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The purpose for this research was to investigate the Riemann zeta function at odd integer values, because there was no simple representation for these results. The research resulted in the closed form expression for representing the zeta function at the odd integer values 2n+1 for n a positive integer. The above representation shows the zeta function at odd positive integers can be represented in terms of the Euler numbers E2n and the polygamma functions ψ(2n)(3/4). This is a new result for this study area. For completeness, this paper presents a review of selected properties of the Riemann zeta function together with how these properties are derived. This paper will summarize how to evaluate zeta (n) for all integers n different from 1. Also as a result of this research, one can obtain a closed form expression for the Dirichlet beta series evaluated at positive even integers. The results presented enable one to construct closed form expressions for the Dirichlet eta, lambda and beta series evaluated at odd and even integers. Closed form expressions for Apery’s constant zeta (3) and Catalan’s constant beta (2) are also presented.

The protection of astronauts and instrumentation from galactic cosmic rays and solar particle events is one of the primary constraints associated with mission planning in low earth orbit or deep space. To help satisfy this constraint, several computational tools have been developed to analyze the effectiveness of various shielding materials and structures exposed to space radiation. These tools are now being carefully scrutinized through a systematic effort of verification, validation, and uncertainty quantification. In this benchmark study, the deterministic transport code HZETRN is compared to the Monte Carlo transport codes HETC-HEDS and FLUKA for a 30 g/cm 2 water target protected by a 20 g/cm 2 aluminum shield exposed to a parameterization of the February 1956 solar particle event. Neutron and proton fluences as well as dose and dose equivalent are compared at various depths in the water target. The regions of agreement and disagreement between the three codes are quantified and discussed, and recommendations for future work are given.

The HZETRN deterministic radiation code is one of several tools developed to analyze the effects of harmful galactic cosmic rays (GCR) and solar particle events on mission planning and shielding for astronauts and instrumentation. This paper is a comparison study involving the two Monte Carlo transport codes, HETC–HEDS and FLUKA and the deterministic transport code, HZETRN. Each code is used to transport an ion from the 1977 solar minimum GCR spectrum impinging upon a 20 g/cm 2 aluminum slab followed by a 30 g/cm 2 water slab. This research is part of a systematic effort of verification and validation to quantify the accuracy of HZETRN and determine areas where it can be improved. Comparisons of dose and dose equivalent values at various depths in the water slab are presented in this report. This is followed by a comparison of the proton and forward, backward and total neutron flux at various depths in the water slab. Comparisons of the secondary light ion 2 H, 3 H, 3 He and 4 He fluxes are also examined.

Exposure estimates inside space vehicles, surface habitats, and high altitude aircraft exposed to space radiation are highly influenced by secondary neutron production. The deterministic transport code HZETRN has been identified as a reliable and efficient tool for such studies, but improvements to the underlying transport models and numerical methods are still necessary. In this paper, the forward-backward (FB) and directionally coupled forward-backward (DC) neutron transport models are derived, numerical methods for the FB model are reviewed, and a computationally efficient numerical solution is presented for the DC model. Both models are compared to the Monte Carlo codes HETCHEDS and FLUKA, and the DC model is shown to agree closely with the Monte Carlo results. Finally, it is found in the development of either model that the decoupling of low energy neutrons from the light ion (A

It has long been recognized that galactic cosmic rays are of such high energy that they tend to pass through available shielding materials resulting in exposure of astronauts and equipment within space vehicles and habitats. Any protection provided by shielding materials result not so much from stopping such particles but by changing their physical character in interaction with shielding material nuclei forming, hopefully, less dangerous species. Clearly, the fidelity of the nuclear cross-sections is essential to correct specification of shield design and sensitivity to cross-section error is important in guiding experimental validation of cross-section models and database. We examine the Boltzmann transport equation which is used to calculate dose equivalent during solar minimum, with units (cSv/yr), associated with various depths of shielding materials. The dose equivalent is a weighted sum of contributions from neutrons, protons, light ions, medium ions and heavy ions using the ICRP-60 LET dependent quality factors. We investigate the sensitivity of dose equivalent calculations due to errors in nuclear fragmentation cross-sections. We do this error analysis for all possible projectile-fragment combinations (14,365 such combinations) to estimate the sensitivity of the shielding calculations to errors in the nuclear fragmentation cross-sections. Numerical differentiation with respect to the cross-sections will be evaluated in a broad class of materials including polyethylene, aluminum and copper. We will identify the most important cross-sections to ensure adequate experimental study and evaluate their impact on propagated errors in shielding estimates.

