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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
NASA NTRS1990

Radio-source structure in astrometric and geodetic very long baseline interferometry

An algorithm to calculate source structure corrections for bandwidth synthesis delays and delay rates is used to refer the position of extragalacitc radio sources to a specific feature of each source's morphology. The delay and delay rate are obtained from VBLI observations. The algorithm and its theoretical basis are described, and simulations for a simple two-component source are discussed. VLBI data for the complex radio source NRAO 140 are analyzed. For this source, structure corrections are calculated with maps from three different VLBI imaging techniques: the hybrid-CLEAN algorithm, the maximum entropy method, and model fitting. The calculated structure corrections are compared with brightness distributions to the actual data. The results are used to map NRAO 140. It is found that the CLEAN map produces the most accurate structure corrections.

Charlot, P.↗
NASA NTRS2022

Bryce Canyon Water Resources: Monitoring Vegetation Health and Water Availability in Bryce Canyon National Park for Drought Stress Mitigation Planning

Bryce Canyon National Park is home to groundwater-dependent ecosystems (GDEs) that are threatened by a multidecadal drought and increased groundwater extraction due to a spike in tourism. These ecosystems contain unique species that are only found in areas where near-surface groundwater is present, such as aspen groves and fens. These species contribute to the high biodiversity found in Bryce Canyon, which boosts an ecosystem’s productivity and the services it provides to the park. Unfortunately, many of these GDEs are too small to identify with traditional Earth observation platforms and are difficult to physically reach for monitoring purposes. This project partnered with the National Park Service to identify springs and seeps as a proxy for GDEs within Bryce Canyon from 2013–2022. Furthermore, this project tested the feasibility of various methods to detect and monitor springs and seeps and therefore facilitate the partner’s efforts to conserve these ecologically valuable GDEs in Bryce Canyon. The team mapped groundwater discharge with high resolution National Agriculture Imagery Program (NAIP) and assessed park vegetation trends with Landsat 8 Operational Land Imager (OLI) and PlanetScope imagery. In-situ precipitation data and the Western Land Data Assimilation System (WLDAS) were used to produce time series of climatic variables. Seeps and spring locations were predicted using random forest classification and maximum entropy machine learning models.

Groundwater dependent ecosystems↗
DOE OSTI2014

Statistical mechanics of collisionless orbits. IV. Distribution of angular momentum

It has been shown in previous work that DARKexp, which is a theoretically derived, maximum entropy, one shape parameter model for isotropic collisionless systems, provides very good fits to simulated and observed dark matter halos. Specifically, it fits the energy distribution, N(E), and the density profiles, including the central cusp. Here, we extend DARKexp N(E) to include the distribution in angular momentum, L {sup 2}, for spherically symmetric systems. First, we argue, based on theoretical, semi-analytical, and simulation results, that while dark matter halos are relaxed in energy, they are not nearly as relaxed in angular momentum, which precludes using maximum entropy to uniquely derive N(E, L {sup 2}). Instead, we require that when integrating N(E, L {sup 2}) over squared angular momenta one retrieves the DARKexp N(E). Starting with a general expression for N(E, L {sup 2}) we show how the distribution of particles in L {sup 2} is related to the shape of the velocity distribution function, VDF, and velocity anisotropy profile, β(r). We then demonstrate that astrophysically realistic halos, as judged by the VDF shape and β(r), must have linear or convex distributions in L {sup 2}, for each separate energy bin. The distribution in energy of the most bound particles must be nearly flat, and become more tilted in favor of radial orbits for less bound particles. These results are consistent with numerical simulations and represent an important step toward deriving the full distribution function for spherically symmetric dark matter halos.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Realizability-preserving discontinuous Galerkin method for spectral two-moment radiation transport in special relativity

