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At least 307 records · Page 17

Local limits of detection for anthropogenic aerosol-cloud interactions

Ship tracks are quasi-linear cloud patterns produced from the interaction of ship emissions with low boundary layer clouds. They are visible throughout the diurnal cycle in satellite images from space-borne assets like the Advanced Baseline Imagers (ABI) aboard the National Oceanic and Atmospheric Administration Geostationary Operational Environmental Satellites (GOES-R). However, complex atmospheric dynamics often make it difficult to identify and characterize the formation and evolution of tracks. Ship tracks have the potential to increase a cloud's albedo and reduce the impact of global warming. Thus, it is important to study these patterns to better understand the complex atmospheric interactions between aerosols and clouds to improve our climate models, and examine the efficacy of climate interventions, such as marine cloud brightening. Over the course of this 3-year project, we have developed novel data-driven techniques that advance our ability to assess the effects of ship emissions on marine environments and the risks of future marine cloud brightening efforts. The three main innovative technical contributions we will document here are a method to track aerosol injections using optical flow, a stochastic simulation model for track formations and an automated detection algorithm for efficient identification of ship tracks in large datasets.

54 ENVIRONMENTAL SCIENCES↗

Burning conditions and transportation pathways determine biomass-burning aerosol properties in the Ascension Island marine boundary layer

Abstract. African biomass-burning aerosol (BBA) in the southeast Atlantic Ocean (SEA) marine boundary layer (MBL) is an important contributor to Earth's radiation budget, yet its representation remains poorly constrained in regional and global climate models. Data from the Layered Atlantic Smoke Interactions with Clouds (LASIC) field campaign on Ascension Island (7.95° S, 14.36° W) provide insight into how burning conditions, fuel type, transport pathways, and atmospheric processing affect the chemical, microphysical, and optical properties of BBA between June and September 2017. A total of 10 individual plume events characterize the seasonal evolution of the BBA properties. Early-season inefficient fires, determined by low refractory black carbon to above-background carbon monoxide mixing ratios (rBC : ΔCO), led to enhanced concentrations of organic- and sulfate-rich aerosols. Mid-season efficient fires, determined by higher rBC : ΔCO values, led to rBC-enriched BBA. A mix of efficient and inefficient fires later in the season resulted in conflicting BBA properties. Prolonged transport (∼ 10 d) through the MBL and lower free troposphere (FT) facilitated chemical and aqueous-phase processing, which led to a reduction in organic aerosol mass concentrations. This resulted in lower organic aerosol (OA) to rBC (OA : rBC) mass ratios (2–5) in the MBL compared to higher values (5–15) in the nearby FT. These atmospheric and cloud oxidation processes yield more light-absorbing BBA and explain the notably low single-scattering albedo at 530 nm (SSA530) values (< 0.80) observed in the MBL. This study establishes a robust correlation between SSA530 and OA : rBC across the MBL and FT, underscoring the dependency of optical properties on chemical composition.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

POPSnet-SGP: A Pilot Aerosol Microphysics Network for Targeting Climate Model Uncertainty Interim Field Campaign Report

Aerosols mediate the radiative fluxes in clear and cloudy skies and dominate the uncertainty in the radiative forcing of climate. Relatively dense networks of aerosol optical depth measurements have been used to effectively constrain simulated aerosol optical properties, but model diversity of aerosol microphysical properties is much larger (Mann et al. 2014, Myhre et al. 2009). A better understanding of aerosol microphysics is essential as they are the fundamental pieces of information required to convert aerosol emissions information to radiative and cloud-nucleating properties that drive radiative effects and forcing of climate. Model representation of aerosol microphysical properties has lagged in part due to their inherently greater spatial variability, but also due to a lack of available observations. This project – the Printed Optical Particle Spectrometer Network (POPSnet)-Southern Great Plains (SGP) Pilot – launched the first spatially dense network of aerosol size distribution measurements over an area the size of a global model grid cell and demonstrated its use in providing valuable information regarding the spatial variability of aerosol microphysical properties and its drivers for improving model constraints.

