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At least 73 records · Page 4

Exploring Whether Super-puffs can be Explained as Ringed Exoplanets

An intriguing, growing class of planets are the “super-puffs,” objects with exceptionally large radii for their masses and thus correspondingly low densities (≲0.3 g cm{sup −3}). Here we consider whether they could have large inferred radii because they are in fact ringed. This would naturally explain why super-puffs have thus far only shown featureless transit spectra. We find that this hypothesis can work in some cases but not all. The close proximity of the super-puffs to their parent stars necessitates rings with a rocky rather than icy composition. This limits the radius of the rings, and makes it challenging to explain the large size of Kepler 51b, 51c, 51d, and 79d unless the rings are composed of porous material. Furthermore, the short tidal locking timescales for Kepler 18d, 223d, and 223e mean that these planets may be spinning too slowly, resulting in a small oblateness and rings that are warped by their parent star. Kepler 87c and 177c have the best chance of being explained by rings. Using transit simulations, we show that testing this hypothesis requires photometry with a precision of somewhere between ∼10 ppm and ∼50 ppm, which roughly scales with the ratio of the planet and star’s radii. We conclude with a note about the recently discovered super-puff HIP 41378f.

79 ASTRONOMY AND ASTROPHYSICS↗

Ring Formation in Protoplanetary Disks Driven by an Eccentric Instability

We find that, under certain conditions, protoplanetary disks may spontaneously generate multiple, concentric gas rings without an embedded planet through an eccentric cooling instability. Using both linear theory and nonlinear hydrodynamics simulations, we show that a variety of background states may trap a slowly processing, one-armed spiral mode that becomes unstable when a gravitationally stable disk rapidly cools. The angular momentum required to excite this spiral comes at the expense of nonuniform mass transport that generically results in multiple rings. For example, one long-term hydrodynamics simulation exhibits four long-lived, axisymmetric gas rings. We verify the instability evolution and ring-formation mechanism from first principles with our linear theory, which shows remarkable agreement with the simulation results. Dust trapped in these rings may produce observable features consistent with observed disks. Additionally, direct detection of the eccentric gas motions may be possible when the instability saturates, and any residual eccentricity left over in the rings at later times may also provide direct observational evidence of this mechanism.

79 ASTRONOMY AND ASTROPHYSICS↗

Advanced storage ring lattice options based on hybrid six-bend achromat for Stanford Synchrotron Radiation Lightsource upgrade

Three storage ring lattices have been designed as options for a future upgrade of the Stanford synchrotron radiation lightsource (SSRL). The three options differ in circumference and targeted future site, with one to be built in the tunnel of the present SPEAR3 ring, one as a green field ring on the SLAC campus, and the third in the tunnel of the decommissioned PEP-II ring. The lattices are based on the newly proposed hybrid 6-bend achromat (H6BA) lattice cells, which is ideal for pushing the photon beam brightness while achieving excellent nonlinear dynamics performance. The transparent matching conditions are enforced to minimize the negative impact of the loss of periodicity due to insertion of various long straight sections. Numerical optimization is performed to further improve the nonlinear dynamics. Finally, in addition to reaching very low emittances, the lattices can accommodate traditional off-axis injection and achieve beam lifetimes similar to or exceeding that of typical third generation rings.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

Ring currents modulate optoelectronic properties of aromatic chromophores at 25 T

The properties of organic molecules can be influenced by magnetic fields, and these magnetic field effects are diverse. They range from inducing nuclear Zeeman splitting for structural determination in NMR spectroscopy to polaron Zeeman splitting organic spintronics and organic magnetoresistance. A pervasive magnetic field effect on an aromatic molecule is the aromatic ring current, which can be thought of as an induction of a circular current of π-electrons upon the application of a magnetic field perpendicular to the π-system of the molecule. While in NMR spectroscopy the effects of ring currents on the chemical shifts of nearby protons are relatively well understood, and even predictable, the consequences of these modified electronic states on the spectroscopy of molecules has remained unknown. In this work, we find that photophysical properties of model phthalocyanine compounds and their aggregates display clear magnetic field dependences up to 25 T, with the aggregates showing more drastic magnetic field sensitivities depending on the intermolecular interactions with the amplification of ring currents in stacked aggregates. These observations are consistent with ring currents measured in NMR spectroscopy and simulated in time-dependent density functional theory calculations of magnetic field-dependent phthalocyanine monomer and dimer absorption spectra. Here, we propose that ring currents in organic semiconductors, which commonly comprise aromatic moieties, may present new opportunities for the understanding and exploitation of combined optical, electronic, and magnetic properties.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Effects of polarization-reversed EMIC waves on the ring current dynamics