Here we present a realizability-preserving numerical method for solving a spectral two-moment model to simulate the transport of massless, neutral particles interacting with a steady background material moving with relativistic velocities. The model is obtained as the special relativistic limit of a four-momentum-conservative general relativistic two-moment model. Using a maximum-entropy closure, we solve for the Eulerian-frame energy and momentum. The proposed numerical method is designed to preserve moment realizability, which corresponds to moments defined by a nonnegative phase-space density. The realizability-preserving method is achieved with the following key components: (i) a discontinuous Galerkin phase-space discretization with specially constructed numerical fluxes in the spatial and energy dimensions; (ii) a strong stability-preserving implicit-explicit time-integration method; (iii) a realizability-preserving conserved to primitive moment solver; (iv) a realizability-preserving implicit collision solver; and (v) a realizability-enforcing limiter. Component (iii) is necessitated by the closure procedure, which closes higher order moments nonlinearly in terms of primitive moments. The nonlinear conserved to primitive and the implicit collision solves are formulated as fixed-point problems, which are solved with custom iterative solvers designed to preserve the realizability of each iterate. With a series of numerical tests, we demonstrate the accuracy and robustness of this discontinuous-Galerkin-implicit-explicit method.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS2001

Modulation of Gravity Waves by Tides as Seen in CRISTA Temperatures

During shuttle missions STS-66 (November, 1994) and STS-85 (August, 1997) the Cryogenic Infrared Spectrometers and Telescopes for the Atmosphere (CRISTA) acquired temperature data with very high spatial resolution. These are analyzed for gravity waves (GW). The altitude range spans the whole middle atmosphere from the tropopause up to the mesopause. In the upper mesosphere tidal amplitudes exceed values of 10 K. Modulation of GW activity by the tides is observed and analyzed using CRISTA temperatures and tidal predictions of the Global Scale Wave Model (GSWM). The modulation process is identified as a tidally-induced change of the background buoyancy frequency. The findings agree well with the expectations for saturated GW and are the first global scale observations of this process.

SHUTTLE MISSIONS↗
DOE OSTI2024

MENT-Flow: maximum-entropy phase space tomography using normalizing flows

Generative models can be trained to reproduce low-dimensional projections of high-dimensional phase space distributions. Normalizing flows are generative models that parameterize invertible transformations, allowing exact probability density evaluation and sampling. Consequently, flows are unbiased entropy estimators and could be used to solve the high-dimensional maximum-entropy tomography (MENT) problem. In this work, we evaluate a flow-based MENT solver (MENT-Flow) against exact maximum-entropy solutions and Minerbo's iterative MENT algorithm in two dimensions.

Hoover, Austin↗
NASA NTRS1989

Stochastic model of the NASA/MSFC ground facility for large space structures with uncertain parameters: The maximum entropy approach, part 2

A validated technology data base is being developed in the areas of control/structures interaction, deployment dynamics, and system performance for Large Space Structures (LSS). A Ground Facility (GF), in which the dynamics and control systems being considered for LSS applications can be verified, was designed and built. One of the important aspects of the GF is to verify the analytical model for the control system design. The procedure is to describe the control system mathematically as well as possible, then to perform tests on the control system, and finally to factor those results into the mathematical model. The reduction of the order of a higher order control plant was addressed. The computer program was improved for the maximum entropy principle adopted in Hyland's MEOP method. The program was tested against the testing problem. It resulted in a very close match. Two methods of model reduction were examined: Wilson's model reduction method and Hyland's optimal projection (OP) method. Design of a computer program for Hyland's OP method was attempted. Due to the difficulty encountered at the stage where a special matrix factorization technique is needed in order to obtain the required projection matrix, the program was successful up to the finding of the Linear Quadratic Gaussian solution but not beyond. Numerical results along with computer programs which employed ORACLS are presented.

Hsia, Wei Shen↗
DOE OSTI2014

Electron density distribution and disordered crystal structure of 8H-SiAlON, Si{sub 3−x}Al{sub 1+x}O{sub x}N{sub 5−x} (x∼2.2)

