arXiv subjects

Philip F. Hopkins

Publications and source records attributed to Philip F. Hopkins.

At least 217 records · Page 12Linked to original sources

arXiv2020

A Stable Finite-Volume Method for Scalar-Field Dark Matter

We describe and test a family of new numerical methods to solve the Schrodinger equation in self-gravitating systems, e.g. Bose-Einstein condensates or 'fuzzy'/ultra-light scalar field dark matter. The methods are finite-volume Godunov schemes with stable, higher-order accurate gradient estimation, based on a generalization of recent mesh-free finite-mass Godunov methods. They couple easily to particle-based N-body gravity solvers (with or without other fluids, e.g. baryons), are numerically stable, and computationally efficient. Different sub-methods allow for manifest conservation of mass, momentum, and energy. We consider a variety of test problems and demonstrate that these can accurately recover solutions and remain stable even in noisy, poorly-resolved systems, with dramatically reduced noise compared to some other proposed implementations (though certain types of discontinuities remain challenging). This is non-trivial because the "quantum pressure" is neither isotropic nor positive-definite and depends on higher-order gradients of the density field. We implement and test the method in the code GIZMO.

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arXiv2020

Radiative Stellar Feedback in Galaxy Formation: Methods and Physics

Radiative feedback (RFB) from stars plays a key role in galaxies, but remains poorly-understood. We explore this using high-resolution, multi-frequency radiation-hydrodynamics (RHD) simulations from the Feedback In Realistic Environments (FIRE) project. We study ultra-faint dwarf through Milky Way mass scales, including H+He photo-ionization; photo-electric, Lyman Werner, Compton, and dust heating; and single+multiple scattering radiation pressure (RP). We compare distinct numerical algorithms: ray-based LEBRON (exact when optically-thin) and moments-based M1 (exact when optically-thick). The most important RFB channels on galaxy scales are photo-ionization heating and single-scattering RP: in all galaxies, most ionizing/far-UV luminosity (~1/2 of lifetime-integrated bolometric) is absorbed. In dwarfs, the most important effect is photo-ionization heating from the UV background suppressing accretion. In MW-mass galaxies, meta-galactic backgrounds have negligible effects; but local photo-ionization and single-scattering RP contribute to regulating the galactic star formation efficiency and lowering central densities. Without some RFB (or other 'rapid' FB), resolved GMCs convert too-efficiently into stars, making galaxies dominated by hyper-dense, bound star clusters. This makes star formation more violent and 'bursty' when SNe explode in these hyper-clustered objects: thus, including RFB 'smoothes' SFHs. These conclusions are robust to RHD methods, but M1 produces somewhat stronger effects. Like in previous FIRE simulations, IR multiple-scattering is rare (negligible in dwarfs, ~10% of RP in massive galaxies): absorption occurs primarily in 'normal' GMCs with A_v~1.

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arXiv2020

But What About... Cosmic Rays, Magnetic Fields, Conduction, & Viscosity in Galaxy Formation

We present a suite of high-resolution cosmological simulations, using the FIRE-2 feedback physics together with explicit treatment of magnetic fields, anisotropic conduction and viscosity, and cosmic rays (CRs) injected by supernovae (including anisotropic diffusion, streaming, adiabatic, hadronic and Coulomb losses). We survey systems from ultra-faint dwarf ($M_{\ast}\sim 10^{4}\,M_{\odot}$, $M_{\rm halo}\sim 10^{9}\,M_{\odot}$) through Milky Way masses, systematically vary CR parameters (e.g. the diffusion coefficient $κ$ and streaming velocity), and study an ensemble of galaxy properties (masses, star formation histories, mass profiles, phase structure, morphologies). We confirm previous conclusions that magnetic fields, conduction, and viscosity on resolved ($\gtrsim 1\,$pc) scales have small effects on bulk galaxy properties. CRs have relatively weak effects on all galaxy properties studied in dwarfs ($M_{\ast} \ll 10^{10}\,M_{\odot}$, $M_{\rm halo} \lesssim 10^{11}\,M_{\odot}$), or at high redshifts ($z\gtrsim 1-2$), for any physically-reasonable parameters. However at higher masses ($M_{\rm halo} \gtrsim 10^{11}\,M_{\odot}$) and $z\lesssim 1-2$, CRs can suppress star formation by factors $\sim 2-4$, given relatively high effective diffusion coefficients $κ\gtrsim 3\times10^{29}\,{\rm cm^{2}\,s^{-1}}$. At lower $κ$, CRs take too long to escape dense star-forming gas and lose energy to hadronic collisions, producing negligible effects on galaxies and violating empirical constraints from $γ$-ray emission. But around $κ\sim 3\times10^{29}\,{\rm cm^{2}\,s^{-1}}$, CRs escape the galaxy and build up a CR-pressure-dominated halo which supports dense, cool ($T\ll 10^{6}$ K) gas that would otherwise rain onto the galaxy. CR heating (from collisional and streaming losses) is never dominant.

