arXiv subjects

Karin Lind

Publications and source records attributed to Karin Lind.

At least 37 records · Page 2Linked to original sources

arXiv2021

The GALAH Survey: Improving our understanding of confirmed and candidate planetary systems with large stellar surveys

Pioneering photometric, astrometric, and spectroscopic surveys are helping exoplanetary scientists better constrain the fundamental properties of stars within our galaxy, and the planets these stars host. In this study, we use the third data release from the stellar spectroscopic GALAH Survey, coupled with astrometric data of eDR3 from the \textit{Gaia} satellite, and other data from NASA's Exoplanet Archive, to refine our understanding of 279 confirmed and candidate exoplanet host stars and their exoplanets. This homogenously analysed data set comprises 105 confirmed exoplanets, along with 146 K2 candidates, 95 TESS Objects of Interest (TOIs) and 52 Community TOIs (CTOIs). Our analysis significantly shifts several previously (unknown) planet parameters while decreasing the uncertainties for others; Our radius estimates suggest that 35 planet candidates are more likely brown dwarfs or stellar companions due to their new radius values. We are able to refine the radii and masses of WASP-47 e, K2-106 b, and CoRoT-7 b to their most precise values yet, to less than 2.3\% and 8.5\% respectively. We also use stellar rotational values from GALAH to show that most planet candidates will have mass measurements that will be tough to obtain with current ground-based spectrographs. With GALAH's chemical abundances, we show through chemo-kinematics that there are five planet-hosts that are associated with the galaxy's thick disc, including NGTS-4, K2-183 and K2-337. Finally, we show there is no statistical difference between the chemical properties of hot Neptune and hot rocky exoplanet hosts, with the possibility that short-period rocky worlds might be the remnant cores of hotter, gaseous worlds.

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arXiv2021

The GALAH survey: Chemical homogeneity of the Orion complex

Due to its proximity, the Orion star forming region is often used as a proxy to study processes related to star formation and to observe young stars in the environment they were born in. With the release of Gaia DR2, the distance measurements to the Orion complex are now good enough that the three dimensional structure of the complex can be explored. Here we test the hypothesis that, due to non-trivial structure and dynamics, and age spread in the Orion complex, the chemical enrichment of youngest stars by early core-collapse supernovae can be observed. We obtained spectra of 794 stars of the Orion complex with the HERMES spectrograph at the Anglo Australian telescope as a part of the GALAH and GALAH-related surveys. We use the spectra of $\sim300$ stars to derive precise atmospheric parameters and chemical abundances of 25 elements for 15 stellar clusters in the Orion complex. We demonstrate that the Orion complex is chemically homogeneous and that there was no self-pollution of young clusters by core-collapse supernovae from older clusters; with a precision of 0.02 dex in relative alpha-elements abundance and 0.06 dex in oxygen abundance we would have been able to detect pollution from a single supernova, given a fortunate location of the SN and favourable conditions for ISM mixing. We estimate that the supernova rate in the Orion complex was very low, possibly producing no supernova by the time the youngest stars of the observed population formed (from around 21 to 8 Myr ago).

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arXiv2021

The GALAH+ Survey: A New Library of Observed Stellar Spectra Improves Radial Velocities and Hints at Motions within M67

GALAH+ is a magnitude-limited survey of high resolution stellar spectra obtained by the HERMES spectrograph at the Australian Astronomical Observatory. Its third data release provides reduced spectra with new derivations of stellar parameters and abundances of 30 chemical elements for 584,015 dwarfs and giants, 88% of them in the Gaia magnitude range 11 < G < 14. Here we use these improved values of stellar parameters to build a library of observed spectra which is useful to study variations of individual spectral lines with stellar parameters. This and other improvements are used to derive radial velocities with uncertainties which are generally within 0.1 km/s or ~25% smaller than in the previous release. Median differences in radial velocities measured here and by the Gaia DR2 or APOGEE DR16 surveys are smaller than 30 m/s, a larger offset is present only for Gaia measurements of giant stars. We identify 4483 stars with intrinsically variable velocities and 225 stars for which the velocity stays constant over >=3 visits spanning more than a year. The combination of radial velocities from GALAH+ with distances and sky plane motions from Gaia enables studies of dynamics within streams and clusters. For example, we estimate that the open cluster M67 has a total mass of ~3300 Msun and its outer parts seem to be expanding, though astrometry with a larger time-span than currently available from Gaia eDR3 is needed to judge if the latter result is real.

