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DOE OSTI2024

Inverse prediction of PuO2 processing conditions using Bayesian seemingly unrelated regression with functional data

Over the past decade, a variety of innovative methodologies have been developed to better characterize the relationships between processing conditions and the physical, morphological, and chemical features of special nuclear material (SNM). Different processing conditions generate SNM products with different features, which are known as “signatures” because they are indicative of the processing conditions used to produce the material. These signatures can potentially allow a forensic analyst to determine which processes were used to produce the SNM and make inferences about where the material originated. This article investigates a statistical technique for relating processing conditions to the morphological features of PuO 2 particles. We develop a Bayesian implementation of seemingly unrelated regression (SUR) to inverse-predict unknown PuO 2 processing conditions from known PuO 2 features. Model results from simulated data demonstrate the usefulness of the technique. Applied to empirical data from a bench-scale experiment specifically designed with inverse prediction in mind, our model successfully predicts nitric acid concentration, while results for Pu concentration and precipitation temperature were equivalent to a simple mean model. Our technique compliments other recent methodologies developed for forensic analysis of nuclear material and can be generalized across the field of chemometrics for application to other materials.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

DOE OSTI2021

Valuation of Wind Energy Turbines Using Volatility of Wind and Price

The limitedness of the nonrenewable local energy resources in Israel, even in the background of the later gas fields’ findings, continues to force the state to devote various efforts towards ‘green’ energy development. These efforts include installations, both for the solar and for wind energy, thus improving the diversity of energy sources. While the standard discounted cash flow (DCF) method using the net present value (NPV) criterion is extensively adopted to evaluate investments, the standard DCF method is inappropriate for the rapidly changing investment climate and for the managerial flexibility in investment decisions. In recent years, the real options analysis (ROA) technique has been widely applied in many studies for the valuation of renewable energy investment projects. Taking into account the above background, we apply, in this study, the real options analysis approach for the valuation of wind energy turbines and apply it to the analysis of wind energy economic potential in Israel, which is the context of our work. We hypothesize that due to nature of wind energy production uncertainties, the ROA method is better than the alternative. The novelty of this paper includes the following: real world wind statistics of the Merom Golan site in Israel (velocity 3.73 m/s, with a standard deviation of 2.03 m/s), a realistic power generation estimation (power generation of 1205.84 kW with a standard deviation of about 0.5% in annual value which is worth about 1.3 M$ per annum), and an economic model to evaluate the profitability of such a project. We thus discuss the existing challenges of diversifying renewable energy sources in Israel by adding wind installations. Our motivation is to introduce a method which will allow investors and officials to take into account uncertainties when deciding in investing in such wind installations. The outcomes of the paper, which are obtained using the method of Weibull statistics and the Black–Scholes ROA technique, include the result that market price volatility adds to the uncertainties much more than any wind fluctuations, provided that the analysis is integrated over a long enough time.

17 WIND ENERGY↗

DOE OSTI2023

High-fidelity Experiments of Turbulent Buoyant Jets from Simultaneous Particle Image Velocimetry and Laser Induced Fluorescence

Accurate models of turbulent buoyant flows are essential for the design of nuclear reactors thermal hydraulics and passive safety systems. However, available models fail to fully capture the physics of turbulent mixing when buoyancy becomes predominant with respect to momentum. Therefore, high-fidelity experiments of well-controlled fundamental flows are needed to develop and validate more accurate models. We performed experiments of positive and negative turbulent buoyant jets, both in uniform and stratified environments, with the aim of understanding the thermal hydraulics of turbulent mixing with variable density and providing high-fidelity data for the development and validation of turbulence models. Non-intrusive, simultaneous Particle Image Velocimetry and Laser Induced Fluorescence measurements were carried out to acquire instantaneous velocity and concentration fields on a vertical section parallel to the axis of a jet in the self-similar region. The Refractive Index Matching method was applied to measure high-resolution buoyant jets with up to 8.6% density difference. These data are free of the typical errors that characterize optical measurements of buoyancy driven flows (e.g., natural and mixed convection) where the refractive index of the fluid is inhomogeneous throughout the measurement domain. Turbulent statistics and entrainment of buoyant jets in uniform and stratified environments are presented. These data are compared with non-buoyant jets in uniform environment, as a reference to investigate the effects of buoyancy and stratification on turbulent mixing. The results will be used for the assessment of current turbulence models and as basis for the development of a new one that captures turbulent mixing.