Electromagnetic ion cyclotron (EMIC) waves are widely believed to play an important role in influencing the radiation belt and ring current dynamics. Most studies have investigated the effects or characteristics of EMIC waves by assuming their left-handed polarization. However, recent studies have found that the reversal of polarization, which occurs at higher latitudes along the wave propagation path, can change the wave-induced pitch angle diffusion coefficients. Whether such a polarization reversal can influence the global ring current dynamics remains unknown. In this study, we investigate the ring current dynamics and proton precipitation loss in association with polarization-reversed EMIC waves by using the ring current–atmosphere interactions model (RAM). The results indicate that the polarization reversal of H-band EMIC waves can truly decrease the scattering rates of protons of 10 to 50 keV or >100 keV in comparison with the scenario in which the EMIC waves are considered purely left-handed polarized. Additionally, the global ring current intensity and proton precipitation may be slightly affected by the polarization reversal, especially during prestorm time and the recovery phase, but the effects are not large during the main phase. This is probably because the H-band EMIC waves contribute to the proton scattering loss primarily at E < 10 keV, an energy range that is not strongly affected by the polarization reversal.

79 ASTRONOMY AND ASTROPHYSICS↗

Multifrequency Models of Black Hole Photon Rings from Low-luminosity Accretion Disks

Images of black holes encode both astrophysical and gravitational properties. Detecting highly lensed features in images can differentiate between these two effects. We present an accretion disk emission model coupled to the Adaptive Analytical Ray Tracing (AART) code that allows a fast parameter space exploration of black hole photon ring images produced from synchrotron emission from 10 to 670 GHz. As an application, we systematically study several disk models and compute their total flux density, average radii, and optical depth. The model parameters are chosen around fiducial values calibrated to general relativistic magnetohydrodynamic (GRMHD) simulations and observations of M87*. For the parameter space studied, we characterize the transition between optically thin and thick regimes and the frequency at which the first photon ring is observable. Our results highlight the need for careful definitions of photon ring radius in the image domain, as in certain models the highly lensed photon ring is dimmer than the direct emission at certain angles. We find that at low frequencies the ring radii are set by the electron temperature, while at higher frequencies the magnetic field strength plays a more significant role, demonstrating how multifrequency analysis can also be used to infer plasma parameters. Lastly, we show how our implementation can qualitatively reproduce multifrequency black hole images from GRMHD simulations when adding time variability to our disk model through Gaussian random fields. This approach provides a new method for simulating observations from the Event Horizon Telescope and the proposed Black Hole Explorer space mission.

79 ASTRONOMY AND ASTROPHYSICS↗

Investigation of Ring-heterogeneous Geometry Approximation for Efficient and Practical SFR Analysis

A novel geometry approximation scheme named ring-heterogeneous approximation for intermediate resolution multiphysics analysis of SFRs is presented. This approach particularly targets neutronics coupled with thermo-mechanics feedback, which dominate the passive safety of SFR designs. The ring-heterogeneous approximation casts an array of pins into volume-preserving bands of materials that constitute the pins. It allows to efficiently simulate SFRs with an unstructured mesh and also to explicitly consider the deformation of each material by retaining their heterogeneity. Verification results with single assembly and core problems of the ABTR benchmark demonstrated that the ring-heterogeneous approximation is a reasonable approximation for SFRs. For a 2D full core problem, the ring-heterogeneous approximation significantly reduced the number of elements by at least a factor of 70 from a fully-heterogeneous mesh while almost exactly preserving the reference fully-heterogeneous solutions with the RMS error of only 0.02% in the ring-wise fission reaction rate.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

Ring-opening and hydrodenitrogenation of indole under hydrothermal conditions over Ni, Pt, Ru, and Ni-Ru bimetallic catalysts

