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At least 37 records · Page 2

Characterize the Rings in Protoplanetary Disks Formed by an Eccentric Instability [Slides]

Typical long-term time evolution of gas surface density in a disk undergoing eccentric mode instability (EMI). In a disk with no initial substructures or external perturbers, EMI generates a growing one arm spiral (top right). After EMI saturates, the disk continues to evolve with a long-lived coherent eccentric pattern. If the disk has strong wave damping, the disk is then left with circular rings. Synthesized dust continuum images of some EMI simulations from this IC project show some characteristics similar to the actual PPD images.

79 ASTRONOMY AND ASTROPHYSICS↗

Probing UV-sensitive Pathways for CN and HCN Formation in Protoplanetary Disks with the Hubble Space Telescope

The UV radiation field is a critical regulator of gas-phase chemistry in surface layers of disks around young stars. In an effort to understand the relationship between photocatalyzing UV radiation fields and gas emission observed at infrared and submillimeter wavelengths, we present an analysis of new and archival Hubble Space Telescope (HST), Spitzer, ALMA, IRAM, and SMA data for five targets in the Lupus cloud complex and 14 systems in Taurus-Auriga. The HST spectra were used to measure Lyα and far-UV (FUV) continuum fluxes reaching the disk surface, which are responsible for dissociating relevant molecular species (e.g., HCN, N{sub 2}). Semi-forbidden C ii] λ2325 and UV-fluorescent H{sub 2} emission were also measured to constrain inner disk populations of C{sup +} and vibrationally excited H{sub 2}. We find a significant positive correlation between 14 μm HCN emission and fluxes from the FUV continuum and C ii] λ2325, consistent with model predictions requiring N{sub 2} photodissociation and carbon ionization to trigger the main CN/HCN formation pathways. We also report significant negative correlations between submillimeter CN emission and both C ii] and FUV continuum fluxes, implying that CN is also more readily dissociated in disks with stronger FUV irradiation. No clear relationships are detected between either CN or HCN and Lyα or UV-H{sub 2} emission. This is attributed to the spatial stratification of the various molecular species, which span several vertical layers and radii across the inner and outer disk. We expect that future observations with the James Webb Space Telescope will build on this work by enabling more sensitive IR surveys than were possible with Spitzer.

79 ASTRONOMY AND ASTROPHYSICS↗

Zonal Flow and Self-regulating Mechanism in a Hydrodynamic Disk

This study addresses key aspects of momentum transport in hydrodynamic disks, which is critical for understanding zonal flow generation and turbulence in compressible hydrodynamic disks. We find that nonlinear momentum/density transport leads to the formation of zonal flows from the Rossby wave instability in disks. We analytically derive the generation and location of zonal flows and describe a modified Taylor identity applicable to compressible disk flows. We further present a self-regulation model, revealing a dynamic interplay between zonal flow and fluctuations driven by Rossby wave instability that regulates the nonlinear saturation state. This theoretical framework contributes insights into the dynamics of disks such as protoplanetary disks, shedding light on the intricate processes governing momentum/density transport and the emergence of zonal flows in the saturation of protoplanetary disks.

Astrophysical fluid dynamics↗

Thermal History of Matrix Forsterite Grains from Murchison Based on High-resolution Tomography

Protoplanetary disks are dust- and gas-rich structures surrounding protostars. Depending on the distance from the protostar, this dust is thermally processed to different degrees and accreted to form bodies of varying chemical compositions. The primordial accretion processes occurring in the early protoplanetary disk such as chondrule formation and metal segregation are not well understood. One way to constrain them is to study the morphology and composition of forsteritic grains from the matrix of carbonaceous chondrites. Here, we present high-resolution ptychographic X-ray nanotomography and multimodal chemical microtomography (X-ray diffraction and X-ray fluorescence) to reveal the early history of forsteritic grains extracted from the matrix of the Murchison CM2.5 chondrite. Here,. the 3D electron density maps revealed, at unprecedented resolution (64 nm), spherical inclusions containing Fe–Ni, very little silica-rich glass and void caps (i.e., volumes where the electron density is consistent with conditions close to vacuum) trapped in forsterite. The presence of the voids along with the overall composition, petrological textures, and shrinkage calculations is consistent with the grains experiencing one or more heating events with peak temperatures close to the melting point of forsterite (~2100 K), and subsequently cooled and contracted, in agreement with chondrule-forming conditions.