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arXiv2021

The GALAH Survey: Accreted stars also inhabit the Spite Plateau

The ESA Gaia astrometric mission has enabled the remarkable discovery that a large fraction of the stars near the Solar neighbourhood appear to be debris from a single in-falling system, the so-called Gaia-Enceladus-Sausage (GSE). One exciting feature of this result is that it gives astronomers for the first time a large sample of easily observable unevolved stars that formed in an extra-Galactic environment, which can be compared to stars that formed within our Milky Way. Here we use these stars to investigate the "Spite Plateau" -- the near-constant lithium abundance observed in metal-poor dwarf stars across a wide range of metallicities (-3<[Fe/H]<-1). In particular our aim is to test whether the stars that formed in the GSE show a different Spite Plateau to other Milky Way stars that inhabit the disk and halo. Individual galaxies could have different Spite Plateaus -- e.g., the ISM could be more depleted in lithium in a lower galactic mass system due to it having a smaller reservoir of gas. We identified 76 GSE dwarf stars observed and analyzed by the GALactic Archeology with HERMES (GALAH) survey as part of its Third Data Release. Orbital actions were used to select samples of Gaia-Enceladus stars, and comparison samples of halo and disk stars. We find that the Gaia-Enceladus stars show the same lithium abundance as other likely accreted stars and in situ Milky Way stars, strongly suggesting that the "lithium problem" is not a consequence of the formation environment. This result fits within the growing consensus that the Spite Plateau, and more generally the "cosmological lithium problem" -- the observed discrepancy between the amount of lithium in warm, metal-poor dwarf stars in our Galaxy, and the amount of lithium predicted to have been produced by Big Bang Nucleosynthesis -- is the result of lithium depletion processes within stars.

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arXiv2021

The GALAH survey: A census of lithium-rich giant stars

We investigate the properties of 1262 red giant stars with high photospheric abundances of lithium observed by the GALAH and \Ktwo-HERMES surveys, and discuss them in the context of proposed mechanisms for lithium enrichment and re-depletion in giant stars. We confirm that Li-rich giants are rare, making up only 1.2 per cent of our giant star sample. We use stellar parameters from the third public data release from the GALAH survey and a Bayesian isochrone analysis to divide the sample into first-ascent red giant branch and red clump stars, and confirm these classifications using asteroseismic data from \Ktwo. We find that red clump stars are 2.5 times as likely to be lithium-rich as red giant branch stars, in agreement with other recent work. The probability for a star to be lithium-rich is affected by a number of factors, though the causality in those correlations is not entirely clear. We show for the first time that primary and secondary red clump stars have distinctly different lithium enrichment patterns. The data set discussed here is large and heterogeneous in terms of evolutionary phase, metallicity, rotation rate and mass. We expect that if the various mechanisms that have been proposed for lithium enrichment in evolved stars are in fact active, they should all contribute to this sample of lithium-rich giants at some level.

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arXiv2021

The GALAH+ Survey: Third Data Release

The ensemble of chemical element abundance measurements for stars, along with precision distances and orbit properties, provides high-dimensional data to study the evolution of the Milky Way. With this third data release of the Galactic Archaeology with HERMES (GALAH) survey, we publish 678 423 spectra for 588 571 mostly nearby stars (81.2% of stars are within <2 kpc), observed with the HERMES spectrograph at the Anglo-Australian Telescope. This release (hereafter GALAH+ DR3) includes all observations from GALAH Phase 1 (bright, main, and faint survey, 70%), K2-HERMES (17%), TESS-HERMES (5%), and a subset of ancillary observations (8%) including the bulge and >75 stellar clusters. We derive stellar parameters $T_\text{eff}$, $\log g$, [Fe/H], $v_\text{mic}$, $v_\text{broad}$ & $v_\text{rad}$ using our modified version of the spectrum synthesis code Spectroscopy Made Easy (SME) and 1D MARCS model atmospheres. We break spectroscopic degeneracies in our spectrum analysis with astrometry from $Gaia$ DR2 and photometry from 2MASS. We report abundance ratios [X/Fe] for 30 different elements (11 of which are based on non-LTE computations) covering five nucleosynthetic pathways. We describe validations for accuracy and precision, flagging of peculiar stars/measurements and recommendations for using our results. Our catalogue comprises 65% dwarfs, 34% giants, and 1% other/unclassified stars. Based on unflagged chemical composition and age, we find 62% young low-$α$, 9% young high-$α$, 27% old high-$α$, and 2% stars with $\mathrm{[Fe/H]} \leq -1$. Based on kinematics, 4% are halo stars. Several Value-Added-Catalogues, including stellar ages and dynamics, updated after $Gaia$ eDR3, accompany this release and allow chrono-chemodynamic analyses, as we showcase.

