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Optimization of the Observing Cadence for the Rubin Observatory Legacy Survey of Space and Time: A Pioneering Process of Community-focused Experimental Design

Vera C. Rubin Observatory is a ground-based astronomical facility under construction, a joint project of the National Science Foundation and the U.S. Department of Energy, designed to conduct a multipurpose 10 yr optical survey of the Southern Hemisphere sky: the Legacy Survey of Space and Time. Significant flexibility in survey strategy remains within the constraints imposed by the core science goals of probing dark energy and dark matter, cataloging the solar system, exploring the transient optical sky, and mapping the Milky Way. The survey’s massive data throughput will be transformational for many other astrophysics domains and Rubin’s data access policy sets the stage for a huge community of potential users. To ensure that the survey science potential is maximized while serving as broad a community as possible, Rubin Observatory has involved the scientific community at large in the process of setting and refining the details of the observing strategy. The motivation, history, and decision-making process of this strategy optimization are detailed in this paper, giving context to the science-driven proposals and recommendations for the survey strategy included in this Focus Issue.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

Geophysical Observations of the 2023 September 24 OSIRIS-REx Sample Return Capsule Reentry

Sample return capsules (SRCs) entering Earth's atmosphere at hypervelocity from interplanetary space are a valuable resource for studying meteor phenomena. The 2023 September 24 arrival of the Origins, Spectral Interpretation, Resource Identification, and Security-Regolith Explorer SRC provided an unprecedented chance for geophysical observations of a well-characterized source with known parameters, including timing and trajectory. A collaborative effort involving researchers from 16 institutions executed a carefully planned geophysical observational campaign at strategically chosen locations, deploying over 400 ground-based sensors encompassing infrasound, seismic, distributed acoustic sensing, and Global Positioning System technologies. Additionally, balloons equipped with infrasound sensors were launched to capture signals at higher altitudes. This campaign (the largest of its kind so far) yielded a wealth of invaluable data anticipated to fuel scientific inquiry for years to come. The success of the observational campaign is evidenced by the near-universal detection of signals across instruments, both proximal and distal. This paper presents a comprehensive overview of the collective scientific effort, field deployment, and preliminary findings. The early findings have the potential to inform future space missions and terrestrial campaigns, contributing to our understanding of meteoroid interactions with planetary atmospheres. Furthermore, the data set collected during this campaign will improve entry and propagation models and augment the study of atmospheric dynamics and shock phenomena generated by meteoroids and similar sources.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

Statistical chronometry of Meteorites: II. Initial abundances and homogeneity of short-lived radionuclides

Astrophysical models of planet formation require accurate radiometric dating of meteoritic components by short-lived (Al-Mg, Mn-Cr, Hf-W) and long-lived (U-Pb) chronometers, to develop a timeline of such events in the solar nebula as formation of Ca-rich, Al-rich Inclusions (CAIs), chondrules, planetesimals, etc. CAIs formed mostly around a time (“t=0”) when the short-lived radionuclide 26 Al (t 1/2 = 0.72 Myr) was present and presumably homogeneously distributed at a known level we define as ( 26 Al/ 27 Al) SS = 5.23 × 10 -5 . The time of formation after t=0 of another object can be found by determining its initial ( 26 Al/ 27 Al) 0 ratio and comparing it to ( 26 Al/ 27 Al) SS . Dating of meteoritic objects using the Mn-Cr or Hf-W systems is hindered because the abundances ( 53 Mn/ 55 Mn) SS and ( 182 Hf/ 180 Hf) SS at t=0 are not known precisely. To constrain these quantities, we compile literature Al-Mg, Mn-Cr, Hf-W and Pb-Pb data for 13 achondrites and use novel statistical techniques to minimize the discrepancies between their times of formation across these systems. We find that for ( 53 Mn/ 55 Mn) SS = (7.80±0.36)×10 -6 , ( 182 Hf/ 180 Hf) SS = (10.41 ± 0.12) × 10-5, t SS = 4568.65 ± 0.10 Myr, and a 53 Mn half-life of 3.98 ± 0.22 Myr, these four free parameters make concordant 18 formation times recorded by the different systems in all six known volcanic achondrites (the quenched angrites D’Orbigny, Sahara 99555, Northwest Africa 1670; the psuedo-eucrites Asuka 881394 and Ibitira; and the basaltic achondrite Northwest Africa 7325). These parameters also make concordant the ages derived for chondrules from CB/CH achondrites, formed simultaneously in an impact. The other seven achondrites are not quite concordant, but are plutonic angrites or ‘carbonaceous achondrites’ for which simultaneous closure of the isotopic systems might not be expected. Our findings provide very strong support for homogeneity of 26 Al, 53 Mn, and 182 Hf in the solar nebula, and our approach offers a path for more precise chronometry

