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NASA NTRS1985

Altimetry data and the elastic stress tensor of subduction zones

The stress field in the lithosphere caused by the distribution of density anomalies associated to the geoidal undulations observed by the GEOS-3 and SEASAT Earth satellites in the Tonga region was studied. Different models of the lithosphere were generated with different assumptions on the density distribution and geometry, all generating a geoid profile almost identical to the observed one. The first model is the Airy isostatic hypothesis which consists of a crust of density 2.85 laying on a lithosphere of density 3.35. The models obtained with different compensation depths give residual shortwavelength anomalies of the order of several tens of mgal and several tens of meters geoidal undulations. It indicates that there is no isostasy of the Airy type in the Tonga region because the observed geoid has very smooth undulation of about 25 m over a distance of 2000 km. The Pratt isostatic hypothesis is used in a model consisting of a crust of variable density laying on a lithosphere of higher density. This model gives smaller residual anomalies but still shows that there is no isostasy of the Pratt type in the Tonga region because the observed geoidal undulation are much smaller and smoother than the residual undulations associated to the Pratt model of isostasy.

Caputo, M.↗

NASA NTRS1989

Geoid height versus topography for oceanic plateaus and swells

Gridded geoid height data (Marsh et al.l, 1986) and gridded bathymetry data (Van Wykhouse, 1973) are used to estimate the average compensation depths of 53 oceanic swells and plateaus. The relationship between geoid height and topography is examined using Airy and thermal compensation models. It is shown that geoid height is linearly related to topography between wavelengths of 400 and 4000 m as predicted by isostatic compensation models. The geoid/topography ratio is dependent on the average depth of compensation. The intermediate geoid/topography ratios of most thermal swells are interpreted as a linear combination of the decaying thermal swell signature and that of the persisting Airy-compensated volcanic edifice.

Sandwell, David T.↗

NASA NTRS1991

Computer program for Bessel and Hankel functions

A set of FORTRAN subroutines for calculating Bessel and Hankel functions is presented. The routines calculate Bessel and Hankel functions of the first and second kinds, as well as their derivatives, for wide ranges of integer order and real or complex argument in single or double precision. Depending on the order and argument, one of three evaluation methods is used: the power series definition, an Airy function expansion, or an asymptotic expansion. Routines to calculate Airy functions and their derivatives are also included.

Kreider, Kevin L.↗

NASA NTRS1996

Scattering and the Point Spread Function of the New Generation Space Telescope

Preliminary design work on the New Generation Space Telescope (NGST) is currently under way. This telescope is envisioned as a lightweight, deployable Cassegrain reflector with an aperture of 8 meters, and an effective focal length of 80 meters. It is to be folded into a small-diameter package for launch by an Atlas booster, and unfolded in orbit. The primary is to consist of an octagon with a hole at the center, and with eight segments arranged in a flower petal configuration about the octagon. The comers of the petal-shaped segments are to be trimmed so that the package will fit atop the Atlas booster. This mirror, along with its secondary will focus the light from a point source into an image which is spread from a point by diffraction effects, figure errors, and scattering of light from the surface. The distribution of light in the image of a point source is called a point spread function (PSF). The obstruction of the incident light by the secondary mirror and its support structure, the trimmed corners of the petals, and the grooves between the segments all cause the diffraction pattern characterizing an ideal point spread function to be changed, with the trimmed comers causing the rings of the Airy pattern to become broken up, and the linear grooves causing diffraction spikes running radially away from the central spot, or Airy disk. Any figure errors the mirror segments may have, or any errors in aligning the petals with the central octagon will also spread the light out from the ideal point spread function. A point spread function for a mirror the size of the NGST and having an incident wavelength of 900 nm is considered. Most of the light is confined in a circle with a diameter of 0.05 arc seconds. The ring pattern ranges in intensity from 10(exp -2) near the center to 10(exp -6) near the edge of the plotted field, and can be clearly discerned in a log plot of the intensity. The total fraction of the light scattered from this point spread function is called the total integrated scattering (TIS), and the fraction remaining is called the Strehl ratio. The angular distribution of the scattered light is called the angle resolved scattering (ARS), and it shows a strong spike centered on a scattering angle of zero, and a broad , less intense distribution at larger angles. It is this scattered light, and its effect on the point spread function which is the focus of this study.

