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Table of contents for issue 1, volume 248, The Astrophysical Journal Supplement Series

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We propose a flexible method for estimating luminosity functions (LFs) based on kernel density estimation (KDE), the most popular nonparametric density estimation approach developed in modern statistics, to overcome issues surrounding the binning of LFs. One challenge in applying KDE to LFs is how to treat the boundary bias problem, as astronomical surveys usually obtain truncated samples predominantly due to the flux-density limits of surveys. We use two solutions, the transformation KDE method (${\hat{\phi }}_{{\rm{t}}}$) and the transformation–reflection KDE method (${\hat{\phi }}_{\mathrm{tr}}$) to reduce the boundary bias. We develop a new likelihood cross-validation criterion for selecting optimal bandwidths, based on which the posterior probability distribution of the bandwidth and transformation parameters for ${\hat{\phi }}_{{\rm{t}}}$ and ${\hat{\phi }}_{\mathrm{tr}}$ are derived within a Markov Chain Monte Carlo sampling procedure. The simulation result shows that ${\hat{\phi }}_{{\rm{t}}}$ and ${\hat{\phi }}_{\mathrm{tr}}$ perform better than the traditional binning method, especially in the sparse data regime around the flux limit of a survey or at the bright end of the LF. To further improve the performance of our KDE methods, we develop the transformation–reflection adaptive KDE approach (${\hat{\phi }}_{\mathrm{tra}}$). Monte Carlo simulations suggest that it has good stability and reliability in performance, and is around an order of magnitude more accurate than using the binning method. By applying our adaptive KDE method to a quasar sample, we find that it achieves estimates comparable to the rigorous determination in a previous work, while making far fewer assumptions about the LF. The KDE method we develop has the advantages of both parametric and nonparametric methods.

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We describe the PDFI_SS software library, which is designed to find the electric field at the Sun’s photosphere from a sequence of vector magnetogram and Doppler velocity measurements and estimates of horizontal velocities obtained from local correlation tracking using the recently upgraded Fourier Local Correlation Tracking code. The library, a collection of FORTRAN subroutines, uses the “PDFI” technique described by Kazachenko et al., but modified for use in spherical, Plate Carrée geometry on a staggered grid. The domain over which solutions are found is a subset of the global spherical surface, defined by user-specified limits of colatitude and longitude. Our staggered grid approach, based on that of Yee, is more conservative and self-consistent compared to the centered, Cartesian grid used by Kazachenko et al. The library can be used to compute an end-to-end solution for electric fields from data taken by the HMI instrument aboard NASA’s SDO mission. This capability has been incorporated into the HMI pipeline processing system operating at SDO’s Joint Science Operations Center. The library is written in a general and modular way so that the calculations can be customized to modify or delete electric field contributions, or used with other data sets. Other applications include “nudging” numerical models of the solar atmosphere to facilitate assimilative simulations. The library includes an ability to compute “global” (whole-Sun) electric field solutions. The library also includes an ability to compute potential magnetic field solutions in spherical coordinates. This distribution includes a number of test programs that allow the user to test the software.

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We performed a radio recombination line (RRL) survey to construct a high-mass star-forming region (HMSFR) sample in the Milky Way based on the all-sky Wide-Field Infrared Survey Explorer point-source catalog. The survey was observed with the Shanghai 65 m Tianma radio telescope covering 10 hydrogen RRL transitions ranging from H98α to H113α (corresponding to the rest frequencies of 4.5–6.9 GHz) simultaneously. Out of 3348 selected targets, we identified an HMSFR sample consisting of 517 sources traced by RRLs; a large fraction of this sample (486) is located near the Galactic Plane (∣b∣ < 2°). In addition to the hydrogen RRLs, we also detected helium and carbon RRLs toward 49 and 23 sources, respectively. We crossmatch the RRL detections with the 6.7 methanol maser sources built up in previous works for the same target sample. As a result, 103 HMSFR sources were found to harbor both emissions. In this paper, we present the HMSFR catalog accompanied by the measured RRL line properties and a correlation with our methanol maser sample, which is believed to trace massive stars at earlier stages. The construction of an HMSFR sample consisting of sources in various evolutionary stages indicated by different tracers is fundamental for future studies of high-mass star formation in such regions.