Here, the activity and selectivity of activated-carbon-supported Ni, Pt, Ru, and Ni-Ru bimetallic catalysts was examined for hydrothermal denitrogenation of indole. The molar yield of pyrrole ring-opening compounds, without an added hydrogen source, are in the order: Ni < Pt < Ni 90 Ru 10 < Ni 75 Ru 25 < Ni 50 Ru 50 ~ Ni 25 Ru 75 ~ Ru. Ru-containing catalysts facilitated production of hydrocarbons (hydrodenitrogenation (HDN) products) when used with added formic acid (hydrogen source). We elucidated catalytic hydrothermal HDN pathways for indole based on the product distributions and the variation of their yields with time. Hydrogenation of indole to indoline is the primary pathway and ring-opening of indoline to form alkyl anilines is faster than forming HDN products (alkyl benzenes). DFT calculations confirmed experimental activity trends, showing Ru is more active than Ni for indole ring opening and for o-toluidine deamination. If no hydrogen source is present, directly breaking the N-C bond in the pyrrole ring is more favorable than breaking the C-N bond with an aromatic carbon. If a H source is provided, the pyrrole ring hydrogenates first, forming indoline, followed by cleavage of the C-N bond.

42 ENGINEERING↗

Intrinsic activation energies for ring contraction of allylic cations in zeolites

Cyclic carbocations are important intermediates in zeolite-catalyzed chemistries such as methanol-to-hydrocarbons conversion, naphthenes ring opening, or coke formation. While information about their thermodynamic stability exists, little is known about the kinetics of formation and transformation of cyclic carbocations in zeolites. To fill this knowledge gap, ring contraction of the 1,3,5,5-tetramethylcyclohexenyl cation (C 10 H 17 + ), a representative of 6-membered ring allylic cations, was investigated by in situ UV–vis and IR spectroscopy. Protonic forms of zeolites served as catalysts, at temperatures from 80 °C to 135 °C. Significant oligomerization and hydride transfer in BEA and FAU hampered kinetics analysis, whereas ring contraction dominated in the channels of MOR. The reactant cation, characterized by an electronic absorption at 314 nm and an allylic stretch at 1549 cm −1 , contracted to both a 1,3-alkyl-substituted cyclopentenyl cation (287 nm and 1506 cm −1 ) and a 1,2,3-alkyl-substituted cyclopentenyl cation (297 nm and 1489 cm −1 ). Collection of time-resolved IR spectra and fitting of the intensities with various kinetic models revealed a third transformation, which is expected from thermodynamics: the 1,3-alkyl-substituted cyclopentenyl cation isomerizes to the 1,2,3-alkyl-substituted cyclopentenyl cation. Series of IR spectra recorded at different temperatures delivered intrinsic activation enthalpies (entropies) in MOR of 67 ± 2 kJ mol −1 (−130 ± 6 J mol −1 K −1 ) and 90 ± 3 kJ mol −1 (−70 ± 8 J mol −1 K −1 ) for the contraction to 1,3- and 1,2,3-substituted species, and of 88 ± 5 kJ mol −1 (−84 ± 12 J mol −1 K −1 ) for the isomerization. The findings characterize one path – via contraction of larger rings – to different cyclopentenyl species in zeolites; and the associated, moderate activation energies suggest such transformations contribute to many complex hydrocarbon reaction networks.