42 ENGINEERING↗

DOE OSTI2021

Characterizing the Sample Selection for Supernova Cosmology

Type Ia supernovae (SNe Ia) are used as distance indicators to infer the cosmological parameters that specify the expansion history of the universe. Parameter inference depends on the criteria by which the analysis SN sample is selected. Only for the simplest selection criteria and population models can the likelihood be calculated analytically, otherwise it needs to be determined numerically, a process that inherently has error. Numerical errors in the likelihood lead to errors in parameter inference. This article presents toy examples where the distance modulus is inferred given a set of SNe at a single redshift. Parameter estimators and their uncertainties are calculated using Monte Carlo techniques. The relationship between the number of Monte Carlo realizations and numerical errors is presented. The procedure can be applied to more realistic models and used to determine the computational and data management requirements of the transient analysis pipeline.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2020

Flare Statistics for Young Stars from a Convolutional Neural Network Analysis of TESS Data

All-sky photometric time-series missions have allowed for the monitoring of thousands of young (t age < 800 Myr) stars in order to understand the evolution of stellar activity. In this work, we developed a convolutional neural network (CNN), stella, specifically trained to find flares in Transiting Exoplanet Survey Satellite (TESS) short-cadence data. We applied the network to 3200 young stars in order to evaluate flare rates as a function of age and spectral type. The CNN takes a few seconds to identify flares on a single light curve. We also measured rotation periods for 1500 of our targets and find that flares of all amplitudes are present across all spot phases, suggesting high spot coverage across the entire surface. Additionally, flare rates and amplitudes decrease for stars t age > 50 Myr across all temperatures T eff ≥ 4000 K, while stars from 2300 ≤ T eff < 4000 K show no evolution across 800 Myr. Stars of T eff ≤ 4000 K also show higher flare rates and amplitudes across all ages. We investigate the effects of high flare rates on photoevaporative atmospheric mass loss for young planets. In the presence of flares, planets lose 4%–7% more atmosphere over the first 1 Gyr. stella is an open-source Python toolkit hosted on GitHub and PyPI.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

DOE OSTI2022

High bias machine learning for antineutrino-based safeguards for small reactors

The statistical methods used for antineutrino detection will need to be improved to effectively monitor the inventory of next-generation nuclear reactors. In this sensitivity study, we evaluate machine learning models compared to previously used statistical approaches to identify diversion scenarios in a simulated Advanced Fast Reactor (AFR)-100. A chi-square goodness-of-fit technique, which individually compares the simulated antineutrino yields to the expected antineutrino yield, resulted in precise but low diversion detection probability. Various support vector machine (SVM) models were applied with diverse training datasets to evaluate the robustness of the method towards unexpected or “unseen” diversion scenarios. Furthermore, our results indicate that while the SVM models significantly improved the detection probability of near-field antineutrino-based safeguards, up to a probability of ~0.04, for the simulated small reactor, the detection system still needs improvements to reach the 0.2 detection limit established by the International Atomic Energy Agency.

98 NUCLEAR DISARMAMENT, SAFEGUARDS, AND PHYSICAL P↗

DOE OSTI2024

Divide and conquer: Learning chaotic dynamical systems with multistep penalty neural ordinary differential equations