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arXiv2020

The Robustness of Cosmological Hydrodynamic Simulation Predictions to Changes in Numerics and Cooling Physics

We test and improve the numerical schemes in our smoothed particle hydrodynamics (SPH) code for cosmological simulations, including the pressure-entropy formulation (PESPH), a time-dependent artificial viscosity, a refined timestep criterion, and metal-line cooling that accounts for photoionisation in the presence of a recently refined Haardt \& Madau (2012) model of the ionising background. The PESPH algorithm effectively removes the artificial surface tension present in the traditional SPH formulation, and in our test simulations it produces better qualitative agreement with mesh-code results for Kelvin-Helmholtz instability and cold cloud disruption. Using a set of cosmological simulations, we examine many of the quantities we have studied in previous work. Results for galaxy stellar and HI mass functions, star formation histories, galaxy scaling relations, and statistics of the Ly$α$ forest are robust to the changes in numerics and microphysics. As in our previous simulations, cold gas accretion dominates the growth of high-redshift galaxies and of low mass galaxies at low redshift, and recycling of winds dominates the growth of massive galaxies at low redshift. However, the PESPH simulation removes spurious cold clumps seen in our earlier simulations, and the accretion rate of hot gas increases by up to an order of magnitude at some redshifts. The new numerical model also influences the distribution of metals among gas phases, leading to considerable differences in the statistics of some metal absorption lines, most notably NeVIII.

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arXiv2019

On the Survival of Cool Clouds in the Circum-Galactic Medium

We explore the survival of cool clouds in multi-phase circum-galactic media. We revisit the "cloud crushing problem" in a large survey of simulations including radiative cooling, self-shielding, self-gravity, magnetic fields, and anisotropic Braginskii conduction and viscosity (with saturation). We explore a wide range of parameters including cloud size, velocity, ambient temperature and density, as well as a variety of magnetic field configurations and cloud turbulence. We find that realistic magnetic fields and turbulence have weaker effects on cloud survival; the most important physics is radiative cooling and conduction. Self-gravity and self-shielding are important for clouds which are initially Jeans-unstable, but largely irrelevant otherwise. Non-self-gravitating, realistically magnetized clouds separate into four regimes: (1) At low column densities, clouds evaporate rapidly via conduction. (2) A "failed pressure confinement" regime, where the ambient hot gas cools too rapidly to provide pressure confinement for the cloud. (3) An "infinitely long-lived" regime, in which the cloud lifetime becomes longer than the cooling time of gas swept up in the leading bow shock, so the cloud begins to accrete and grow. (4) A "classical cloud destruction" regime, where clouds are eventually destroyed by instabilities. In the final regime, the cloud lifetime can exceed the naive cloud-crushing time owing to conduction-induced compression. However, small and/or slow-moving clouds can also evaporate more rapidly than the cloud-crushing time. We develop simple analytic models that explain the simulated cloud destruction times in this regime.

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arXiv2019

Astro2020 APC White Paper: Theoretical Astrophysics 2020-2030

The past two decades have seen a tremendous investment in observational facilities that promise to reveal new and unprecedented discoveries about the universe. In comparison, the investment in theoretical work is completely dwarfed, even though theory plays a crucial role in the interpretation of these observations, predicting new types of phenomena, and informing observing strategies. In this white paper, we argue that in order to reach the promised critical breakthroughs in astrophysics over the next decade and well beyond, the national agencies must take a serious approach to investment in theoretical astrophysics research. We discuss the role of theory in shaping our understanding of the universe, and then we provide a multi-level strategy, from the grassroots to the national, to address the current underinvestment in theory relative to observational work.