58 GEOSCIENCES↗

Titanium and chromium isotopic compositions of calcium-aluminum-rich inclusions: Implications for the sources of isotopic anomalies and the formation of distinct isotopic reservoirs in the early Solar System

As the earliest-dated solids in our Solar System, calcium-aluminum-rich inclusions (CAIs) provide a record of their formation environment near the young Sun and hold clues to the formation of planetary-scale isotopic reservoirs in the solar protoplanetary disk. Although CAIs from several CV, CK, CM, CO, and ordinary chondrites have been analyzed previously for their Ti isotopic compositions, CAIs from just three CV chondrites have been analyzed for their Cr isotopic compositions, and only a handful of CAIs have been measured for both their Ti and Cr isotopic compositions. We report mass-independent Ti and Cr isotopic anomalies in several CAIs from CV and CK chondrites; this is the first report of the Cr isotopic composition of a CAI from a CK chondrite. With this data, we aim to better constrain the compositional range of CAIs in ε 50 Ti versus ε 54 Cr space, thereby facilitating the isotopic characterization of the material inherited by the solar protoplanetary disk and the role of CAIs in the formation of distinct planetary-scale isotopic reservoirs in our early Solar System. The narrow range in isotopic anomalies in CAIs when compared to other early-formed refractory inclusions such as platy hibonite crystals (PLACs) and spinel-hibonite inclusions (SHIBs) suggests that CAIs record the mixing of these precursor materials and the averaging of their larger isotopic anomalies. The isotopic composition of CAIs is therefore likely the result of a combination of factors, including mixing of material inherited from their formation region, heterogeneous carrier phase distribution, and thermal processing in the disk. The ε 50 Ti and ε 54 Cr isotopic compositions of CAIs are not correlated, further demonstrating that these isotopic anomalies have different carrier phases. The Ti and Cr isotopic compositions of CAIs additionally show that CAIs alone cannot be responsible for the compositional difference between the non-carbonaceous chondritic (NC) and carbonaceous chondritic (CC) isotopic reservoirs but nevertheless do play a role in the formation of these large-scale isotopic reservoirs in the early Solar System.

58 GEOSCIENCES↗

The variety and origin of materials accreted by Bennu’s parent asteroid

The first bodies to form in the Solar System acquired their materials from stars, the presolar molecular cloud and the protoplanetary disk. Asteroids that have not undergone planetary differentiation retain evidence of these primary accreted materials. However, geologic processes such as hydrothermal alteration can dramatically change their bulk mineralogy, isotopic compositions and chemistry. Here we analyse the elemental and isotopic compositions of samples from asteroid Bennu to uncover the sources and types of material accreted by its parent body. We show that some primary accreted materials escaped the extensive aqueous alteration that occurred on the parent asteroid, including presolar grains from ancient stars, organic matter from the outer Solar System or molecular cloud, refractory solids that formed close to the Sun, and dust enriched in neutron-rich Ti isotopes. We find Bennu to be richer in isotopically anomalous organic matter, anhydrous silicates, and light isotopes of K and Zn than its closest compositional counterparts, asteroid Ryugu and Ivuna-type (CI) carbonaceous chondrite meteorites. We propose that the parent bodies of Bennu, Ryugu and CI chondrites formed from a common but spatially and/or temporally heterogeneous reservoir of materials in the outer protoplanetary disk.

Kuiper belt↗

Disk Evolution Study through Imaging of Nearby Young Stars (DESTINYS): A Panchromatic View of DO Tau’s Complex Kilo-astronomical-unit Environment

While protoplanetary disks are often treated as isolated systems in planet formation models, observations increasingly suggest that vigorous interactions between Class II disks and their environments are not rare. DO Tau is a T Tauri star that has previously been hypothesized to have undergone a close encounter with the HV Tau system. As part of the DESTINYS ESO Large Programme, we present new Very Large Telescope (VLT)/SPHERE polarimetric observations of DO Tau and combine them with archival Hubble Space Telescope (HST) scattered-light images and Atacama Large Millimeter/submillimeter Array (ALMA) observations of CO isotopologues and CS to map a network of complex structures. The SPHERE and ALMA observations show that the circumstellar disk is connected to arms extending out to several hundred astronomical units. HST and ALMA also reveal stream-like structures northeast of DO Tau, some of which are at least several thousand astronomical units long. These streams appear not to be gravitationally bound to DO Tau, and comparisons with previous Herschel far-IR observations suggest that the streams are part of a bridge-like structure connecting DO Tau and HV Tau. We also detect a fainter redshifted counterpart to a previously known blueshifted CO outflow. While some of DO Tau's complex structures could be attributed to a recent disk–disk encounter, they might be explained alternatively by interactions with remnant material from the star formation process. These panchromatic observations of DO Tau highlight the need to contextualize the evolution of Class II disks by examining processes occurring over a wide range of size scales.

