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NASA NTRS2017

Incorporation of Failure Into an Orthotropic Three-Dimensional Model with Tabulated Input Suitable for Use in Composite Impact Problems

The need for accurate material models to simulate the deformation, damage and failure of polymer matrix composites under impact conditions is becoming critical as these materials are gaining increased use in the aerospace and automotive communities. The aerospace community has identified several key capabilities which are currently lacking in the available material models in commercial transient dynamic finite element codes. To attempt to improve the predictive capability of composite impact simulations, a next generation material model is being developed for incorporation within the commercial transient dynamic finite element code LS-DYNA. The material model, which incorporates plasticity, damage and failure, utilizes experimentally based tabulated input to define the evolution of plasticity and damage and the initiation of failure as opposed to specifying discrete input parameters such as modulus and strength. The plasticity portion of the orthotropic, three-dimensional, macroscopic composite constitutive model is based on an extension of the Tsai-Wu composite failure model into a generalized yield function with a non-associative flow rule. For the damage model, a strain equivalent formulation is used to allow for the uncoupling of the deformation and damage analyses. In the damage model, a semi-coupled approach is employed where the overall damage in a particular coordinate direction is assumed to be a multiplicative combination of the damage in that direction resulting from the applied loads in various coordinate directions. For the failure model, a tabulated approach is utilized in which a stress or strain based invariant is defined as a function of the location of the current stress state in stress space to define the initiation of failure. Failure surfaces can be defined with any arbitrary shape, unlike traditional failure models where the mathematical functions used to define the failure surface impose a specific shape on the failure surface. In the current paper, the complete development of the failure model is described and the generation of a tabulated failure surface for a representative composite material is discussed.

Impact↗

NASA NTRS2017

Implementation of a Tabulated Failure Model Into a Generalized Composite Material Model Suitable for Use in Impact Problems

The need for accurate material models to simulate the deformation, damage and failure of polymer matrix composites under impact conditions is becoming critical as these materials are gaining increased use in the aerospace and automotive communities. The aerospace community has identified several key capabilities which are currently lacking in the available material models in commercial transient dynamic finite element codes. To attempt to improve the predictive capability of composite impact simulations, a next generation material model is being developed for incorporation within the commercial transient dynamic finite element code LS-DYNA. The material model, which incorporates plasticity, damage and failure, utilizes experimentally based tabulated input to define the evolution of plasticity and damage and the initiation of failure as opposed to specifying discrete input parameters such as modulus and strength. The plasticity portion of the orthotropic, three-dimensional, macroscopic composite constitutive model is based on an extension of the Tsai-Wu composite failure model into a generalized yield function with a non-associative flow rule. For the damage model, a strain equivalent formulation is used to allow for the uncoupling of the deformation and damage analyses. For the failure model, a tabulated approach is utilized in which a stress or strain based invariant is defined as a function of the location of the current stress state in stress space to define the initiation of failure. Failure surfaces can be defined with any arbitrary shape, unlike traditional failure models where the mathematical functions used to define the failure surface impose a specific shape on the failure surface. In the current paper, the complete development of the failure model is described and the generation of a tabulated failure surface for a representative composite material is discussed.

polymer matrix composites↗

NASA NTRS2019

Verification and Validation of a Three-Dimensional Composite Impact Model with Tabulated Input

A material model which incorporates several key capabilities which have been identified by the aerospace community as lacking in the composite impact models currently available in the commercial transient dynamic finite element code LS-DYNA® has been developed. The material model utilizes experimentally based tabulated input to define the evolution of plasticity and damage as opposed to specifying discrete input parameters (such as modulus and strength. The plasticity portion of the orthotropic, three-dimensional, macroscopic composite constitutive model is based on an extension of the Tsai-Wu composite failure model into a generalized yield function with a non-associative flow rule. The capability to account for the rate and temperature dependent deformation response of composites has also been incorporated into the material model. For the damage model, a strain equivalent formulation is utilized to allow for the uncoupling of the deformation and damage analyses. In the damage model, a diagonal damage tensor is defined to account for the directionally dependent variation of damage. However, the terms in the damage matrix are semi-coupled such that the damage in a particular coordinate direction is a function of the stresses and plastic strains in all of the coordinate directions. For the failure model, a tabulated approach is utilized in which a stress or strain based invariant is defined as a function of the location of the current stress state in stress space to define the initiation of failure, which allows an arbitrarily shaped failure surface to be defined. A systematic series of validation and verification studies, at a variety of length scales ranging from single element simulations to simulations of a flat panel impact test, have been performed to fully exercise and evaluate the capabilities of the developed model.

