arXiv subjects

J. Ted Mackereth

Publications and source records attributed to J. Ted Mackereth.

At least 19 recordsLinked to original sources

arXiv2025

SF-R You Sure? The Conflicting Role of Star Formation Rates in Constraining the Evolution of Milky Way Analogues in Cosmological Simulations

Milky Way analogues (MWAs) have long been studied by astronomers to place our Galaxy within an extragalactic context. With the power of cosmological simulations, we are now able to not only characterize MWAs today, but also watch as they evolve through cosmic time. We use the EAGLE and IllustrisTNG simulations to study a group of MWAs defined by their stellar mass (SM) and star formation rate (SFR). We trace these galaxies back along their evolution to investigate the star forming and mass assembly tracks taken by a galaxy to become a MWA today in light of these chosen parameters. We also take mock-observations of "MWAs" at $z>0$ and trace them forwards in time to determine if galaxies that looked similar to the Milky Way earlier in their evolution still look like the Milky Way today, thus quantifying a selection efficiency which could inform future observational studies of MWAs. We find that most galaxies with Milky Way-SM follow a similar evolution regardless of present-day SFR, although MWAs in IllustrisTNG generally have not quenched, leading to star formation histories that produce "too-blue" galaxies today. Additionally, we find contamination by MWA-"imposters" in our mock-observations, with low selection efficiency at high redshift due to the tight constraint requiring convergence to the Milky Way's present-day SFR. Our work suggests present-day SM may suffice as a stand-alone selection parameter and helps to clarify how MWAs should be selected, and thus will be an important reference for future studies of both simulated and observed MWAs.

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arXiv2025

A Galactic Self-Portrait: Density Structure and Integrated Properties of the Milky Way Disk

The evolution history of the Milky Way disk is imprinted in the ages, positions, and chemical compositions of individual stars. In this study, we derive the intrinsic density distribution of different stellar populations using the final data release of the Apache Point Observatory Galactic Evolution Experiment (APOGEE) survey. A total of 203,197 red giant branch stars are used to sort the stellar disk ($R \leq 20$ kpc) into sub-populations of metallicity ($Δ$[M/H]$= 0.1$ dex), age ($Δ\log(\frac{\textrm{age}}{\textrm{yr}})= 0.1$), and $α$-element abundances ([$α$/M]). We fit the present-day structural parameters and density distribution of each stellar sub-population after correcting for the survey selection function. The low-$α$ disk is characterized by longer scale lengths and shorter scale heights, and is best fit by a broken exponential radial profile for each population. The high-$α$ disk is characterized by shorter scale lengths and larger scale heights, and is generally well-approximated by a single exponential radial profile. These results are applied to produce new estimates of the integrated properties of the Milky Way from early times to the present day. We measure the total stellar mass of the disk to be $5.27^{+0.2}_{-1.5} \times 10^{10}$ M$_\odot$ and the average mass-weighted scale length is $R_{d} = 2.37 \pm 0.2$ kpc. The Milky Way's present-day color of $(g-r) = 0.72 \pm 0.02$ is consistent with the classification of a red spiral galaxy, although it has only been in the "green valley" region of the galaxy color-mass diagram for the last $\sim 3$ Gyr.

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arXiv2023

The Galactic Interstellar Object Population: A Framework for Prediction and Inference

The Milky Way is thought to host a huge population of interstellar objects (ISOs), numbering approximately $10^{15}\mathrm{pc}^{-3}$ around the Sun, which are formed and shaped by a diverse set of processes ranging from planet formation to galactic dynamics. We define a novel framework: firstly to predict the properties of this Galactic ISO population by combining models of processes across planetary and galactic scales, and secondly to make inferences about the processes modelled, by comparing the predicted population to what is observed. We predict the spatial and compositional distribution of the Galaxy's population of ISOs by modelling the Galactic stellar population with data from the APOGEE survey and combining this with a protoplanetary disk chemistry model. Selecting ISO water mass fraction as an example observable quantity, we evaluate its distribution both at the position of the Sun and averaged over the Galactic disk; our prediction for the Solar neighbourhood is compatible with the inferred water mass fraction of 2I/Borisov. We show that the well-studied Galactic stellar metallicity gradient has a corresponding ISO compositional gradient. We also demonstrate the inference part of the framework by using the current observed ISO composition distribution to constrain the parent star metallicity dependence of the ISO production rate. This constraint, and other inferences made with this framework, will improve dramatically as the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST) progresses and more ISOs are observed. Finally, we explore generalisations of this framework to other Galactic populations, such as that of exoplanets.