Acid catalysis↗

Facile polymerization in a bicellar template to produce polymer nano-rings

We report a well-defined discoidal bicelle composed of three lipids, specifically zwitterionic long-chain 1,2-dipalmitoyl phosphocholine (DPPC) and short-chain 1,2-dihexanoyl phosphocholine (DHPC) doped with anionic 1,2-dipalmitoyl phosphoglycerol (DPPG) provides a generalized template for the synthesis of hydrophobic polymer nano-rings. The lipid molar ratio of DPPC/DHPC/DPPG is 0.71/0.25/0.04. The detailed investigation and discussion were based on styrene but tested on three other vinyl monomers. The structure of nano-rings is identified through the detailed analysis of small angle X-ray/neutron scattering (SAXS and SANS) data and transmission electron micrographs (TEM), supported by the differential scanning calorimetric (DSC) data before and after polymerization. The investigation covers samples with a styrene-to-lipid ratio ranged varied from 1:50 to 1:10.The styrene monomers are initially located at both the discoidal planar (long-chain lipid rich) and rim (short-chain lipid rich) regions. During polymerization, they migrate to the more fluid rim region-section. The formation mechanism involves the interplay of hydrophobic interaction, mismatched miscibility of polystyrene between the ordered and disordered phases, and crystallinity of the long lipid acyl chains. This facile synthesis is proven applicable for several hydrophobic monomers. The well-defined nano-rings greatly enhance the interfacial area and have the potential to be the building blocks for functional materials, if monomers are incorporated with desirable functions, for future applications. Crosslinked polymer nano-rings are produced in well-defined discoidal bicelles. The enhanced immiscibility in the planar region drives the hydrophobic polymers to the fluidic rim. The platform can be generalized for synthesis of other hydrophobic polymer nano-rings.

36 MATERIALS SCIENCE↗

On the stability of a pair of vortex rings

The growth of perturbations subject to the Crow instability along two vortex rings of equal and opposite circulation undergoing a head-on collision is examined. Unlike the planar case for semi-infinite line vortices, the zero-order geometry of the flow (i.e. the ring radius, core thickness and separation distance) and by extension the growth rates of perturbations vary in time. The governing equations are therefore temporally integrated to characterize the perturbation spectrum. The analysis, which considers the effects of ring curvature and the distribution of vorticity within the vortex cores, explains several key flow features observed in experiments. First, the zero-order motion of the rings is accurately reproduced. Next, the predicted emergent wavenumber, which sets the number of secondary vortex structures emerging after the cores come into contact, agrees with experiments, including the observed increase in the number of secondary structures with increasing Reynolds number. Finally, the analysis predicts an abrupt transition at a critical Reynolds number to a regime dominated by a higher-frequency, faster-growing instability mode that may be consistent with the experimentally observed rapid generation of a turbulent puff following the collision of rings at high Reynolds numbers.

42 ENGINEERING↗

The Role of Methylaryl Radicals in the Growth of Polycyclic Aromatic Hydrocarbons: The Formation of Five-Membered Rings

The regions of the C 13 H 11 potential energy surface (PES) related to the unimolecular isomerization and decomposition of the 1-methylbiphenylyl radical and accessed by the 1-/2-methylnaphthyl + C 2 H 2 reactions have been explored by ab initio G3(MP2,CC)//B3LYP/6-311G(d,p) calculations. The kinetics of these reactions relevant to the growth of polycyclic aromatic hydrocarbons (PAH) under high-temperature conditions in circumstellar envelopes and in combustion flames has been studied employing the RRKM-Master Equation approach. The unimolecular reaction of 1-methylbiphenylyl proceeding via a five-membered ring closure followed by H elimination is predicted to be very fast, on a submicrosecond scale above 1000 K and to result in the formation of an embedded five-membered ring in the 9H-fluorene product. The 1-/2-methylnaphthyl + C 2 H 2 reaction mechanism involves acetylene addition to the radical on the methylene group followed by a six- or five-membered ring closure and aromatization via an H atom loss. Despite of the complexity of the C 13 H 11 PES, these straightforward pathways are dominant in the high-temperature regime (above ~1000 K), with the prevailing products being phenalene, with a significant contribution of 1H-cyclopenta(a)naphthalene, for 1-methylnaphthyl + C 2 H 2 , and 1H-cyclopenta(b)naphthalene and 3H-cyclopenta(a)naphthalene, for 2-methylnaphthyl + C 2 H 2 . The methylnaphthyl reactions with acetylene represent a clean source of the three-ring PAHs, but they are relatively slow owing to the high entrance barriers of ~10 kcal/mol, with the rate constants of about an order of magnitude lower as compared to those for naphthyl + allene and σ-aryl + C 2 H 2 . Here, the 1-methylnaphthyl + C 2 H 2 and 2-methylnaphthyl + C 2 H 2 reactions represent prototypes for PAH growth by an extra six- and five-membered ring on a zigzag edge or a corner of PAH and the generated modified Arrhenius expressions are recommended for kinetic modeling of PAH expansion by the mechanism of acetylene addition to methylaryl radicals.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Microscopic Theory of Long-Time Center-of-Mass Self-Diffusion and Anomalous Transport in Ring Polymer Liquids