Forecasting high-dimensional dynamical systems is a fundamental challenge in various fields, such as geosciences and engineering. Neural Ordinary Differential Equations (NODEs), which combine the power of neural networks and numerical solvers, have emerged as a promising algorithm for forecasting complex nonlinear dynamical systems. However, classical techniques used for NODE training are ineffective for learning chaotic dynamical systems. In this work, we propose a novel NODE-training approach that allows for robust learning of chaotic dynamical systems. Here, our method addresses the challenges of non-convexity and exploding gradients associated with underlying chaotic dynamics. Training data trajectories from such systems are split into multiple, non-overlapping time windows. In addition to the deviation from the training data, the optimization loss term further penalizes the discontinuities of the predicted trajectory between the time windows. The window size is selected based on the fastest Lyapunov time scale of the system. Multi-step penalty(MP) method is first demonstrated on Lorenz equation, to illustrate how it improves the loss landscape and thereby accelerates the optimization convergence. MP method can optimize chaotic systems in a manner similar to least-squares shadowing with significantly lower computational costs. Our proposed algorithm, denoted the Multistep Penalty NODE, is applied to chaotic systems such as the Kuramoto-Sivashinsky equation, the two-dimensional Kolmogorov flow, and ERA5 reanalysis data for the atmosphere. It is observed that MP-NODE provide viable performance for such chaotic systems, not only for short-term trajectory predictions but also for invariant statistics that are hallmarks of the chaotic nature of these dynamics.

Chaotic dynamical systems↗

DOE OSTI2020

A corrected method for Coulomb scattering in arbitrarily weighted particle-in-cell plasma simulations

A physically and statistically accurate method to perform binary, pairwise Coulomb collisions for arbitrarily-weighted macroparticles in particle-in-cell simulations is described. The method is shown to conserve global energy and momentum on average and to produce physically accurate scattering frequencies. This is justified using the statistically expected values of the method and through a series of numerical tests showing that the method is valid independently of macroparticle weight. This novel method is required due to an error in previous work on binary scattering, initially by Nanbu and Yonemura [J. Comp. Phys. 145, 639 (1998)] and applied to subsequent work, that causes inaccuracies in the scattering frequencies for unequally weighted simulations; a problem that occurs both for scattering within the group and between different groups.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2024

High-Fidelity Velocity and Concentration Measurements of Turbulent Buoyant Jets

Accurate models of turbulent buoyant flows are essential for the design of the cooling circuit of nuclear reactors and passive safety systems. However, available models fail to fully capture the physics of turbulent mixing when buoyancy becomes predominant with respect to momentum. Therefore, high-fidelity experiments of well-controlled fundamental flows are needed to develop and validate more accurate models. We analyze experiments of positive and negative turbulent buoyant jets, both in uniform and stratified environments, with the aim of understanding the thermal hydraulics of turbulent mixing with variable density and providing high-fidelity data for the development and validation of turbulence models. Non-intrusive, simultaneous particle image velocimetry and laser-induced fluorescence measurements were carried out to acquire instantaneous velocity and concentration fields on a vertical section parallel to the axis of a jet in the self-similar region. The refractive index matching method was applied to measure high-resolution buoyant jets with up to 8.6% density difference. These data are free of the typical errors that characterize optical measurements of buoyancy-driven flows (e.g. natural and mixed convection) where the refractive index of the fluid is inhomogeneous throughout the measurement domain. Turbulent statistics and entrainment of buoyant jets in uniform and stratified environments are presented. These data are compared with non-buoyant jets in a uniform environment, as a reference to investigate the effects of buoyancy and stratification on turbulent mixing. The results will be used for the assessment of current turbulence models and as a basis for the development of a new model that captures turbulent mixing.

laser induced fluorescence↗

DOE OSTI2025

Measurement of the work statistics of an open quantum system using a quantum computer

Abstract We report on the experimental measurement of the work statistics of a genuinely open quantum system using a quantum computer. Such measurement has remained elusive thus far due to the inherent difficulty in measuring the total energy change of a system-bath compound (which is the work) in the open quantum system scenario. We overcome this difficulty by applying the interferometric scheme, originally conceived for closed systems, to the open system case and implement it on a superconducting quantum computer, taking advantage of the relatively high levels of noise on current quantum hardware to realize an open quantum system. We demonstrate that the method can be used as a diagnostic tool to probe physical properties of the system-bath compound, such as its temperature and specific transition frequencies in its spectrum. Our experiments corroborate that the interferometric scheme is a promising tool to achieve the long-sought experimental validation of the Jarzynski equality for arbitrary open quantum systems.