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arXiv2023

The evolution of the Milky Way's thin disc radial metallicity gradient with K2 asteroseismic ages

The radial metallicity distribution of the Milky Way's disc is an important observational constraint for models of the formation and evolution of our Galaxy. It informs our understanding of the chemical enrichment of the Galactic disc and the dynamical processes therein, particularly radial migration. We investigate how the metallicity changes with guiding radius in the thin disc using a sample of red-giant stars with robust astrometric, spectroscopic and asteroseismic parameters. Our sample contains $668$ stars with guiding radii $4$ kpc < $R_\mathrm{g}$ < $11$ kpc and asteroseismic ages covering the whole history of the thin disc with precision $\approx 25\%$. We use MCMC analysis to measure the gradient and its intrinsic spread in bins of age and construct a hierarchical Bayesian model to investigate the evolution of these parameters independently of the bins. We find a smooth evolution of the gradient from $\approx -0.07$ dex/kpc in the youngest stars to $\approx -0.04$ dex/kpc in stars older than $10$ Gyr, with no break at intermediate ages. Our results are consistent with those based on asteroseismic ages from CoRoT, with that found in Cepheid variables for stars younger than $1$ Gyr, and with open clusters for stars younger than $6$ Gyr. For older stars we find a significantly lower metallicity in our sample than in the clusters, suggesting a survival bias favouring more metal-rich clusters. We also find that the chemical evolution model of Chiappini (2009) is too metal-poor in the early stages of disc formation. Our results provide strong new constraints for the growth and enrichment of the thin disc and radial migration, which will facilitate new tests of model conditions and physics.

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arXiv2023

Disentangling Stellar Age Estimates from Galactic Chemodynamical Evolution

Stellar ages are key for determining the formation history of the Milky Way, but are difficult to measure precisely. Furthermore, methods that use chemical abundances to infer ages may entangle the intrinsic evolution of stars with the chemodynamical evolution of the Galaxy. In this paper, we present a framework for making probabilistic predictions of stellar ages, and then quantify the contribution of both stellar evolution and Galactic chemical evolution to those predictions using SHAP values. We apply this interpretable prediction framework to both a simulated Milky Way sample containing stars in a variety of evolutionary stages and an APOGEE-mocked sample of red clump stars. We find that in the former case, stellar evolution is the dominant driver for age estimates, while in the latter case, the more restricted evolutionary information causes the model to proxy ages through the chemical evolution model. We show that as a result of the use of non-intrinsic Galactic chemical information, trends estimated with the predicted ages, such as the age-metallicity relation, can deviate from the truth.

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arXiv2023

A variational encoder-decoder approach to precise spectroscopic age estimation for large Galactic surveys

Constraints on the formation and evolution of the Milky Way Galaxy require multi-dimensional measurements of kinematics, abundances, and ages for a large population of stars. Ages for luminous giants, which can be seen to large distances, are an essential component of studies of the Milky Way, but they are traditionally very difficult to estimate precisely for a large dataset and often require careful analysis on a star-by-star basis in asteroseismology. Because spectra are easier to obtain for large samples, being able to determine precise ages from spectra allows for large age samples to be constructed, but spectroscopic ages are often imprecise and contaminated by abundance correlations. Here we present an application of a variational encoder-decoder on cross-domain astronomical data to solve these issues. The model is trained on pairs of observations from APOGEE and Kepler of the same star in order to reduce the dimensionality of the APOGEE spectra in a latent space while removing abundance information. The low dimensional latent representation of these spectra can then be trained to predict age with just $\sim$ 1,000 precise seismic ages. We demonstrate that this model produces more precise spectroscopic ages ($\sim$ 22% overall, $\sim$ 11% for red-clump stars) than previous data-driven spectroscopic ages while being less contaminated by abundance information (in particular, our ages do not depend on [$α$/M]). We create a public age catalog for the APOGEE DR17 data set and use it to map the age distribution and the age-[Fe/H]-[$α$/M] distribution across the radial range of the Galactic disk.