Impact↗

NASA NTRS2021

Ultraviolet line profiles of slowly rotating massive star winds using the ‘analytic dynamical magnetosphere’ formalism

Recent large-scale spectropolarimetric surveys have established that a small but significant percentage of massive stars host stable, surface dipolar magnetic fields with strengths on the order of kG. These fields channel the dense, radiatively driven stellar wind into circumstellar magnetospheres, whose density and velocity structure can be probed using ultraviolet (UV) spectroscopy of wind-sensitive resonance lines. Coupled with appropriate magnetosphere models, UV spectroscopy provides a valuable way to investigate the wind–field interaction, and can yield quantitative estimates of the wind parameters of magnetic massive stars. We report a systematic investigation of the formation of UV resonance lines in slowly rotating magnetic massive stars with dynamical magnetospheres. We pair the analytic dynamical magnetosphere (ADM) formalism with a simplified radiative transfer technique to produce synthetic UV line profiles. Using a grid of models, we examine the effect of magnetosphere size, the line strength parameter, and the cooling parameter on the structure and modulation of the line profile. We find that magnetic massive stars uniquely exhibit redshifted absorption at most viewing angles and magnetosphere sizes, and that significant changes to the shape and variation of the line profile with varying line strengths can be explained by examining the individual wind components described in the ADM formalism. Finally, we show that the cooling parameter has a negligible effect on the line profiles.

C Erba↗

NASA NTRS2001

Probabilistic Structural Analysis and Reliability Using NESSUS With Implemented Material Strength Degradation Model

This project included both research and education objectives. The goal of this project was to advance innovative research and education objectives in theoretical and computational probabilistic structural analysis, reliability, and life prediction for improved reliability and safety of structural components of aerospace and aircraft propulsion systems. Research and education partners included Glenn Research Center (GRC) and Southwest Research Institute (SwRI) along with the University of Texas at San Antonio (UTSA). SwRI enhanced the NESSUS (Numerical Evaluation of Stochastic Structures Under Stress) code and provided consulting support for NESSUS-related activities at UTSA. NASA funding supported three undergraduate students, two graduate students, a summer course instructor and the Principal Investigator. Matching funds from UTSA provided for the purchase of additional equipment for the enhancement of the Advanced Interactive Computational SGI Lab established during the first year of this Partnership Award to conduct the probabilistic finite element summer courses. The research portion of this report presents the cumulation of work performed through the use of the probabilistic finite element program, NESSUS, Numerical Evaluation and Structures Under Stress, and an embedded Material Strength Degradation (MSD) model. Probabilistic structural analysis provided for quantification of uncertainties associated with the design, thus enabling increased system performance and reliability. The structure examined was a Space Shuttle Main Engine (SSME) fuel turbopump blade. The blade material analyzed was Inconel 718, since the MSD model was previously calibrated for this material. Reliability analysis encompassing the effects of high temperature and high cycle fatigue, yielded a reliability value of 0.99978 using a fully correlated random field for the blade thickness. The reliability did not change significantly for a change in distribution type except for a change in distribution from Gaussian to Weibull for the centrifugal load. The sensitivity factors determined to be most dominant were the centrifugal loading and the initial strength of the material. These two sensitivity factors were influenced most by a change in distribution type from Gaussian to Weibull. The education portion of this report describes short-term and long-term educational objectives. Such objectives serve to integrate research and education components of this project resulting in opportunities for ethnic minority students, principally Hispanic. The primary vehicle to facilitate such integration was the teaching of two probabilistic finite element method courses to undergraduate engineering students in the summers of 1998 and 1999.