54 ENVIRONMENTAL SCIENCES↗

Bridging New Observational Capabilities and Process-Level Simulation: Insights into Aerosol Roles in the Earth System

The spatial distribution of ambient aerosol particles significantly impacts aerosol–radiation–cloud interactions, which contribute to the largest uncertainty in global anthropogenic radiative forcing estimations. However, the atmospheric boundary layer and lower free troposphere have not been adequately sampled in terms of spatiotemporal resolution, hindering a comprehensive characterization of various atmospheric processes and impeding our understanding of the Earth system. To address this research data gap, we have leveraged the development of uncrewed aerial systems (UAS) and advanced measurement techniques to obtain mesoscale spatial data on aerosol microphysical and optical properties around the U.S. Southern Great Plains (SGP) atmospheric observatory. Our study also benefits from state-of-the-art laboratory facilities that include three-dimensional molecular imaging techniques enabled by secondary ion mass spectrometry and nanogram-level chemical composition analysis via micronebulization aerosol mass spectrometry. Through our study, we have developed a framework for observation–modeling integration, enabling an examination of how various assumptions about the organic–inorganic components mixing state, inferred from chemical analysis, affect clouds and radiation in observation-constrained model simulations. By integrating observational constraints (derived from offline chemical analysis of the aerosol surface using collected samples) with in situ UAS observations, we have identified a prominent role of organic-enriched nanometer layers located at the surface of aerosol particles in determining profiles of aerosol optical and hygroscopic properties over the SGP observatory. Furthermore, we have improved the agreement between predicted clouds and ground-based cloud lidar measurements. This UAS–model–laboratory integration exemplifies how these new advanced capabilities can significantly enhance our understanding of aerosol–radiation–cloud interactions.

54 ENVIRONMENTAL SCIENCES↗

Cloud, Aerosol, and Radiative Properties Over the Western North Atlantic Ocean

This study examines the atmospheric properties of weather states (WSs) derived from the International Satellite Cloud Climatology Project (ISCCP) over the Western North Atlantic Ocean. In particular, radiation and aerosol data corresponding to two sites in the study domain, Pennsylvania State University and Bermuda, were examined to characterize the atmospheric properties of the various satellite-derived WSs. At both sites, the fair weather WS was most prevalent, followed by the cirrus WS. Differences in the seasonality of the various WSs were observed at the two sites. Fractional sky cover and effective shortwave cloud transmissivity derived from ground-based radiation measurements were able to capture differences among the satellite-derived WSs. Speciated aerosol optical thicknesses (AOT) from the Modern-Era Retrospective Analysis for Research and Applications, version 2 (MERRA-2) were used to investigate potential differences in aerosol properties among the WSs. The clear sky WS exhibited below-average seasonal values of AOT at both sites year-round, as well as relatively high rates of occurrence with low AOT events. In addition, the clear sky WS showed above-average contributions from dust and black carbon to the total AOT year-round. Finally, transitions between various WSs were examined under low, high, and mid-range AOT conditions. The most common pathway was for the WSs to remain in the same state after a 3 hr interval. Some WSs, such as midlatitude storms, deep convection, middle top, and shallow cumulus, were more prevalent as ending states under high AOT conditions. This work motivates examining differences in aerosol properties between WSs in other regions.

54 ENVIRONMENTAL SCIENCES↗

Increased range and contrast in fog with circularly polarized imaging

Fogs, low lying clouds, and other highly scattering environments pose a challenge for many commercial and national security sensing systems. Current autonomous systems rely on optical sensors for navigation whose performance is degraded by highly scattering environments. In our previous simulation work, we have shown that polarized light can penetrate through a scattering environment such as fog. We have demonstrated that circularly polarized light maintains its initial polarization state better than linearly polarized light, even through large numbers of scattering events and thus ranges. This has recently been experimentally verified by other researchers. In this work, we present the design, construction, and testing of active polarization imagers at short-wave infrared and visible wavelengths. We explore multiple polarimetric configurations for the imagers, focusing on linear and circular polarization states. The polarized imagers were tested at the Sandia National Laboratories Fog Chamber under realistic fog conditions. Here, we show that active circular polarization imagers can increase range and contrast in fog better than linear polarization imagers. We show that when imaging typical road sign and safety retro-reflective films, circularly polarized imaging has enhanced contrast throughout most fog densities/ranges compared to linearly polarized imaging and can penetrate over 15 to 25 m into the fog beyond the range limit of linearly polarized imaging, with a strong dependence on the interaction of the polarization state with the target materials.

