Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Statistical simulation”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 1,099 records · Page 61

Open-Loop Control of Adjustable Tuned Mass Dampers for Floating Wind Turbine Platforms

In this study, we demonstrate the ability of adjustable tuned mass dampers (TMDs) to reduce the platform motion of floating offshore wind turbines (FOWTs). The TMDs are located in the hull and provide control authority by varying the amount of water ballast and compressed air in a reservoir over time. The optimal TMD settings depend on the wind and wave conditions of the FOWT. We present an open-loop control scheme that changes the TMD natural frequency based on sea state and a simple method for estimating the peak wave period and significant wave height. The performance of this open-loop control is compared to ideal control of the TMDs (set knowing the exact sea state), a TMD with a constant natural frequency targeting the design load case with the greatest platform motion, and a baseline set of simulations of the platform without TMDs. The constant natural frequency case performs nearly as effectively as the actively controlled case, suggesting that the TMDs can be simply designed for extreme load conditions and provide consistent control over a range of environments; however, the methods used to find the optimal TMD parameters and estimate sea state statistics could be used in other aspects of FOWT control and design.

artificial intelligence↗

Mapping the structural–mechanical landscape of amorphous carbon with ReaxFF molecular dynamics

We use ReaxFF molecular dynamics (MD) to investigate the relationship between structural and mechanical properties in bulk and nanostructured amorphous carbon (a-C). The liquid-quench MD method is used to generate isotropic bulk samples with mass densities ranging from 0.96 to 3.29 g/cm3. Structural analysis identifies two types of structures with distinct short- and medium-range order: lower-density sp2-dominated a-C, which is characterized by a bimodal ring-size distribution, and higher-density sp3-dominated tetrahedral amorphous carbon (ta-C), exhibiting a unimodal ring-size distribution. Stress–strain MD simulations and analysis reveal how an atomistic structure impacts elastic properties and post-yield atomic rearrangements. All stretched structures demonstrate elastic isotropy and plasticity driven by a ring-size expansion mechanism reflected in changes in ring statistics. The plastic region is substantially larger in ta-C than in a-C due to the post-yield shift from sp3 to sp2 C dominant bonding. In both a-C and ta-C, ultimate failure occurs when a reactive crack, traversed by long sp chains, forms and propagates predominantly perpendicular to the direction of the applied strain. Oxygen infiltration into the fractured region significantly reduces stress resistance, primarily through the early rupture of long sp chains. MD simulations and analysis are extended to a-C slabs, a-C nanotubes, and partially a-C nanotubes. The latter nanostructure highlights the differences between the elastically isotropic a-C walls, which develop circumferential cracking, and the crystalline walls, which tear along crystallographic directions. These results provide a strong foundation for further computational characterization of a-C materials.

Dernov, A. (ORCID:0009000004220973)↗

Evaluation of probability density function descriptions for three-component Rayleigh–Taylor mixing

Results from simulations of a three-component Rayleigh–Taylor (RT) mixing problem are presented. These simulations are conducted in heavy–light–heavy and heavy–intermediate–light configurations, and each of these configurations are further considered in high- and low-Reynolds-number regimes. This results in RT-unstable flow with one or both interfaces initially unstable, permitting the influence of problem configuration on the statistical description of three-component RT-driven mixing to be considered. Mass fraction covariances are observed to undergo a sign change through the mixing layer in all four configurations considered. This appears to be unique to the multi-component case and represents another way in which multi-component RT mixing differs from the two-component case. Qualitative and quantitative comparisons of joint and marginal probability density function (PDF) descriptions of species concentration are made. Three-, five-, and six-parameter model PDFs are compared against simulation data to assess how accurately they describe the mixing, and it is found that three-component mixing requires at least a five-parameter model PDF to accurately describe the mixing. Notably, the marginal distributions of three-component mixing do not appear to conform to a beta distribution, representing a departure from the classical two-component RT case. In conclusion, statistical neutrality also appears to influence the optimal choice of model PDF, which is found to be a function of problem configuration.