The 8H-SiAlON crystal with general formula Si{sub 3−x}Al{sub 1+x}O{sub x}N{sub 5−x} was characterized using laboratory X-ray powder diffraction (CuKα{sub 1}), transmission electron microscopy and energy dispersive X-ray spectroscopy. The [Si: Al] molar ratios were determined to be [0.21(1): 0.79(1)], corresponding to x=2.2(1). The Si{sub 0.8(1)}Al{sub 3.2(1)}O{sub 2.2(1)}N{sub 2.8(1)} compound is hexagonal with space group P6{sub 3}/mmc (Z=2). The unit-cell dimensions are a=0.298877(8) nm, c=2.30872(5) nm and V=0.178602(8) nm{sup 3}. The initial structural model was successfully derived by the charge-flipping method and further refined by the Rietveld method. The final structural model showed the positional disordering of two of the three (Si,Al) sites. The maximum-entropy method-based pattern fitting (MPF) method was used to confirm the validity of the split-atom model, in which conventional structure bias caused by assuming intensity partitioning was minimized. The reliability indices calculated from the MPF were R{sub wp}=4.74%, S (=R{sub wp}/R{sub e})=1.30, R{sub p}=3.40%, R{sub B}=1.04% and R{sub F}=0.74%. The disordered crystal structure was successfully described by overlapping four types of domains with ordered atom arrangements. The distribution of atomic positions in each of the domains can be achieved in the space group P6{sub 3}mc. Two of the four types of domains are related by a pseudo-symmetry inversion, and the two remaining domains also have each other the inversion pseudo-symmetry. - Graphical abstract: Bird's eye view of electron densities up to 49.3% (0.071 nm{sup −3}) of the maximum on the plane parallel to (110) with the corresponding atomic arrangements of Si{sub 0.8}Al{sub 3.2}O{sub 2.2}N{sub 2.8}. - Highlights: • Crystal structure of Si{sub 0.8}Al{sub 3.2}O{sub 2.2}N{sub 2.8} is determined by laboratory X-ray powder diffraction. • The atom arrangements are represented by the split-atom model. • The maximum-entropy method-based pattern fitting method is used to confirm the validity of the model. • The disordered structure is described by overlapping four types of domains with ordered atom arrangements.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2014

Electron density distribution and disordered crystal structure of 15R-SiAlON, SiAl{sub 4}O{sub 2}N{sub 4}

The crystal structure of SiAl{sub 4}O{sub 2}N{sub 4} was characterized by laboratory X-ray powder diffraction (CuKα{sub 1}). The title compound is trigonal with space group R3-bar m. The hexagonal unit-cell dimensions (Z=3) are a=0.301332(3) nm, c=4.18616(4) nm and V=0.3291825(5) nm{sup 3}. The initial structural model was successfully derived by the charge-flipping method and further refined by the Rietveld method. The final structural model showed the positional disordering of one of the three (Si,Al) sites. The maximum-entropy method-based pattern fitting (MPF) method was used to confirm the validity of the split-atom model, in which conventional structure bias caused by assuming intensity partitioning was minimized. The reliability indices calculated from the MPF were R{sub wp}=5.05%, S (=R{sub wp}/R{sub e})=1.21, R{sub p}=3.77%, R{sub B}=1.29% and R{sub F}=1.01%. The disordered crystal structure was successfully described by overlapping three types of domains with ordered atom arrangements. The distribution of atomic positions in one of the three types of domains can be achieved in the space group R3-bar m. The atom arrangements in the other two types of domains are noncentrosymmetrical with the space group R3m. These two structural configurations are related by the pseudo-symmetry inversion. -- Graphical abstract: A bird's eye view of electron densities up to 75.3% (0.133 nm{sup −3}) of the maximum on the plane parallel to (110) with the corresponding atomic arrangements of SiAl{sub 4}O{sub 2}N{sub 4}. Highlights: • Crystal structure of SiAl{sub 4}O{sub 2}N{sub 4} is determined by laboratory X-ray powder diffraction. • The atom arrangements are represented by the split-atom model. • The maximum-entropy method-based pattern fitting method is used to confirm the validity of the model. • The disordered structure is described by overlapping three types of domains with ordered atom arrangements.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2015

ENTROPY PRODUCTION IN COLLISIONLESS SYSTEMS. III. RESULTS FROM SIMULATIONS

The equilibria formed by the self-gravitating, collisionless collapse of simple initial conditions have been investigated for decades. We present the results of our attempts to describe the equilibria formed in N-body simulations using thermodynamically motivated models. Previous work has suggested that it is possible to define distribution functions for such systems that describe maximum entropy states. These distribution functions are used to create radial density and velocity distributions for comparison to those from simulations. A wide variety of N-body code conditions are used to reduce the chance that results are biased by numerical issues. We find that a subset of initial conditions studied lead to equilibria that can be accurately described by these models, and that direct calculation of the entropy shows maximum values being achieved.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2015