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arXiv2022

The chemical characterisation of halo substructure in the Milky Way based on APOGEE

Galactic haloes in a $Λ$-CDM universe are predicted to host today a swarm of debris resulting from cannibalised dwarf galaxies. The chemo-dynamical information recorded in their stellar populations helps elucidate their nature, constraining the assembly history of the Galaxy. Using data from APOGEE and \textit{Gaia}, we examine the chemical properties of various halo substructures, considering elements that sample various nucleosynthetic pathways. The systems studied are Heracles, \textit{Gaia}-Enceladus/Sausage (GES), the Helmi stream, Sequoia, Thamnos, Aleph, LMS-1, Arjuna, I'itoi, Nyx, Icarus, and Pontus. Abundance patterns of all substructures are cross-compared in a statistically robust fashion. Our main findings include: {\it i)} the chemical properties of most substructures studied match qualitatively those of dwarf Milky Way satellites, such as the Sagittarius dSph. Exceptions are Nyx and Aleph, which are chemically similar to disc stars, implying that these substructures were likely formed \textit{in situ}; {\it ii)} Heracles differs chemically from {\it in situ} populations such as Aurora and its inner halo counterparts in a statistically significant way. The differences suggest that the star formation rate was lower in Heracles than in the early Milky Way; {\it iii)} the chemistry of Arjuna, LMS-1, and I'itoi is indistinguishable from that of GES, suggesting a possible common origin; {\it iv)} all three Sequoia samples studied are qualitatively similar. However, only two of those samples present chemistry that is consistent with GES in a statistically significant fashion; {\it v)} the abundance patterns of the Helmi stream and Thamnos are different from all other halo substructures.

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arXiv2022

The contribution of N-rich stars to the Galactic stellar halo using APOGEE red giants

The contribution of dissolved globular clusters (GCs) to the stellar content of the Galactic halo is a key constraint on models for GC formation and destruction, and the mass assembly history of the Milky Way. Earlier results from APOGEE pointed to a large contribution of destroyed GCs to the stellar content of the inner halo, by as much as 25$\%$, which is an order of magnitude larger than previous estimates for more distant regions of the halo. We set out to measure the ratio between N-rich and normal halo field stars, as a function of distance, by performing density modelling of halo field populations in APOGEE DR16. Our results show that at 1.5 kpc from the Galactic Centre, N-rich stars contribute a much higher 16.8$^{+10.0}_{-7.0}$$\%$ fraction to the total stellar halo mass budget than the 2.7$^{+1.0}_{-0.8}$$\%$ ratio contributed at 10 kpc. Under the assumption that N-rich stars are former GC members that now reside in the stellar halo field, and assuming the ratio between first-and second-population GC stars being 1:2, we estimate a total contribution from disrupted GC stars of the order of 27.5$^{+15.4}_{-11.5}$$\%$ at r = 1.5 kpc and 4.2$^{+1.5}_{-1.3}$$\%$ at r = 10 kpc. Furthermore, since our methodology requires fitting a density model to the stellar halo, we integrate such density within a spherical shell from 1.5-15 kpc in radius, and find a total stellar mass arising from dissolved and/or evaporated GCs of $M_{\mathrm{GC,total}}$ = 9.6$^{+4.0}_{-2.6}$ $\times$ 10$^{7}$ M$\odot$.

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arXiv2022

The Kinematic Properties of Milky Way Stellar Halo Populations

In the $Gaia$ era stellar kinematics are extensively used to study Galactic halo stellar populations, to search for halo structures, and to characterize the interface between the halo and hot disc populations. We use distribution function-based models of modern datasets with 6D phase space data to qualitatively describe a variety of kinematic spaces commonly used in the study of the Galactic halo. Furthermore, we quantitatively assess how well each kinematic space can separate radially anisotropic from isotropic halo populations. We find that scaled action space (the ``action diamond'') is superior to other commonly used kinematic spaces at this task. We present a new, easy to implement selection criterion for members of the radially-anisotropic $Gaia$-Enceladus merger remnant. Assuming a 1:1 ratio of $Gaia$-Enceladus stars to more isotropic halo, we find our selection achieves a sample purity of 82 per cent in our models with respect to contamination from the more isotropic halo. We compare this criterion to literature criteria, finding that it produces the highest purity in the resulting samples, at the expense of a modest reduction in completeness. We also show that selection biases that underlie nearly all contemporary spectroscopic datasets can noticeably impact the $E-L_{z}$ distribution of samples in a manner that may be confused for real substructure. We conclude by providing recommendations for how authors should use stellar kinematics in the future to study the Galactic stellar halo.