47 OTHER INSTRUMENTATION↗

Cloud drop number concentrations over the western North Atlantic Ocean: seasonal cycle, aerosol interrelationships, and other influential factors

Cloud drop number concentrations (N d ) over the western North Atlantic Ocean (WNAO) are generally highest during the winter (DJF) and lowest in summer (JJA), in contrast to aerosol proxy variables (aerosoloptical depth, aerosol index, surface aerosol mass concentrations, surface cloud condensation nuclei (CCN) concentrations) that generally peak inspring (MAM) and JJA with minima in DJF. Using aircraft, satellite remote sensing, ground-based in situ measurement data, and reanalysis data, we characterize factors explaining the divergent seasonal cycles and furthermore probe into factors influencing N d on seasonal timescales. The results can be summarized well by features most pronounced in DJF, including features associated with cold-air outbreak (CAO) conditions such as enhanced values of CAO index, planetary boundary layer height (PBLH),low-level liquid cloud fraction, and cloud-top height, in addition to winds aligned with continental outflow. Data sorted into high- and low-N d days in each season, especially in DJF, revealed that all of these conditions were enhanced on the high-N d days, including reduced sea level pressure and stronger wind speeds. Although aerosols may be more abundant in MAM and JJA, the conditions needed to activate those particles into cloud droplets are weaker than in colder months, which is demonstrated by calculations of the strongest (weakest) aerosol indirect effects in DJF (JJA) based on comparing N d to perturbations in four different aerosol proxy variables (total and sulfate aerosol optical depth, aerosol index, surface mass concentration of sulfate). We used three machine learning models and up to 14 input variables to infer about most influential factors related to N d for DJF and JJA, with the best performance obtained with gradient-boosted regression tree (GBRT) analysis. The model results indicated that cloud fraction was the most important input variable, followed by some combination (depending on season) of CAO index and surface mass concentrations of sulfate and organic carbon. Future work is recommended to further understand aspects uncovered here such as impacts of free tropospheric aerosol entrainment on clouds, degree of boundary layer coupling, wet scavenging, and giant CCN effects on aerosol–N d relationships, updraft velocity, and vertical structure of cloud properties such as adiabaticity that impact the satellite estimation of N d .

54 ENVIRONMENTAL SCIENCES↗

Into the UV: A Precise Transmission Spectrum of HAT-P-41b Using Hubble’s WFC3/UVIS G280 Grism

The ultraviolet–visible wavelength range holds critical spectral diagnostics for the chemistry and physics at work in planetary atmospheres. To date, time-series studies of exoplanets to characterize their atmospheres have relied on several combinations of modes on the Hubble Space Telescope’s STIS/COS instruments to access this wavelength regime. Here for the first time, we apply the Hubble WFC3/UVIS G280 grism mode to obtain exoplanet spectroscopy from 200 to 800 nm in a single observation. We test the G280 grism mode on the hot Jupiter HAT-P-41b over two consecutive transits to determine its viability for the characterization of exoplanet atmospheres. We obtain a broadband transit depth precision of 29–33 ppm and a precision of on average 200 ppm in 10 nm spectroscopic bins. Spectral information from the G280 grism can be extracted from both the positive and negative first-order spectra, resulting in a 60% increase in the measurable flux. Additionally, the first Hubble Space Telescope orbit can be fully utilized in the time-series analysis. We present detailed extraction and reduction methods for use by future investigations with this mode, testing multiple techniques. We find the results to be fully consistent with STIS measurements of HAT-P-41b from 310 to 800 nm, with the G280 results representing a more observationally efficient and precise spectrum. HAT-P-41b’s transmission spectrum is best fit with a model with T {sub eq} = 2091 K, high metallicity, and significant scattering and cloud opacity. With these first-of-their-kind observations, we demonstrate that WFC3/UVIS G280 is a powerful new tool to obtain UV–optical spectra of exoplanet atmospheres, adding to the UV legacy of Hubble and complementing future observations with the James Webb Space Telescope.