General multi-group macroscopic modeling for thermo-chemical non-equilibrium gas mixtures

This paper opens a new door to macroscopic modeling for thermal and chemical non-equilibrium. In a game-changing approach, we discard conventional theories and practices stemming from the separation of internal energy modes and the Landau-Teller relaxation equation. Instead, we solve the fundamental microscopic equations in their moment forms but seek only optimum representations for the microscopic state distribution function that provides converged and time accurate solutions for certain macroscopic quantities at all times. The modeling makes no ad hoc assumptions or simplifications at the microscopic level and includes all possible collisional and radiative processes; it therefore retains all non-equilibrium fluid physics. We formulate the thermal and chemical non-equilibrium macroscopic equations and rate coefficients in a coupled and unified fashion for gases undergoing completely general transitions. All collisional partners can have internal structures and can change their internal energy states after transitions. The model is based on the reconstruction of the state distribution function. The internal energy space is subdivided into multiple groups in order to better describe non-equilibrium state distributions. The logarithm of the distribution function in each group is expressed as a power series in internal energy based on the maximum entropy principle. The method of weighted residuals is applied to the microscopic equations to obtain macroscopic moment equations and rate coefficients succinctly to any order. The model’s accuracy depends only on the assumed expression of the state distribution function and the number of groups used and can be self-checked for accuracy and convergence. We show that the macroscopic internal energy transfer, similar to mass and momentum transfers, occurs through nonlinear collisional processes and is not a simple relaxation process described by, e.g., the Landau-Teller equation. Unlike the classical vibrational energy relaxation model, which can only be applied to molecules, the new model is applicable to atoms, molecules, ions, and their mixtures. Numerical examples and model validations are carried out with two gas mixtures using the maximum entropy linear model: one mixture consists of nitrogen molecules undergoing internal excitation and dissociation and the other consists of nitrogen atoms undergoing internal excitation and ionization. Results show that the original hundreds to thousands of microscopic equations can be reduced to two macroscopic equations with almost perfect agreement for the total number density and total internal energy using only one or two groups. We also obtain good prediction of the microscopic state populations using 5-10 groups in the macroscopic equations.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2017

Modelling bivariate extreme precipitation distribution for data-scarce regions using Gumbel-Hougaard copula with maximum entropy estimation

A new method of parameter estimation in data scarce regions is valuable for bivariate hydrological extreme frequency analysis. Here, this paper proposes a new method of parameter estimation (maximum entropy estimation, MEE) for both Gumbel and Gumbel–Hougaard copula in situations when insufficient data are available. MEE requires only the lower and upper bounds of two hydrological variables. To test our new method, two experiments to model the joint distribution of the maximum daily precipitation at two pairs of stations on the tributaries of Heihe and Jinghe River, respectively, were performed and compared with the method of moments, correlation index estimation, and maximum likelihood estimation, which require a large amount of data. Both experiments show that for the Ye Niugou and Qilian stations, the performance of MEE is nearly identical to those of the conventional methods. For the Xifeng and Huanxian stations, MEE can capture information indicating that the maximum daily precipitation at the Xifeng and Huanxian stations has an upper tail dependence, whereas the results generated by correlation index estimation and maximum likelihood estimation are unreasonable. Moreover, MEE is proved tobe generally reliable and robust by many simulations under three different situations. TheGumbel–Hougaard copula with MEE can also be applied to the bivariate frequency analysis ofother extreme events in data-scarce regions.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2018

MULTI-PRED: A Software Module for Predictive Modeling of Coupled Multi-Physics Systems: User's Manual