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The total solar irradiance at the top of the atmosphere is the primary source of energy of the Earth’s highly coupled atmosphere–land–ocean system. Small fluctuations of the solar flux density in scales from years to millennia could impact the energy balance of this system due to nonlinear effects. The quantification of this variability depends on absolute radiometers on board of space-based platforms. Although there has been significant improvement in the design and calibration of absolute radiometers during the last decades, the uncertainties in the measurements have not allowed us to untangle the natural and anthropogenic drivers of the observed changes of the climatic patterns appropriately. One of the critical components of the absolute radiometers is the coating of the sensor elements, which should absorb the radiation efficiently. Here we discuss the optical characteristics of ultra-black Nickel–Phosphorus (Ni–P) and its relations with the surface morphology. The ultra-black Ni–P has important unique properties such as low reflectance and uniformity of deposition in complex geometries. Ni–P multilayer was deposited by electroless on aluminum substrates. The surface was etched by oxidizing acid to produce ultra-black Ni–P. Characterization techniques were used to describe the properties of the material. We describe the directional reflectance employing the bidirectional reflectance distribution function. Additionally, we used reflectance maps to show the influence of the pores on the reflectance. Ultra-black Ni–P exhibited a high absorptance and dependence with the light incidence angle. Based on the results, the material demonstrated the opportunity of many terrestrial and space applications as a black coating absorber.

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Early Results from Parker Solar Probe: Ushering a New Frontier in Space Exploration

Parker Solar Probe (PSP), NASA’s latest and closest mission to the Sun, is on a journey to investigate fundamental enigmas of the inner heliosphere. This paper reports initial observations made by the Solar Probe Analyzer for Ions (SPAN-I), one of the instruments in the Solar Wind Electrons Alphas and Protons instrument suite. We address the presence of secondary proton beams in concert with ion-scale waves observed by FIELDS, the electromagnetic fields instrument suite. We show two events from PSP’s second orbit that demonstrate signatures consistent with wave–particle interactions. We showcase 3D velocity distribution functions (VDFs) measured by SPAN-I during times of strong wave power at ion scales. From an initial instability analysis, we infer that the VDFs departed far enough away from local thermodynamic equilibrium to provide sufficient free energy to locally generate waves. These events exemplify the types of instabilities that may be present and, as such, may guide future data analysis characterizing and distinguishing between different wave–particle interactions.

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We consider the body of published distance moduli to the Fornax and Coma galaxy clusters, with specific emphasis on the period since 1990. We have carefully homogenized our final catalogs of distance moduli onto the distance scale established in the previous papers in this series. We assessed systematic differences associated with the use of specific tracers and consequently discarded results based on application of the Tully–Fisher relation and of globular cluster and planetary nebula luminosity functions. We recommend “best” weighted relative distance moduli for the Fornax and Coma clusters with respect to the Virgo cluster benchmark of ${\rm{\Delta }}(m-M{)}_{0}^{\mathrm{Fornax}-\mathrm{Virgo}}=0.18\pm 0.28$ mag and ${\rm{\Delta }}(m-M{)}_{0}^{\mathrm{Coma}-\mathrm{Virgo}}=3.75\pm 0.23$ mag. The set of weighted mean distance moduli (distances) we derived as most representative of the clusters’ distances is\begin{eqnarray*}\begin{array}{c}\begin{array}{c}\begin{array}{rcl}{\left(m-M\right)}_{0}^{{\rm{F}}{\rm{o}}{\rm{r}}{\rm{n}}{\rm{a}}{\rm{x}}} & = & 31.41\pm 0.15\,{\rm{m}}{\rm{a}}{\rm{g}}\,(D={19.1}_{-1.2}^{+1.4}\,{\rm{M}}{\rm{p}}{\rm{c}})\,{\rm{a}}{\rm{n}}{\rm{d}}\\ & = & 31.21\pm 0.28\,{\rm{m}}{\rm{a}}{\rm{g}}\,(D={17.5}_{-2.2}^{+2.4}\,{\rm{M}}{\rm{p}}{\rm{c}}),\\ {\left(m-M\right)}_{0}^{{\rm{C}}{\rm{o}}{\rm{m}}{\rm{a}}} & = & 34.99\pm 0.38\,{\rm{m}}{\rm{a}}{\rm{g}}\,(D={99.5}_{-15.9}^{+19.0}\,{\rm{M}}{\rm{p}}{\rm{c}})\,{\rm{a}}{\rm{n}}{\rm{d}}\\ & = & 34.78\pm 0.27\,{\rm{m}}{\rm{a}}{\rm{g}}\,(D={90.4}_{-10.6}^{+11.9}\,{\rm{M}}{\rm{p}}{\rm{c}}),\end{array}\end{array}\end{array}\end{eqnarray*}where the first distance modulus for each cluster is the result of our analysis of the direct, absolute distance moduli, while the second modulus is based on distance moduli relative to the Virgo cluster. While the absolute and relative distance moduli for both clusters are mutually consistent within the uncertainties, the relative distance moduli yield shorter absolute distances by ∼1σ. It is unclear what may have caused this small difference for both clusters; lingering uncertainties in the underlying absolute distance scale appear to have given rise to a systematic uncertainty of the order of 0.20 mag.