We construct a microscopic theory at the level of segment-scale correlated space–time intermolecular forces for the long-time center-of-mass (CM) diffusion constant and intermediate-time non-Fickian transport in dense solutions and melts of ring polymers. The approach combines ideas of polymer, colloid, and liquid-state statistical mechanics to quantify how the multifractal intra-ring conformational structure and inter-ring packing correlations determine dynamic caging constraints and time-dependent friction. Breakdown of Rouse theory is predicted to occur due to length scale-dependent temporal correlation of forces exerted on pairs of tagged ring segments from surrounding polymers. At large enough degrees of polymerization (N), a stronger scaling of the CM diffusion constant (D ∝ N –2 ) is predicted, with a crossover N D proportional to the product of the system-specific macromolecular volume fraction and dimensionless compressibility. In analogy with caging effects in glass-forming fluids, the theory does appear to begin to fail at sufficiently high N/N D for the center-of-mass diffusivity, likely due to another crossover to an even slower activated transport regime. However, use of N/N D with the theoretically predicted N D , in conjunction with dynamic blob scaling ideas for local Rouse friction, collapses semidilute and concentrated solution simulation data onto a master curve. Based on the same physical ideas employed to predict the diffusion constant, a generalized Langevin equation description is formulated for intermediate-time CM transport. It predicts two subdiffusive regimes that emerge due to the self-similar nature of internal ring structure. Analytic and numerical predictions for the apparent non-Fickian exponent as a function of time, N/N D and dimensionless compressibility, the evolution of the maximum degree of subdiffusive motion with N/N D , and the time scale for recovering Fickian diffusion are made, all of which are in good accordance with melt simulations. Furthermore, the present work sets the stage to address activated dynamics and glass formation on the macromolecular scale.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Ring Size Effects on the Structures of Sandwich Compounds with a Stoichiometry of C 12 H 12 M (M = Ti–Ni)

Ring size effects on geometries and electronic structures were investigated for the (C n H n )M(C m H m ) (n = 4, 5, or 6; m = 8, 7, or 6; m + n = 12; M = Ti–Ni) systems using density functional theory. The lowest-energy C 12 H 12 M structures for the early transition metals titanium, vanadium, and chromium are the experimentally known singlet (η 5 -C 5 H 5 )Ti(η 7 -C 7 H 7 ), doublet (η 5 -C 5 H 5 )V(η 7 -C 7 H 7 ), and singlet (η 6 -C 6 H 6 ) 2 Cr, respectively. The likewise experimentally known singlet (η 6 -C 6 H 6 ) 2 Ti, doublet (η 6 -C 6 H 6 ) 2 V, and singlet (η 5 -C 5 H 5 )Cr(η 7 -C 7 H 7 ) are the secondlowest- energy structures with only a small energy difference between the two vanadium structures. For the later transition metals, dibenzenemetal complexes are the lowest-energy C 12 H 12 M species with two fully bonded hexahapto benzene rings in the lowest-energy manganese and iron derivatives and one hexahapto and one dihapto benzene ring in the lowest-energy cobalt and nickel derivatives. The lowest-energy (C 5 H 5 )M(C 7 H 7 ) structures for the later transition metals iron, cobalt, and nickel have partially bonded nonplanar C 7 H 7 rings with one or two uncomplexed C=C bonds. The (C 4 H 4 )M(C 8 H 8 ) (M = Ti–Ni) structures with the metal sandwiched between four- and eight-membered rings were found to be much higher in energy than their (C 5 H 5 )M(C 7 H 7 ) and (C 6 H 6 ) 2 M isomers.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Elastomer Mechanics of Cross-Linked Linear-Ring Polymer Blends

