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DOE OSTI2025

Evolution of in-medium baryon-baryon scattering cross sections and stiffness of dense nuclear matter from Bayesian analyses of FOPI proton-flow excitation functions

Within a Bayesian statistical framework using a Gaussian process (GP) emulator for an isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU) transport model simulator of heavy-ion reactions with momentum-independent Skyrme interactions, we infer from the proton directed and elliptical flow in mid-central Au + Au reactions, measured at beam energies from 150 to 1200 MeV/nucleon by the FOPI Collaboration, the posterior probability distribution functions (PDFs) of the in-medium baryon-baryon scattering cross section (BBSCS) modification factor 𝑋 (with respect to their free-space values) and the stiffness parameter 𝐾 of dense nuclear matter. Here, we find that the most probable value of 𝑋 evolves from around 0.7 to 1.0 as the beam energy 𝐸 beam /𝐴 increases. On the other hand, the posterior PDF⁡(𝐾) may have dual peaks having roughly the same height or extended shoulders at high 𝐾 values. More quantitatively, the posterior PDF⁡(𝐾) changes from having a major peak around 220 MeV, characterizing a soft EOS in the reaction at 𝐸 beam /𝐴 = 150 MeV, to one that peaks around 320 MeV, indicating a stiff EOS in the reactions at 𝐸 beam /𝐴 higher than about 600 MeV. The transition from soft to stiff happens in mid-central Au + Au reactions at beam energies around 250 MeV/nucleon in which 𝐾 = 220 MeV and 𝐾 = 320 MeV are approximately equally probable. Altogether, the FOPI proton-flow excitation function data indicate a gradual hardening of hot and dense nuclear matter as its density and temperature increase in reactions with higher beam energies.

nuclear matter↗

DOE OSTI2021

Dark Energy Survey Year 3 Results: Galaxy mock catalogs for BAO analysis

The calibration and validation of scientific analysis in simulations is a fundamental tool to ensure unbiased and robust results in observational cosmology. In particular, mock galaxy catalogs are a crucial resource to achieve these goals in the measurement of baryon acoustic oscillation (BAO) in the clustering of galaxies. Here we present a set of 1952 galaxy mock catalogs designed to mimic the Dark Energy Survey Year 3 BAO sample over its full photometric redshift range 0.6 < zphoto < 1.1. The mocks are based upon 488 ICE-COLA fast N-body simulations of full-sky light cones and were created by populating halos with galaxies, using a hybrid halo occupation distribution – halo abundance matching model. This model has ten free parameters, which were determined, for the first time, using an automatic likelihood minimization procedure. We also introduced a novel technique to assign photometric redshift for simulated galaxies, following a two-dimensional probability distribution with VIMOS Public Extragalactic Redshift Survey data. The calibration was designed to match the observed abundance of galaxies as a function of photometric redshift, the distribution of photometric redshift errors, and the clustering amplitude on scales smaller than those used for BAO measurements. An exhaustive analysis was done to ensure that the mocks reproduce the input properties. Finally, mocks were tested by comparing the angular correlation function w(θ), angular power spectrum Cℓ, and projected clustering ξp(r⊥) to theoretical predictions and data. The impact of volume replication in the estimate of the covariance is also investigated. The success in accurately reproducing the photometric redshift uncertainties and the galaxy clustering as a function of redshift render this mock creation pipeline as a benchmark for future analyses of photometric galaxy surveys.Key words: catalogs / large-scale structure of Universe / galaxies: distances and redshifts / Galaxy: halo / methods: numerical

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2020

The Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: exploring the halo occupation distribution model for emission line galaxies