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We present modifications to the Athena++ framework to enable the use of general equations of state (EOS). A part of our motivation for doing so is to model transient astrophysics phenomena, as these types of events are often not well approximated by an ideal gas. This necessitated changes to the Riemann solvers implemented in Athena++. We discuss the adjustments made to the Harten-Lax-van Leer-Contact and Harten-Lax-van Leer-Discontinuities solvers and EOS calls required for arbitrary EOS. We demonstrate the reliability of our code in a number of tests that utilize a relatively simple, but nontrivial, EOS based on hydrogen ionization, appropriate for the transition from atomic to ionized hydrogen. Additionally, we perform tests using an electron–positron Helmholtz EOS, appropriate for regimes where nuclear statistical equilibrium is a good approximation. These new complex EOS tests overall show that our modifications to Athena++ accurately solve the Riemann problem with linear convergence and linear wave tests with quadratic convergence. We provide our test solutions as a means to check the accuracy of other hydrodynamic codes. Our tests and additions to Athena++ will enable further research into (magneto)hydrodynamic problems where realistic treatments of the EOS are required.

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3C 454.3 is frequently observed in the flaring state. The long-term light curve of this source has been analyzed with 9 yr (2008 August–2017 July) of data from the Fermi-LAT detector. We have identified five flares and one quiescent state. The flares have substructures with many peaks during the flaring phase. We have estimated the rise and decay time of the flares and compared with flares of other similar sources. The modeling of gamma-ray spectral energy distributions shows in most cases that a log-parabola function gives the best fit to the data. We have done time-dependent leptonic modeling of two of the flares, for which simultaneous multiwavelength data are available. These two long-lasting flares, Flare-2A and Flare-2D, continued for 95 and 133 days, respectively. We have used the average values of Doppler factor, injected luminosity in electrons, size of the emission region, and the magnetic field in the emission region in modeling these flares. The emission region is assumed to be in the broad-line region in our single-zone model. The energy losses (synchrotron, synchrotron self-Compton, external Compton) and escape of electrons from the emission region have been included while doing the modeling. Although the total jet powers required to model these flares with the leptonic model are higher compared to other sources, they are always found to be lower than the Eddington luminosity of 3C 454.3. We also select some flaring peaks and show that the time variation of the Doppler factor or the injected luminosity in electrons over short timescales can explain their light curves.

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The calcium monohydroxide radical (CaOH) is an important astrophysical molecule relevant to cool stars and rocky exoplanets, among other astronomical environments. Here, we present a consistent set of highly accurate rovibronic (rotation-vibration-electronic) energy levels for the five lowest electronic states ($\tilde{X}{}^{2}{{\rm{\Sigma }}}^{+}$, $\tilde{A}{}^{2}{\rm{\Pi }}$, $\tilde{B}{}^{2}{{\rm{\Sigma }}}^{+}$, $\tilde{C}{}^{2}{\rm{\Delta }}$, and $\tilde{D}{}^{2}{{\rm{\Sigma }}}^{+}$) of CaOH. A comprehensive analysis of the published spectroscopic literature on this system has allowed 1955 energy levels to be determined from 3204 rovibronic experimental transitions, all with unique quantum number labeling and measurement uncertainties. The data set covers rotational excitation up to J = 62.5 for molecular states below 29,000 cm−1. The analysis was performed using the Measured Active Rotational-Vibrational Energy Levels algorithm, which is a robust procedure based on the theory of spectroscopic networks. The data set provided will significantly aid future interstellar, circumstellar, and atmospheric detections of CaOH, as well as assist in the design of efficient laser cooling schemes in ultracold molecule research and precision tests of fundamental physics.