Schreur, Julian J.↗

NASA NTRS1999

A Post-Pathfinder Evaluation of Areocentric Solar Coordinates with Improved Timing Recipes for Mars Seasonal/Diurnal Climate Studies

The accurate determination of the Mars pole vector derived from Pathfinder and Viking Lander radio data, together with the VSOP87 representation of planetary orbits, have been applied to a new evaluation of the right ascension of the "fictitious mean sun" (FMS) at Mars. With DELTA t (sub J2000) the elapsed time in days from the J2000 epoch (J.D.2451545.0 (sup TT), alpha FMS = 270 degrees.3863 + 0.52403840(degrees/d) (raised dot) DELTA T (sub j2000) - 4 x 10 (exp -13) (degrees/d (sup 2)) (raised dot) DELTA t (sup 2) (sub J2000) represents a best least-squares quadratic fit of the FMS, including aberration, to each instance of the four equinox and solstice passages for each of 134 Mars orbits spanning the calendar years 1874-2127. The implied tropical orbit period for Mars, 686.9726 (sup d), closely agrees with the recent evaluations. Together with the Pathfinder radio determination of the Mars sidereal rotation, the derived FMS rate corresponds to a mean solar day (or "sol") of 1.027491251 (sup d). The new FMS determination would serve to define the Mean Solar Time at Mars to the nearest tenth-second, according to historical conventions originally established for terrestrial time keeping, once the Mars prime meridian defined by the crater Airy-O is navigated to the same accuracy. For convenient reference to current epochs, 2000 Jan 06 00:00 UTC (= MJD 51549.000 (sup UTC)) corresponds to a coincidence of (alpha (sub FMS)) and the rotation angle of the crater Airy-O measured with respect to the Mars equinox (i.e. "mean solar midnight" on the planet's prime meridian), to within the current uncertainty of several seconds in the locational definition of the planet's cartographic grid. As a further result of the analysis, the consistently derived Mars obliquity of date is epsilon = 25 degrees.192 + 3.45 x l0 (exp -7)(degrees/d)(raised dot) DELTA t (sub J2000). An improved analytic recipe for the calculation of the solar areocentric longitude (L (sub s)) of Mars to an accuracy of 0 degrees.01 is also provided, accounting for the primary perturbations of Earth, Jupiter, and Venus, which may in turn be applied to an efficient evaluation of Mars local true solar time (LTST) to within the uncertainty of the inertial position of the Mars prime meridian. For specific applications to the data archives for landed Mars spacecraft, simple conversion formulae are given for the determination of the Viking "Local Lander Time" and the Pathfinder "Local True Solar Time" in terms of the terrestrial calendar date and UTC.

Allison, Michael↗

NASA NTRS2003

Accurate Thermal Stresses for Beams: Normal Stress

Formulations for a general theory of thermoelasticity to generate accurate thermal stresses for structural members of aeronautical vehicles were developed in 1954 by Boley. The formulation also provides three normal stresses and a shear stress along the entire length of the beam. The Poisson effect of the lateral and transverse normal stresses on a thermally loaded beam is taken into account in this theory by employing an Airy stress function. The Airy stress function enables the reduction of the three-dimensional thermal stress problem to a two-dimensional one. Numerical results from the general theory of thermoelasticity are compared to those obtained from strength of materials. It is concluded that the theory of thermoelasticity for prismatic beams proposed in this paper can be used instead of strength of materials when precise stress results are desired.

Johnson, Theodore F.↗

NASA NTRS2002

Accurate Thermal Stresses for Beams: Normal Stress

Johnson, Theodore F.↗

DOE OSTI2024

Subsurface Spectroscopy in Heterogeneous Materials Using Self-Healing Laser Beams

Self-healing optical beams are a class of propagation modes that can recover their beam shapes after distortion or partial blockage. This self-healing property makes them attractive for use in applications involving turbid media as they can—in theory—penetrate further into these materials than standard Gaussian beams. In this paper, we characterize the propagation of two different self-healing beams (Bessel and Airy) through a solid scattering material with different scatterer concentrations and find that both beams do recover after scattering for samples below a threshold scatterer concentration. Additionally, we test the applicability of both beam shapes for improved sub-surface spectroscopy in heterogeneous materials using fluorescent particles and find that there is an average fluorescence intensity enhancement of 1.3× using self-healing beams versus a standard Gaussian beam.

47 OTHER INSTRUMENTATION↗

NASA NTRS1967

The foundations of beam theory

Airys stress function analysis of supported beam bending under load

STRUCTURAL BEAM↗

DOE OSTI2021

Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on S 1

We investigate the exact-WKB analysis for quantum mechanics in a periodic potential, with N minima on S 1 . We describe the Stokes graphs of a general potential problem as a network of Airy-type or degenerate Weber-type building blocks, and provide a dictionary between the two. The two formulations are equivalent, but with their own pros and cons. Exact-WKB produces the quantization condition consistent with the known conjectures and mixed anomaly. The quantization condition for the case of N-minima on the circle factorizes over the Hilbert sub-spaces labeled by discrete theta angle (or Bloch momenta), and is consistent with ’t Hooft anomaly for even N and global inconsistency for odd N. By using Delabaere-Dillinger-Pham formula, we prove that the resurgent structure is closed in these Hilbert subspaces, built on discrete theta vacua, and by a transformation, this implies that fixed topological sectors (columns of resurgence triangle) are also closed under resurgence.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2020