We study the modelling of the halo occupation distribution (HOD) for the eBOSS DR16 emission line galaxies (ELGs). Motivated by previous theoretical and observational studies, we consider different physical effects that can change how ELGs populate haloes. We explore the shape of the average HOD, the fraction of satellite galaxies, their probability distribution function (PDF), and their density and velocity profiles. Our baseline HOD shape was fitted to a semi-analytical model of galaxy formation and evolution, with a decaying occupation of central ELGs at high halo masses. We consider Poisson and sub/super-Poissonian PDFs for satellite assignment. We model both Navarro–Frenk–White and particle profiles for satellite positions, also allowing for decreased concentrations. We model velocities with the virial theorem and particle velocity distributions. Additionally, we introduce a velocity bias and a net infall velocity. We study how these choices impact the clustering statistics while keeping the number density and bias fixed to that from eBOSS ELGs. The projected correlation function, w p , captures most of the effects from the PDF and satellites profile. The quadrupole, ξ 2 , captures most of the effects coming from the velocity profile. We find that the impact of the mean HOD shape is subdominant relative to the rest of choices. Here, we fit the clustering of the eBOSS DR16 ELG data under different combinations of the above assumptions. The catalogues presented here have been analysed in companion papers, showing that eBOSS RSD+BAO measurements are insensitive to the details of galaxy physics considered here. These catalogues are made publicly available.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2021

Monte Carlo simulation of the passage of $\gamma$-rays and $\alpha$-particles in CsI

In this work, theoretical and computational methods for simulating the creation of ionization tracks by fast ions in solids were applied to the passage of a-particles in CsI, an inorganic scintillator commonly used for radiation detection. The methods were implemented in a Monte Carlo program to simulate the interaction of $\alpha$-particles, with incident energies of up to 1 MeV, with CsI. The simulations followed the fate of individual electron-hole pairs and thus allowed for a detailed description of the microscopic structure of ionization tracks created by incident radiation. Simulations were also performed with $\gamma$-rays of the same energy to compare and contrast the ionization tracks obtained with both types of particle. Intrinsic properties such as the mean energy per electron-hole pair, Fano factor, maximum theoretical light yield, and spatial distributions of electron-hole pairs were computed for both $\alpha$-particles and $\gamma$-rays. $\alpha$-particles created cylindrical tracks that were initially aligned with the incident direction and with initial radii of a few nanometers, whereas $\gamma$-rays showed significant scattering, resulting in probability distributions with lower intensities and much greater radial extents.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

DOE OSTI2024

Assessing correlated truncation errors in modern nucleon-nucleon potentials

We test the BUQEYE model of correlated effective field theory (EFT) truncation errors on Reinert, Krebs, and Epelbaum's semilocal momentum-space implementation of the chiral EFT (𝜒⁢EFT ) expansion of the nucleon-nucleon (NN) potential. This Bayesian model hypothesizes that dimensionless coefficient functions extracted from the order-by-order corrections to NN observables can be treated as draws from a Gaussian process (GP). We combine a variety of graphical and statistical diagnostics to assess when predicted observables have a 𝜒⁢EFT convergence pattern consistent with the hypothesized GP statistical model. Our conclusions are that, first, the BUQEYE model is generally applicable to the potential investigated here, which enables statistically principled estimates of the impact of higher EFT orders on observables. Second, parameters defining the extracted coefficients such as the expansion parameter 𝑄 must be well chosen for the coefficients to exhibit a regular convergence pattern—a property we exploit to obtain posterior distributions for such quantities. Third, the assumption of GP stationarity across lab energy and scattering angle is not generally met; this necessitates adjustments in future work. We provide a workflow and interpretive guide for our analysis framework, and show what can be inferred about probability distributions for 𝑄, the EFT breakdown scale Λ 𝑏 , the scale associated with soft physics in the 𝜒⁢EFT potential 𝑚 eff , and the GP hyperparameters. All our results can be reproduced using a publicly available Jupyter notebook, which can be straightforwardly modified to analyze other 𝜒⁢EFT NN potentials.

Bayesian methods↗

DOE OSTI2026

The Poisson tensor completion parametric estimator

We introduce the Poisson tensor completion (PTC) estimator that exploits inter-sample relationships to compute a low-rank Poisson tensor decomposition of the frequency histogram for samples of a multivariate distribution. Our crucial observation is that the histogram bins are an instance of a space partitioning of counts and thus can be identified with a spatial non-homogeneous Poisson process. The Poisson tensor decomposition leads to a completion of the mean measure over all bins—including those containing few to no samples—and leads to our proposed estimator. A Poisson tensor decomposition models the underlying distribution of the count data and guarantees non-negative estimated values obviating the need for additional constraints to ensure non-negativity. Furthermore, we demonstrate that our PTC estimator is a substantial improvement over standard histogram-based estimators for sub-Gaussian probability distributions because of the concentration of norm phenomenon.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Tidal Stream Energy Resource Characterization in the Salish Sea