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arXiv2019

Stars made in outflows may populate the stellar halo of the Milky Way

We study stellar-halo formation using six Milky Way-mass galaxies in FIRE-2 cosmological zoom simulations. We find that $5-40\%$ of the outer ($50-300$ kpc) stellar halo in each system consists of $\textit{in-situ}$ stars that were born in outflows from the main galaxy. Outflow stars originate from gas accelerated by super-bubble winds, which can be compressed, cool, and form co-moving stars. The majority of these stars remain bound to the halo and fall back with orbital properties similar to the rest of the stellar halo at $z=0$.In the outer halo, outflow stars are more spatially homogeneous, metal rich, and alpha-element-enhanced than the accreted stellar halo. At the solar location, up to $\sim 10 \%$ of our kinematically-identified halo stars were born in outflows; the fraction rises to as high as $\sim 40\%$ for the most metal-rich local halo stars ([Fe/H] $> -0.5$). We conclude that the Milky Way stellar halo could contain local counterparts to stars that are observed to form in molecular outflows in distant galaxies. Searches for such a population may provide a new, near-field approach to constraining feedback and outflow physics. A stellar halo contribution from outflows is a phase-reversal of the classic halo formation scenario of Eggen, Lynden-Bell $\&$ Sandange, who suggested that halo stars formed in rapidly $\textit{infalling}$ gas clouds. Stellar outflows may be observable in direct imaging of external galaxies and could provide a source for metal-rich, extreme velocity stars in the Milky Way.

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arXiv2019

Reconciling observed and simulated stellar halo masses

We use cosmological hydrodynamical simulations of Milky-Way-mass galaxies from the FIRE project to evaluate various strategies for estimating the mass of a galaxy's stellar halo from deep, integrated-light images. We find good agreement with integrated-light observations if we mimic observational methods to measure the mass of the stellar halo by selecting regions of an image via projected radius relative to the disk scale length or by their surface density in stellar mass . However, these observational methods systematically underestimate the accreted stellar component, defined in our (and most) simulations as the mass of stars formed outside of the host galaxy, by up to a factor of ten, since the accreted component is centrally concentrated and therefore substantially obscured by the galactic disk. Furthermore, these observational methods introduce spurious dependencies of the estimated accreted stellar component on the stellar mass and size of galaxies that can obscure the trends in accreted stellar mass predicted by cosmological simulations, since we find that in our simulations the size and shape of the central galaxy is not strongly correlated with the assembly history of the accreted stellar halo. This effect persists whether galaxies are viewed edge-on or face-on. We show that metallicity or color information may provide a way to more cleanly delineate in observations the regions dominated by accreted stars. Absent additional data, we caution that estimates of the mass of the accreted stellar component from single-band images alone should be taken as lower limits.

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arXiv2019

Swirls of FIRE: Spatially Resolved Gas Velocity Dispersions and Star Formation Rates in FIRE-2 Disk Environments

We study the spatially resolved (sub-kpc) gas velocity dispersion ($σ$)--star formation rate (SFR) relation in the FIRE-2 (Feedback in Realistic Environments) cosmological simulations. We specifically focus on Milky Way mass disk galaxies at late times. In agreement with observations, we find a relatively flat relationship, with $σ\approx 15-30$ km/s in neutral gas across 3 dex in SFRs. We show that higher dense gas fractions (ratios of dense gas to neutral gas) and SFRs are correlated at constant $σ$. Similarly, lower gas fractions (ratios of gas to stellar mass) are correlated with higher $σ$ at constant SFR. The limits of the $σ$-$Σ_{\rm SFR}$ relation correspond to the onset of strong outflows. We see evidence of "on-off" cycles of star formation in the simulations, corresponding to feedback injection timescales of 10-100 Myr, where SFRs oscillate about equilibrium SFR predictions. Finally, SFRs and velocity dispersions in the simulations agree well with feedback-regulated and marginally stable gas disk (Toomre's $Q =1$) model predictions, and the data effectively rule out models assuming that gas turns into stars at (low) constant efficiency (i.e., ${\rm 1\%}$ per free-fall time). And although the simulation data do not entirely exclude gas accretion/gravitationally powered turbulence as a driver of $σ$, it appears to be strongly subdominant to stellar feedback in the simulated galaxy disks.