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arXiv2022

A direct measurement of the distance to the Galactic center using the kinematics of bar stars

The distance to the Galactic center $R_0$ is a fundamental parameter for understanding the Milky Way, because all observations of our Galaxy are made from our heliocentric reference point. The uncertainty in $R_0$ limits our knowledge of many aspects of the Milky Way, including its total mass and the relative mass of its major components, and any orbital parameters of stars employed in chemo-dynamical analyses. While measurements of $R_0$ have been improving over a century, measurements in the past few years from a variety of methods still find a wide range of $R_0$ being somewhere within $8.0$ to $8.5\,\mathrm{kpc}$. The most precise measurements to date have to assume that Sgr A$^*$ is at rest at the Galactic center, which may not be the case. In this paper, we use maps of the kinematics of stars in the Galactic bar derived from APOGEE DR17 and Gaia EDR3 data augmented with spectro-photometric distances from the \texttt{astroNN} neural-network method. These maps clearly display the minimum in the rotational velocity $v_T$ and the quadrupolar signature in radial velocity $v_R$ expected for stars orbiting in a bar. From the minimum in $v_T$, we measure $R_0 = 8.23 \pm 0.12\,\mathrm{kpc}$. We validate our measurement using realistic $N$-body simulations of the Milky Way. We further measure the pattern speed of the bar to be $Ω_\mathrm{bar} = 40.08\pm1.78\,\mathrm{km\,s}^{-1}\mathrm{kpc}^{-1}$. Because the bar forms out of the disk, its center is manifestly the barycenter of the bar+disc system and our measurement is therefore the most robust and accurate measurement of $R_0$ to date.

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arXiv2021

Predicting the water content of interstellar objects from galactic star formation histories

Planetesimals inevitably bear the signatures of their natal environment, preserving in their composition a record of the metallicity of their system's original gas and dust, albeit one altered by the formation process. When planetesimals are dispersed from their system of origin, this record is carried with them. As each star is likely to contribute at least $10^{12}$ interstellar objects, the Galaxy's drifting population of interstellar objects (ISOs) provides an overview of the properties of its stellar population through time. Using the EAGLE cosmological simulation and models of protoplanetary formation, our modelling predicts an ISO population with a bimodal distribution in their water mass fraction. Objects formed in low-metallicity, typically older, systems have a higher water fraction than their counterparts formed in high-metallicity protoplanetary disks, and these water-rich objects comprise the majority of the population. Both detected ISOs seem to belong to the lower water fraction population; these results suggest they come from recently formed systems. We show that the population of ISOs in galaxies with different star formation histories will have different proportions of objects with high and low water fractions. This work suggests that it is possible that the upcoming Vera C. Rubin Observatory Legacy Survey of Space and Time will detect a large enough population of ISOs to place useful constraints on models of protoplanetary disks, as well as galactic structure and evolution.

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arXiv2021

APOGEE Detection of N-rich stars in the tidal tails of Palomar 5

Recent results from chemical tagging studies using APOGEE data suggest a strong link between the chemical abundance patterns of stars found within globular clusters, and chemically peculiar populations in the Galactic halo field. In this paper we analyse the chemical compositions of stars within the cluster body and tidal streams of Palomar 5, a globular cluster that is being tidally disrupted by interaction with the Galactic gravitational potential. We report the identification of nitrogen-rich (N-rich) stars both within and beyond the tidal radius of Palomar 5, with the latter being clearly aligned with the cluster tidal streams; this acts as confirmation that N-rich stars are lost to the Galactic halo from globular clusters, and provides support to the hypothesis that field N-rich stars identified by various groups have a globular cluster origin.