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Accurate and complete atomic data are required for the determination of accurate stellar photospheric abundances. This includes hyperfine structure (HFS) for some elements—those with high nuclear spin and/or large nuclear magnetic moments. Lutetium is one such element. In this study, spectra of a commercial Lu–Ne hollow cathode lamp operating at a low current were recorded with the high-resolution University of Wisconsin 3 m focal length echelle spectrograph. These spectra of resolved and partially resolved hyperfine patterns for 35 ultraviolet and blue transitions of Lu ii have been studied for the purpose of extracting hyperfine constants. We present HFS constants for 16 levels of singly ionized lutetium, 10 of which have been measured for the first time.

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We provide a detailed description of the Chimera code, a code developed to model core collapse supernovae (CCSNe) in multiple spatial dimensions. The CCSN explosion mechanism remains the subject of intense research. Progress to date demonstrates that it involves a complex interplay of neutrino production, transport, and interaction in the stellar core, three-dimensional stellar core fluid dynamics and its associated instabilities, nuclear burning, and the fundamental physics of the neutrino–stellar core weak interactions and the equations of state of all stellar core constituents—particularly, the nuclear equation of state associated with core nucleons, both free and bound in nuclei. Chimera, by incorporating detailed neutrino transport, realistic neutrino–matter interactions, three-dimensional hydrodynamics, realistic nuclear, leptonic, and photonic equations of state, and a nuclear reaction network, along with other refinements, can be used to study the role of neutrino radiation, hydrodynamic instabilities, and a variety of input physics in the explosion mechanism itself. It can also be used to compute observables such as neutrino signatures, gravitational radiation, and the products of nucleosynthesis associated with CCSNe. The code contains modules for neutrino transport, multidimensional compressible hydrodynamics, nuclear reactions, a variety of neutrino interactions, equations of state, and modules to provide data for post-processing observables such as the products of nucleosynthesis, and gravitational radiation. Chimera is an evolving code, being updated periodically with improved input physics and numerical refinements. We detail here the current version of the code, from which future improvements will stem, which can in turn be described as needed in future publications.

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We develop a new numerical scheme for ideal magnetohydrodynamic (MHD) simulations, which is robust against one- and multidimensional shocks, and is accurate for low Mach number flows and discontinuities. The scheme belongs to a family of the advection upstream splitting method employed in computational aerodynamics, and it splits the inviscid flux in MHD equations into advection, pressure, and magnetic tension parts, and then individually evaluates mass, pressure, and magnetic tension fluxes at the interface of a computational cell. The mass flux is designed to avoid numerical shock instability in multidimensions, while preserving contact discontinuity. The pressure flux possesses a proper scaling for low Mach number flows, allowing reliable simulations of nearly incompressible flows. The magnetic tension flux is built to be consistent with the HLLD approximate Riemann solver to preserve rotational discontinuity. We demonstrate various benchmark tests to verify the novel performance of the scheme. Our results indicate that the scheme must be a promising tool to tackle astrophysical systems that include both low and high Mach number flows, as well as magnetic field inhomogeneities.

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A majority of Er in the universe is synthesized by the r-process, which can occur in the mergers of neutron stars (NSs). The contribution of this element to the opacity of NS ejecta should be tested, but even the energy levels of first excited configuration have not been fully presented. The main aim of this paper is to present accurate energy levels of the ground [Xe]4f12 and first excited [Xe]4f115d configurations of Er2+. The energy level structure of the Er2+ ion was computed using the multiconfiguration Dirac–Hartree–Fock and relativistic configuration interaction (RCI) methods, as implemented in the GRASP2018 program package. The Breit interaction, self-energy, and vacuum polarization corrections were included in the RCI computations. The zero-first-order approach was used in the computations. Energy levels with the identification in LS coupling for all (399) states belonging to the [Xe]4f12 and [Xe]4f115d configurations are presented. Electric dipole (E1) transition data between the levels of these two configurations are computed. The accuracy of these data is evaluated by studying the behavior of the transition rates as functions of the gauge parameter, as well as by evaluating the cancellation factors. The core electron correlations were studied using different strategies. The rms deviations obtained in this study for states of the ground and excited configurations from the available experimental data are 649 and 754 cm−1, respectively.