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arXiv2019

High-redshift Galaxy Formation with Self-consistently Modeled Stars and Massive Black Holes: Stellar Feedback and Quasar Growth

As computational resolution of modern cosmological simulations reach ever so close to resolving individual star-forming clumps in a galaxy, a need for "resolution-appropriate" physics for a galaxy-scale simulation has never been greater. To this end, we introduce a self-consistent numerical framework that includes explicit treatments of feedback from star-forming molecular clouds (SFMCs) and massive black holes (MBHs). In addition to the thermal supernovae feedback from SFMC particles, photoionizing radiation from both SFMCs and MBHs is tracked through full 3-dimensional ray tracing. A mechanical feedback channel from MBHs is also considered. Using our framework, we perform a state-of-the-art cosmological simulation of a quasar-host galaxy at z~7.5 for ~25 Myrs with all relevant galactic components such as dark matter, gas, SFMCs, and an embedded MBH seed of ~> 1e6 Ms. We find that feedback from SFMCs and an accreting MBH suppresses runaway star formation locally in the galactic core region. Newly included radiation feedback from SFMCs, combined with feedback from the MBH, helps the MBH grow faster by retaining gas that eventually accretes on to the MBH. Our experiment demonstrates that previously undiscussed types of interplay between gas, SFMCs, and a MBH may hold important clues about the growth and feedback of quasars and their host galaxies in the high-redshift Universe.

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arXiv2019

Dwarf Galaxies in CDM, WDM, and SIDM: Disentangling Baryons and Dark Matter Physics

We present a suite of FIRE-2 cosmological zoom-in simulations of isolated field dwarf galaxies, all with masses of $M_\mathrm{halo} \approx 10^{10}\,$M$_\odot$ at $z=0$, across a range of dark matter models. For the first time, we compare how both self-interacting dark matter (SIDM) and/or warm dark matter (WDM) models affect the assembly histories as well as the central density structure in fully hydrodynamical simulations of dwarfs. Dwarfs with smaller stellar half-mass radii (r$_{1/2}<500$ pc) have lower $σ_\star/V_\mathrm{max}$ ratios, reinforcing the idea that smaller dwarfs may reside in halos that are more massive than is naively expected. The majority of dwarfs simulated with self-interactions actually experience contraction of their inner density profiles with the addition of baryons relative to the cores produced in dark-matter-only runs, though the simulated dwarfs are always less centrally dense than in $Λ$CDM. The V$_{1/2}-$r$_{1/2}$ relation across all simulations is generally consistent with observations of Local Field dwarfs, though compact objects such as Tucana provide a unique challenge. Spatially-resolved rotation curves in the central regions ($<400$ pc) of small dwarfs could provide a way to distinguish between CDM, WDM, and SIDM, however: at the masses probed in this simulation suite, cored density profiles in dwarfs with small r$_{1/2}$ values can only originate from dark matter self-interactions.

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arXiv2019

Star formation at the edge of the Local Group: a rising star formation history in the isolated galaxy WLM

We present the star formation history (SFH) of the isolated (D~970 kpc) Local Group dwarf galaxy WLM measured from color-magnitude diagrams constructed from deep Hubble Space Telescope imaging. Our observations include a central (0.5 $r_h$) and outer field (0.7 $ r_h$) that reach below the oldest main sequence turnoff. WLM has no early dominant episode of star formation: 20% of its stellar mass formed by ~12.5 Gyr ago (z~5). It also has an SFR that rises to the present with 50% of the stellar mass within the most recent 5 Gyr (z<0.7). There is evidence of a strong age gradient: the mean age of the outer field is 5 Gyr older than the inner field despite being only 0.4 kpc apart. Some models suggest such steep gradients are associated with strong stellar feedback and dark matter core creation. The SFHs of real isolated dwarf galaxies and those from the the Feedback In Realistic Environment suite are in good agreement for $M_{\star}(z=0) \sim 10^7-10^9 M_{\odot}$, but in worse agreement at lower masses ($M_{\star}(z=0) \sim 10^5-10^7 M_{\odot}$). These differences may be explainable by systematics in the models (e.g., reionization model) and/or observations (HST field placement). We suggest that a coordinated effort to get deep CMDs between HST/JWST (crowded central fields) and WFIRST (wide-area halo coverage) is the optimal path for measuring global SFHs of isolated dwarf galaxies.