79 ASTRONOMY AND ASTROPHYSICS↗

Atmospheric processing and aerosol aging responsible for observed increase in absorptivity of long-range-transported smoke over the southeast Atlantic

Biomass burning aerosol (BBA) from agricultural fires in southern Africa contributes about one-third of the global carbonaceous aerosol load. These particles have strong radiative effects in the southeast Atlantic (SEA), which depend in part on the radiative contrast between the aerosol layer in the free troposphere (FT) and the underlying cloud layer. However, there is large disagreement in model estimates of aerosol-driven climate forcing due to uncertainties in the vertical distribution, optical properties, and life cycle of these particles. This study applies a novel method combining remote sensing observations with regional model outputs to investigate the aging of the BBA and its impact on the optical properties during transatlantic transport from emission sources in Africa to the SEA. Results show distinct variations in extinction Ångström exponent (EAE) and single-scattering albedo (SSA) as aerosols age. Near the source, fresh aerosols are characterized by low mean SSA (0.84) and high EAE (1.85), indicating smaller, highly absorbing particles. By isolating marine contributions from the total column during BBA transport across the SEA, our analysis reveals an initial decrease in BBA absorptivity, with mean FT SSA of 0.87 after 6–7 d, followed by increased absorptivity with mean FT SSA of 0.84 after 10 d, suggesting enhanced absorption due to chemical aging. These findings indicate that BBA becomes more absorbing during extended transport across the SEA, with implications for reducing model uncertainties. Our remote-sensing-based results agree well with previous in situ studies and offer new insights into aerosol–radiation interactions and the energy balance over the SEA.

54 ENVIRONMENTAL SCIENCES↗

A multi-scale temperature-based strategy to map hydrologic exchange flows in highly dynamic systems

Mapping and quantifying hydrologic exchange flows (HEFs) is critical to environmental monitoring and remediation at contaminated sites; however, these objectives are challenging in highly dynamic systems, e.g., along dam-regulated rivers, where HEFs vary rapidly. Direct seepage measurements are labor-intensive and difficult to automate, whereas indirect (e.g., thermal) and remote sensing methods have potential to allow continuous monitoring with limited field effort. We present a preliminary assessment of a multi-scale temperature-based strategy for monitoring HEFs along the Hanford Reach of the Columbia River, in eastern WA, United States. Five thermal methods were assessed. First, a vertical temperature profile (VTP) was installed into the streambed. The VTP data were analyzed using a data assimilation algorithm designed for automated real-time estimation in dynamic systems. Second, a thermal infrared (TIR) camera was used in roving surveys to identify seeps. Third, a TIR camera was stationed at the VTP site to collect images at 1-h intervals. Together, the two TIR datasets provided a basis to assess the potential for drone-based TIR. Fourth, temperature was measured at the sediment/water interface to assess fiber-optic distributed temperature sensing. Fifth, imagery from the ECOSTRESS satellite mission was acquired to assess the potential of spaceborne thermal monitoring. Based on our preliminary assessment, VTP, TIR, and bed temperature measurements provide complementary spatial coverage, temporal sampling, and resolution; these methods have potential for long-term, automated monitoring of HEFs. The publicly available spaceborne imagery, however, proved inadequate because of insufficient spatial resolution and data gaps resulting from cloud cover and revisit frequency.

54 ENVIRONMENTAL SCIENCES↗

Tropospheric and stratospheric wildfire smoke profiling with lidar: mass, surface area, CCN, and INP retrieval

We present retrievals of tropospheric and stratospheric height profiles of particle mass, volume, surface area, and number concentrations in the case of wildfire smoke layers as well as estimates of smoke-related cloud condensation nuclei (CCN) and ice-nucleating particle (INP) concentrations from backscatter lidar measurements on the ground and in space. Conversion factors used to convert the optical measurements into microphysical properties play a central role in the data analysis, in addition to estimates of the smoke extinction-to-backscatter ratios required to obtain smoke extinction coefficients. The set of needed conversion parameters for wildfire smoke is derived from AERONET observations of major smoke events, e.g., in western Canada in August 2017, California in September 2020, and southeastern Australia in January–February 2020 as well as from AERONET long-term observations of smoke in the Amazon region, southern Africa, and Southeast Asia. The new smoke analysis scheme is applied to CALIPSO observations of tropospheric smoke plumes over the United States in September 2020 and to ground-based lidar observation in Punta Arenas, in southern Chile, in aged Australian smoke layers in the stratosphere in January 2020. These case studies show the potential of spaceborne and ground-based lidars to document large-scale and long-lasting wildfire smoke events in detail and thus to provide valuable information for climate, cloud, and air chemistry modeling efforts performed to investigate the role of wildfire smoke in the atmospheric system.