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The Dst index is a commonly geomagnetic index used to measure the strength of geomagnetic activity. The accurate prediction of the Dst index is one of the main subjects of space weather studies. In this study, we use the Bagging ensemble-learning algorithm, which combines three algorithms—the artificial neural network, support vector regression, and long short-term memory network—to predict the Dst index 1–6 hr in advance. Taking solar wind parameters (including the interplanetary total magnetic field, magnetic field Bz component, total electric field, solar wind speed, plasma temperature, and proton pressure) as inputs, we establish the Dst index models and complete not only the point prediction but also the interval prediction in forecasting the Dst index. The results show that the root mean square error (rmse) of the point prediction is always lower than 8.0936 nT, the correlation coefficient (R) is always higher than 0.8572 and the accuracy of interval prediction is always higher than 90%, implying that our model can improve the accuracy of point prediction and significantly promote the accuracy of interval prediction. In addition, an new proposed metric shows that the Bagging algorithm brings better stability to the model. Our model was also used to predicate a magnetic storm event from 2016 October 12–17. The most accurate prediction of this storm event is the 1 hr ahead prediction, which holds a result with the rmse of 3.7327 nT, the correlation coefficient of 0.9928, and the interval prediction accuracy of 96.69%. Moreover, we also discuss the balance in the Bagging ensemble model in this paper.

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We present a survey of molecular outflows across the dark cloud complex in the Cygnus region, based on a 46.75 deg2 field of CO isotopologue data from the Milky Way Imaging Scroll Painting survey. A supervised machine-learning algorithm, the support vector machine, is introduced to accelerate our visual assessment of outflow features in the data cube of 12CO and 13CO J = 1−0 emission. A total of 130 outflow candidates are identified, 77 of which show bipolar structures and 118 are new detections. Spatially, these outflows are located inside dense molecular clouds, and some of them are found in clusters or in elongated linear structures tracing the underlying gas filament morphology. Along the line of sight, 97, 31, and 2 candidates reside in the Local, Perseus, and Outer Arms, respectively. Young stellar objects as outflow drivers are found near most outflows, while 36 candidates show no associated source. The clusters of outflows that we detect are inhomogeneous in their properties; nevertheless, we show that the outflows cannot inject turbulent energy on cloud scales. Instead, at best, they are restricted to affecting the so-called “clump” and “core” scales, and only on short (∼0.3 Myr) estimated timescales. Combined with outflow samples in the literature, our work shows a tight outflow mass–size correlation.

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The X-ray emission from a supernova remnant (SNR) is a powerful diagnostic of the state of the shocked plasma. The temperature (kT) and the emission measure (EM) of the shocked gas are related to the energy of the explosion, the age of the SNR, and the density of the surrounding medium. Progress in X-ray observations of SNRs has resulted in a significant sample of Galactic SNRs with measured kT and EM values. We apply spherically symmetric SNR evolution models to a new set of 43 SNRs to estimate ages, explosion energies, and circumstellar medium densities. The distribution of ages yields an SNR birth rate. The energies and densities are well fit with lognormal distributions, with wide dispersions. SNRs with two emission components are used to distinguish between SNR models with uniform interstellar medium and with stellar wind environment. We find Type Ia SNRs to be consistent with a stellar wind environment. Inclusion of stellar wind SNR models has a significant effect on estimated lifetimes and explosion energies of SNRs. This reduces the discrepancy between the estimated SNR birth rate and the SN rate of the Galaxy.

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Lanthanide elements play important roles as an opacity source in the ejected material from neutron star mergers. Accurate and complete atomic data are necessary to evaluate the opacities and to analyze the observed data. In this paper, we perform extended, ab initio atomic calculations from Pr ii (Z = 59) to Gd ii (Z = 64). By using multiconfiguration Dirac–Hartree–Fock and relativistic configuration-interaction methods, implemented in the general-purpose relativistic atomic structure package (GRASP2K), we calculate the energy levels and transition data of electric dipole transitions. These computations are based on strategies (with small variations) of Nd ii published by Gaigalas et al. Accuracy of data is evaluated by comparing computed energy levels with the National Institute of Standards and Technology (NIST) database or other works. For the energy levels, we obtain the average relative accuracy of 8%, 12%, 6%, 8%, and 7% for Pr ii, Pm ii, Sm ii, Eu ii, and Gd ii ions, respectively, as compared with the NIST data. Accuracy of energy transfer to the wavelength is 3%, 14%, and 11% for Pr ii, Eu ii, and Gd ii. Our computed E1 type transition probabilities are in good agreement with experimental values presented by other authors especially for strong transitions.