The large- N partition function for non-parity-invariant Chern-Simons-matter theories

We extend the Fermi gas approach to a class of ABJM-like necklace quiver theories without parity invariance. The resulting partition function on S 3 retains the form of an Airy function, but now includes a phase that scales as Nk in the large- N limit where k is an overall Chern-Simons level. We demonstrate the presence of this phase both analytically and numerically in the case of a three node quiver.

Supersymmetric Gauge Theory↗

DOE OSTI2021

Dynamics of disordered mechanical systems with large connectivity, free probability theory, and quasi-Hermitian random matrices

Disordered mechanical systems with high connectivity represent a limit opposite to the more familiar case of disordered crystals. Individual ions in a crystal are subjected essentially to nearest-neighbor interactions. In contrast, the systems studied in this paper have all their degrees of freedom coupled to each other. Thus, the problem of linearized small oscillations of such systems involves two full positive-definite and non-commuting matrices, as opposed to the sparse matrices associated with disordered crystals. Consequently, the familiar methods for determining the averaged vibrational spectra of disordered crystals, introduced many years ago by Dyson and Schmidt, are inapplicable for highly connected disordered systems. In this paper we apply random matrix theory (RMT) to calculate the averaged vibrational spectra of such systems, in the limit of infinitely large system size. At the heart of our analysis lies a calculation of the average spectrum of the product of two positive definite random matrices by means of free probability theory techniques. We also show that this problem is intimately related with quasi-hermitian random matrix theory (QHRMT), which means that the ‘hamiltonian’ matrix is hermitian with respect to a non-trivial metric. This extends ordinary hermitian matrices, for which the metric is simply the unit matrix. The analytical results we obtain for the spectrum agree well with our numerical results. The latter also exhibit oscillations at the high-frequency band edge, which fit well the Airy kernel pattern. We also compute inverse participation ratios of the corresponding amplitude eigenvectors and demonstrate that they are all extended, in contrast with conventional disordered crystals. Finally, we compute the thermodynamic properties of the system from its spectrum of vibrations. In addition to matrix model analysis, we also study the vibrational spectra of various multi-segmented disordered pendula, as concrete realizations of highly connected mechanical systems. A universal feature of the density of vibration modes, common to both pendula and the matrix model, is that it tends to a non-zero constant at vanishing frequency.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2021

The response field and the saddle points of quantum mechanical path integrals

Highlights: • Moyal quantum mechanics and Marinov’s path integral. • Classical and semiclassical limits of Marinov’s path integral. • Oscillating functional integrals. • Instantons of the Marinov’s path integral. In quantum statistical mechanics, Moyal’s equation governs the time evolution of Wigner functions and of more general Weyl symbols that represent the density matrix of arbitrary mixed states. A formal solution to Moyal’s equation is given by Marinov’s path integral. In this paper we demonstrate that this path integral can be regarded as the natural link between several conceptual, geometric, and dynamical issues in quantum mechanics. A unifying perspective is achieved by highlighting the pivotal role which the response field, one of the integration variables in Marinov’s integral, plays for pure states even. The discussion focuses on how the integral’s semiclassical approximation relates to its strictly classical limit; unlike for Feynman type path integrals, the latter is well defined in the Marinov case. The topics covered include a random force representation of Marinov’s integral based upon the concept of “Airy averaging”, a related discussion of positivity-violating Wigner functions describing tunneling processes, and the role of the response field in maintaining quantum coherence and enabling interference phenomena. The double slit experiment for electrons and the Bohm–Aharonov effect are analyzed as illustrative examples. Furthermore, a surprising relationship between the instantons of the Marinov path integral over an analytically continued (“Wick rotated”) response field, and the complex instantons of Feynman-type integrals is found. The latter play a prominent role in recent work towards a Picard–Lefschetz theory applicable to oscillatory path integrals and the resurgence program.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2024

Stress-hybrid virtual element method on six-noded triangular meshes for compressible and nearly-incompressible linear elasticity