Large-eddy simulation↗

Statistical uncertainties of the v n { 2 k } harmonics from Q cumulants

Analytic formulas to calculate statistical uncertainties of v n { 2 k } harmonics extracted from the Q cumulants are presented. The Q cumulants are multivariate polynomial functions of the weighted means of 2m-particle azimuthal correlations, $\langle\langle 2 m \rangle\rangle$. Variances and covariances of the $\langle\langle 2 m \rangle\rangle$ are included in the analytic formulas of the uncertainties that can be calculated simultaneously with the calculations of the v n { 2 k } harmonics. The calculations are performed using a simple toy model, which roughly simulates elliptic flow azimuthal anisotropy with magnitudes around 0.05. The results are compared with the results obtained by the many data subsets, and by the bootstrapping method. The first one is a common way of estimation of the statistical uncertainties of the v n { 2 k } harmonics in a real experiment. In order to increase precision in the measurement of the v n { 2 k } harmonics, a large number of 15 000 subsets and the same number of the resampling in the bootstrap method is used. Unlike the other ways of the analytic calculation of the v n { 2 k } statistical uncertainties, our proposal that includes the use of squared weights in the calculation of both the variances and covariances, gives the best agreement with the results obtained from the subsets and bootstrap method. Additionally, a recurrence equation between Q cumulants of any order is also presented.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Large-eddy simulations of turbulent flows in arrays of helical- and straight-bladed vertical-axis wind turbines

Effects of helical-shaped blades on the flow characteristics and power production of finite-length wind farms composed of vertical-axis wind turbines (VAWTs) are studied numerically using large-eddy simulation (LES). Two helical-bladed VAWTs (with opposite blade twist angles) are studied against one straight-bladed VAWT in different array configurations with coarse, intermediate, and tight spacings. Statistical analysis of the LES data shows that the helical-bladed VAWTs can improve the mean power production in the fully developed region of the array by about 4.94%–7.33% compared with the corresponding straight-bladed VAWT cases. The helical-bladed VAWTs also cover the azimuth angle more smoothly during the rotation, resulting in about 47.6%–60.1% reduction in the temporal fluctuation of the VAWT power output. Using the helical-bladed VAWTs also reduces the fatigue load on the structure by significantly reducing the spanwise bending moment (relative to the bottom base), which may improve the longevity of the VAWT system to reduce the long-term maintenance cost.

17 WIND ENERGY↗

Effects of overlapping sources on cosmic shear estimation: Statistical sensitivity and pixel-noise bias

The next generation of dark-energy imaging surveys — so called “Stage-IV” surveys, such as that of the Rubin Observatory Legacy Survey of Space and Time (LSST) — will cross a threshold in the number density of detected sources on the sky that requires qualitatively different image analysis and measurement techniques compared to the current generation of Stage-III surveys. In Stage-IV surveys, a significant amount of the cosmologically useful information is due to sources whose images overlap with those of other sources on the sky. Here, we focus on the weak gravitational lensing probe, for which we expect the largest impact since the cosmic shear signal is primarily encoded in the estimated shapes of observed galaxies and thus directly impacted by overlaps. We introduce a framework based on the Fisher formalism to analyze the effect of the overlapping sources (“blending”) on the estimation of cosmic shear. This method gives concrete predictions for the minimum loss of information due to noise and blending for any choice of “deblending” scheme and shape-measurement algorithm. Our studies account for undetected sources but do not address their full effects and biases they may introduce. We use simulated images and predict this impact of blending for three surveys: the Dark Energy Survey (DES), the Hyper-Suprime Cam Subaru Strategic Program (HSC-SSP), and the Rubin LSST. Our methodology successfully estimates the statistical sensitivity to weak lensing for DES and HSC early results. For LSST, we present the expected loss in statistical sensitivity for the ten-year survey due to blending. We find that for approximately 62% of galaxies that are likely to be detected in full-depth LSST images, at least 1% of the flux in their pixels is from overlapping sources. We also find that the statistical correlations between measures of overlapping galaxies and, to a much lesser extent (0.2%) the higher shot noise level due to their presence, decrease the effective number density of galaxies, N eff , by ~ 18%. We calculate an upper limit on N eff of 39.4 galaxies per arcmin 2 in r band. We study the impact of stars on as a function of stellar density and illustrate the diminishing returns of extending the survey into lower Galactic latitudes. We extend the simulation-based Fisher formalism to predict the expected increase in pixel-noise bias due to blending for maximum-likelihood (ML) shape estimators. We find that noise bias depends sensitively on the particular shape estimator and measure of ensemble-average shape that is used, and properties of the galaxy that include redshift-dependent quantities such as size and luminosity. The source code for these studies is available online.[The documented software developed for the catalog-level studies are available in the open-source LSST DESC github repository https://github.com/LSSTDESC/WeakLensingDeblending. The software for analyzing one or two galaxies with user-defined parameters is in the open-source github repository https://github.com/ismael-mendoza/ShapeMeasurementFisherFormalism.]