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arXiv2021

Identifying resonances of the Galactic bar in Gaia DR2: I. Clues from action space

Action space synthesizes the orbital information of stars and is well-suited to analyse the rich kinematic substructure of the disc in the \emph{Gaia} DR2 radial velocity sample (RVS). We revisit the strong perturbation induced in the Milky Way (MW) disc by an $m=2$ bar, using test particle simulations and the actions $(J_R,L_z,J_z)$ estimated in an axisymmetric potential. These make three useful diagnostics cleanly visible. (1.) We use the well-known characteristic flip from outward to inward motion at the Outer Lindblad Resonance (OLR, $l=+1,m=2$), which occurs along the axisymmetric resonance line (ARL) in $(L_z,J_R)$, to identify in the \emph{Gaia} action data three candidates for the bar's OLR and pattern speed $Ω_\text{bar}$: $1.85Ω_0$, $1.20Ω_0$, and $1.63Ω_0$ (with $\sim0.1Ω_0$ systematic uncertainty). The \emph{Gaia} data is therefore consistent with both slow and fast bar models in the literature, but disagrees with recent measurements of $\sim1.45Ω_0$. (2.) For the first time, we demonstrate that bar resonances -- especially the OLR -- cause a gradient in vertical action $\langle J_z \rangle$ with $L_z$ around the ARL via "$J_z$-sorting" of stars. This could contribute to the observed coupling of $\langle v_R \rangle$ and $\langle | v_z | \rangle$ in the Galactic disc. (3.) We confirm prior results that the behaviour of resonant orbits is well approximated by scattering and oscillation in $(L_z,J_R)$ along a slope $ΔJ_R/ΔL_z = l/m$ centered on the $l$:$m$ ARL. Overall, we demonstrate that axisymmetrically estimated actions are a powerful diagnostic tool even in non-axisymmetric systems.

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arXiv2021

Chemical Cartography with APOGEE: Mapping Disk Populations with a Two-Process Model and Residual Abundances

We apply a novel statistical analysis to measurements of 16 elemental abundances in 34,410 Milky Way disk stars from the final data release (DR17) of APOGEE-2. Building on recent work, we fit median abundance ratio trends [X/Mg] vs. [Mg/H] with a 2-process model, which decomposes abundance patterns into a "prompt" component tracing core collapse supernovae and a "delayed" component tracing Type Ia supernovae. For each sample star, we fit the amplitudes of these two components, then compute the residuals Δ[X/H] from this two-parameter fit. The rms residuals range from ~0.01-0.03 dex for the most precisely measured APOGEE abundances to ~0.1 dex for Na, V, and Ce. The correlations of residuals reveal a complex underlying structure, including a correlated element group comprised of Ca, Na, Al, K, Cr, and Ce and a separate group comprised of Ni, V, Mn, and Co. Selecting stars poorly fit by the 2-process model reveals a rich variety of physical outliers and sometimes subtle measurement errors. Residual abundances allow comparison of populations controlled for differences in metallicity and [α/Fe]. Relative to the main disk (R=3-13 kpc, |Z|<2 kpc), we find nearly identical abundance patterns in the outer disk (R=15-17 kpc), 0.05-0.2 dex depressions of multiple elements in LMC and Gaia Sausage/Enceladus stars, and wild deviations (0.4-1 dex) of multiple elements in ωCen. Residual abundance analysis opens new opportunities for discovering chemically distinctive stars and stellar populations, for empirically constraining nucleosynthetic yields, and for testing chemical evolution models that include stochasticity in the production and redistribution of elements.

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arXiv2021

New families in our Solar neighborhood: applying Gaussian Mixture models for objective classification of structures in the Milky Way and in simulations

The standard picture of galaxy formation motivates the decomposition of the Milky Way into 3--4 stellar populations with distinct kinematic and elemental abundance distributions: the thin disk, thick disk, bulge, and stellar halo. To test this idea, we construct a Gaussian mixture model (GMM) for both simulated and observed stars in the Solar neighborhood, using measured velocities and iron abundances (i.e., an augmented Toomre diagram) as the distributions to be decomposed. We compare results for the Gaia-APOGEE DR16 crossmatch catalog of the Solar neighborhood with those from a suite of synthetic Gaia-APOGEE crossmatches constructed from FIRE-2 cosmological simulations of Milky Way-mass galaxies. We find that in both the synthetic and real data, the best-fit GMM uses five independent components, some of whose properties resemble the standard populations predicted by galaxy formation theory. Two components can be identified unambiguously as the thin disk and another as the halo. However, instead of a single counterpart to the thick disk, there are three intermediate components with different age and alpha abundance distributions (although these data are not used to construct the model). We use decompositions of the synthetic data to show that the classified components indeed correspond to stars with different origins. By analogy with the simulated data, we show that our mixture model of the real Gaia-APOGEE crossmatch distinguishes the following components: (1) a classic thin disk of young stars on circular orbits (46%), (2) thin disk stars heated by interactions with satellites (22%), (3, 4) two components representing the velocity asymmetry of the alpha-enhanced thick disk (27%), and (5) a stellar halo consistent with early, massive accretion (4%).