54 ENVIRONMENTAL SCIENCES↗

Cloud Droplet Measurement System for the ARM Tethered Balloon System (TBS) Field Campaign Report

The Mesa Photonics' cloud droplet measurement system (CDMS) performs in situ measurement of droplet size distribution and droplet number density in clouds. These characteristics are important cloud microphysical properties that are critical input parameters for atmospheric models and are also useful for proper calibration and validation of performance of other atmospheric measurement instrumentation. This small campaign was the first project (out of two) that involved integration of the CDMS into the U.S. Department of Energy (DOE) Atmospheric Radiation Measurement (ARM) user facility’s tethered balloon system (TBS) and its initial testing at the ARM Southern Great Plains (SGP) atmospheric observatory. The follow-up campaign (AFC07016) involved testing of the CDMS under relevant conditions at the third ARM Mobile Facility (AMF3) at Oliktok Point, Alaska and making in situ measurements in clouds. The goal of this small field campaign was to deploy the CDMS on the ARM TBS, demonstrate its integrity and compatibility with the TBS, and test the wireless telecommunication system in preparation for subsequent in-cloud measurements at ARM’s North Slope of Alaska (NSA) observatory at Utqiagvik (formerly Barrow). The specific technical objectives included: (1) Integration of the Mesa Photonics' CDMS into the ARM TBS, (2) testing of the wireless telecommunication system of the CDMS, (3) evaluating the data acquisition software and image processing algorithms especially under high-background-illumination conditions, and (4) testing the ruggedness of the instrument's optical alignment and other performance characteristics. All objectives have been successfully met. The mounting hardware was designed and built mostly at Sandia National Laboratories (SNL), which allowed reliable mounting of the CDMS on the TBS and co-locating the CDMS with other instruments. The field tests were performed by the principal investigator (PI) in collaboration with the TBS operational crew led by D. Dexheimer. The field testing demonstrated good compatibility of the CDMS with the TBS. During the TBS flights, the crew did not experience any operational issues with the CDMS. The instrument demonstrated reliable operational characteristics, including sufficient battery life and good thermal management. The wireless telecommunication system has been successfully tested at TBS flight altitudes of up to 1000 m. During all flights, the CDMS raw data were wirelessly transmitted to the ground station and processed in real time. The ruggedness of the instrument's optical alignment was evaluated by performing calibration of the CDMS (using Mesa Photonics' fixed-size monodisperse droplet generator) before and after the campaign. The calibration was stable within the instrument's measurement precision. In summary, the campaign demonstrated good compatibility of the Mesa Photonics' CDMS with the ARM TBS. The flight tests confirmed the mechanical integrity of the CDMS and stability of its optical layout, as well as reliability of the wireless telecommunication system. Successful completion of this campaign provided a solid basis for the next campaign (AFC07016, November 2020), that involved testing of the CDMS during in-cloud TBS flights at AMF3 at Oliktok Point.

54 ENVIRONMENTAL SCIENCES↗

Aerosol Inlets for a Mid-Sized Uncrewed Aerial System (UAS)

The purpose of this technical report is to document the efforts to design and test two inlet systems for aerosol sampling suitable for deployment on a medium-sized fixed-wing Uncrewed Aerial System (UAS). This work, which was supported by the U.S. Department of Energy’s Atmospheric Radiation Measurement (ARM) user facility, was conducted at the Pacific Northwest National Laboratory (PNNL) for the ARM Aircraft Facility (AAF) starting in November 2017. The current work is a part of AAF efforts to instrument the ArcticShark (a mid-sized UAS owned and operated by the AAF) for atmospheric research and develop a scientific payload for deployment on a similar-sized UAS with minimal adaptation and integration. An aerosol inlet system is necessary to sample and transport ambient air sample to the scientific instrumentation with minimal distortions to the aerosols. Two isokinetic aerosol inlets were designed: the first is a simple passive system for a single instrument suitable to be installed in a wing pylon; the second is a system with active control designed to sample and distribute air among several heterogeneous instruments and to provide basic humidity control of the air sample so that the measured aerosol parameters should correspond to “dry” conditions (a common requirement). Both systems could be easily adapted for deployment on another platform and/or with a different set of instrumentation. Several conducted flight tests showed that the inlets’ performance met our design goals.