79 ASTRONOMY AND ASTROPHYSICS↗

Windowing artefacts likely account for recent claimed detection of oscillating cosmic scale factor

ABSTRACT Using the Pantheon data set of Type Ia supernovae, a recent publication (R20 in this work) reports a 2σ detection of oscillations in the expansion history of the Universe. The study conducted by R20 is wholly worthwhile. However, we demonstrate that there is a $\gt 10{{\ \rm per\ cent}}$ chance of statistical fluctuations in the Pantheon data producing a false oscillatory signal larger than the oscillatory signal that R20 report. Their results are a less than 2σ detection. Applying the R20 methodology to simulated Pantheon data, we determine that these oscillations could arise due to analysis artefacts. The uneven spacing of Type Ia supernovae in redshift space and the complicated analysis method of R20 impose a structured throughput function. When analysed with the R20 prescription, about $11{{\ \rm per\ cent}}$ of artificial ΛCDM data sets produce a stronger oscillatory signal than the actual Pantheon data. Our results underscore the importance of understanding the false ‘signals’ that can be introduced by complicated data analyses.

79 ASTRONOMY AND ASTROPHYSICS↗

High-Burnup LOCA Burst Susceptibility BISON Analysis in PWRs and BWRs

Accurately assessing high-burnup fuel behavior during loss-of-coolant accidents (LOCAs) is essential for understanding fuel fragmentation, relocation, and dispersal (FFRD) risks across the US light-water reactor fleet. This work updates previous Nuclear Energy Advanced Modeling and Simulation (NEAMS) Program multiphysics LOCA analyses for a pressurized water reactor (PWR) and a boiling water reactor (BWR) by incorporating recent model and material property advancements in the BISON fuel performance code, including a high-burnup structure (HBS) model, revised cladding burst criteria, and updated thermal–mechanical correlations. This update was needed to support ongoing industry initiatives and upcoming regulatory changes. Full-core, rod-resolved operating histories generated using Virtual Environment for Reactor Analysis (VERA) and system-level LOCA conditions obtained from TRACE were applied to statistically representative rod samples in BISON to evaluate burst behavior and FFRD susceptibility. These calculations used two cladding burst correlations and three fuel pulverization models so that the predictions of these models could be compared. The updated PWR simulations show markedly improved numerical stability as the number of crashed simulations decreased by 95% compared to the previous study, and hence higher confidence in results. The updated PWR simulations predicted cladding bursts exclusively among once-burned, high-power rods, with two different cladding burst models identifying the same burst-susceptible population. Resulting FFRD susceptibility estimates are significantly reduced compared with earlier studies, driven by cooler predicted fuel and plenum temperatures, lower hoop strains, and reduced fission gas release in the updated models. In contrast, none of the BWR rods were predicted to burst under either burst criterion, reaffirming minimal BWR FFRD susceptibility even with updated HBS and material models. Comparisons between the PWR and BWR end-of-cycle predictions are made. Comparison with prior work highlights significant shifts in PWR fuel performance metrics and confirmation of earlier BWR conclusions. Overall, the updated results underscore the importance of having high-resolution detailed modeling capability and continuously integrating evolving material models and physics into high-resolution multiphysics simulations. The unified assessment presented here strengthens confidence in predicting high-burnup LOCA behavior by improving agreement between different cladding burst correlations. These results also provide an improved foundation for future BISON model development, FFRD susceptibility calculations.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