This User’s Manual describes the code module MULTI-PRED, written in FORTRAN which implements the methodology for “predictive modeling of coupled multi-physics systems (PM-CMPS)” formulated by Cacuci (2014). This methodology fully takes into account the coupling terms between the systems but requires only the computational resources that would be needed to perform predictive modeling on each system separately. The PM-CMPS methodology uses the maximum entropy principle to construct an optimal approximation of the unknown a priori distribution based on a priori known mean values and uncertainties characterizing the experimental and computational parameters and results of interest responses, called for the multi-physics models under consideration. This “maximum entropy” a priori distribution is combined, using Bayes’ theorem, with the “likelihood” provided by the multi-physics simulation models to obtain a formal posterior distribution. Subsequently, the posterior distribution thus obtained is evaluated using the saddle-point method to obtain analytical expressions for the optimally predicted values for the multi-physics models parameters and responses along with corresponding reduced uncertainties. Noteworthy, the predictive modeling methodology for the coupled systems is constructed such that the systems can be considered sequentially rather than simultaneously, while preserving exactly the same results as if the systems were treated simultaneously. Consequently, very large coupled systems, which could perhaps exceed available computational resources if treated simultaneously, can be treated with the PM-CMPS methodology presented in this work sequentially and without any loss of generality or information, requiring just the resources that would be needed if the systems were treated sequentially. Three illustrative demonstration problems are also provided. The first problem presents the application of the PM-CMPS methodology to a simple particle diffusion problem which admits a closed-form analytical solution which facilitates a rapid understanding of this methodology and its predicted results. The second demonstration problem presents the application of the PM-CMPS methodology to the problem of inverse prediction, from detector responses in the presence of counting uncertainties, of the thickness of a homogeneous slab of material containing uniformly distributed gamma-emitting sources, for optically thin and thick slabs. This problem highlights the essential role played by the relative uncertainties (or, conversely, accuracies) of measured and computed responses. The third demonstration problem presents the application of the PM-CMPS methodology to the F-area cooling towers at the Savannah River National Lab. This problem demonstrates that the PM-CMPS methodology reduces the predicted response uncertainties not only at locations where measurements are available, but also at locations where measurements are not available.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

Upscaling ab-initio chemistry models to non-equilibrium flow simulations

Thermochemical nonequilibrium is generated by hypersonic cruise vehicles, high speed combustion and detonations. Nonequilibrium interactions affect reaction rates and energy transfer that are critical for many engineering applications. Ab-initio potential energy surfaces can be used in quasi-classical trajectory (QCT) calculations to calculate state-specific dissociation rates and cross sections, as well as state-to-state energy exchange rates and cross sections. These rates and cross sections are needed for simulating such nonequilibrium flows. However, there are two main challenges in extending QCT calculations to modeling all chemical processes found in typical engineering flowfields. First, ab-initio surfaces for many collision partners are not available and when available their fidelity is not quantified. Second, state-to-state cross section datasets are typically on the order of 1000 4, , and are too large to be efficiently implemented in engineering simulations. This work focuses on O 2 +O interactions because of prevalence of oxygen in many engineering application; however, all of the models developed here may be applied to other diatom-atom interactions and can be extended to other diatom-diatom combinations. It is shown in this work that at temperatures between 2,500 and 20,000K, state-specific cross sections are not highly sensitive to potentials and a Morse additive pairwise (MAP) potential is proposed for modeling collisions for which ab-initio potentials are not available. Next, this work suggests an approach to model state-to-state cross sections based on maximum entropy considerations. The calculated cross sections are fit to a new model with just 8 parameters, which can be easily scaled up to flow simulations.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS1989

Spectral and correlation analysis with applications to middle-atmosphere radars

The correlation and spectral analysis methods for uniformly sampled stationary random signals, estimation of their spectral moments, and problems arising due to nonstationary are reviewed. Some of these methods are already in routine use in atmospheric radar experiments. Other methods based on the maximum entropy principle and time series models have been used in analyzing data, but are just beginning to receive attention in the analysis of radar signals. These methods are also briefly discussed.