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We report a simultaneous 44 and 95 GHz class I methanol maser survey toward 144 sources from the 95 GHz class I methanol maser catalog. The observations were made with the three telescopes of the Korean very long baseline interferometry network operating in single-dish mode. The detection rates are 89% at 44 GHz and 77% at 95 GHz. There are 106 new discoveries at 44 GHz. Comparing the previous 95 GHz detections with new observations of the same transitions made using the Purple Mountain Observatory 13.7 m radio telescope shows no clear evidence of variability on a timescale of six years. Emission from the 44 and 95 GHz transitions shows strong correlations in peak velocity, peak flux density, and integrated flux density, indicating that they are likely cospatial. We found that the peak flux density ratio ${S}_{\mathrm{pk},95}$/${S}_{\mathrm{pk},44}$ decreases as the 44 GHz peak flux density increases. We found that some class I methanol masers in our sample might be associated with infrared dark clouds, while others are associated with H ii regions, indicating that some sources occur at an early stage of high-mass star formation, while others are located toward more evolved sources.

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Accurate atmospheric parameters and chemical composition of stars play a vital role in characterizing physical parameters of exoplanetary systems and understanding of their formation. A full asteroseismic characterization of a star is also possible if its main atmospheric parameters are known. The NASA Transiting Exoplanet Survey Satellite (TESS) space telescope will play a very important role in searching of exoplanets around bright stars and stellar asteroseismic variability research. We have observed all 302 bright (V < 8 mag) and cooler than F5 spectral class stars in the northern TESS continuous viewing zone with a 1.65 m telescope at the Molėtai Astronomical Observatory of Vilnius University and the high-resolution Vilnius University Echelle Spectrograph. We uniformly determined the main atmospheric parameters, ages, orbital parameters, velocity components, and precise abundances of 24 chemical species (C(C2), N(CN), [O i], Na i, Mg i, Al i, Si i, Si ii, Ca i, Ca ii, Sc i, Sc ii, Ti i, Ti ii, V i, Cr i, Cr ii, Mn i, Fe i, Fe ii, Co i, Ni i, Cu i, and Zn i) for 277 slowly rotating single stars in the field. About 83% of the sample stars exhibit the Mg/Si ratios greater than 1.0 and may potentially harbor rocky planets in their systems.

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We present Morpheus, a new model for generating pixel-level morphological classifications of astronomical sources. Morpheus leverages advances in deep learning to perform source detection, source segmentation, and morphological classification pixel-by-pixel via a semantic segmentation algorithm adopted from the field of computer vision. By utilizing morphological information about the flux of real astronomical sources during object detection, Morpheus shows resiliency to false-positive identifications of sources. We evaluate Morpheus by performing source detection, source segmentation, morphological classification on the Hubble Space Telescope data in the five CANDELS fields with a focus on the GOODS South field, and demonstrate a high completeness in recovering known GOODS South 3D-HST sources with H < 26 AB. We release the code publicly, provide online demonstrations, and present an interactive visualization of the Morpheus results in GOODS South.

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The exact relationship between the long gamma-ray burst (LGRB) rate and the cosmic star formation rate (CSFR) is essential for using LGRBs as cosmological probes. In this work, we collect a large sample composed of 371 Swift LGRBs with known redshifts and prompt emission properties. We first compare the rest-frame prompt properties of these bursts in different redshift bins, finding negligible redshift evolution of the luminosity of LGRBs with ${L}_{\mathrm{iso}}\gtrsim {10}^{51}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ between z ∼ 1 and z ∼ 4. Then, by utilizing the CSFR obtained from the large-scale cosmological hydrodynamical simulation, the Illustris simulation, we calculate the cumulative redshift distribution of LGRBs under different metallicity thresholds. After comparing with our sample, we find that the predictions with a moderate threshold between $0.3\,{Z}_{\odot }\leqslant {Z}_{\mathrm{th}}\leqslant 1.0\,{Z}_{\odot }$ are consistent with the sample between redshift 0 < z < 3, while at higher redshifts, between 3 < z < 5, all metallicity thresholds fit the data well. When changing to an empirical model based on observations, the predictions show similar results as well. After comparing with the metallicity distribution of the observed LGRB host galaxies between 0 < z < 1, we confirm that the production of LGRBs in galaxies with super-solar metallicity is suppressed. Nevertheless, considering that a significant fraction of stars are born in sub-solar metallicity environments at z ≳ 3, we suggest that, as a first approximation, LGRBs can be used as direct tracers of the CSFR in this redshift range.