Oftelie, Lindsay Bassman (ORCID:0000000335421553)↗

DOE OSTI2025

ForceFinder

SAND2025-11750O ForceFinder extends the Structural Dynamics Python Libraries (SDynPy) with comprehensive tools for inverse source estimation (ISE) tasks via frequency response function (FRF) matrix inversion. The software is designed for transfer path analysis and multiple-input/multiple-output (MIMO) vibration control problems. It allows users to estimate sources through various algorithms, from the basic Moore-Penrose pseudo-inverse to statistical learning methods such as Tikhonov regularization via an L-curve and elastic net regularization via an information criterion. ForceFinder uses an object-oriented framework, where all components of the ISE problem—such as FRFs, responses, and transformations—are stored in a "SourcePathReceiver" object. This software can be applied to any noise and vibration problem. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

Carter, Steven [Sandia National Lab. (SNL-CA), Liv↗

DOE OSTI2021

Design-Space Data: Informing Common Design Decisions with Pre-Simulated Data

Design Space Exploration (DSE) analysis techniques represent a data-centric approach to integrating performance analysis in early design phases when there is the greatest potential to cheaply improve the energy efficiency of a building. We focus on a novel extension of DSE called Universal Design Space Exploration (UDSE), which leverages massive databases of pre-simulated analysis that represent all possible outcomes of common analysis workflows. These databases, called Design Spaces, become “universal” when a single pre-simulated design space can be re-applied to future unknown projects. Unlike current simulation methods, which require a design to exist before it can be analyzed and often take minutes or hours to simulate, UDSE leverages pre-simulation to deliver rapid and relevant insight as new designs are conceptualized. The data underpinning UDSE enables advanced statistical and Artificial Intelligence methods, allowing UDSE to deliver a greater understanding of the larger problem being explored, rather than simply delivering analysis of several pre-conceived design options. We believe that UDSE can provide instantaneous, relevant analysis for all building design projects at negligible cost. This paper has two main goals, to develop a relevant Universal Design Space that showcases the potential of UDSE and to release this data freely to industry and academia; thereby lowering the barrier to entry to digital literacy in statistics, ML and AI within the architecture, engineering, construction (AEC) industry.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

DOE OSTI2023

Evaluation of four ground-based retrievals of cloud droplet number concentration in marine stratocumulus with aircraft in situ measurements

Abstract. Cloud droplet number concentration (Nd) is crucial for understanding aerosol–cloud interactions (ACI) and associated radiative effects. We present evaluations of four ground-based Nd retrievals based on comprehensive datasets from the Atmospheric Radiation Measurement (ARM) Aerosol and Cloud Experiments in the Eastern North Atlantic (ACE-ENA) field campaign. The Nd retrieval methods use ARM ENA observatory ground-based remote sensing observations from a micropulse lidar, Raman lidar, cloud radar, and the ARM NDROP (Droplet Number Concentration) value-added product (VAP), all of which also retrieve cloud effective radius (re). The retrievals are compared against aircraft measurements from the fast cloud droplet probe (FCDP) and the cloud and aerosol spectrometer (CAS) obtained from low-level marine boundary layer clouds on 12 flight days during summer and winter seasons. Additionally, the in situ measurements are used to validate the assumptions and characterizations used in the retrieval algorithms. Statistical comparisons of the probability distribution function (PDF) of the Nd and cloud re retrievals with aircraft measurements demonstrate that these retrievals align well with in situ measurements for overcast clouds, but they may substantially differ for broken clouds or clouds with low liquid water path (LWP). The retrievals are applied to 4 years of ground-based remote sensing measurements of overcast marine boundary layer clouds at the ARM ENA observatory to find that Nd (re) values exhibit seasonal variations, with higher (lower) values during the summer season and lower (higher) values during the winter season. The ensemble of various retrievals using different measurements and retrieval algorithms such as those in this paper can help to quantify Nd retrieval uncertainties and identify reliable Nd retrieval scenarios. Of the retrieval methods, we recommend using the micropulse lidar-based method. This method has good agreement with in situ measurements, less sensitivity to issues arising from precipitation and low cloud LWP and/or optical depth, and broad applicability by functioning for both daytime and nighttime conditions.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2026

ResStock Measure Documentation: Residential Two-Stage Geothermal Heat Pump (4.0 COP, 20.5 EER) With Envelope Improvements and Advanced Air Sealing