Tidal stream energy holds great potential as a renewable energy source in regions of the world where tidal currents are strong and predictable. The tidal stream energy resource of a region is strongly controlled by its tidal wave characteristics, the local bathymetry, and coastal geometric features. The Salish Sea, a large estuary on the Pacific Northwest coast, represents a great tidal stream resource because of its strong tidal currents in many tidal channels. However, the tidal resource in the Salish Sea has not been systematically characterized, primarily because of its large area and complex bathymetry and coastlines. This paper presents a modeling study conducted to characterize the tidal energy resource of the Salish Sea based on a high-resolution three-dimensional tidal hydrodynamic model of the Salish Sea, which was extensively validated using data derived from 10 tidal gauge stations and 132 Acoustic Doppler Current Profiler stations. Model validation results indicated the Salish Sea hydrodynamic model is skillful in simulating tidal wave propagation and velocity distributions in the Salish Sea. Based on model results, a total of 16 tidal channels with strong tidal currents, 9 in the San Juan Islands and Rosario Strait regions and 7 in the Puget Sound, were identified as hotspots for potential tidal energy development. Velocity probability distributions and exceedance curves of cross-channel average velocity magnitudes were calculated at all 16 channels based on the recommendations of the International Electrotechnical Commission Technical Specifications for tidal energy resource characterization. The tidal energy resource at the 16 hotspots was also characterized using power density distributions and kinetic energy fluxes. The ranking of the kinetic energy fluxes suggested that Admiralty Inlet, Rosario Strait, and Middle Channel are the top three hotspot sites in the Salish Sea for tidal stream energy development. The study demonstrated the need for a high-resolution 3-D modeling framework for accurate simulation of tidal currents in large complex estuarine systems, in the context of tidal energy resource characterization.

tidal stream energy, resource characterization, Sa↗

DOE OSTI2024

Innovative dead-time correction and background subtraction for neutron multiplicity measurements using neural networks

Abstract The number of neutrons emitted from a nuclear reaction plays a crucial role in various fields, including nuclear theory, nuclear nonproliferation, nuclear energy and nuclear criticality safety. Accurate determination of neutron multiplicities requires the application of several corrections, with dead-time correction and background subtraction being particularly significant. These corrections become more challenging for neutron detectors with time-dependent neutron capture. In this work, we perform a comprehensive study of three existing methods used for dead-time correction and background subtraction in neutron detectors with time-dependent neutron capture. The methods were tested for dead-times in the range from 0 to 1 μs using a Monte Carlo model simulating the dead-time and background effects in the standard neutron multiplicity probability distribution of $$^{252}$$ 252 Cf. The previous methods showed larger than desired uncertainty or systematic trade off. Those uncertainties prompted the development of a novel approach using neural networks trained with data from Monte Carlo simulations. The Neural Network method enabled the correction of neutron multiplicity probabilities more accurately than the other methods with fractional errors smaller than 3% for multiplicities around the peak of $$^{252}$$ 252 Cf. A similar approach using neural networks could be applied to problems where the system being studied can be accurately simulated without having an accurate analytical description available. The neural network method presented in this paper can be easily expanded if multiplicities greater than 10 are expected.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2025

The Poisson tensor completion non-parametric differential entropy estimator

We introduce the Poisson tensor completion (PTC) estimator, a non-parametric differential entropy estimator. The PTC estimator leverages inter-sample relationships to compute a low-rank Poisson tensor decomposition of the frequency histogram. Our crucial observation is that the histogram bins are an instance of a space partitioning of counts and thus can be identified with a spatial Poisson process. The Poisson tensor decomposition leads to a completion of the intensity measure over all bins—including those containing few to no samples—and leads to our proposed PTC differential entropy estimator. A Poisson tensor decomposition models the underlying distribution of the count data and guarantees non-negative estimated values and so can be safely used directly in entropy estimation. Our estimator is the first tensor-based estimator that exploits the underlying spatial Poisson process related to the histogram explicitly when estimating the probability density with low-rank tensor decompositions for the purpose of tensor completion. Furthermore, we demonstrate that our PTC estimator is a substantial improvement over standard histogram-based estimators for sub-Gaussian probability distributions because of the concentration of norm phenomenon.