A Broad Grid of 2D Kilonova Emission Models

Depending upon the properties of their compact remnants and the physics included in the models, simulations of neutron star mergers can produce a broad range of ejecta properties. The characteristics of this ejecta, in turn, define the kilonova emission. To explore the effect of ejecta properties, we present a grid of 2-component 2D axisymmetric kilonova simulations that vary mass, velocity, morphology, and composition. The masses and velocities of each component vary, respectively, from 0.001 to 0.1 M ⊙ and 0.05 to 0.3c, covering much of the range of results from the neutron star merger literature. The set of 900 models is constrained to have a toroidal low electron fraction (Y e ) ejecta with a robust r-process composition and either a spherical or lobed high-Y e ejecta with two possible compositions. We simulate these models with the Monte Carlo radiative transfer code SuperNu using a full suite of lanthanide and 4th row element opacities. We examine the trends of these models with parameter variation, show how it can be used with statistical tools, and compare the model light curves and spectra to those of AT2017gfo, the electromagnetic counterpart of GW170817.

79 ASTRONOMY AND ASTROPHYSICS↗

Beyond the Condon limit: Condensed phase optical spectra from atomistic simulations

While dark transitions made bright by molecular motions determine the optoelectronic properties of many materials, simulating such non-Condon effects in condensed phase spectroscopy remains a fundamental challenge. We derive a Gaussian theory to predict and analyze condensed phase optical spectra beyond the Condon limit. Our theory introduces novel quantities that encode how nuclear motions modulate the energy gap and transition dipole of electronic transitions in the form of spectral densities. By formulating the theory through a statistical framework of thermal averages and fluctuations, we circumvent the limitations of widely used microscopically harmonic theories, allowing us to tackle systems with generally anharmonic atomistic interactions and non-Condon fluctuations of arbitrary strength. We show how to calculate these spectral densities using first-principles simulations, capturing realistic molecular interactions and incorporating finite-temperature, disorder, and dynamical effects. Our theory accurately predicts the spectra of systems known to exhibit strong non-Condon effects (phenolate in various solvents) and reveals distinct mechanisms for electronic peak splitting: timescale separation of modes that tune non-Condon effects and spectral interference from correlated energy gap and transition dipole fluctuations. We further introduce analysis tools to identify how intramolecular vibrations, solute–solvent interactions, and environmental polarization effects impact dark transitions. Moreover, we prove an upper bound on the strength of cross correlated energy gap and transition dipole fluctuations, thereby elucidating a simple condition that a system must follow for our theory to accurately predict its spectrum.

Chemistry↗

Coupling localized Noah-MP-Crop model with the WRF model improved dynamic crop growth simulation across Northeast China

Croplands play a critical role in regulating the energy and moisture exchanges between the land surface and atmosphere. However, the interactions between cropland and climate are usually poorly represented due to a lack of detailed representation in crop types and field management. Here, we coupled the Noah-MP-Crop model with the state-of-the-art Weather Research and Forecasting (WRF) model to explore and evaluate the crop growth dynamics in response to climate variations across Northeast China. The default parameters of the crop model were not exactly suitable for the agricultural ecosystems in Northeast China. The detailed cropland distribution, and crop phenology parameters including growing degree days (GDD) and planting (harvesting) date were first created using multi-source remote sensing products and reanalysis data, and was then successfully used to simulate the growth and yield for corn and soybean and associated energy exchanges. We also optimized and calibrated other crop parameters using the time-series of the Moderate Resolution Imaging Spectroradiometer (MODIS) land surface products. The modified crop model substantially improved the simulation of crop growth, plant physiology, and biomass accumulation for both corn and soybean. Coupling the localized dynamic crop model into the WRF led to considerable decreases in the simulated mean-absolute-errors (MAEs) and biases of the leaf area index, evapotranspiration, and gross primary production compared with the MODIS observed values. Compared with the statistical yield from each province, the modified crop model underestimated the corn yield from 11.1% to 48.6%, whereas overestimated the soybean yield from 16.5% to 162.6%.