The goal of this work is to develop energy efficiency, demand flexibility, and other retrofit end-use load shapes (electricity, gas, propane, or fuel oil) that cover a majority of the high-impact, market-ready (or nearly market-ready) measures. "Measures" refers to retrofits that can be applied to buildings during modeling. An "end-use savings shape" is the difference in energy consumption between a baseline building and a building with an energy efficiency, demand flexibility, or other retrofit measure applied. It results in a time-series profile that is broken down by end use and fuel (electricity or on-site gas, propane, or fuel oil use) at each time step. ResStock is a highly granular, physics-based, bottom-up model that uses multiple data sources, statistical sampling methods, and advanced building energy simulations to estimate the annual subhourly energy consumption of the residential building stock across the United States. The baseline model intends to represent the U.S. residential building stock as it existed in 2018. Technical documentation for the inputs and assumptions in the baseline building stock model is available in Reyna et al. (2025). Calibration and validation of the baseline model results are available in the final technical report of the End-Use Load Profiles project (Wilson, et al. 2022). This document focuses on a single end-use savings shape measure: Residential Two-Stage Geothermal Heat Pump (GHP) (4.0 COP, 20.5 EER) With Envelope Improvements. This measure combines a two-stage GHP with envelope improvements as a single package. As this package is a combination of two other measures, this document focused on documenting the results associated with this combination of technologies, with individual measure documents for two-stage GHPs and envelope improvements providing the information on the details of these measures. When the two technologies are combined, envelope improvements can modestly reduce energy consumption by a further 10%-15%, but also reduce the required size of the ground heat exchanger and heat pump by approximately 33% on average across all sites. The cost of installing envelope improvements in these homes is likely to be more than paid for by the reduction in equipment and drilling costs in these buildings for the majority of the stock.

15 GEOTHERMAL ENERGY↗

DOE OSTI2026

ResStock Measure Documentation: Residential Single-Stage Geothermal Heat Pump (3.8 COP, 18.6 EER)

The goal of this work is to develop energy efficiency, demand flexibility, and other retrofit end-use load shapes (electricity, gas, propane, or fuel oil) that cover a majority of the high-impact, market-ready (or nearly market-ready) measures. "Measures" refers to retrofits that can be applied to buildings during modeling. An "end-use savings shape" is the difference in energy consumption between a baseline building and a building with an energy efficiency, demand flexibility, or other retrofit measure applied. It results in a time-series profile that is broken down by end use and fuel (electricity or on-site gas, propane, or fuel oil use) at each time step. ResStock is a highly granular, physics-based, bottom-up model that uses multiple data sources, statistical sampling methods, and advanced building energy simulations to estimate the annual subhourly energy consumption of the residential building stock across the United States. The baseline model intends to represent the U.S. residential building stock as it existed in 2018. Technical documentation for the inputs and assumptions in the baseline building stock model is available in Reyna et al. (2025). Calibration and validation of the baseline model results are available in the final technical report of the End-Use Load Profiles project (Wilson et al. 2022). This documentation focuses on a single end-use savings shape measure: Residential Single-Stage Geothermal Heat Pump (GHP).?Single-stage GHPs are able to reduce energy consumption by 31% for the entire stock. Additional results provided below detail how savings changes for sections of the housing stock with different base heating fuel and in different climate zones, as well as the savings potential by state for both heating and cooling. Utility bills and electric panel impacts are also shown and discussed.

15 GEOTHERMAL ENERGY↗

DOE OSTI2026

ResStock Measure Documentation: Residential Two-Stage Geothermal Heat Pump (4.0 COP, 20.5 EER)