42 ENGINEERING↗

DOE OSTI2022

Economics of Crop Rotations With and Without Carinata for Sustainable Aviation Fuel Production in the SE United States

In 2019, the aviation sector in the United States emitted 255 million metric tons of carbon dioxide (CO 2 ) emissions, i.e., about five percent of the total domestic CO 2 emissions from the energy sector. The sustainable aviation fuel (SAF) derived from carinata (Brassica carinata) could reduce CO 2 emissions of the aviation sector in the United States. Therefore, it is important to estimate changes in farm economics with and without carinata for ascertaining its production feasibility. In this context, the current study first assesses a combination of 12 popular rotations of corn, cotton, peanut, and soybean with winter crops of winter wheat and carinata in South Georgia over 4 years. Then, the net present values (NPVs) of 292 feasible cropping systems over 4 years are calculated. Finally, this study develops a risk model for ascertaining the probability distributions of NPVs for selected cropping systems subject to uncertainties related to prices and yields of summer and winter crops. Carinata in the corn-corn-soybean rotation has the highest NPV ($\$$2,996/ha). The least risky rotation is cotton-cotton-peanut, with a 58.9% probability of a positive NPV. Carinata can decrease the risk level of crop rotations by 8.1%, only if a contract price of $\$$440.9/t is offered. Therefore, a risk averse, risk neutral, or risk acceptant farmer can potentially include carinata in the rotation. Overall, carinata would increase the profitability of farm operations and decrease risk in the SE United States, and therefore, a high likelihood exists, that farmers would adopt it for meeting the growing demand for SAF in the United States.

09 BIOMASS FUELS↗

DOE OSTI2023

Search for low mass dark matter in DarkSide-50: the bayesian network approach

Abstract We present a novel approach for the search of dark matter in the DarkSide-50 experiment, relying on Bayesian Networks. This method incorporates the detector response model into the likelihood function, explicitly maintaining the connection with the quantity of interest. No assumptions about the linearity of the problem or the shape of the probability distribution functions are required, and there is no need to morph signal and background spectra as a function of nuisance parameters. By expressing the problem in terms of Bayesian Networks, we have developed an inference algorithm based on a Markov Chain Monte Carlo to calculate the posterior probability. A clever description of the detector response model in terms of parametric matrices allows us to study the impact of systematic variations of any parameter on the final results. Our approach not only provides the desired information on the parameter of interest, but also potential constraints on the response model. Our results are consistent with recent published analyses and further refine the parameters of the detector response model.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2024

From Latent Dynamics to Meaningful Representations

While representation learning has been central to the rise of machine learning and artificial intelligence, a key problem remains in making the learnt representations meaningful. For this the typical approach is to regularize the learned representation through prior probability distributions. However such priors are usually unavailable or are ad hoc. To deal with this, recent efforts have shifted towards leveraging the insights from physical principles to guide the learning process. In this spirit, we propose a purely dynamics-constrained representation learning framework. Instead of relying on predefined probabilities, we restrict the latent representation to follow overdamped Langevin dynamics with a learnable transition density — a prior driven by statistical mechanics. We show this is a more natural constraint for representation learning in stochastic dynamical systems, with the crucial ability to uniquely identify the ground truth representation. We validate our framework for different systems including a real-world fluorescent DNA movie dataset. Here, we show that our algorithm can uniquely identify orthogonal, isometric and meaningful latent representations.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