Bast, Callie C.↗

NASA NTRS1988

Radar investigation of asteroids and planetary satellites

The aim is to make radar reconnaissance of near-Earth asteroids, mainbelt ateroids, the Galilean satellites, the Martian satellites, and the largest Saturnian satellites, using the Arecibo 13-cm and the Goldstone 3.5-cm systems. Measurements of echo strength, polarization, and delay/Doppler distribution of echo power provide information about dimensions, spin vector, large-scale topography, cm-to-m-scale morphology, and surface bulk density. The observations also yield refined estimates of target orbital elements. Radar signatures were measured for 31 mainbelt asteroids and 16 near-Earth asteroids since this task began eight years ago. The dispersion in asteroid radar albedoes and circular polarization ratios is extreme, revealing huge differences in surface morphologies, bulk densities, and metal concentration. For the most part, correction between radar signature and VIS/IR class is not high. Many near-Earth asteroids have extremely irregular, nonconvex shapes, but some have polar silhouettes that appear only slightly noncircular. The signatures of 1627 Ivar, 1986 DA, and the approximately 180-km mainbelt asteroid 216 Kleopatra suggest bifurcated shapes. Observational milestones during 1987 and 1988 are noted.

Ostro, Steven J.↗

NASA NTRS1991

Rubber-toughened cyanate composites - Properties and toughening mechanism

Earlier work by Young et al. (1990) has shown that Dow experimental cyanate ester resin XU71787.02 is readily toughenable by rubber. A particularly effective rubber for this purpose is an experimental core-shell rubber which toughens the polymer by inducing shear yielding. This paper describes an investigation into the toughening mechanism in the corresponding carbon-fiber composite systems and the effect of fibers on composite properties. Resin-fiber interfacial shear strengths have been successfully correlated to the compressive strengths after impact and other key properties of composites based on rubber-toughened cyanate and several carbon fibers. The differences in the properties are explained by the difference in the functioning of the rubber particles during the fracture process.

Yang, P. C.↗

NASA NTRS1993

Bone geometry, structure and mineral distribution using Dual energy X ray Absorptiometry (DXA)

Dual energy x-ray absorptiometry (DXA) is currently the most widely used method of analyzing regional and whole body changes in bone mineral content (BMC) and areal (g/sq cm) bone mineral density (BMD). However, BMC and BMD do not provide direct measures of long bone geometry, structure, or strength nor do regional measurements detect localized changes in other regions of the same bone. The capabilities of DXA can be enhanced significantly by special processing of pixel BMC data which yields cross-sectional geometric and structural information. We have extended this method of analysis in order to develop non-uniform structural beam models of long bones.

Whalen, Robert↗

NASA NTRS1984

Response of the annual and zonal mean winds and temperatures to variations in the heat and momentum sources

Comparisons are made among solutions to zonal-mean equations obtained with parameterized friction and radiative cooling and those forced from specified heat and momentum sources. Budget equations are defined for zonally averaged steady state responses in a thin spherically rotating atmosphere. The heat sources and sinks and mean meridional circulation that maintain observed annual and zonal mean temperatures are identified with a diagnostic calculation. Estimates are made of the surface sensible heating, atmospheric latent heating and vertical flux eddy divergences. The heat and moisture sources and sinks are varied to obtain the steady state responses. The Hadley circulation is fairly insensitive to changes in the strength of the eddy momentum flux when sufficient internal friction is present. Varying the width of the total precipitation of the intertropical convergence zone with fixed eddy fluxes and extratropical heat sources yields conditions similar to El Nino. Finally, a minimum speed is found for the jet stream after varying the horizontal eddy momentum fixing latent and eddy heat sources.

Schneider, E. K.↗

NASA NTRS1987

Impact experiments in low-temperature ice

Cubic and cylindrical water ice targets at 257 and 81 K have been subjected to impact velocities between 0.1 and 0.64 km/sec and impact energies of 10 to the 9th-10 to the 10th ergs, yielding craters that are 2-3 times larger than those obtained through equal energy impacts in basaltic targets. On the basis of a similarity analysis, general scaling laws are derived for strength-controlled crater formation and applied in a consideration of crater formation on the icy Galilean and Saturnian satellites. Surface ages are predicted by the analysis to appear greater than those for a silicate crust experiencing the same impact history, on the basis of icy crust impact crater statistics.