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Tests of Dark MACHOs: lensing, accretion, and glow

Dark matter could take the form of dark massive compact halo objects (dMACHOs); i.e., composite objects that are made up of dark-sector elementary particles, that could have a macroscopic mass from the Planck scale to above the solar mass scale, and that also admit a wide range of energy densities and sizes. Concentrating on the gravitational interaction of dMACHOs with visible matter, we map out the mass-radius parameter space that is consistent with gravitational lensing experiments, as well as anisotropies of the cosmic microwave background (CMB) based on the spherical accretion of matter onto a dMACHO in the hydrostatic approximation. For dMACHOs with a uniform-density mass profile and total mass in the range of ~ 10 -12 - 10 M , we find that a dMACHO could explain 100% of the dark matter if its radius is above ≈ 3 times the Einstein radius of the lensing system. For a larger mass above 10 M , a dMACHO with radius above ~ 1×10 8 cm×(M/100 M ) 9/2 is consistent with CMB observables. For a lighter dMACHO with mass below ~ 10 -12 M , there still is not a good experimental probe. Finally, we point out that heavier dMACHOs with masses ~ 0.1 M may be observed by X-ray and optical telescopes if they reside at rest in a large molecular cloud, nearby to our solar system, and accrete ordinary matter to emit photons.

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The Tidal Clock Effects of the Lunisolar Gravitational Field and the Earth’s Tidal Deformation

State-of-the-art atomic and optical clocks have the great potential to precisely test fundamental physical assumptions and enhance our understanding of nature. Their widespread applications require us to rigorously deduce the relativistic frequency shift in the framework of general relativity. One interesting question for clocks is tidal field effect which contains variously periodic variations. By introducing tidally deformed Earth, the general algorithms for clock and frequency comparisons are derived. We investigate the effects of external gravitational bodies and Earth’s tidal deformation on the ground- and space-based clocks. The orbital elements of satellite and locations of laboratory are introduced for corresponding clocks, in which the secular and long-period terms and short-period terms are subsequently presented. These effects called tidal clock effects produce the non-negligible contributions in the modern clock experiments and can be directly evaluated from our parameterized formulas. In addition, we also demonstrate for tidal clock effects the position and distance dependences, as well as periodicity. These tidal effects and dependences can provide valuable information for the clock comparison experiments.

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A Deep and Wide Twilight Survey for Asteroids Interior to Earth and Venus

We are conducting a survey using twilight time on the Dark Energy Camera with the Blanco 4 m telescope in Chile to look for objects interior to Earth's and Venus' orbits. To date we have discovered two rare Atira/Apohele asteroids, 2021 LJ4 and 2021 PH27, which have orbits completely interior to Earth's orbit. We also discovered one new Apollo-type Near Earth Object (NEO) that crosses Earth's orbit, 2022 AP7. Two of the discoveries have diameters ≳1 km. 2022 AP7 is likely the largest Potentially Hazardous Asteroid (PHA) discovered in about eight years. To date we have covered 624 square degrees of sky near to and interior to the orbit of Venus. The average images go to 21.3 mag in the r band, with the best images near 22nd mag. Our new discovery 2021 PH27 has the smallest semimajor axis known for an asteroid, 0.4617 au, and the largest general relativistic effects (53 arcsec/century) known for any body in the solar system. The survey has detected ~15% of all known Atira NEOs. We put strong constraints on any stable population of Venus co-orbital resonance objects existing, as well as the Atira and Vatira asteroid classes. These interior asteroid populations are important to complete the census of asteroids near Earth, including some of the most likely Earth impactors that cannot easily be discovered in other surveys. Comparing the actual population of asteroids found interior to Earth and Venus with those predicted to exist by extrapolating from the known population exterior to Earth is important to better understand the origin, composition, and structure of the NEO population.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