Cross-linking a blend of linear and ring polymers creates a new topology-based dual-network elastomer in which the two components differ significantly in their topology. We use molecular simulations and topological analysis to examine key mechanical properties as functions of ring polymer volume fraction Φ R . For Φ R < Φ R *, where the rings begin to overlap, the network shear modulus G and the maximum stretch ratio λ p are weakly dependent on Φ R . For Φ R > Φ R *, entanglements trapped in the network are diluted as the rings overlap, leading to a significant decrease in G and an increase in λ p with increasing Φ R . Here, the peak tensile stress, σ p , exhibits a maximum around Φ R *, indicating an enhancement of network strength due to the stronger cohesion from the entanglements between linear and ring polymers.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Quantifying the Effect of Magnetic Field Line Curvature Scattering on the Loss of Ring Current Ions

During geomagnetic storms, the ring current ions sometimes exhibit rapid loss as suggested by the fast recovery of the Dst index on a time scale of a few hours. Here, the effects of magnetic field line curvature (FLC) scattering on the loss of ring current ions, which have not been well quantified, are studied here by test particle simulations under the T89c magnetic field model. Our simulation results show that the prediction of ion loss based on a single–value cutoff of the κ parameter or maximum of δμ/μ of a single FLC scattering is not accurate. Instead, the e–folding lifetime (τ) for the loss of ring current ions due to cumulative FLC scattering has been calculated for different initial ion energies, equatorial pitch angles, and L shells under different geomagnetic conditions. The results show that in general the FLC scattering loss is faster for ions of higher energy, higher mass, smaller pitch angle, higher L, and at high Kp level. Specifically, we find that at Kp = 6 the lifetime can be <10 h at L > 5 for 100 s keV protons and at L > 4 for 100 s keV O + , which demonstrates that FLC scattering can be an important mechanism for the observed fast loss (τ < 10 h) of ring current ions during geomagnetic storms. Furthermore, we formulate an empirical formula for τ as a function of ion energy, pitch angle, position, species, and Kp. The empirical formula can be directly included in ring current models to account for the FLC scattering effects.

79 ASTRONOMY AND ASTROPHYSICS↗

Comparison of the Distribution of the In Situ Ring Current Density Based on Magnetic Field Observations From THEMIS, MMS and Cluster

Abstract We review and extend previous work on the ring current density distribution by applying the curlometer technique to multi‐spacecraft magnetic field data on different spatial scales, directly comparing the results of three space missions: Time History of Events and Macroscale Interactions during Substorms (THEMIS), The Magnetospheric Multiscale (MMS) and Cluster, that have so far accrued decades of data on a range of spatial separation scales. We suggest that the statistical results of Cluster data over two decades can illustrate well the overall form of the ring current (in the ring plane), but can underestimate the current density, while THEMIS (partially) and MMS are able to observe the detailed structure of the ring current and indeed show different behavior on some (smaller) spatial scales. These results suggest that, in addition to the local time and radial form, reported previously, the morphology of the ring current may also depend on magnetic latitude, as influenced by the tilt angle between the inner and outer magnetic fields.

Tan, X. [School of Space and Earth Sciences Beihan↗

The Milky Way’s middle-aged inner ring

Aims. We investigate the metallicity, age, and orbital anatomy of the inner Milky Way, specifically focussing on the outer bar region. Methods. We integrated a sample of APOGEE DR16 inner Galaxy stars in a state of the art bar-bulge potential with a slow pattern speed and investigated the link between the resulting orbits and their [Fe/H] and ages. By superimposing the orbits, we built density, [Fe/H], and age maps of the inner Milky Way, which we divided further using the orbital parameters eccentricity, |Xmax|, and |Zmax|. Results. We find that at low heights from the Galactic plane, the Galactic bar gradually transitions into a radially thick, vertically thin, elongated inner ring with average solar [Fe/H]. This inner ring is mainly composed of stars with AstroNN ages between 4 and 9 Gyr with a peak in age between 6 and 8 Gyr, making the average age of the ring ~6 Gyr. The vertical thickness of the ring decreases markedly towards younger ages. We also find very large L4 Lagrange orbits that have average solar to super-solar metallicities and intermediate ages. Lastly, we confirm a clear X-shape in the [Fe/H] and density distributions at large Galactic heights. Conclusions. The orbital structure obtained for the APOGEE stars reveals that the Milky Way hosts an inner ring-like structure between the planar bar and corotation. This structure is on average metal rich, intermediately aged, and enhances the horizontal metallicity gradient along the bar’s major axis.

79 ASTRONOMY AND ASTROPHYSICS↗