Multi-variate model of T cell clonotype competition and homeostasis

Abstract Diversity of the naive T cell repertoire is maintained by competition for stimuli provided by self-peptides bound to major histocompatibility complexes (self-pMHCs). We extend an existing bi-variate competition model to a multi-variate model of the dynamics of multiple T cell clonotypes which share stimuli. In order to understand the late-time behaviour of the system, we analyse: (i) the dynamics until the extinction of the first clonotype, (ii) the time to the first extinction event, (iii) the probability of extinction of each clonotype, and (iv) the size of the surviving clonotypes when the first extinction event takes place. We also find the probability distribution of the number of cell divisions per clonotype before its extinction. The mean size of a new clonotype at quasi-steady state is an increasing function of the stimulus available to it, and a decreasing function of the fraction of stimuli it shares with other clonotypes. Thus, the probability of, and time to, extinction of a new clonotype entering the pool of T cell clonotypes is determined by the extent of competition for stimuli it experiences and by its initial number of cells.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2021

Energy sector portfolio analysis with uncertainty

Governments are dealing with the challenge of how to efficiently invest in research and development portfolios related to energy technologies. Research and development investment decisions in the energy space are especially difficult due to numerous risks and uncertainties, and due to the complexity of energy's interactions with the broad economy. Historically, much of the U.S. Department of Energy's in-depth research and development analyses focused on assessing the impact of a research and development activity in isolation from other available opportunities and did not substantially consider risk and uncertainty. Endeavoring to combine integrated energy-economy modeling with uncertainty analysis and technology-specific research and development activities, the U.S. Department of Energy commissioned the development of the Stochastic Energy Deployment System to support and improve public energy research and development decision-making. The Stochastic Energy Deployment System draws from expert-elicited probability distributions for research and development-driven improvements in technology cost and performance, and it uses Monte Carlo simulations to evaluate the likelihood of outcomes within a system dynamics energy-economy model. The framework estimates the uncertain benefits and costs of various research and development portfolios and provides insight into the probability of meeting national technology goals, while accounting for interactions with the larger economy and for interactions among research and development investments spanning many energy sectors.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗

DOE OSTI2023

Projecting Future Energy Production from Operating Wind Farms in North America. Part I: Dynamical Downscaling

Abstract New simulations at 12-km grid spacing with the Weather and Research Forecasting (WRF) Model nested in the MPI Earth System Model (ESM) are used to quantify possible changes in wind power generation potential as a result of global warming. Annual capacity factors (CF; measures of electrical power production) computed by applying a power curve to hourly wind speeds at wind turbine hub height from this simulation are also used to illustrate the pitfalls in seeking to infer changes in wind power generation directly from low-spatial-resolution and time-averaged ESM output. WRF-derived CF are evaluated using observed daily CF from operating wind farms. The spatial correlation coefficient between modeled and observed mean CF is 0.65, and the root-mean-square error is 5.4 percentage points. Output from the MPI-WRF Model chain also captures some of the seasonal variability and the probability distribution of daily CF at operating wind farms. Projections of mean annual CF (CF A ) indicate no change to 2050 in the southern Great Plains and Northeast. Interannual variability of CF A increases in the Midwest, and CF A declines by up to 2 percentage points in the northern Great Plains. The probability of wind droughts (extended periods with anomalously low production) and wind bonus periods (high production) remains unchanged over most of the eastern United States. The probability of wind bonus periods exhibits some evidence of higher values over the Midwest in the 2040s, whereas the converse is true over the northern Great Plains. Significance Statement Wind energy is playing an increasingly important role in low-carbon-emission electricity generation. It is a “weather dependent” renewable energy source, and thus changes in the global atmosphere may cause changes in regional wind power production (PP) potential. We use PP data from operating wind farms to demonstrate that regional simulations exhibit skill in capturing actual power production. Projections to the middle of this century indicate that over most of North America east of the Rocky Mountains annual expected PP is largely unchanged, as is the probability of extended periods of anomalously high or low production. Any small declines in annual PP are of much smaller magnitude than changes due to technological innovation over the last two decades.