Bioverse: A Comprehensive Assessment of the Capabilities of Extremely Large Telescopes to Probe Earth-like O 2 Levels in Nearby Transiting Habitable-zone Exoplanets

Molecular oxygen is a strong indicator of life on Earth and may indicate biological processes on exoplanets too. Recent studies proposed that Earth-like O 2 levels might be detectable on nearby exoplanets using high-resolution spectrographs on future extremely large telescopes (ELTs). However, these studies did not consider constraints like relative velocities, planet occurrence rates, and target observability. We expanded on past studies by creating a homogeneous catalog of 286,391 main-sequence stars within 120 pc using Gaia DR3 and used the Bioverse framework to simulate the likelihood of finding nearby transiting Earth analogs. We also simulated a survey of M dwarfs within 20 pc accounting for η ⊕ estimates, transit probabilities, relative velocities, and target observability to determine how long ELTs and theoretical 50–100 m ground-based telescopes need to observe to probe for Earth-like O 2 levels with an R = 100,000 spectrograph. This would only be possible within 50 yr for up to ~21% of nearby M-dwarf systems if a suitable transiting habitable-zone Earth analog was discovered, assuming signals from every observable partial transit from each ELT can be combined. If so, Earth-like O 2 levels could be detectable on TRAPPIST-1 d–g within 16–55 yr, respectively, and about half that time with an R = 500,000 spectrograph. These results have important implications for whether ELTs can survey nearby habitable-zone Earth analogs for O 2 via transmission spectroscopy. Our work provides the most comprehensive assessment to date of the ground-based capabilities to search for life beyond the solar system.

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The radioactive nuclei 26 Al and 60 Fe in the Cosmos and in the solar system

The cosmic evolution of the chemical elements from the Big Bang to the present time is driven by nuclear fusion reactions inside stars and stellar explosions. A cycle of matter recurrently re-processes metal-enriched stellar ejecta into the next generation of stars. The study of cosmic nucleosynthesis and this matter cycle requires the understanding of the physics of nuclear reactions, of the conditions at which the nuclear reactions are activated inside the stars and stellar explosions, of the stellar ejection mechanisms through winds and explosions, and of the transport of the ejecta towards the next cycle, from hot plasma to cold, star-forming gas. Due to the long timescales of stellar evolution, and because of the infrequent occurrence of stellar explosions, observational studies are challenging, as they have biases in time and space as well as different sensitivities related to the various astronomical methods. Here, we describe in detail the astrophysical and nuclear-physical processes involved in creating two radioactive isotopes useful in such studies, 26 Al and 60 Fe. Due to their radioactive lifetime of the order of a million years, these isotopes are suitable to characterise simultaneously the processes of nuclear fusion reactions and of interstellar transport. We describe and discuss the nuclear reactions involved in the production and destruction of 26 Al and 60 Fe, the key characteristics of the stellar sites of their nucleosynthesis and their interstellar journey after ejection from the nucleosynthesis sites. This allows us to connect the theoretical astrophysical aspects to the variety of astronomical messengers presented here, from stardust and cosmic-ray composition measurements, through observation of γ rays produced by radioactivity, to material deposited in deep-sea ocean crusts and to the inferred composition of the first solids that have formed in the Solar System. We show that considering measurements of the isotopic ratio of 26 Al to 60 Fe eliminate some of the unknowns when interpreting astronomical results, and discuss the lessons learned from these two isotopes on cosmic chemical evolution. This review paper has emerged from an ISSI-BJ Team project in 2017–2019, bringing together nuclear physicists, astronomers, and astrophysicists in this inter-disciplinary discussion.

