arXiv subjects

Philip F. Hopkins

Publications and source records attributed to Philip F. Hopkins.

At least 253 records · Page 14Linked to original sources

arXiv2018

Modelling chemical abundance distributions for dwarf galaxies in the Local Group: the impact of turbulent metal diffusion

We investigate stellar metallicity distribution functions (MDFs), including Fe and $α$-element abundances, in dwarf galaxies from the Feedback in Realistic Environments (FIRE) project. We examine both isolated dwarf galaxies and those that are satellites of a Milky Way-mass galaxy. In particular, we study the effects of including a sub-grid turbulent model for the diffusion of metals in gas. Simulations that include diffusion have narrower MDFs and abundance ratio distributions, because diffusion drives individual gas and star particles toward the average metallicity. This effect provides significantly better agreement with observed abundance distributions of dwarf galaxies in the Local Group, including the small intrinsic scatter in [$α$/Fe] vs. [Fe/H] (less than 0.1 dex). This small intrinsic scatter arises in our simulations because the interstellar medium (ISM) in dwarf galaxies is well-mixed at nearly all cosmic times, such that stars that form at a given time have similar abundances to within 0.1 dex. Thus, most of the scatter in abundances at z = 0 arises from redshift evolution and not from instantaneous scatter in the ISM. We find similar MDF widths and intrinsic scatter for satellite and isolated dwarf galaxies, which suggests that environmental effects play a minor role compared with internal chemical evolution in our simulations. Overall, with the inclusion of metal diffusion, our simulations reproduce abundance distribution widths of observed low-mass galaxies, enabling detailed studies of chemical evolution in galaxy formation.

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arXiv2018

A General Class of Lagrangian Smoothed Particle Hydrodynamics Methods and Implications for Fluid Mixing Problems

Various formulations of smooth-particle hydrodynamics (SPH) have been proposed, intended to resolve certain difficulties in the treatment of fluid mixing instabilities. Most have involved changes to the algorithm which either introduce artificial correction terms or violate arguably the greatest advantage of SPH over other methods: manifest conservation of energy, entropy, momentum, and angular momentum. Here, we show how a class of alternative SPH equations of motion (EOM) can be derived self-consistently from a discrete particle Lagrangian (guaranteeing manifest conservation) in a manner which tremendously improves treatment of instabilities and contact discontinuities. Saitoh & Makino recently noted that the volume element used to discretize the EOM does not need to explicitly invoke the mass density (as in the 'standard' approach); we show how this insight, and the resulting degree of freedom, can be incorporated into the rigorous Lagrangian formulation that retains ideal conservation properties and includes the 'Grad-h' terms that account for variable smoothing lengths. We derive a general EOM for any choice of volume element (particle 'weights') and method of determining smoothing lengths. We then specify this to a 'pressure-entropy formulation' which resolves problems in the traditional treatment of fluid interfaces. Implementing this in a new version of the GADGET code, we show it leads to good performance in mixing experiments (e.g. Kelvin-Helmholtz & blob tests). And conservation is maintained even in strong shock/blastwave tests, where formulations without manifest conservation produce large errors. This also improves the treatment of sub-sonic turbulence, and lessens the need for large kernel particle numbers. The code changes are trivial and entail no additional numerical expense. This provides a general framework for self-consistent derivation of different 'flavors' of SPH.

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arXiv2018

The Local Group on FIRE: Dwarf galaxy populations across a suite of hydrodynamic simulations

We present a new set of high-resolution hydrodynamic cosmological zoom-in simulations that apply the Feedback In Realistic Environments (FIRE) physics to both Local Group (LG)-like and isolated Milky Way (MW)-like volumes (ten host systems in total with baryonic particle mass $\simeq 3,500-7,000\,M_\odot$). We study the stellar mass functions, circular velocity or mass profiles, and velocity dispersions of the dwarf galaxy populations. The simulations reproduce the stellar mass function and central densities of MW satellite dwarfs for $M_\ast \geq10^{5.5}\,M_\odot$ and predict the existence of $\sim3$ unidentified galaxies with $M_\ast\sim10^5\,M_\odot$ within $300$ kpc of the MW. Overall, we find no evidence for the classical missing satellites or too-big-to-fail (TBTF) problems for satellite galaxies in our sample. Among the satellites, TBTF is resolved primarily by subhalo disruption and overall mass loss; central density profiles of subhalos are of secondary importance. For non-satellite galaxies, our LG-like simulations predict as many as $\sim10$ as-of-yet unseen galaxies at distances $0.3-1$ Mpc from both hosts, with $M_\ast\simeq10^{5-6}\,M_\odot$ (in halos with $V_\mathrm{max}\sim 20~\mathrm{km\,s}^{-1}$), albeit with large halo-to-halo variance. None of our simulations produces a compact, baryon-dominated, high-density dwarf elliptical-type galaxy (with $V_\mathrm{circ} \gtrsim 35~\mathrm{km\,s}^{-1}$ at $r<1\,$kpc), of which six may appear in the LG (but none in the MW). It may therefore remain a challenge to reproduce the full diversity of the dwarf population, including both the highest and lowest density systems.