Lange, Manfred A.↗

NASA NTRS1995

Far-ultravoilet and optical spectrscopy of the detached white dwarf-M binary RE J1629+780

We present optical and far-ultraviolet observations, including IUE high- and low-dispersion spectroscopy, of the (DA2+dM4-5) close binary, RE J1629+780, a very bright ROSAT EUV source. The optical data reveal emission reversals in the cores of the Balmer lines, which vary in strength but reveal no evidence of wavelength shifts reflecting orbital motion, suggesting that the red dwarf and the white dwarf are widely separated. Thus it is unlikely that irradiation plays a dominant role in the activity of the red dwarf. One dramatic increase in the intensity of the H-beta emission strength in the course of one hour indicates that the observed intensity variations are most likely related to flare activity on the red dwarf. Our analysis of Balmer and Lyman alpha features yield T(sub eff) = 42,500 +/- 1300 K and log g = 7.6 +/- 0.3 for the DA2 white dwarf, which is consistent with the overall 900 to 3200 A energy distribution derived from Voyager and IUE low-dispersion spectra. Our analysis of three IUE echelle spectra reveal numerous lines of interstellar origin. Using a total of 13 interstellar matter (ISM) features we find a mean radial velocity of -13.57 +/- 1.66 km/s for the interstellar features. However, we found no evidence of trace metal absorbers levitated or accreted in the white-dwarf photosphere nor any evidence of circumstellar or wind features. We estimate a lower limit hydrogen layer mass of 2.1 x 10(exp -15) solar mass for a stratified atmosphere based upon the nondetection of He II (1640). Tighter constraints on the helium abundance and the hydrogen layer mass, from EUV and X-ray data, are presented for both homogeneous and stratified atmospheric configurations.

Sion, Edward M.↗

NASA NTRS1998

Evaluation of 3D Inverse Code Using Rotor 67 as Test Case

A design modification of Rotor 67 is carried out with a full 3D inverse method. The blade camber surface is modified to produce a prescribed pressure loading distribution, with the blade tangential thickness distribution and the blade stacking line at midchord kept the same as the original Rotor 67 design. Because of the inviscid-flow assumption used in the current version of the method, Rotor 67 geometry is modified for use at a design point different from the original design value. A parametric study with the prescribed pressure loading distribution yields the following results. In the subsonic section, smooth pressure loading shapes generally produce blades with well-behaved blade surface pressure distributions. In the supersonic section, the study shows that the strength and position of the passage shock correlate with the characteristics of the blade pressure loading shape. In general, "smooth" prescribed blade pressure loading distributions generate blade designs with reverse cambers which have the effect of weakening the passage shock.

Dang, T.↗

NASA NTRSDate not supplied

Development of Laser Powder Bed Fusion NASA HR-2 for Hydrogen Sensitive Liquid Rocket Engine Applications

The National Aeronautics and Space Administration (NASA) has been involved in the development and maturation of metal additive manufacturing (AM) for space applications since the late 2000s. AM has provided new design and manufacturing opportunities to reduce cost and schedule, consolidate parts, and optimize performance. Laser Powder Bed Fusion (L-PBF) is one of the most commonly used AM processes to fabricate components of complex shape requiring fine feature resolution. Due to exposure to high-pressure gaseous hydrogen, mechanical property degradation caused by hydrogen environment embrittlement (HEE) is a critical concern for many materials in liquid hydrogen propulsion systems. NASA has identified the need to develop and advance new materials in unique engine applications using liquid hydrogen as a propellant. One such material being developed at NASA Marshall Space Flight Center is L-PBF NASA HR-2 (Hydrogen Resistant-2), a high-strength Fe-Ni-based superalloy resistant to HEE. The chemistry of NASA HR-2 was formulated to meet requirements for key liquid rocket engine (LRE) components that operate in high-pressure hydrogen environments. Initial development and material characterization found that NASA HR-2 has excellent L-PBF printability, and its microstructure evolves well after heat treatment. This new alloy has undergone fundamental metallurgical evaluations, heat treatment studies, detailed microstructure characterization, and mechanical testing across a range of temperatures. Tensile testing was performed in a pressurized gaseous hydrogen (GH2) environment to assess its resistance to HEE. L-PBF NASA HR-2 has an average yield stress of 95 ksi, an ultimate tensile stress of 165 ksi, and a very high fracture elongation of 34 - 36% when tested in a high-pressure (5 ksi) hydrogen environment. The tensile property data confirms hydrogen has little influence on HR-2’s ductility, strength, and fracture behavior. L-PBF NASA HR-2 is a promising option for many hydrogen-sensitive LRE components that require exceptional resistance to HEE. This paper will provide an overview of the L-PBF process development, material characterization, mechanical and thermophysical properties, and LRE hardware development for NASA HR-2.