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arXiv2021

An enquiry on the origins of N-rich stars in the inner Galaxy basedon APOGEE chemical compositions

Recent evidence based on APOGEE data for stars within a few kpc of the Galactic centre suggests that dissolved globular clusters (GCs) contribute significantly to the stellar mass budget of the inner halo. In this paper we enquire into the origins of tracers of GC dissolution, N-rich stars, that are located in the inner 4 kpc of the Milky Way. From an analysis of the chemical compositions of these stars we establish that about 30% of the N-rich stars previously identified in the inner Galaxy may have an accreted origin. This result is confirmed by an analysis of the kinematic properties of our sample. The specific frequency of N-rich stars is quite large in the accreted population, exceeding that of its in situ counterparts by near an order of magnitude, in disagreement with predictions from numerical simulations. We hope that our numbers provide a useful test to models of GC formation and destruction.

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arXiv2020

Prospects for Galactic and stellar astrophysics with asteroseismology of giant stars in the $\it{TESS}$ Continuous Viewing Zones and beyond

The NASA-$\it{TESS}$ mission presents a treasure trove for understanding the stars it observes and the Milky Way, in which they reside. We present a first look at the prospects for Galactic and stellar astrophysics by performing initial asteroseismic analyses of bright ($G < 11$) red giant stars in the $\it{TESS}$ Southern Continuous Viewing Zone (SCVZ). Using three independent pipelines, we detect $ν_{\mathrm{max}}$ and $Δν$ in 41% of the 15,405 star parent sample (6,388 stars), with consistency at a level of $\sim 2\%$ in $ν_{\mathrm{max}}$ and $\sim 5\%$ in $Δν$. Based on this, we predict that seismology will be attainable for $\sim 3\times10^{5}$ giants across the whole sky, subject to improvements in analysis and data reduction techniques. The best quality $\it{TESS}$-CVZ data, for 5,574 stars where pipelines returned consistent results, provide high quality power spectra across a number of stellar evolutionary states. This makes possible studies of, for example, the Asymptotic Giant Branch bump (AGBb). We demonstrate that mixed $\ell=1$ modes and rotational splitting are cleanly observed in the 1-year data set. By combining $\it{TESS}$-CVZ data with $\it{TESS}$-HERMES, $\it{SkyMapper}$, APOGEE and $\it{Gaia}$ we demonstrate the potential for Galactic archaeology studies using the data, which provides good age precision and accuracy that reproduces the age of high $\mathrm{[α/Fe]}$ stars and relationships between mass and kinematics from studies based on $\it{Kepler}$. Better quality astrometry and simpler target selection than the $\it{Kepler}$ sample makes this data ideal for studies of the local star formation history and evolution of the Galactic disc. These results provide a strong case for detailed spectroscopic follow-up in the CVZs to complement that which has been (or will be) collected by current surveys. [Abridged]

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arXiv2020

Evidence from APOGEE for the presence of a major building block of the halo buried in the inner Galaxy

We report evidence from APOGEE for the presence of a new metal-poor stellar structure located within $\sim$4~kpc of the Galactic centre. Characterised by a chemical composition resembling those of low mass satellites of the Milky Way, this new inner Galaxy structure (IGS) seems to be chemically and dynamically detached from more metal-rich populations in the inner Galaxy. We conjecture that this structure is associated with an accretion event that likely occurred in the early life of the Milky Way. Comparing the mean elemental abundances of this structure with predictions from cosmological numerical simulations, we estimate that the progenitor system had a stellar mass of $\sim5\times10^8M_\odot$, or approximately twice the mass of the recently discovered Gaia-Enceladus/Sausage system. We find that the accreted:{\it in situ} ratio within our metal-poor ([Fe/H]$<$--0.8) bulge sample is somewhere between 1:3 and 1:2, confirming predictions of cosmological numerical simulations by various groups.

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