Early space radiation shield code development relied on Monte Carlo methods and made important contributions to the space program. Monte Carlo methods have resorted to restricted one-dimensional problems leading to imperfect representation of appropriate boundary conditions. Even so, intensive computational requirements resulted and shield evaluation was made near the end of the design process. Resolving shielding issues usually had a negative impact on the design. Improved spacecraft shield design requires early entry of radiation constraints into the design process to maximize performance and minimize costs. As a result, we have been investigating high-speed computational procedures to allow shield analysis from the preliminary concept to the final design. For the last few decades, we have pursued deterministic solutions of the Boltzmann equation allowing field mapping within the International Space Station (ISS) in tens of minutes using standard Finite Element Method (FEM) geometry common to engineering design methods. A single ray trace in such geometry requires 14 milliseconds and limits application of Monte Carlo methods to such engineering models. A potential means of improving the Monte Carlo efficiency in coupling to spacecraft geometry is given.

A low-energy neutron transport algorithm for use in space-radiation protection is developed. The algorithm is based upon a multiple energy group analysis of the straight ahead Boltzmann equation utilizing a mean value theorem for integrals. The algorithm developed is then verified by using a collocation method solution on the same straight ahead Boltzmann equation. This algorithm was then coupled to the existing NASA Langley HZETRN (high charge and energy transport) code through the evaporation source term. Evaluation of the neutron fluence generated by the February 23, 1956 solar particle event for an aluminum-water shield-target configuration is then compared with the LAHET Monte Carlo calculation for the same shield-target configuration. The algorithm developed showed a great improvement in results over the unmodified HZETRN solution. A bidirectional modification of the evaporation source produced further improvement of the fluence. PACS Nos.; 87.50N, 25.40D, 28.20G

Improved spacecraft shield design requires early entry of radiation constraints into the design process to maximize performance and minimize costs. As a result, we have been investigating high-speed computational procedures to allow shield analysis from the preliminary design concepts to the final design. In particular, we will discuss the progress towards a full three-dimensional and computationally efficient deterministic code for which the current HZETRN evaluates the lowest-order asymptotic term. HZETRN is the first deterministic solution to the Boltzmann equation allowing field mapping within the International Space Station (ISS) in tens of minutes using standard finite element method (FEM) geometry common to engineering design practice enabling development of integrated multidisciplinary design optimization methods. A single ray trace in ISS FEM geometry requires 14 ms and severely limits application of Monte Carlo methods to such engineering models. A potential means of improving the Monte Carlo efficiency in coupling to spacecraft geometry is given in terms of re-configurable computing and could be utilized in the final design as verification of the deterministic method optimized design. Published by Elsevier Ltd on behalf of COSPAR.

High-speed computational procedures for space radiation shielding have relied on asymptotic expansions in terms of the off-axis scatter and replacement of the general geometry problem by a collection of flat plates. This type of solution was derived for application to human rated systems in which the radius of the shielded volume is large compared to the off-axis diffusion limiting leakage at lateral boundaries. Over the decades these computational codes are relatively complete and lateral diffusion effects are now being added. The analysis for developing a practical full 3D space shielding code is presented.

The interaction of high-energy space radiation with spacecraft materials generates a host of secondary particles, some, such as neutrons, are more biologically damaging and penetrating than the original primary particles. Before committing astronauts to long term exposure in such high radiation environments, a quantitative understanding of the exposure and estimates of the associated risks are required. Energetic neutrons are traditionally difficult to measure due to their neutral charge. Measurement methods have been limited by mass and weight requirements in space to nuclear emulsion, activation foils, a limited number of Bonner spheres, and TEPCs. Such measurements have had limited success in quantifying the neutron component relative to the charged components. We will show that a combination of computational models and experimental measurements can be used as a quantitative tool to evaluate the radiation environment within the Shuttle, including neutrons. Comparisons with space measurements are made with special emphasis on neutron sensitive and insensitive devices.

The HZETRN code has been developed over the past decade to evaluate the local radiation fields within sensitive materials on spacecraft in the space environment. Most of the more important nuclear and atomic processes are now modeled and evaluation within a complex spacecraft geometry with differing material components, including transition effects across boundaries of dissimilar materials, are included. The atomic/nuclear database and transport procedures have received limited validation in laboratory testing with high energy ion beams. The codes have been applied in design of the SAGE-III instrument resulting in material changes to control injurious neutron production, in the study of the Space Shuttle single event upsets, and in validation with space measurements (particle telescopes, tissue equivalent proportional counters, CR-39) on Shuttle and Mir. The present paper reviews the code development and presents recent results in laboratory and space flight validation.