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arXiv2021

The GALAH Survey: No chemical evidence of an extragalactic origin for the Nyx stream

The results from the ESA Gaia astrometric mission and deep photometric surveys have revolutionized our knowledge of the Milky Way. There are many ongoing efforts to search these data for stellar substructure to find evidence of individual accretion events that built up the Milky Way and its halo. One of these newly identified features, called Nyx, was announced as an accreted stellar stream traveling in the plane of the disk. Using a combination of elemental abundances and stellar parameters from the GALAH and APOGEE surveys, we find that the abundances of the highest likelihood Nyx members are entirely consistent with membership of the thick disk, and inconsistent with a dwarf galaxy origin. We conclude that the postulated Nyx stream is most probably a high-velocity component of the Milky Way's thick disk. With the growing availability of large data sets including kinematics, stellar parameters, and detailed abundances, the probability of detecting chance associations increases, and hence new searches for substructure require confirmation across as many data dimensions as possible.

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arXiv2021

The GALAH survey: tracing the Galactic disk with Open Clusters

Open clusters are unique tracers of the history of our own Galaxy's disk. According to our membership analysis based on \textit{Gaia} astrometry, out of the 226 potential clusters falling in the footprint of GALAH or APOGEE, we find that 205 have secure members that were observed by at least one of the survey. Furthermore, members of 134 clusters have high-quality spectroscopic data that we use to determine their chemical composition. We leverage this information to study the chemical distribution throughout the Galactic disk of 21 elements, from C to Eu. The radial metallicity gradient obtained from our analysis is $-$0.076$\pm$0.009 dex kpc$^{-1}$, which is in agreement with previous works based on smaller samples. Furthermore, the gradient in the [Fe/H] - guiding radius (r$_{\rm guid}$) plane is $-$0.073$\pm$0.008 dex kpc$^{-1}$. We show consistently that open clusters trace the distribution of chemical elements throughout the Galactic disk differently than field stars. In particular, at given radius, open clusters show an age-metallicity relation that has less scatter than field stars. As such scatter is often interpreted as an effect of radial migration, we suggest that these differences are due to the physical selection effect imposed by our Galaxy: clusters that would have migrated significantly also had higher chances to get destroyed. Finally, our results reveal trends in the [X/Fe]$-$r$_{\rm guid}$$-$age space, which are important to understand production rates of different elements as a function of space and time.

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arXiv2020

The GALAH Survey: Dependence of elemental abundances on age and metallicity for stars in the Galactic disc

Using data from the GALAH survey, we explore the dependence of elemental abundances on stellar age and metallicity among Galactic disc stars. We find that the abundance of most elements can be predicted from age and [Fe/H] with an intrinsic scatter of about 0.03 dex. We discuss the possible causes for the existence of the abundance-age-metallicity relations. Using a stochastic chemical enrichment scheme based on the size of Supernovae remnants, we show the intrinsic scatter is expected to be small, about 0.05 dex or even smaller if there is additional mixing in the ISM. Elemental abundances show trends with both age and metallicity and the relationship is well described by a simple model in which the dependence of abundance ([X/Fe]) on age and [Fe/H] are additively separable. Elements can be grouped based on the direction of their abundance gradient in the (age,[Fe/H]) plane and different groups can be roughly associated with three distinct nucleosynthetic production sites, the exploding massive stars, the exploding white dwarfs and the AGB stars. However, the abundances of some elements, like Co, La, and Li, show large scatter for a given age and metallicity, suggesting processes other than simple Galactic chemical evolution are at play. We also compare the abundance trends of main-sequence turn-off stars against that of giants, whose ages were estimated using asteroseismic information from the K2 mission. For most elements, the trends of main-sequence turn-off stars are similar to that of giants. The existence of abundance relations implies that we can estimate the age and birth radius of disc stars, which is important for studying the dynamic and chemical evolution of the Galaxy.