79 ASTRONOMY AND ASTROPHYSICS↗

Early High-contrast Imaging Results with Keck/NIRC2-PWFS: The SR 21 Disk

High-contrast imaging of exoplanets and protoplanetary disks depends on wave front sensing and correction made by adaptive optics instruments. Classically, wave front sensing has been conducted at optical wavelengths, which made high-contrast imaging of red targets such as M-type stars or extincted T Tauri stars challenging. Keck/NIRC2 has combined near-infrared (NIR) detector technology with the pyramid wave front sensor (PWFS). With this new module we observed SR 21, a young star that is brighter at NIR wavelengths than at optical wavelengths. Compared with the archival data of SR 21 taken with the optical wave front sensing we achieved ~20% better Strehl ratio in similar natural seeing conditions. Further post-processing utilizing angular differential imaging and reference-star differential imaging confirmed the spiral feature reported by the Very Large Telescope/Spectro-Polarimetric High-contrast Exoplanet REsearch instrument polarimetric observation, which is the first detection of the SR 21 spiral in total intensity at L' band. We also compared the contrast limit of our result (10 -4 at 0.'4 and 2 × 10 -5 at 1.'0) with the archival data that were taken with optical wave front sensing and confirmed the improvement, particularly at ≤0.'5. Our observation demonstrates that the NIR PWFS improves AO performance and will provide more opportunities for red targets in the future.

79 ASTRONOMY AND ASTROPHYSICS↗

A Laboratory-driven Multiscale Investigation of X-Ray Induced Mass Loss and Photochemical Evolution in Cosmic Carbon and Silicate Dust

We present the results of an integrated laboratory and modeling investigation into the impact of stellar X-rays on cosmic dust. Carbonaceous grains were prepared in a cooled (< 200 K) supersonic expansion from aromatic molecular precursors, and were later irradiated with 970 eV X-rays. Silicate (enstatite) grains were prepared via laser ablation, thermally annealed, and later irradiated with 500 eV X-rays. Infrared spectra of the 3.4 μm band of the carbon sample prepared with benzene revealed 84% ± 5% band area loss for an X-ray dose of 5.2 x 10 23 eV.cm -2 . Infrared spectra of the 8 -12 μm Si-O band of the silicate sample revealed band area loss up to 63% ± 5% for doses of 2.3 x 10 23 eV.cm -2 . A hybrid Monte Carlo particle trajectory approach was used to model the impact of X-rays and ensuing photoelectrons, Auger and collisionally ionized electrons through the bulk. As a result of x-ray ionization and ensuing Coulomb explosions on surface molecules, the calculated mass loss is 60% for the carbonaceous sample and 46% for the silicate sample, within a factor of 2 of the IR band loss, supporting an X-ray induced mass-loss mechanism. We apply the laboratory X-ray destruction rates to estimate the lifetimes of dust grains in protoplanetary disks surrounding 1 M ⊙ and 0.1 M ⊙ G and M stars. In both cases, X-ray destruction timescales are short (a few million years) at the disk surface, but are found to be much longer than typical disk lifetimes (≳10 Myr) over the disk bulk.

79 ASTRONOMY AND ASTROPHYSICS↗

The Origin of the Doppler Flip in HD 100546: A Large-scale Spiral Arm Generated by an Inner Binary Companion

Companions at subarcsecond separation from young stars are difficult to image. However, their presence can be inferred from the perturbations they create in the dust and gas of protoplanetary disks. Here we present a new interpretation of SPHERE polarized observations that reveal the previously detected inner spiral in the disk of HD 100546. The spiral coincides with a newly detected 12 CO inner spiral and the previously reported CO emission Doppler flip, which has been interpreted as the signature of an embedded protoplanet. Comparisons with hydrodynamical models indicate that this Doppler flip is instead the kinematic counterpart of the spiral, which is likely generated by an inner companion inside the disk cavity.