The projected radiation levels within the International Space Station (ISS) have been criticized by the Aerospace Safety Advisory Panel in their report to the NASA Administrator. Methods for optimal reconfiguration and augmentation of the ISS shielding are now being developed. The initial steps are to develop reconfigurable and realistic radiation shield models of the ISS modules, develop computational procedures for the highly anisotropic radiation environment, and implement parametric and organizational optimization procedures. The targets of the redesign process are the crew quarters where the astronauts sleep and determining the effects of ISS shadow shielding of an astronaut in a spacesuit. The ISS model as developed will be reconfigurable to follow the ISS. Swapping internal equipment rack assemblies via location mapping tables will be one option for shield optimization. Lightweight shield augmentation materials will be optimally fit to crew quarter areas using parametric optimization procedures to minimize the augmentation shield mass. The optimization process is being integrated into the Intelligence Synthesis Environment''s (ISE''s) immersive simulation facility at the Langley Research Center and will rely on High Performance Computing and Communication (HPCC) for rapid evaluation of shield parameter gradients.

The long term exposure of astronauts on the developing International Space Station (ISS) requires an accurate knowledge of the internal exposure environment for human risk assessment and other onboard processes. The natural environment is moderated by the solar wind, which varies over the solar cycle. The HZETRN high charge and energy transport code developed at NASA Langley Research Center can be used to evaluate the neutron environment on ISS. A time dependent model for the ambient environment in low earth orbit is used. This model includes GCR radiation moderated by the Earth s magnetic field, trapped protons, and a recently completed model of the albedo neutron environment formed through the interaction of galactic cosmic rays with the Earth s atmosphere. Using this code, the neutron environments for space shuttle missions were calculated and comparisons were made to measurements by the Johnson Space Center with onboard detectors. The models discussed herein are being developed to evaluate the natural and induced environment data for the Intelligence Synthesis Environment Project and eventual use in spacecraft optimization.

The normal working and living areas of the astronaut are designed to provide an acceptable level of protection against the hazards of ionizing space radiation. Attempts to reduce the exposures require intervening shield materials to reduce the transmitted radiation. An unwelcome side effect of the shielding is the production of neutrons, which are themselves dangerous particles that can be (but are not always) more hazardous than the particles that produced them. This is especially true depending on the choice of shield materials. Although neutrons are not a normal part of the space environment, this paper focuses on them as principle component of astronaut exposure in the massive spacecraft's required for human space travel and habitation near planetary surfaces or other large bodies of material in space.

The components of the physical forces between noble gas atoms, alkali ions, and halogen ions are analyzed and a data base developed from analysis of the two‐body potential data, the alkali–halide molecular data, and the noble gas crystal and salt crystal data. A satisfactory global fit to this molecular and crystal data is then reproduced by the model to within several percent. Surface potentials are evaluated for noble gas atoms on noble gas surfaces and salt crystal surfaces with surface tension neglected. Within this context, the noble gas surface potentials on noble gas and salt crystals are considered to be accurate to within several percent.

Resonanzphanomene und Energieubertragung, assoziiert mit einem Paar gekoppelter nichtlinearer Differentialgleichungen, werden unter Anwendung asymptotischer Verfahren analysiert. Diese Gleichungen beschreiben die Bewegung des dehnbaren Pendels, das autoparametrische Erregung aufweist. Genaue Bedingungen werden gefunden, unter denen resonante oder nichtresonante Schwingungen entstehen. Diese Bedingungen hangen nicht nur von den physikalischen Parametern des Systems ab, sondern auch von der Energie der Schwingungen (das heisst von den Anfangsbedingungen). Im allgemeinen ist die Umhullung der Resonanzschwingungen periodisch mit langen Perioden, und es besteht periodische Energieubertragung zwischen den zwei Freiheitsgraden.

Der Warmeubergang bei laminarer Rohrstromung mit konstanter Wandtemperatur wurde analytisch untersucht. Das Resultat ist eine Darstellung der Nusseltschen Zahl als Funktion von der Pecletschen Zahl.

A correction to the usual Madelung form for the electrostatic energy of cubic ionic crystals due to the finite size of real atomic ions is derived.