The goal of this work is to develop energy efficiency, demand flexibility, and other retrofit end-use load shapes (electricity, gas, propane, or fuel oil) that cover a majority of the high-impact, market-ready (or nearly market-ready) measures. "Measures" refers to retrofits that can be applied to buildings during modeling. An "end-use savings shape" is the difference in energy consumption between a baseline building and a building with an energy efficiency, demand flexibility, or other retrofit measure applied. It results in a time-series profile that is broken down by end use and fuel (electricity or on-site gas, propane, or fuel oil use) at each time step. ResStock is a highly granular, physics-based, bottom-up model that uses multiple data sources, statistical sampling methods, and advanced building energy simulations to estimate the annual subhourly energy consumption of the residential building stock across the United States. The baseline model intends to represent the U.S. residential building stock as it existed in 2018. Technical documentation for the inputs and assumptions in the baseline building stock model is available in Reyna et al. (2025). Calibration and validation of the baseline model results are available in the final technical report of the End-Use Load Profiles project (Wilson et al. 2022). This document focuses on a single end-use savings shape measure: Residential Two-Stage Geothermal Heat Pump (4.0 COP, 20.5 EER). This document builds on details established in the single-stage document (Maguire et al. 2025) to detail differences in the approach to modeling this higher efficiency, but more commonly deployed, type of geothermal heat pump. Specific EnergyPlus objects and product specific curves used are highlighted along with showing the results of this measure compared to the baseline and single-speed geothermal heat pumps. Two-speed geothermal heat pumps are able to save even more energy and on utility bills than single-speed products, albeit at the expense of a higher first cost.

15 GEOTHERMAL ENERGY↗

DOE OSTI2026

Accelerating multicanonical sampling with irreversibility

Flat-histogram Monte Carlo simulations are well-established, robust methods to perform random walks in a physical observable or parameter space, making them suitable for finding ground states or studying phase transitions in complex systems in statistical physics. However, their efficiency can be limited by the time to attain the desired flat distribution, which is generally unknown prior to the simulations. In particular, they might suffer from slowing down towards the end of a simulation due to the diffusive nature of random walks. In this work we apply irreversibility to the multicanonical Monte Carlo method via the lifting approach to alleviate this behavior. We achieve a 2–4 times speedup in ground-state search for a two-dimensional (2D) Ising model, and up to an order of magnitude of speedup for finding the ground-state energy in an Edwards–Anderson spin glass, compared to traditional multicanonical sampling. In conclusion, the round-trip times between ground states show a narrower distribution and are significantly shorter compared to the reversible counterpart, suggesting that a lower convergence time with a smaller time variance is feasible.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2020

The completed SDSS-IV extended baryon oscillation spectroscopic survey: pairwise-inverse probability and angular correction for fibre collisions in clustering measurements

ABSTRACT The completed extended Baryon Oscillation Spectroscopic Survey (eBOSS) catalogues contain redshifts of 344 080 quasars at 0.8 < z < 2.2, 174 816 luminous red galaxies between 0.6 < z < 1.0, and 173 736 emission-line galaxies over 0.6 < z < 1.1 in order to constrain the expansion history of the Universe and the growth rate of structure through clustering measurements. Mechanical limitations of the fibre-fed spectrograph on the Sloan telescope prevent two fibres being placed closer than 62 arcsec in a single pass of the instrument. These ‘fibre collisions’ strongly correlate with the intrinsic clustering of targets and can bias measurements of the two-point correlation function resulting in a systematic error on the inferred values of the cosmological parameters. We combine the new techniques of pairwise-inverse probability and the angular upweighting (PIP+ANG) to correct the clustering measurements for the effect of fibre collisions. Using mock catalogues, we show that our corrections provide unbiased measurements, within data precision, of both the projected $\rm {\mathit{ w}_p}\left(\mathit{ r}_p\right)$ and the redshift-space multipole ξ(ℓ = 0, 2, 4)(s) correlation functions down to $0.1\, h^{-1}{\rm Mpc}$, regardless of the tracer type. We apply the corrections to the eBOSS DR16 catalogues. We find that, on scales $s\gtrsim 20\, h^{-1}{\rm Mpc}$ for ξℓ, as used to make baryon acoustic oscillation and large-scale redshift-space distortion measurements, approximate methods such as nearest-neighbour upweighting are sufficiently accurate given the statistical errors of the data. Using the PIP method, for the first time for a spectroscopic program of the Sloan Digital Sky Survey, we are able to successfully access the one-halo term in the clustering measurements down to $\sim 0.1\, h^{-1}{\rm Mpc}$ scales. Our results will therefore allow studies that use the small-scale clustering to strengthen the constraints on both cosmological parameters and the halo occupation distribution models.

79 ASTRONOMY AND ASTROPHYSICS↗