79 ASTRONOMY AND ASTROPHYSICS↗

X-Ray Performance of Critical-angle Transmission Grating Prototypes for the Arcus Mission

Arcus is a proposed Explorer Class soft X-ray grating spectrometer. It aims to explore cosmic feedback by mapping hot gases within and between galaxies and galaxy clusters and characterizing jets and winds from supermassive black holes and to investigate the dynamics of protoplanetary disks and stellar accretion. Arcus features 12 m focal-length grazing-incidence silicon pore optics (SPO) developed for the Athena mission. Critical-angle transmission (CAT) gratings efficiently disperse high diffraction orders onto CCDs. We report new and improved X-ray performance results for Arcus-like CAT gratings, including a record resolving power for two coaligned CAT gratings. Multiple Arcus prototype grating facets were illuminated by an SPO at the PANTER facility. The facets consist of 32 × 32.5 mm 2 patterned silicon membranes, bonded to metal frames. The bonding angle is adjusted according to the measured average tilt angle of the grating bars in the membrane. Two simultaneously illuminated facets show a minor broadening of the Al-K α doublet in the 18th and 21st orders with the best-fit record effective resolving power of ${R}_{G}\approx {1.3}_{-0.5}^{+\infty }\times {10}^{4}$ (3σ), about three to four times the Arcus requirement. We measured the diffraction efficiency of quasi-fully illuminated gratings at O-K wavelengths in orders 4–7 in an Arcus-like configuration and compare results with synchrotron spot measurements. After corrections for geometrical effects and bremsstrahlung continuum, we find agreement between full and spot illumination at the two different facilities, as well as with the models used for Arcus effective area predictions. We find that these flight-like gratings meet the diffraction efficiency and greatly exceed the resolving power Arcus requires.

79 ASTRONOMY AND ASTROPHYSICS↗

Observation of Nonaxisymmetric Standard Magnetorotational Instability Induced by a Free-Shear Layer

The standard magnetorotational instability (SMRI) with a magnetic field component parallel to the rotation axis is widely believed to be responsible for the fast accretion in astronomical disks. In conventional base flows with a Keplerian profile or an ideal Couette profile, most studies focus on axisymmetric SMRI, since excitation of nonaxisymmetric SMRI in such flows requires a magnetic Reynolds number (Rm) more than an order of magnitude larger. Here, we report that, in a magnetized Taylor-Couette flow, nonaxisymmetric SMRI with an azimuthal mode number m = 1 can be triggered by a free-shear layer in the base flow at Rm ≳1, the same threshold as for axisymmetric SMRI. Global linear analysis reveals that the free-shear layer reduces the required Rm, possibly by introducing an extremum in the vorticity of the base flow. Nonlinear simulations validate the results from linear analysis and confirm that a novel instability recently discovered experimentally [Wang et al., Nat. Commun. 13, 4679 (2022)] is the nonaxisymmetric m =1 SMRI. Further, our finding has astronomical implications as free-shear layers are ubiquitous in celestial systems, such as the disk-star boundary layer, the solar tachocline, and the edge of planet-opened gaps in protoplanetary disks.

79 ASTRONOMY AND ASTROPHYSICS↗

Does the Sun have a dark disk?

The Sun is not quite a perfect sphere, and its oblateness, thought to be induced through its rotation, has been measured using optical observations of its radius. Its gravitational quadrupole moment can then be deduced using solar models, or through helioseismology, and it can also be determined from measurements of its gravitational effects on Mercury’s orbit. The various assessments do not appear to agree, with the most complete and precise orbital assessments being in slight excess of other determinations. This may speak to the existence of a nonluminous disk or ring, where we also note evidence for a circumsolar dust ring within Mercury’s orbit from the Solar TErrestrial RElations Observatory (STEREO) mission. Historically, too, a protoplanetary disk may have been key to reconciling the Sun’s metallicity with its neutrino yield. The distribution of the nonluminous mass within Mercury’s orbit can modify the relative size of the optical and orbital quadrupole moments in different ways. We develop how we can use these findings to limit the mass of a dark disk, ring, or halo in the immediate vicinity of the Sun, and we note how future observational studies of the inner Solar System can not only refine these constraints but can also help to identify and to assess the mass of its dark-matter component. Published by the American Physical Society 2025