54 ENVIRONMENTAL SCIENCES↗

Statistical characterization of a shock interacting with an inclined gas column

We present a statistical characterization of the interaction between a planar shock and a finite-diameter, cylindrical column of dense gas based on three-dimensional, large-eddy simulation results. In the simulation, the column of gas is initially inclined at an angle $\alpha _0$ with respect to the shock plane. Effects of the initial column angle on the mixing characteristics are examined at Mach number 2.0 for column incline angles $1^\circ$ , $5^\circ$ , $10^\circ$ and $30^\circ$ . Mean velocity profiles show that the column angle affects the gas velocity components in the vertical plane, but not in the spanwise direction. The gas undergoes higher initial upward acceleration at larger initial column incline angles. With time, the gas motion tends to become one-dimensional in the streamwise direction. Initially, velocity fluctuations are most intense within the interior of the column, but concentrate near the column leading edge over time. At high wavenumbers $\kappa$ , the turbulent kinetic energy spectra follow a power-law scaling of $\kappa ^{-1}$ . The structure functions of the mass fraction do not clearly demonstrate power-law scaling except at early times for $\alpha _0=30^\circ$ , manifesting overall trends very similar to those observed in earlier experiments. Probability distributions of the mass fraction show independence of the mean and the standard deviation of the mixed gas on $\alpha _0$ . The column angle was also found to have little effect on the mixing efficiency characterized by the molecular mixedness. Velocity components in the streamwise and transverse directions tend towards a bimodal distribution for larger $\alpha _{0}$ .

42 ENGINEERING↗

Pressure Effects on the Relaxation of an Excited Ethane Molecule in High-Pressure Bath Gases

Here, we use molecular dynamics to calculate the rotational and vibrational energy relaxation of C 2 H 6 in Ar, Kr, and Xe bath gases over a pressure range of 10 to 400 atm and at temperatures of 300 K and 800 K. The C 2 H 6 is instantaneously excited by 80 kcal/mol randomly distributed into both vibrational and rotational modes. The computed relaxation rates show little sensitivity to the identity of the noble gas in the bath. Vibrational relaxation rates show a non-linear pressure dependence at 300 K. At 800 K the reduced range of bath gas densities covered by the range of pressures do not yet show any non-linearity in the pressure dependence. Rotational relaxation is characterized with two relaxation rates. The slower rate is comparable to the vibrational relaxation rate. The faster rate has a linear pressure dependence at 300 K but an irregular, nonlinear pressure dependence at 800 K. To understand this, a model was developed based on approximating the periodic box used in the molecular dynamics simulations by an equal-volume collection of cubes where each cube is sized to allow only single occupancy by the noble gas or the molecule. Combinatorial statistics then leads to a pressure and temperature dependent analytic distribution of the bath gas species the molecule encounters in a collision. This distribution, the dissociation energy of molecule/bath gas complexes and bath gas clusters, and the computed energy release per collision combine to show that only at 300 K is the energy release sufficient to dissociate likely complexes and clusters. This suggests that persistent and pressure-dependent clusters and complexes at 800 K may be responsible for the non-linear pressure dependence of rotational relaxation.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Deep potential generation scheme and simulation protocol for the Li 10 GeP 2 S 12 -type superionic conductors

We report solid-state electrolyte materials with superior lithium ionic conductivities are vital to the next-generation Li-ion batteries. Molecular dynamics could provide atomic scale information to understand the diffusion process of Li-ion in these superionic conductor materials. Here, we implement the deep potential generator to set up an efficient protocol to automatically generate interatomic potentials for Li 10 GeP 2 S 12 -type solid-state electrolyte materials (Li 10 GeP 2 S 12 , Li 10 SiP 2 S 12 , and Li 10 SnP 2 S 12 ). The reliability and accuracy of the fast interatomic potentials are validated. With the potentials, we extend the simulation of the diffusion process to a wide temperature range (300 K–1000 K) and systems with large size (~1000 atoms). Important technical aspects such as the statistical error and size effect are carefully investigated, and benchmark tests including the effect of density functional, thermal expansion, and configurational disorder are performed. The computed data that consider these factors agree well with the experimental results, and we find that the three structures show different behaviors with respect to configurational disorder. Our work paves the way for further research on computation screening of solid-state electrolyte materials.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Bringing discrete-time Langevin splitting methods into agreement with thermodynamics