Po S Chen↗

NASA NTRS1997

Environmental modification of yield and nutrient composition of 'Waldmann's Green' leaf lettuce

Leaf number, dry weight, and nutrient composition of Lactuca sativa L. cv. Waldmann's Green leaves were compared following 9 days of treatment in a controlled environment room under various combinations of photosynthetic photon flux (PPF:350 vs 800 micromoles m-2 s-1), atmospheric CO2 level (ambient vs 1500 micromoles mol-1), and single-strength (1X:15 mM) vs double-strength (2X:30 mM) nitrogen (N) as NO3- alone or as NH4(+) + NO3- (1:5 molar ratio). CO2 enrichment greatly enhanced leaf number under all PPF and N conditions, but increased leaf dry weight only at high PPF. Conditions favoring high photosynthesis enhanced leaf starch content 3-fold, and protein content increased as much as 64% with 2X NH4(+)+NO3-. Free sugar content was 6 to 9% of leaf dry weight for all treatment combinations, while fat was 1.5 to 3.5%. Ash content varied from 15 to 20% of leaf dry weight. Modified controlled environments can be used to enhance the nutritional content as well as the yield of crops to be used for life support in space-deployed, self-sustaining human habitats. Leaf lettuce is a useful model crop for demonstrating the potential of nutritional value added by environmental manipulation.

Non-NASA Center↗

NASA NTRS1993

Galileo imaging results from the second Earth-Moon flyby: Lunar Maria and related units

The second flyby of the Earth-Moon System by Galileo occurred on December 7, 1992, on its trajectory toward Jupiter. The flyby took the spacecraft over the lunar north polar region from the dark farside and continued across the illuminated nearside. This provided the first opportunity to observe northern and northeastern limb regions with a modern, multispectral imaging system with high spatial resolution (up to 1.1 km/pixel). Scientific objectives included compositional assessment of previously uncharacterized mare regions, study of various light plains materials, and assessment of dark mantle deposits (DMD) and dark halo craters (DHC). Color composite images were prepared from ratios of Galileo SSI filter data (0.76/0.41 yields red; 0.76/0.99 yields green; 0.41/0.76 yields blue) and used for preliminary comparison of units. The 0.41/0.76 ratio has been empirically correlated to Ti content of mare soils (blue is relatively high, red is relatively low). The relative strengths of the ferrous one micron absorption in mafic minerals can be compared using the 0.76/0.99 ratio. In addition, relative ages of units analyzed spectrally were determined from crater statistics using Lunar Orbiter images following the techniques of Neukum et al. Mare deposits analyzed include Mare Humboldtianum, central and eastern Mare Frigoris, Mare Crisium and other deposits in the Crisium Basin, and isolated mare patches on the northeastern lunar limb. Preliminary results show a diversity of 0.41/0.76 micron signatures, implying a wide range of titanium contents. Some light plains units are similar to units found at the Apollo 16 site; others may be ancient mare materials. Dark mantle deposits (DMD) analyzed also are available.

Greeley, R.↗

NASA NTRS2006

Mechanically Strong, Polymer Cross-linked Aerogels (X-Aerogels)