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arXiv2019

On the dust temperatures of high redshift galaxies

Dust temperature is an important property of the interstellar medium (ISM) of galaxies. It is required when converting (sub)millimeter broadband flux to total infrared luminosity (L_IR), and hence star formation rate, in high-z galaxies. However, different definitions of dust temperatures have been used in the literature, leading to different physical interpretations of how ISM conditions change with, e.g., redshift and star formation rate. In this paper, we analyse the dust temperatures of massive (M* > 10^10 Msun) z=2-6 galaxies with the help of high-resolution cosmological simulations from the Feedback in Realistic Environments (FIRE) project. At z~2, our simulations successfully predict dust temperatures in good agreement with observations. We find that dust temperatures based on the peak emission wavelength increase with redshift, in line with the higher star formation activity at higher redshift, and are strongly correlated with the specific star formation rate. In contrast, the mass-weighted dust temperature does not strongly evolve with redshift over z=2-6 at fixed IR luminosity but is tightly correlated with L_IR at fixed z. The mass-weighted temperature is important for accurately estimating the total dust mass. We also analyse an 'equivalent' dust temperature for converting (sub)millimeter flux density to total IR luminosity, and provide a fitting formula as a function of redshift and dust-to-metal ratio. We find that galaxies of higher equivalent (or higher peak) dust temperature ('warmer dust') do not necessarily have higher mass-weighted temperatures. A 'two-phase' picture for interstellar dust can explain the different scaling relations of the various dust temperatures.

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arXiv2019

Nonlinear Evolution of Instabilities Between Dust and Sound Waves

We study the non-linear evolution of the acoustic 'Resonant Drag Instability' (RDI) using numerical simulations. The acoustic RDI is excited in a dust-gas mixture when dust grains stream through gas, interacting with sound waves to cause a linear instability. We study this process in a periodic box by accelerating neutral dust with an external driving force. The instability grows as predicted by linear theory, eventually breaking into turbulence and saturating. As in linear theory, the non-linear behavior is characterized by three regimes - high, intermediate, and low wavenumbers - the boundary between which is determined by the dust-gas coupling strength and the dust-to-gas mass ratio. The high and intermediate wavenumber regimes behave similarly to one another, with large dust-to-gas ratio fluctuations while the gas remains largely incompressible. The saturated state is highly anisotropic: dust is concentrated in filaments, jets, or plumes along the direction of acceleration, with turbulent vortex-like structures rapidly forming and dissipating in the perpendicular directions. The low-wavenumber regime exhibits large fluctuations in gas and dust density, but the dust and gas remain more strongly coupled in coherent 'fronts' perpendicular to the acceleration. These behaviors are qualitatively different from those of dust 'passively' driven by external hydrodynamic turbulence, with no back-reaction force from dust onto gas. The virulent nature of these instabilities has interesting implications for dust-driven winds in a variety of astrophysical systems, including around cool-stars, in dusty torii around active-galactic-nuclei, and in and around giant molecular clouds.

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arXiv2019

Dust attenuation, dust emission, and dust temperature in galaxies at z>=5: a view from the FIRE-2 simulations

We present a suite of 34 high-resolution cosmological zoom-in simulations consisting of thousands of halos up to M_halo~10^12 M_sun (M_star~10^10.5 M_sun) at z>=5 from the Feedback in Realistic Environments project. We post-process our simulations with a three-dimensional Monte Carlo dust radiative transfer code to study dust extinction, dust emission, and dust temperature within these simulated z>=5 galaxies. Our sample forms a tight correlation between infrared excess (IRX=F_IR/F_UV) and ultraviolet (UV)-continuum slope (beta_UV), despite the patchy, clumpy dust geometry shown in our simulations. We find that the IRX-beta_UV relation is mainly determined by the shape of the extinction curve and is independent of its normalization (set by the dust-to-gas ratio). The bolometric IR luminosity (L_IR) correlates with the intrinsic UV luminosity and the star formation rate (SFR) averaged over the past 10 Myr. We predict that at a given L_IR, the peak wavelength of the dust spectral energy distributions for z>=5 galaxies is smaller by a factor of 2 (due to higher dust temperatures on average) than at z=0. The higher dust temperatures are driven by higher specific SFRs and SFR surface densities with increasing redshift. We derive the galaxy UV luminosity functions (LFs) at z=5-10 from our simulations and confirm that a heavy attenuation is required to reproduce the observed bright-end UVLFs. We also predict the IRLFs and UV luminosity densities at z=5-10. We discuss the implications of our results on current and future observations probing dust attenuation and emission in z>=5 galaxies.

