile U. According to the convergence results [Hancke, ‘Calculating large spectra in hydrodynamic stability: a p FEM approach to solve the Orr–Sommerfeld equation’, Diploma Thesis, Swiss Federal Institute of Technology Zürich, Seminar for Applied Mathematics, 1998; Hancke, Melenk and Schwab, ‘A spectral Galerkin method for hydrodynamic stability problems’, Research Report No. 98-06, Seminar for Applied Mathematics, Swiss Federal Institute of Technology, Zürich], no spurious eigenvalue has been found. Numerical experiments with spectral orders up to p=600 illustrate the results. Copyright © 2000 John Wiley & Sons, Ltd.

Stabilizedhp-Finite Element Methods for First-Order Hyperbolic Problems

We analyze the hp-version of the streamline-diffusion finite element method (SDFEM) and of the discontinuous Galerkin finite element method (DGFEM) for first-order linear hyperbolic problems. For both methods, we derive new error estimates on general finite element meshes which are sharp in the mesh-width h and in the spectral order p of the method, assuming that the stabilization parameter is O(h/p). For piecewise analytic solutions, exponential convergence is established on quadrilateral meshes. For the DGFEM we admit very general irregular meshes and for the SDFEM we allow meshes which contain hanging nodes. Numerical experiments confirm the theoretical results.

NMR and molecular structure of partially oriented mono and para methyl- and chlorobenzenes dissolved in nematic liquid crystals

Spectral parameters, order parameters, and structural parameters, including the vibrationally corrected ralpha structure of the partially oriented solutes p-xylene, p-chlorotoluene, p-dichlorobenzene, toluene, and chlorobenzene dissolved in three liquid crystal mixtures, are reported. For samples containing the three solutes p-xylene, p-dichlorobenzene, and 1,3,5-trichlorobenzene, multiple quantum (MQ) nuclear magnetic resonance was used to aid in the analysis of the complex high-resolution spectra of the p-xylene. The high-resolution spectra of 1,3,5-trichlorobenzene and p-dichlorobenzene were easily identified and analyzed once the calculated p-xylene spectrum was subtracted from the experimental one. The methyl groups of p-xylene, p-chlorotoluene, and toluene have similar geometries when the structure is determined from dipolar couplings corrected for harmonic vibrations.Key words: nematic liquid crystal, NMR, molecular structure, order parameter.

An hp finite element method for convection-diffusion problems in one dimension

We analyse an hp FEM for convection-diffusion problems. Stability is achieved by suitably upwinded test functions, generalizing the classical α-quadratically upwinded and the Hemker test-functions for piecewise linear trial spaces (see, e.g., Morton 1995 Numerical Solutions of Convection-Diffusion Problems, Oxford: Oxford University Press, and the references therein). The method is proved to be stable independently of the viscosity. Further, the stability is shown to depend only weakly on the spectral order. We show how sufficiently accurate, approximate upwinded test functions can be computed on each element by a local least-squares FEM. Under the assumption of analyticity of the input data, we prove robust exponential convergence of the method. Numerical experiments confirm our convergence estimates and show robust exponential convergence of the hp FEM even for viscosities of the order of the machine precision, i.e., for the limiting transport problem.