Rastogi, Prabhat K.↗
NASA NTRS2003

First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Foreground Emission

The WMAP mission has mapped the full sky to determine the geometry, content, and evolution of the universe. Full sky maps are made in five microwave frequency bands to separate the temperature anisotropy of the cosmic microwave background (CMB) from foreground emission, including diffuse Galactic emission and Galactic and extragalactic point sources. We define masks that excise regions of high foreground emission, so CMB analyses can became out with minimal foreground contamination. We also present maps and spectra of the individual emission components, leading to an improved understanding of Galactic astrophysical processes. The effectiveness of template fits to remove foreground emission from the WMAP data is also examined. These efforts result in a CMB map with minimal contamination and a demonstration that the WMAP CMB power spectrum is insensitive to residual foreground emission. We use a Maximum Entropy Method to construct a model of the Galactic emission components. The observed total Galactic emission matches the model to less than 1% and the individual model components are accurate to a few percent. We find that the Milky Way resembles other normal spiral galaxies between 408 MHz and 23 GHz, with a synchrotron spectral index that is flattest (beta(sub s) approx. -2.5) near star-forming regions, especially in the plane, and steepest (beta(sub s) approx. -3) in the halo. This is consistent with a picture of relativistic cosmic ray electron generation in star-forming regions and diffusion and convection within the plane. The significant synchrotron index steepening out of the plane suggests a diffusion process in which the halo electrons are trapped in the Galactic potential long enough to suffer synchrotron and inverse Compton energy losses and hence a spectral steepening. The synchrotron index is steeper in the WMAP bands than in lower frequency radio surveys, with a spectral break near 20 GHz to beta(sub s) less than -3. The modeled thermal dust spectral index is also steep in the WMAP bands, with beta(sub d) approx. = 2.2. Our model is driven to these conclusions by the low level of total foreground contamination at approx. 60 GHz. Microwave and Ha measurements of the ionized gas agree well with one another at about the expected levels. Spinning dust emission is limited to less than 5% of the Ka-band foreground emission. A catalog of 208 point sources is presented. The reliability of the catalog is 98%, i.e., we expect five of the 208 sources to be statistically spurious. The mean spectral index of the point sources is alpha approx. 0(beta approx. -2). Derived source counts suggest a contribution to the anisotropy power from unresolved sources of (15.0 +/- 1.4) x 10(exp -3)micro sq K sr at Q-band and negligible levels at V-band and W-band. The Sunyaev-Zeldovich effect is shown to be a negligible "contamination" to the maps.

Bennett, C. L.↗
NASA NTRS1986

Stochastic modeling and control system designs of the NASA/MSFC Ground Facility for large space structures: The maximum entropy/optimal projection approach

In the Control Systems Division of the Systems Dynamics Laboratory of the NASA/MSFC, a Ground Facility (GF), in which the dynamics and control system concepts being considered for Large Space Structures (LSS) applications can be verified, was designed and built. One of the important aspects of the GF is to design an analytical model which will be as close to experimental data as possible so that a feasible control law can be generated. Using Hyland's Maximum Entropy/Optimal Projection Approach, a procedure was developed in which the maximum entropy principle is used for stochastic modeling and the optimal projection technique is used for a reduced-order dynamic compensator design for a high-order plant.

Hsia, Wei-Shen↗
NASA NTRS2003

Macroscopic Models of Radiative Transfer as Applied to Computation of the Radiation Field in the Solar Atmosphere

In this paper a new three dimensional half-moment model for radiative transfer is presented for a gray medium. It describes the evolution of the zeroth and first directional half moments of the radiative intensity. The closure is provided, similarly to Dubroca and Klar, by the maximum entropy concept. This work generalizes that model to three dimensions. The model presented here (the derivation being done in Ripoll and Wray, called the M(sup 1/2)(sub 1) model, is a hyperbolic system consisting of a total of eight equations in three dimensions, four equations for each direction. Each half model has the classical form of a macroscopic moment model in which the pressure tensor is constructed from the well-known Eddington tensor with a particular Eddington factor. Moreover, different source and border terms occur. The latter introduce couplings between the macroscopic and microscopic quantities and between the + and - streams, through the intensity in the plane perpendicular to the flux. The main theoretical application of the half moment model, treated in this paper, is its reduction to a full moment model, called M(sup +)(sub 1), for the particular but important case of a hot, opaque source radiating in a cold transparent (or semi-transparent) medium for very specific applications, such as stellar interiors or atmospheres, or combustion problems. The structure of the paper is as follows. In section 2, the model M(sup 1/2)(sub 1) is presented. In section 3, for the particular case of a hot, opaque source radiating into a cold medium, the half moment model is reduced to the M(sup +)(sub 1) model. In section 4, we first solve a simple and academic problem to validate the models, followed by a simplified solar atmosphere.

Ripoll, J.-F.↗