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A Detailed Characterization of HR 8799's Debris Disk with ALMA in Band 7

The exoplanetary system of HR 8799 is one of the rare systems in which multiple planets have been directly imaged. Its architecture is strikingly similar to that of the solar system, with the four imaged giant planets surrounding a warm dust belt analogous to the Asteroid Belt, and themselves being surrounded by a cold dust belt analog to the Kuiper Belt. Previous observations of this cold belt with ALMA in Band 6 (1.3 mm) revealed its inner edge, but analyses of the data differ on its precise location. It was therefore unclear whether or not the outermost planet HR 8799 b was dynamically sculpting it. We present here new ALMA observations of this debris disk in Band 7 (340 GHz, 880 μm). These are the most detailed observations of this disk obtained so far, with a resolution of 1″ (40 au) and sensitivity of 9.8 μJy beam{sup −1}, which allowed us to recover the disk structure with high confidence. In order to constrain the disk morphology, we fit its emission using radiative transfer models combined with a Markov Chain Monte Carlo procedure. We find that this disk cannot be adequately represented by a single power law with sharp edges. It exhibits a smoothly rising inner edge and smoothly falling outer edge, with a peak in between, as expected from a disk that contains a high-eccentricity component, hence confirming previous findings. Whether this excited population and inner edge shape stem from the presence of an additional planet remains, however, an open question.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Earth as an Exoplanet. I. Time Variable Thermal Emission Using Spatially Resolved Moderate Imaging Spectroradiometer Data

Among the more than 4000 exoplanets known today, some terrestrial planets have been detected in the so-called habitable zone of their host stars and their number is expected to increase in the near future, energizing a drive to understand and interpret the eagerly awaited wealth of data to identify signs of life beyond our solar system. So far, Earth remains the best and only example of a habitable (and inhabited) world. Although, it seems extremely unlikely that any other exoplanets will be true Earth twins, it is important to explore and understand the full range of spectral signatures and variability of Earth in order to inform the design of future instruments and missions, and understand their diagnostic power as well as potential limitations. In this work we use Earth observation data collected by the MODIS instrument aboard the Aqua satellite. The complete data set comprises 15 years of thermal emission observations in the 3.66–14.40 μm range for five different locations on Earth (Amazon Rainforest, Antarctica, Arctic, Indian Ocean, and the Sahara Desert). We then determine flux levels and variations as a function of wavelength and surface type (i.e., climate zone and surface thermal properties) and investigate whether periodic signals indicating Earth’s tilted rotation axis can be detected. Our findings suggest that (1) viewing geometry plays an important role when thermal emission data is analyzed as Earth’s spectrum varies by a factor of three and more depending on the dominant surface type underneath; (2) typically strong absorption bands from CO 2 (15 μm) and O 3 (9.65 μm) are significantly less pronounced and partially absent in data from the polar regions implying that estimating correct abundance levels for these molecules might be challenging in these cases; and (3) the time-resolved thermal emission spectrum encodes information about seasons/planetary obliquity, but the significance depends on the viewing geometry and spectral band considered.

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C{sub 2}N{sub 2} Vertical Profile in Titan’s Stratosphere

In this paper, we present the first measurements of the vertical distribution of cyanogen (C{sub 2}N{sub 2}) in Titan’s lower atmosphere at different latitudes and seasons, using Cassini’s Composite Infrared Spectrometer far-infrared data. We also study the vertical distribution of three other minor species detected in our data: methylacetylene (C{sub 3}H{sub 4}), diacetylene (C{sub 4}H{sub 2}), and H{sub 2}O, in order to compare them to C{sub 2}N{sub 2}, but also to get an overview of their seasonal and meridional variations in Titan’s lower stratosphere from 85 km to 225 km. We measured an average volume mixing ratio of C{sub 2}N{sub 2} of 6.2±0.8×10{sup −11} at 125 km at the equator, but poles exhibit a strong enrichment in C{sub 2}N{sub 2} (up to a factor 100 compared to the equator), greater than what was measured for C{sub 3}H{sub 4} or C{sub 4}H{sub 2}. Measuring C{sub 2}N{sub 2} profiles provides constraints on the processes controlling its distribution, such as bombardment by Galactic Cosmic Rays which seem to have a smaller influence on C{sub 2}N{sub 2} than predicted by photochemical models.