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Constraints on Chondrule Generation, Disk Dynamics, and Asteroid Accretion from the Compositions of Carbonaceous Meteorites

The elemental and isotopic compositions of meteorites are expected to reflect several key processes that occurred in the early solar system, including the migration of gas and dust throughout the protoplanetary disk, the formation of chondrules, and the accretion of the first planetary bodies. However, the specific origins of the various compositions measured among these rocks are currently poorly constrained, limiting our understanding of these processes. Here, we use previously measured elemental and isotopic compositions of chondrites and iron meteorites to identify that carbonaceous (CC) meteorites are mixtures of noncarbonaceous (NC) material, calcium–aluminum-rich inclusion (CAI) material, and CI (Ivuna-like) material, in varying proportions. These trends indicate that chondrules in CO (Ornans-like), CM (Mighei-like), CV (Vigarano-like), and TL (Tagish Lake) chondrites share near-identical average proportions of CI material, arguing that they were generated through the remelting of preexisting NC chondrules all in the same disk environment. Because this proportion likely evolved over space and time throughout the disk, this similarity argues that these chondrules originate from a restricted spatial region and time interval, favoring their generation through a localized event. Moreover, the compositions of CR (Renazzo-like) chondrites indicate that their constituents formed through mechanisms different from those in CO, CM, CV, and TL chondrites. Furthermore, the recovered proportions of CI material in CC iron meteorites and chondrites together also argue for evolution in either the predominant direction of dust and gas motion in the first ~10 au of the disk or the radial distance of planetesimal accretion throughout the CC reservoir.

58 GEOSCIENCES↗

Elementary Composite Binary and Grain Alignment Locked in Dust Growth

Abstract Planets are known to grow out of a star-encircling disk of the gas and dust inherited from an interstellar cloud; their formation is thought to begin with coagulation of submicron dust grains into aggregates, the first foundational stage of planet formation. However, with nanoscale and submicron solids unobservable directly in the interstellar medium (ISM) and protoplanetary disks, how dust grains grow is unclear, as are the morphology and structure of interstellar grains and the whereabouts and form of “missing iron.” Here we show an elementary composite binary in 3D sub-10 nm detail—and the alignments of its two subunits and nanoinclusions and a population of elongated composite grains locked in a primitive cosmic dust particle—noninvasively uncovered with phase-contrast X-ray nanotomography. The binary comprises a pair of oblate, quasi-spheroidal grains whose alignment and shape meet the astrophysical constraints on polarizing interstellar grains. Each member of the pair contains a high-density core of octahedral nanocrystals whose twin relationship is consistent with the magnetite’s diagnostic property at low temperatures, with a mantle exhibiting nanoscale heterogeneities, rounded edges, and pitted surfaces. This elongated binary evidently formed from an axially aligned collision of the two similar composite grains whose core–mantle structure and density gradients are consistent with interstellar processes and astronomical evidence for differential depletion. Our findings suggest that the ISM is threaded with dust grains containing preferentially oriented iron-rich magnetic nanocrystals that hold answers to astronomical problems from dust evolution, grain alignment, and the structure of magnetic fields to planetesimal growth.

Hu, Z. W. (ORCID:0000000278648830)↗

Evidence for the Late Arrival of Hot Jupiters in Systems with High Host-star Obliquities

It has been shown that hot Jupiters systems with massive, hot stellar primaries exhibit a wide range of stellar obliquities. On the other hand, hot Jupiter systems with low-mass, cool primaries often have stellar obliquities close to zero. Efficient tidal interactions between hot Jupiters and the convective envelopes present in lower-mass main-sequence stars have been a popular explanation for these observations. If this explanation is accurate, then aligned systems should be older than misaligned systems. Likewise, the convective envelope mass of a hot Jupiter's host star should be an effective predictor of its obliquity. We derive homogeneous stellar parameters—including convective envelope masses—for hot Jupiter host stars with high-quality sky-projected obliquity inferences. Using a thin-disk stellar population's Galactic velocity dispersion as a relative age proxy, we find that hot Jupiter host stars with larger-than-median obliquities are older than hot Jupiter host stars with smaller-than-median obliquities. The relative age difference between the two populations is larger for hot Jupiter host stars with smaller-than-median fractional convective envelope masses and is significant at the 3.6σ level. We identify stellar mass, not convective envelope mass, as the best predictor of stellar obliquity in hot Jupiter systems. The best explanation for these observations is that many hot Jupiters in misaligned systems arrived in the close proximity of their host stars long after their parent protoplanetary disks dissipated. The dependence of observed age offset on convective envelope mass suggests that tidal realignment contributes to the population of aligned hot Jupiters orbiting stars with convective envelopes.