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arXiv2020

The GALAH Survey: Chemical Clocks

Previous studies have found that the elemental abundances of a star correlate directly with its age and metallicity. Using this knowledge, we derive ages for a sample of 250,000 stars taken from GALAH DR3 using only their overall metallicity and chemical abundances. Stellar ages are estimated via the machine learning algorithm $XGBoost$, using main sequence turnoff stars with precise ages as our input training set. We find that the stellar ages for the bulk of the GALAH DR3 sample are accurate to 1-2 Gyr using this method. With these ages, we replicate many recent results on the age-kinematic trends of the nearby disk, including the age-velocity dispersion relationship of the solar neighborhood and the larger global velocity dispersion relations of the disk found using $Gaia$ and GALAH. The fact that chemical abundances alone can be used to determine a reliable age for a star have profound implications for the future study of the Galaxy as well as upcoming spectroscopic surveys. These results show that the chemical abundance variation at a given birth radius is quite small, and imply that strong chemical tagging of stars directly to birth clusters may prove difficult with our current elemental abundance precision. Our results highlight the need of spectroscopic surveys to deliver precision abundances for as many nucleosynthetic production sites as possible in order to estimate reliable ages for stars directly from their chemical abundances. Applying the methods outlined in this paper opens a new door into studies of the kinematic structure and evolution of the disk, as ages may potentially be estimated for a large fraction of stars in existing spectroscopic surveys. This would yield a sample of millions of stars with reliable age determinations, and allow precise constraints to be put on various kinematic processes in the disk, such as the efficiency and timescales of radial migration.

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arXiv2020

Atomic data for the Gaia-ESO Survey

We describe the atomic and molecular data that were used for the abundance analyses of FGK-type stars carried out within the Gaia-ESO Survey. We present an unprecedented effort to create a homogeneous line list, which was used by several abundance analysis groups to calculate synthetic spectra and equivalent widths. The atomic data are accompanied by quality indicators and detailed references to the sources. The atomic and molecular data are made publicly available in electronic form. In general experimental transition probabilities were preferred but theoretical values were also used. Astrophysical gf-values were avoided due to the model-dependence of such a procedure. For elements whose lines are significantly affected by hyperfine structure or isotopic splitting a concerted effort has been made to collate the necessary data for the individual line components. We also performed a detailed investigation of available data for line broadening due to collisions with neutral hydrogen atoms. Synthetic spectra calculated for the Sun and Arcturus were used to assess the blending properties of the lines. Among a subset of over 1300 lines of 35 elements in the wavelength ranges from 475 nm to 685 nm and from 850 nm to 895 nm we identified about 200 lines of 24 species which have accurate gf-values and are free of blends in the spectra of the Sun and Arcturus. For the broadening due to collisions with neutral hydrogen we recommend data based on Anstee-Barklem-O'Mara theory, where available, and to avoid lines of neutral species otherwise. Theoretical broadening data by R.L. Kurucz should be used for Sc II, Ti II, and Y II lines. For ionised rare-earth species the Unsöld approximation with an enhancement factor of 1.5 for the line width can be used. Desirable improvements in atomic data were identified for a number of species, including Al I, S I, Cr II, Na I, Si I, Ca II, and Ni I.

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arXiv2020

The GALAH Survey: Using Galactic Archaeology to Refine our Knowledge of TESS Target Stars

An unprecedented number of exoplanets are being discovered by the Transiting Exoplanet Survey Satellite (TESS). Determining the orbital parameters of these exoplanets, and especially their mass and radius, will depend heavily upon the measured physical characteristics of their host stars. We have cross-matched spectroscopic, photometric, and astrometric data from GALAH Data Release 2, the TESS Input Catalog and Gaia Data Release 2, to create a curated, self-consistent catalog of physical and chemical properties for 47,285 stars. Using these data we have derived isochrone masses and radii that are precise to within 5\%. We have revised the parameters of three confirmed, and twelve candidate, TESS planetary systems. These results cast doubt on whether CTOI-20125677 is indeed a planetary system since the revised planetary radii are now comparable to stellar sizes. Our GALAH-TESS catalog contains abundances for up to 23 elements. We have specifically analysed the molar ratios for C/O, Mg/Si, Fe/Si and Fe/Mg, to assist in determining the composition and structure of planets with $R_p < 4R_\oplus$. From these ratios, 36% fall within 2 sigma of the Sun/Earth values, suggesting that these stars may host rocky exoplanets with geological compositions similar to planets found within our own Solar system.