In light of the recently published complete set of statistically correct Grønbech–Jensen (GJ) methods for discrete-time thermodynamics, we revise a differential operator splitting method for the Langevin equation in order to comply with the basic GJ thermodynamic sampling features, namely, the Boltzmann distribution and Einstein diffusion, in linear systems. This revision, which is based on the introduction of time scaling along with flexibility of a discrete-time velocity attenuation parameter, provides a direct link between the ABO splitting formalism and the GJ methods. This link brings about the conclusion that any GJ method has at least weak second order accuracy in the applied time step. It further helps identify a novel half-step velocity, which simultaneously produces both correct kinetic statistics and correct transport measures for any of the statistically sound GJ methods. Explicit algorithmic expressions are given for the integration of the new half-step velocity into the GJ set of methods. Finally, numerical simulations, including quantum-based molecular dynamics (QMD) using the QMD suite Los Alamos Transferable Tight-Binding for Energetics, highlight the discussed properties of the algorithms as well as exhibit the direct application of robust, time-step-independent stochastic integrators to QMD.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

The picasso map-making code: application to a simulation of the QUIJOTE northern sky survey

ABSTRACT Map-making is an important step for the data analysis of cosmic microwave background (CMB) experiments. It consists of converting the data, which are typically a long, complex, and noisy collection of measurements, into a map, which is an image of the observed sky. We present in this paper a new map-making code named picasso (Polarization and Intensity CArtographer for Scanned Sky Observations), which was implemented to construct intensity and polarization maps from the Multi Frequency Instrument (MFI) of the QUIJOTE (Q-U-I Joint TEnerife) CMB polarization experiment. picasso is based on the destriping algorithm, and is suited to address specific issues of ground-based microwave observations, with a technique that allows the fit of a template function in the time domain, during the map-making step. This paper describes the picasso code, validating it with simulations and assessing its performance. For this purpose, we produced realistic simulations of the QUIJOTE-MFI survey of the northern sky (approximately ∼20 000 deg2), and analysed the reconstructed maps with picasso, using real and harmonic space statistics. We show that, for this sky area, picasso is able to reconstruct, with high fidelity, the injected signal, recovering all the scales with ℓ > 10 in TT, EE, and BB. The signal error is better than 0.001 per cent at 20 < ℓ < 200. Finally, we validated some of the methods that will be applied to the real wide-survey data, like the detection of the CMB anisotropies via cross-correlation analyses. Despite that the implementation of picasso is specific for QUIJOTE-MFI data, it could be adapted to other experiments.

79 ASTRONOMY AND ASTROPHYSICS↗

Neural network emulation of flow in heavy-ion collisions at intermediate energies

Applications of new techniques in machine learning are speeding up progress in research in various fields. In this work, we construct and evaluate a deep neural network (DNN) to be used within a Bayesian statistical framework as a faster and more reliable alternative to the Gaussian process (GP) emulator of an isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU) transport model simulator of heavy-ion reactions at intermediate beam energies. We found strong evidence of the DNN being able to emulate the IBUU simulator's prediction on the strengths of protons' directed and elliptical flow very efficiently even with small training datasets and with accuracy about ten times higher than the GP. Here, limitations of our present work and future improvements are also discussed.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Fast radio burst dispersion measure distribution as a probe of helium reionization

Fast radio burst (FRB) discoveries are occurring rapidly, with thousands expected from upcoming surveys. The dispersion measures (DM) observed for FRB include important information on cosmological distances and the ionization state of the universe from the redshift of emission until today. Rather than considering the DM–redshift relation, we investigate the statistical ensemble of the distribution of dispersion measures. Here, we explore the use of this abundance information, with and without redshift information, to probe helium reionization through simulated data to redshift z = 6. Carrying out Monte Carlo simulations of FRB survey samples, we examine the effect of different source redshift distributions, host galaxy models, sudden vs gradual reionization, and covariance with cosmological parameters on determination of helium reionization properties. We find that a fluence limited survey with 10 4 FRBs can discriminate different helium reionization histories at ~6σ using the DM-distribution of bursts, without redshift information (and ~10σ with redshifts).

79 ASTRONOMY AND ASTROPHYSICS↗