In this paper, we present a first-order Stress-Hybrid Virtual Element Method (SH-VEM) on six-noded triangular meshes for linear plane elasticity. Here, we adopt the Hellinger–Reissner variational principle to construct a weak equilibrium condition and a stress based projection operator. In each element, the stress projection operator is expressed in terms of the nodal displacements, which leads to a displacement based formulation. This stress-hybrid approach assumes a globally continuous displacement field while the stress field is discontinuous across each element. The stress field is initially represented by divergence-free tensor polynomials based on Airy stress functions, but we also present a formulation that uses a penalty term to enforce the element equilibrium conditions, referred to as the Penalty Stress-Hybrid Virtual Element Method (PSH-VEM). Numerical results are presented for PSH-VEM and SH-VEM, and we compare their convergence to the composite triangle FEM and B-bar VEM on benchmark problems in linear elasticity. The SH-VEM converges optimally in the L 2 norm of the displacement, energy seminorm, and the L 2 norm of hydrostatic stress. Furthermore, the results reveal that PSH-VEM converges in most cases at a faster rate than the expected optimal rate, but it requires the selection of a suitably chosen penalty parameter.

42 ENGINEERING↗

DOE OSTI2025

Modular-Invariant Random Matrix Theory and AdS 3 Wormholes

We develop a nonperturbative definition of RMT 2 : a generalization of random matrix theory that is compatible with the symmetries of two-dimensional conformal field theory. Given any random matrix ensemble, its 𝑛-point spectral correlations admit a prescribed modular-invariant lift to RMT 2 , which moreover reduce to the original random matrix correlators in a near-extremal limit. Central to the prescription is a presentation of random matrix theory in Mellin space, which lifts to two dimensions via the SL⁡(2,ℤ) spectral decomposition employed in previous work. As a demonstration we perform the explicit RMT 2 lift of two-point correlations of the GUE Airy model. We propose that in AdS 3 pure gravity, semiclassical amplitudes for off-shell 𝑛-boundary torus wormholes with topology Σ 0,𝑛 × 𝑆 1 are given by the RMT 2 lift of JT gravity wormhole amplitudes. For the three-boundary case, we identify a gravity calculation which matches the RMT 2 result.

conformal field theory↗

DOE OSTI2021

Steepest-descent algorithm for simulating plasma-wave caustics via metaplectic geometrical optics

The design and optimization of radiofrequency-wave systems for fusion applications is often performed using ray-tracing codes, which rely on the geometrical-optics (GO) approximation. However, GO fails at wave cutoffs and caustics. To accurately model the wave behavior in these regions, more advanced and computationally expensive “full-wave” simulations are typically used, but this is not strictly necessary. A new generalized formulation called metaplectic geometrical optics (MGO) has been proposed that reinstates GO near caustics. The MGO framework yields an integral representation of the wavefield that must be evaluated numerically in general. We present an algorithm for computing these integrals using Gauss-Freud quadrature along the steepest-descent contours. Benchmarking is performed on the standard Airy problem, for which the exact solution is known analytically. Furthermore, the numerical MGO solution provided by the new algorithm agrees remarkably well with the exact solution and significantly improves on previously derived analytical approximations of the MGO integral.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2021

WKB Estimate of Bilayer Graphene’s Magic Twist Angles

We report that graphene bilayers exhibit zero-energy flatbands at a discrete series of magic twist angles. In the absence of intrasublattice interlayer hopping, zero-energy states satisfy a Dirac equation with a non-Abelian SU(2) gauge potential that cannot be diagonalized globally. We develop a semiclassical WKB approximation scheme for this Dirac equation by introducing a dimensionless Planck’s constant proportional to the twist angle, solving the linearized Dirac equation around AB and BA turning points, and connecting Airy function solutions via bulk WKB wave functions. We find zero-energy solutions at a discrete set of values of the dimensionless Planck’s constant, which we obtain analytically. Our analytic flatband twist angles correspond closely to those determined numerically in previous work.

36 MATERIALS SCIENCE↗

DOE OSTI2026

The Scaling and Units of the Elastic Response Term for Rayleigh Waves that is Output by Computer Programs in Seismology (CPS)

We report on the scaling and units of the Rayleigh wave elastic response function A R (ω) that is output from the widely used Computer Programs in Seismology (CPS) to aid in the modeling of ground motion sourced by atmospheric explosions. The program uses mixed units (km, second, km/s, gm/cc) to keep A R (ω) near 10 0 and prevent any numerical underflow or overflow. We compare two models for the response of an elastic half space to the output from CPS. Our application inputs the recommended, mixed unit geological models to determine how researchers must scale this output to obtain physical units for A R (ω) that represents the amplitude scaling for the minimum group velocity (Airy phase) contribution to Rayleigh waves. We determine that a CPS user must scale the output for A R (ω) by 10 -12 (m/km) 2 (g/cc/m 3 /kg) to obtain MKS (meter, kg, second) units and then must multiply this result by the vertical component eigenfunction squared, that is evaluated at the free surface.

45 MILITARY TECHNOLOGY, WEAPONRY, AND NATIONAL DEF↗