42 ENGINEERING↗

A remote sensing algorithm for vertically resolved cloud condensation nuclei number concentrations from airborne and spaceborne lidar observations

Cloud condensation nuclei (CCN) are mediators of aerosol–cloud interactions (ACIs), contributing to the largest uncertainties in the understandings of global climate change. We present a novel remote-sensing-based algorithm that quantifies the vertically resolved CCN number concentrations (N CCN ) using aerosol optical properties measured by a multiwavelength lidar. The algorithm considers five distinct aerosol subtypes with bimodal size distributions. The inversion used the lookup tables developed in this study, based on the observations from the Aerosol Robotic Network, to efficiently retrieve optimal particle size distributions from lidar measurements. The method derives dry aerosol optical properties by implementing hygroscopic enhancement factors in lidar measurements. The retrieved optically equivalent particle size distributions and aerosol-type-dependent particle composition are utilized to calculate critical diameters using κ-Köhler theory and N CCN at six supersaturations ranging from 0.07 % to 1.0 %. Sensitivity analyses indicate that uncertainties in extinction coefficients and relative humidity greatly influence the retrieval error in N CCN . The potential of this algorithm is further evaluated by retrieving N CCN using airborne lidar from the NASA ObseRvations of Aerosols above CLouds and their intEractionS (ORACLES) campaign and is validated against simultaneous measurements from the CCN counter. The independent validation with robust correlation demonstrates promising results. Furthermore, the N CCN has been retrieved for the first time using a proposed algorithm from spaceborne lidar – Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) – measurements. The application of this new capability demonstrates the potential for constructing a 3D CCN climatology at a global scale, which helps to better quantify ACI effects and thus reduce the uncertainty in aerosol climate forcing.

54 ENVIRONMENTAL SCIENCES↗

High Compression Blue-Detuned Magneto-Optical Trap of Polyatomic Molecules

We demonstrate a blue-detuned magneto-optical trap (MOT) of a polyatomic molecule, calcium monohydroxide (CaOH). We identify a novel MOT frequency configuration that produces high spatial compression of the molecular cloud. This high compression MOT achieves a cloud radius of 59⁢(5) μ⁢m and a peak density of 8⁢(2) ×10 8 cm −3 . We experimentally compare several blue-detuned MOT schemes for CaOH and find good agreement between our data and Monte Carlo simulations.

Cold and ultracold molecules↗

LiDAR For Heliostat Optical Assessment (Final Report)

This project has sought to develop new uses for surveying-quality Light Detecting and Ranging (LiDAR) 3D scanning sensors in the automatic/autonomous assessment of optical errors in largescale concentrating solar power heliostat fields. Past experiments have demonstrated the ability of a 3D-LiDAR to acquire highly accurate point cloud measurements across several Sandia NSTTF heliostats. The goal of this project is to expand upon this work to see if and how it can be used in large commercial heliostat fields.

14 SOLAR ENERGY↗

The Hubble Space Telescope PanCET Program: An Optical to Infrared Transmission Spectrum of HAT-P-32Ab

We present a 0.3−5 μm transmission spectrum of the hot Jupiter HAT-P-32Ab observed with the Space Telescope Imaging Spectrograph and Wide Field Camera 3 instruments mounted on the Hubble Space Telescope, combined with Spitzer Infrared Array Camera photometry. The spectrum is composed of 51 spectrophotometric bins with widths ranging between 150 and 400 Å, measured to a median precision of 215 ppm. Comparisons of the observed transmission spectrum to a grid of 1D radiative-convective equilibrium models indicate the presence of clouds/hazes, consistent with previous transit observations and secondary eclipse measurements. To provide more robust constraints on the planet’s atmospheric properties, we perform the first full optical to infrared retrieval analysis for this planet. The retrieved spectrum is consistent with a limb temperature of 1248{sub −92}{sup +92} K, a thick cloud deck, enhanced Rayleigh scattering, and ∼10× solar H{sub 2}O abundance. We find log(Z/Z {sub ⊙}) = 2.41{sub −0.07}{sup +0.06}, and compare this measurement with the mass–metallicity relation derived for the solar system.

79 ASTRONOMY AND ASTROPHYSICS↗