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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.

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.