79 ASTRONOMY AND ASTROPHYSICS↗

A collective trigger for widespread planetesimal formation revealed by accretion ages

The formation of planetesimals was an integral part of the cascading series of processes that built the terrestrial planets. To illuminate planetesimal formation, here we develop a refined thermal evolution model to calculate the formation ages of meteorite parent planetesimals. This model includes chemical reactions and phase changes during heating, as well as natural variations in the proportions of the constituent phases of these planetesimals. We find that the parent bodies of non-carbonaceous (NC) and carbonaceous (CC) iron meteorites start forming at very similar times ( ∼ 0.95 Myr after calcium-aluminium-rich inclusion [CAI] formation) and occupy overlapping time windows. NC and CC chondrite parent bodies formed later during non-overlapping periods. We combine these ages with proportions of isotopic end-members we recover from mixing models to construct records of motion throughout the protoplanetary disk. These records argue that NC and CC material traversed the barrier in the disk after ∼ 0.95 Myr after CAI formation. The onset of this motion coincided with planetesimal formation, indicating that the phenomenon that drove motion also triggered planetesimal formation. We argue that this feature also served as the semi-permeable barrier in the disk. Although its identity is uncertain, the effects this phenomenon had on the timing of planetesimal formation and motion through the disk can now serve as constraints on models of disk evolution. Models that reproduce these effects would elucidate the nature and implications of this phenomenon, which is key to unlocking a holistic model of terrestrial planet building.

Astronomy and AstroPhysics↗

Titanium isotope systematics of refractory inclusions: Echoes of molecular cloud heterogeneity

Calcium-aluminum-rich inclusions (CAIs) are highly refractory objects found in different chondrite groups and represent some of the oldest known solids of the Solar System. As such, CAIs provide key information regarding the conditions prevailing in the solar protoplanetary disk as well as subsequent mixing and transport processes. Many studies have investigated CAIs for their isotopic compositions and reported nucleosynthetic isotope anomalies in numerous elements, which are typically explained by the variable incorporation of isotopically highly anomalous presolar phases. However, with the exception of 54 Cr-enriched nanospinels, the exact presolar phases responsible for the isotopic heterogeneities are yet to be identified. To address this issue, we here present in-situ Ti isotopic analyses obtained on a diverse set of CAIs from various CV3 chondrites. In this work, the in-situ measurements were performed by targeting individual mineral phases of 15 CAIs with laser-ablation mass spectrometry and indicate significant inter- and intra-CAI isotopic heterogeneity in the neutron-rich isotope 50 Ti. This is particularly pronounced for primitive fine-grained CAIs, whereas coarse-grained CAIs, which have been subject to melting, exhibit smaller degrees of Ti isotopic heterogeneity. To further investigate this Ti isotopic heterogeneity, we additionally obtained Ti isotopic compositions of sequential acid leachates from two fine-grained and two coarse-grained CAIs derived from CV3 chondrites. In contrast to potential expectations from the first part of the study, we do not observe any significant intra-CAI Ti isotopic heterogeneity between the different leaching steps. The lack of intra-CAI Ti isotopic heterogeneity in the acid leachate samples of this study likely reflects that the leaching procedure is unable to efficiently separate the carriers of isotopically anomalous Ti in CAIs. By comparing the bulk CAI Ti isotope compositions with Ti isotope data for hibonite-rich objects from the literature, we find that the range of Ti isotope compositions recorded by CAIs from various chondrite groups can be accounted for by the averaging of hibonite grains. In turn, the variable Ti isotope compositions of hibonite grains can be explained by the averaging of isotopically diverse presolar grains present in the Sun’s parental molecular cloud. This effect of averaging is statistically supported by the central limit theorem, and the concept has the potential to be useful for other isotopic systems.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