Meteorology & Atmospheric Sciences↗

DOE OSTI2022

Learning generative neural networks with physics knowledge

Deep generative neural networks have enabled modeling complex distributions, but incorporating physics knowledge into the neural networks is still challenging and is at the core of current physics-based machine learning research. To this end, we propose a physics generative neural network (PhysGNN), a new class of generative neural networks for learning unknown distributions in a physical system described by partial differential equations (PDE). PhysGNN couples PDE systems with generative neural networks. It is a fully differentiable model that allows back-propagation of gradients through both numerical PDE solvers and generative neural networks, and is trained by minimizing the discrete Wasserstein distance between generated and observed probability distributions of the PDE outputs using the stochastic gradient descent method. Moreover, PhysGNN does not require adversarial training like standard generative neural networks, which offers better stability than adversarial training. We show that PhysGNN can learn complex distributions in stochastic inverse problems, where conventional methods such as maximum likelihood estimation and momentum matching methods may be inapplicable when little knowledge is known about the form of unknown distributions or the physical model is too complex. Furthermore, our method allows physics-based generative neural network training for learning complex distributions in the context of differential equations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Correlated $\ n-γ$ angular distributions from the $\ Q$ = 4.4398 MeV 12 C ($\ n, n' γ$) reaction for incident neutron energies from 6.5 MeV to 16.5 MeV

Neutron scattering cross sections and angular distributions represent one of the most glaring sources of uncertainty in calculations of nuclear systems. Even simple nuclei like 12 C show indications of errors in nuclear databases for scattering reactions. Measurements of inelastic neutron scattering have historically measured either the scattered neutrons or the nuclear deexcitation $\ γ$ emission. Only a very small number of experiments attempted correlated measurements of both the neutron and $\ γ$ data simultaneously, even though these $\ n-γ$ correlations could be essential for understanding particle transport in nuclear systems. In this work we describe a measurement of the $\ n, γ$, and correlated $\ n-γ$ angular distributions from the $\ Q$ = 4.4398 MeV 12 C ($\ n, n'γ$) reaction in a single experiment using an EJ-309 liquid scintillator detector array with wide angular coverage, and with a continuous incident neutron energy range from 6.5 to 16.5 MeV. We also provide a thorough covariance description of these results, including normalization of the probability distribution. While the measured n distributions agree well with the relatively large number of available literature measurements, there are comparatively very few measurements of the γ distributions from this reaction. However, our data support the presence of a nonzero α 4 Legendre polynomial component of the γ angular distribution suggested in past measurements, which is currently not incorporated in the ENDF/B-VIII.0 library despite the use of these same literature data for evaluation of the 12 C ($\ n, n'γ$) cross section. The correlated $\ n-γ$ distribution measurements are limited to three measurements at incident neutron energies near 14 MeV. Our results do not generally agree with any of these literature measurements. We observe clear indications of significant changes in the $\ n$ distribution for specific $\ γ$-detection angles and vice versa especially near thresholds for other reaction channels, which shows the potential for significant bias in experiments that, for example, tag on inelastic scattering using a single or small number of $\ γ$ -detection angles and could impact particle transport calculations.

6 ≤ A ≤ 19↗

DOE OSTI2022

How Many Elements Matter?

Some studies of stars' multielement abundance distributions suggest at least 5–7 significant dimensions, but others show that many elemental abundances can be predicted to high accuracy from [Fe/H] and [Mg/Fe] (or [Fe/H] and age) alone. We show that both propositions can be, and are, simultaneously true. We adopt a machine-learning technique known as normalizing flow to reconstruct the probability distribution of Milky Way disk stars in the space of 15 elemental abundances measured by APOGEE. Conditioning on T eff and $\mathrm{log}\,g$ minimizes the differential systematics. After further conditioning on [Fe/H] and [Mg/Fe], the residual scatter for most abundances is σ [X/H] ≲ 0.02 dex, consistent with APOGEE's reported statistical uncertainties of ~0.01–0.015 dex and intrinsic scatter of 0.01–0.02 dex. Despite the small scatter, residual abundances display clear correlations between elements, which we show are too large to be explained by measurement uncertainties or by the finite sampling noise. We must condition on at least seven elements to reduce the correlations to a level consistent with the observational uncertainties. Our results demonstrate that cross-element correlations are a much more sensitive probe of a hidden structure than dispersion, and they can be measured precisely in a large sample even if the star-by-star measurement noise is comparable to the intrinsic scatter. We conclude that many elements have an independent story to tell, even for the mundane disk stars and elements produced by the core-collapse and Type Ia supernovae. The only way to learn these lessons is to measure the abundances directly, and not merely infer them.

79 ASTRONOMY AND ASTROPHYSICS↗