Aerogels comprise a class of low-density, high porous solid objects consisting of dimensionally quasi-stable self-supported three-dimensional assemblies of nanoparticles. Aerogels are pursued because of properties above and beyond those of the individual nanoparticles, including low thermal conductivity, low dielectric constant and high acoustic impedance. Possible applications include thermal and vibration insulation, dielectrics for fast electronics, and hosting of functional guests for a wide variety of optical, chemical and electronic applications. Aerogels, however, are extremely fragile materials, hence they have found only limited application in some very specialized environments, for example as Cerenkov radiation detectors in certain types of nuclear reactors, aboard spacecraft as collectors of hypervelocity particles (refer to NASA's Stardust program) and as thermal insulators on planetary vehicles on Mars (refer to Sojourner Rover in 1997 and Spirit and Opportunity in 2004). Along these lines, the X-Aerogel is a new NASA-developed strong lightweight material that has resolved the fragility problem of traditional (native) aerogels. X-Aerogels are made by applying a conformal polymer coating on the surfaces of the skeletal nanoparticles of native aerogels (see Scanning Electron Micrographs). Since the relative amounts of the polymeric crosslinker and the backbone are comparable, X-Aerogels can be viewed either as aerogels modified by the templated accumulation of polymer on the skeletal nanoparticles, or as nanoporous polymers made by remplated casting of polymer on a nanostructured framework. The most striking feature of X-Aerogels is that for a nominal 3-fold increase in density (still a ultralighweight material), the mechanical strength can be up to 300 times higher than the strength of the underlying native aerogel. Thus, X-Aerogels combine a multiple of the specific compressive strength of steel, with the the thermal conductivity of styrofoam. X-Aerogels have been demonstrated with several polymers such as polyurethanes/polyureas, epoxies and polyolefins, while crosslinking of approximately 35 different oxide aerogels yields a range of dimensionally stable, porous lightweight materials with unique combinations of structural, magnetic and optical properties. The main theme in materials development for space exploration is multifunction. For example, use of one material for thermal insulation/structural component will free weight for useful payload. In that regard, X-aerogels are evaluated at NASA for cryogenic fuel storage tanks and for spacesuits. Along the same lines, major impact fro X-Aerogels is also expected in commercial applications for thermal/acoustic insulation, in catalytic reformers and converters, in filtration membranes and membranes for fuel cells, as platforms for optical, electrical and magnetic sensors, and as lightweight structural component for aircraft and satellites.

Leventis, Nicholas↗

NASA NTRS2011

The Potential of Spaced-based High-Energy Neutrino Measurements via the Airshower Cherenkov Signal

Future space-based experiments, such as (Orbiting Wide-angle Light Collectors (OWL) and JEM-EUSO, view large atmospheric and terrestrial neutrino targets. With energy thresholds slightly above 10(exp 19) eV for observing airshowers via air fluorescence, the potential for observing the cosmogenic neutrino flux associated with the GZK effect is limited. However, the forward Cherenkov signal associated with the airshower can be observed at much lower energies. A simulation was developed to determine the Cherenkov signal strength and spatial extent at low-Earth orbit for upward-moving airshowers. A model of tau neutrino interactions in the Earth was employed to determine the event rate of interactions that yielded a tau lepton which would induce an upward-moving airshower observable by a space-based instrument. The effect of neutrino attenuation by the Earth forces the viewing of the Earth's limb to observe the vT-induced Cherenkov airshower signal at above the OWL Cherenkov energy threshold of approximately 10(exp 16.5) eV for limb-viewed events. Furthermore, the neutrino attenuation limits the effective terrestrial neutrino target area to approximately 3 x 10(exp 5) square km at 10(exp 17) eV, for an orbit of 1000 km and an instrumental full Field-of-View of 45 deg. This translates into an observable cosmogenic neutrino event rate of approx. l/year based upon two different models of the cosmogenic neutrino flux, assuming neutrino oscillations and a 10% duty cycle for observation.

Krizmanic, John F.↗

NASA NTRS2014

NuSTAR Discovery Of A Young, Energetic Pulsar Associated with the Luminous Gamma-Ray Source HESS J1640-465

We report the discovery of a 206 ms pulsar associated with the TeV gamme-ray source HESS J1640−465 using the Nuclear Spectroscopic Telescope Array (NuSTAR) X-ray observatory. PSR J1640−4631 lies within the shelltype supernova remnant (SNR) G338.3−0.0, and coincides with an X-ray point source and putative pulsar wind nebula (PWN) previously identified in XMM-Newton and Chandra images. It is spinning down rapidly with period derivative P = 9.758(44) × 10(exp −13), yielding a spin-down luminosity E = 4.4 × 10(exp 36) erg s(exp −1), characteristic age tau(sub c) if and only if P/2 P = 3350 yr, and surface dipole magnetic field strength B(sub s) = 1.4×10(exp 13) G. For the measured distance of 12 kpc to G338.3−0.0, the 0.2-10 TeV luminosity of HESS J1640−465 is 6% of the pulsar's present E. The Fermi source 1FHL J1640.5−4634 is marginally coincident with PSR J1640−4631, but we find no gamma-ray pulsations in a search using five years of Fermi Large Area Telescope (LAT) data. The pulsar energetics support an evolutionary PWN model for the broadband spectrum of HESS J1640−465, provided that the pulsar's braking index is n approximately equal to 2, and that its initial spin period was P(sub 0) approximately 15 ms.

pulsars: individual↗