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arXiv2020

The GALAH Survey: A new constraint on cosmological lithium and Galactic lithium evolution from warm dwarf stars

Lithium depletion and enrichment in the cosmos is not yet well understood. To help tighten constraints on stellar and Galactic evolution models, we present the largest high-resolution analysis of Li abundances A(Li) to date, with results for over 100 000 GALAH field stars spanning effective temperatures $5900\,\mathrm{K} \lesssim \rm{T_{eff}} \lesssim7000\,\mathrm{K}$ and metallicities $-3 \lesssim \rm[Fe/H] \lesssim +0.5$. We separated these stars into two groups, on the warm and cool side of the so-called Li-dip, a localised region of the Kiel diagram wherein lithium is severely depleted. We discovered that stars in these two groups show similar trends in the A(Li)-[Fe/H] plane, but with a roughly constant offset in A(Li) of 0.4 dex, the warm group having higher Li abundances. At $\rm[Fe/H]\gtrsim-0.5$, a significant increasing in Li abundance with increasing metallicity is evident in both groups, signalling the onset of significant Galactic production. At lower metallicity, stars in the cool group sit on the Spite plateau, showing a reduced lithium of around 0.4 dex relative to the primordial value predicted from Big Bang nucleosynthesis (BBN). However, stars in the warm group between [Fe/H] = -1.0 and -0.5, form an elevated plateau that is largely consistent with the BBN prediction. This may indicate that these stars in fact preserve the primordial Li produced in the early Universe.

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arXiv2020

K2-HERMES II. Planet-candidate properties from K2 Campaigns 1-13

Accurate and precise radius estimates of transiting exoplanets are critical for understanding their compositions and formation mechanisms. To know the planet, we must know the host star in as much detail as possible. We present complete results for planet-candidate hosts from the K2-HERMES survey, which uses the HERMES multi-object spectrograph on the Anglo-Australian Telescope to obtain R$\sim$28,000 spectra for more than 30,000 K2 stars. We present complete host-star parameters and planet-candidate radii for 224 K2 candidate planets from C1-C13. Our results cast severe doubt on 30 K2 candidates, as we derive unphysically large radii, larger than 2 $R_{Jup}$. This work highlights the importance of obtaining accurate, precise, and self-consistent stellar parameters for ongoing large planet search programs - something that will only become more important in the coming years, as TESS begins to deliver its own harvest of exoplanets.

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arXiv2020

Fundamental relations for the velocity dispersion of stars in the Milky Way

We explore the fundamental relations governing the radial and vertical velocity dispersions of stars in the Milky Way, from combined studies of complementary surveys including GALAH, LAMOST, APOGEE, the NASA $Kepler$ and K2 missions, and $Gaia$ DR2. We find that different stellar samples, even though they target different tracer populations and employ a variety of age estimation techniques, follow the same set of fundamental relations. We provide the clearest evidence to date that, in addition to the well-known dependence on stellar age, the velocity dispersions of stars depend on orbital angular momentum $L_z$, metallicity and height above the plane $|z|$, and are well described by a multiplicatively separable functional form. The dispersions have a power-law dependence on age with exponents of 0.441$\pm 0.007$ and 0.251$\pm 0.006$ for $σ_z$ and $σ_R$ respectively, and the power law is valid even for the oldest stars. For the solar neighborhood stars, the apparent break in the power law for older stars, as seen in previous studies, is due to the anti-correlation of $L_z$ with age. The dispersions decrease with increasing $L_z$ until we reach the Sun's orbital angular momentum, after which $σ_z$ increases (implying flaring in the outer disc) while $σ_R$ flattens. The dispersions increase with decreasing metallicity, suggesting that the dispersions increase with birth radius. The dispersions also increase linearly with $|z|$. The same set of relations that work in the solar neighborhood also work for stars between $3<R/{\rm kpc}<20$. Finally, the high-[$α$/Fe] stars follow the same relations as the low-[$α$/Fe] stars.