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A Dynamic Trajectory Fit to Multisensor Fireball Observations

Meteorites with known orbital origins are key to our understanding of solar system formation and the source of life on Earth. Fireball networks have been developed globally in a unified effort to record and ultimately retrieve these cosmic samples. However, the accuracy of the determined orbit and the likelihood of meteorite recovery depend directly on the accuracy of the chosen meteoroid triangulation method. There are three leading techniques for meteoroid triangulation discussed in the literature: the method of planes, the straight-line least-squares method, and the multiparameter fit method. Here we describe an alternative method to meteoroid triangulation, called the dynamic trajectory fit. This approach uses the meteoroid’s 3D dynamic equations of motion to fit a realistic trajectory directly to multisensor line-of-sight observations. This method has the ability to resolve fragmentation events, fit systematic observatory timing offsets, and determine mass estimates of the meteoroid along its observable trajectory. Through a comprehensive Monte Carlo analysis of over 100,000 trajectory simulations, we find this new method to more accurately estimate meteoroid trajectories of slow entry events (<25 km s{sup −1}) and events observed from low convergence angles (<10°) compared to existing meteoroid triangulation techniques. Additionally, we triangulate an observed fireball event with visible fragmentation using the various triangulation methods to show that the proposed dynamic trajectory fit implementing fragmentation to best match the captured multisensor line-of-sight data.

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Tuning the Legacy Survey of Space and Time (LSST) Observing Strategy for Solar System Science

Abstract The Vera C. Rubin Observatory is expected to start the Legacy Survey of Space and Time (LSST) in early to mid-2025. This multiband wide-field synoptic survey will transform our view of the solar system, with the discovery and monitoring of over five million small bodies. The final survey strategy chosen for LSST has direct implications on the discoverability and characterization of solar system minor planets and passing interstellar objects. Creating an inventory of the solar system is one of the four main LSST science drivers. The LSST observing cadence is a complex optimization problem that must balance the priorities and needs of all the key LSST science areas. To design the best LSST survey strategy, a series of operation simulations using the Rubin Observatory scheduler have been generated to explore the various options for tuning observing parameters and prioritizations. We explore the impact of the various simulated LSST observing strategies on studying the solar system’s small body reservoirs. We examine what are the best observing scenarios and review what are the important considerations for maximizing LSST solar system science. In general, most of the LSST cadence simulations produce ±5% or less variations in our chosen key metrics, but a subset of the simulations significantly hinder science returns with much larger losses in the discovery and light-curve metrics.

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Tycho: Realistically Simulating Exoplanets within Stellar Clusters. I. Improving the Monte Carlo Approach

To fully understand the diverse population of exoplanets, we must study their early lives within open clusters, the birthplace of most stars with masses >0.5M {sub ⊙} (including those currently in the field). Indeed, when we observe planets within clustered environments, we notice highly eccentric and odd systems that suggest the importance of dynamical pathways created by interactions with additional bodies (as in the case of HD 285507b). However, it has proven difficult to investigate these effects, as many current numerical solvers for the multi-scale N-body problem are simplified and limited in scope. To remedy this, we aim to create a physically complete computational solution to explore the role of stellar close encounters and interplanetary interactions in producing the observed exoplanet populations for both open cluster stars and field stars. We present a new code, Tycho, which employs a variety of different computational techniques, including multiple N-body integration methods, close encounter handling, modified Monte Carlo scattering experiments, and a variety of empirically informed initial conditions. We discuss the methodology in detail, and its implementation within the AMUSE software framework. Approximately 1% of our systems are promptly disrupted by star-star encounters contributing to the rogue planets occurrence rate. Additionally, we find that close encounters which that perturb long-period planets lead to 38.3% of solar-system-like planetary systems becoming long-term unstable.

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An Automated Probabilistic Asteroid Prediscovery Pipeline

We present an automated and probabilistic method to make prediscovery detections of near-Earth asteroids (NEAs) in archival survey images, with the goal of reducing orbital uncertainty immediately after discovery. We refit the Minor Planet Center's astrometry and propagate the full six-parameter covariance to survey epochs to define search regions. We build low-threshold source catalogs for viable images and evaluate every detected source in a search region as a candidate prediscovery. We eliminate false positives by refitting a new orbit to each candidate and probabilistically linking detections across images using a likelihood ratio. Applied to the Zwicky Transient Facility's (ZTF) imaging, we identify approximately 3000 recently discovered NEAs with prediscovery potential, including a doubling of the observational arc for about 500. We use archival ZTF imaging to make prediscovery detections of the potentially hazardous asteroid 2021 DG1, extending its arc by 2.5 yr and reducing future apparition sky plane uncertainty from many degrees to arcseconds. We also recover 2025 FU24 nearly 7 yr before its first known observation, when its sky plane uncertainty covers hundreds of square degrees across thousands of ZTF images. The method is survey agnostic and scalable, enabling rapid orbit refinement for new discoveries from Rubin, NEO Surveyor, and NEOMIR.