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arXiv2019

A Simple Non-equilibrium Feedback Model for Galaxy-Scale Star Formation: Delayed Feedback and SFR Scatter

We explore a class of simple non-equilibrium star formation models within the framework of a feedback-regulated model of the ISM, applicable to kiloparsec-scale resolved star formation relations (e.g. Kennicutt-Schmidt). Combining a Toomre-Q-dependent local star formation efficiency per free-fall time with a model for delayed feedback, we are able to match the normalization and scatter of resolved star formation scaling relations. In particular, this simple model suggests that large ($\sim$dex) variations in star formation rates (SFRs) on kiloparsec scales may be due to the fact that supernova feedback is not instantaneous following star formation. The scatter in SFRs at constant gas surface density in a galaxy then depends on the properties of feedback and when we observe its star-forming regions at various points throughout their collapse/star formation "cycles". This has the following important observational consequences: (1) the scatter and normalization of the Kennicutt-Schmidt relation are relatively insensitive to the local (small-scale) star formation efficiency, (2) but gas depletion times and velocity dispersions are; (3) the scatter in and normalization of the Kennicutt-Schmidt relation is a sensitive probe of the feedback timescale and strength; (4) even in a model where $\tilde Q_{\rm gas}$ deterministically dictates star formation locally, time evolution, variation in local conditions (e.g., gas fractions and dynamical times), and variations between galaxies can destroy much of the observable correlation between SFR and $\tilde Q_{\rm gas}$ in resolved galaxy surveys. Additionally, this model exhibits large scatter in SFRs at low gas surface densities, in agreement with observations of flat outer HI disk velocity dispersion profiles.

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arXiv2019

Formation, vertex deviation and age of the Milky Way's bulge: input from a cosmological simulation with a late-forming bar

We present the late-time evolution of m12m, a cosmological simulation of a Milky Way-like galaxy from the FIRE project. The simulation forms a bar after redshift z = 0.2. We show that the evolution of the model exhibits behaviours typical of kinematic fractionation, with a bar weaker in older populations, an X-shape traced by the younger, metal-rich populations and a prominent X-shape in the edge-on mean metallicity map. Because of the late formation of the bar in m12m, stars forming after 10 Gyr (z = 0.34) significantly contaminate the bulge, at a level higher than is observed at high latitudes in the Milky Way, implying that its bar cannot have formed as late as in m12m. We also study the model's vertex deviation of the velocity ellipsoid as a function of stellar metallicity and age in the equivalent of Baade's Window. The formation of the bar leads to a non-zero vertex deviation. We find that metal-rich stars have a large vertex deviation (~ 40 degrees), which becomes negligible for metal-poor stars, a trend also found in the Milky Way, despite not matching in detail. We demonstrate that the vertex deviation also varies with stellar age and is large for stars as old as 9 Gyr, while 13 Gyr old stars have negligible vertex deviation. When we exclude stars that have been accreted, the vertex deviation is not significantly changed, demonstrating that the observed variation of vertex deviation with metallicity is not necessarily due to an accreted population.

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arXiv2019

Is it possible to reconcile extragalactic IMF variations with a universal Milky Way IMF?

One of the most robust observations of the stellar initial mass function (IMF) is its near-universality in the Milky Way and neighboring galaxies. But recent observations of early-type galaxies can be interpreted to imply a bottom-heavy IMF, while others of ultra-faint dwarfs could imply a top-heavy IMF. This would impose powerful constraints on star formation models. We explore what sort of cloud-scale IMF models could possibly satisfy these constraints. We utilize simulated galaxies which reproduce (broadly) the observed galaxy properties, while they also provide the detailed star formation history and properties of each progenitor star-forming cloud. We then consider generic models where the characteristic mass of the IMF is some arbitrary power-law function of progenitor cloud properties, along with well-known literature IMF models, which scale with Jeans mass, turbulent Bonnor-Ebert mass, temperature, the opacity limit, metallicity, or the protostellar heating mass. We show that no IMF models currently in the literature - nor any model where the turnover mass is an arbitrary power-law function of a combination of cloud temperature/density/size/metallicity/velocity dispersion/magnetic field - can reproduce the claimed IMF variation in ellipticals or dwarfs without severely violating observational constraints in the Milky Way. Specifically, they predict too much variation in the extreme environments of the Galaxy, compared to that observed. Either the IMF varies in a more complicated manner, or alternative interpretations of the extragalactic observations must be explored.

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