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arXiv2019

The GALAH Survey: Chemically tagging the Fimbulthul stream to the globular cluster $ω$ Centauri

Using kinematics from Gaia and the large elemental abundance space of the second data release of the GALAH survey, we identify two new members of the Fimbulthul stellar stream, and chemically tag them to massive, multi-metallic globular cluster $ω$ Centauri. Recent analysis of the second data release of Gaia had revealed the Fimbulthul stellar stream in the halo of the Milky Way. It had been proposed that the stream is associated with the $ω$ Centauri, but this proposition relied exclusively upon the kinematics and metallicities of the stars to make the association. In this work, we find our two new members of the stream to be metal-poor stars that are enhanced in sodium and aluminium, typical of second population globular cluster stars, but not otherwise seen in field stars. Furthermore, the stars share the s-process abundance pattern seen in $ω$ Centauri, which is rare in field stars. Apart from one star within 1.5 deg of $ω$ Centauri, we find no other stars observed by GALAH spatially near $ω$ Centauri or the Fimbulthul stream that could be kinematically and chemically linked to the cluster. Chemically tagging stars in the Fimbulthul stream to $ω$ Centauri confirms the earlier work, and further links this tidal feature in the Milky Way halo to $ω$ Centauri.

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arXiv2019

Abundance Estimates for 16 Elements in 6 Million Stars from LAMOST DR5 Low-Resolution Spectra

We present the determination of stellar parameters and individual elemental abundances for 6 million stars from $\sim$8 million low-resolution ($R\sim1800$) spectra from LAMOST DR5. This is based on a modeling approach that we dub $The$ $Data$--$Driven$ $Payne$ ($DD$--$Payne$), which inherits essential ingredients from both {\it The Payne} \citep{Ting2019} and $The$ $Cannon$ \citep{Ness2015}. It is a data-driven model that incorporates constraints from theoretical spectral models to ensure the derived abundance estimates are physically sensible. Stars in LAMOST DR5 that are in common with either GALAH DR2 or APOGEE DR14 are used to train a model that delivers stellar parameters ($T_{\rm eff}$, $\log g$, $V_{\rm mic}$) and abundances for 16 elements (C, N, O, Na, Mg, Al, Si, Ca, Ti, Cr, Mn, Fe, Co, Ni, Cu, and Ba) when applied to LAMOST spectra. Cross-validation and repeat observations suggest that, for ${\rm S/N}_{\rm pix}\ge 50$, the typical internal abundance precision is 0.03--0.1\,dex for the majority of these elements, with 0.2--0.3\,dex for Cu and Ba, and the internal precision of $T_{\rm eff}$ and $\log g$ is better than 30\,K and 0.07\,dex, respectively. Abundance systematics at the $\sim$0.1\,dex level are present in these estimates, but are inherited from the high-resolution surveys' training labels. For some elements, GALAH provides more robust training labels, for others, APOGEE. We provide flags to guide the quality of the label determination and to identify binary/multiple stars in LAMOST DR5. The abundance catalogs are publicly accessible via \href{url}{http://dr5.lamost.org/doc/vac}.

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arXiv2019

Spatial variations in the Milky Way disc metallicity-age relation

Stellar ages are a crucial component to studying the evolution of the Milky Way. Using Gaia DR2 distance estimates, it is now possible to estimate stellar ages for a larger volume of evolved stars through isochrone matching. This work presents [M/H]-age and [$α$/M]-age relations derived for different spatial locations in the Milky Way disc. These relations are derived by hierarchically modelling the star formation history of stars within a given chemical abundance bin. For the first time, we directly observe that significant variation is apparent in the [M/H]-age relation as a function of both Galactocentric radius and distance from the disc mid-plane. The [M/H]-age relations support claims that radial migration has a significant effect in the plane of the disc. Using the [M/H] bin with the youngest mean age at each radial zone in the plane of the disc, the present-day metallicity gradient is measured to be $-0.059 \pm 0.010$ dex kpc$^{-1}$, in agreement with Cepheids and young field stars. We find a vertically flared distribution of young stars in the outer disc, confirming predictions of models and previous observations. The mean age of the [M/H]-[$α$/M] distribution of the solar neighborhood suggests that the high-[M/H] stars are not an evolutionary extension of the low-$α$ sequence. Our observational results are important constraints to Galactic simulations and models of chemical evolution.

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