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A Numerical Method for Determining the Elements of Circumbinary Orbits and Its Application to Circumbinary Planets and the Satellites of Pluto-Charon

Planets and satellites orbiting a binary system exist in the solar system and extrasolar planetary systems. Their orbits can be significantly different from Keplerian orbits, if they are close to the binary and the secondary-to-primary mass ratio is high. A proper description of a circumbinary orbit is in terms of the free eccentricity e {sub free} at the epicyclic frequency κ {sub 0}, forced eccentricity e {sub forced} at the mean motion n {sub 0}, and oscillations at higher frequencies forced by the non-axisymmetric components of the binary’s potential. We show that accurate numerical values for the amplitudes and frequencies of these terms can be extracted from numerical orbit integrations by applying fast Fourier transformation (FFT) to the cylindrical distance between the circumbinary object and the center of mass of the binary as a function of time. We apply this method to three Kepler circumbinary planets and the satellites of Pluto-Charon. For the satellite Styx of Pluto-Charon, the FFT results for κ {sub 0} and e {sub free} differ significantly from the first-order analytic value and the value reported by Showalter & Hamilton, respectively. We show that the deviation in κ {sub 0} is likely due to the effect of the 3:1 mean-motion resonance and discuss the implications of the lower value for e {sub free}.

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Carbon isotope evidence for the substrates and mechanisms of prebiotic synthesis in the early solar system

Meteorites contain prebiotic, bio-relevant organic compounds including amino acids. Their syntheses could result from diverse sources and mechanisms and provide a window on the conditions and materials present in the early solar system. Here we constrain alanine’s synthetic history in the Murchison meteorite using site-specific 13 C/ 12 C measurements, reported relative to the VPDB standard. The δ 13 C VPDB values of -29 ± 10‰, 142 ± 20‰, and -36 ± 20‰ for the carboxyl, amine-bound, and methyl carbons, respectively, are consistent with Strecker synthesis of interstellar-medium-derived aldehydes, ammonia, and low-δ 13 C nebular or interstellar-medium-derived CN. We report experimentally measured isotope effects associated with Strecker synthesis, and use them to constrain the δ 13 C values of the alanine precursors, which we then use to construct a model that predicts the molecular-average δ 13 C values of 19 other organic compounds of prebiotic significance found in Murchison if they were made by our proposed synthetic network. Finally, most of these predictions agree with previous measurements, suggesting that interstellar-medium-derived aldehydes and nebular and/or pre-solar CN could have served as substrates for synthesis of a wide range of prebiotic compounds in the early solar system.

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Chemical Analysis of the Ultrafaint Dwarf Galaxy Grus II. Signature of High-mass Stellar Nucleosynthesis

We present a detailed abundance analysis of the three brightest member stars at the top of the giant branch of the ultrafaint dwarf (UFD) galaxy Grus II. All stars exhibit a higher than expected [Mg/Ca] ratio compared to metal-poor stars in other UFD galaxies and in the Milky Way (MW) halo. Nucleosynthesis in high-mass ( 20 M ⊙) core-collapse supernovae has been shown to create this signature. The abundances of this small sample (three) stars suggests the chemical enrichment of Grus II could have occurred through substantial high-mass stellar evolution, and is consistent with the framework of a top-heavy initial mass function. However, with only three stars it cannot be ruled out that the abundance pattern is the result of a stochastic chemical enrichment at early times in the galaxy. The most metal-rich of the three stars also possesses a small enhancement in rapid neutron-capture (r-process) elements. The abundance pattern of the r-process elements in this star matches the scaled r-process pattern of the solar system and r-process enhanced stars in other dwarf galaxies and in the MW halo, hinting at a common origin for these elements across a range of environments. All current proposed astrophysical sites of r-process element production are associated with high-mass stars, thus the possible top-heavy initial mass function of Grus II would increase the likelihood of any of these events occurring. The time delay between the α and r-process element enrichment of the galaxy favors a neutron star merger as the origin of the r-process elements in Grus II.

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