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arXiv2018

Isothermal Fragmentation: Is there a low-mass cut-off?

The evolution of self-gravitating clouds of isothermal gas forms the basis of many star formation theories. Therefore it is important to know under what conditions such a cloud will undergo homologous collapse into a single, massive object, or will fragment into a spectrum of smaller ones. And if it fragments, do initial conditions (e.g. Jeans mass, sonic mass) influence the mass function of the fragments, as predicted by many theories of star formation? In this paper we show that the relevant parameter separating homologous collapse from fragmentation is not the Mach number of the initial turbulence (as suspected by many), but the infall Mach number $\mathcal{M}_{\rm infall}\sim\sqrt{G M/(R c_s^2)}$, equivalent to the number of Jeans masses in the initial cloud $N_J$. We also show that fragmenting clouds produce a power-law mass function with slopes close to the expected -2 (i.e. equal mass in all logarithmic mass intervals). However, the low-mass cut-off of this mass function is entirely numerical; the initial properties of the cloud have no effect on it. In other words, if $\mathcal{M}_{\rm infall}\gg 1$, fragmentation proceeds without limit to masses much smaller than the initial Jeans mass.

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arXiv2018

Submillimeter flux as a probe of molecular ISM mass in high-$z$ galaxies

Recent long wavelength observations on the thermal dust continuum suggest that the Rayleigh-Jeans (RJ) tail can be used as a time-efficient quantitative probe of the dust and ISM mass in high-$z$ galaxies. We use high-resolution cosmological simulations from the Feedback in Realistic Environment (FIRE) project to analyze the dust emission of $M_*>10^{10}\;M_{\odot}$ galaxies at $z=2-4$. Our simulations (MassiveFIRE) explicitly include various forms of stellar feedback, and they produce the stellar masses and star formation rates of high-$z$ galaxies in agreement with observations. Using radiative transfer modelling, we show that sub-millimeter (sub-mm) luminosity and molecular ISM mass are tightly correlated and that the overall normalization is in quantitative agreement with observations. Notably, sub-mm luminosity traces molecular ISM mass even during starburst episodes as dust mass and mass-weighted temperature evolve only moderately between $z=4$ and $z=2$, including during starbursts. Our finding supports the empirical approach of using broadband sub-mm flux as a proxy for molecular gas content in high-$z$ galaxies. We thus expect single-band sub-mm observations with ALMA to dramatically increase the sample size of high-$z$ galaxies with reliable ISM masses in the near future.

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arXiv2018

Resonant Drag Instabilities in protoplanetary disks: the streaming instability and new, faster-growing instabilities

We identify and study a number of new, rapidly growing instabilities of dust grains in protoplanetary disks, which may be important for planetesimal formation. The study is based on the recognition that dust-gas mixtures are generically unstable to a Resonant Drag Instability (RDI), whenever the gas, absent dust, supports undamped linear modes. We show that the "streaming instability" is an RDI associated with epicyclic oscillations; this provides simple interpretations for its mechanisms and accurate analytic expressions for its growth rates and fastest-growing wavelengths. We extend this analysis to more general dust streaming motions and other waves, including buoyancy and magnetohydrodynamic oscillations, finding various new instabilities. Most importantly, we identify the disk "settling instability," which occurs as dust settles vertically into the midplane of a rotating disk. For small grains, this instability grows many orders of magnitude faster than the standard streaming instability, with a growth rate that is independent of grain size. Growth timescales for realistic dust-to-gas ratios are comparable to the disk orbital period, and the characteristic wavelengths are more than an order of magnitude larger than the streaming instability (allowing the instability to concentrate larger masses). This suggests that in the process of settling, dust will band into rings then filaments or clumps, potentially seeding dust traps, high-metallicity regions that in turn seed the streaming instability, or even overdensities that coagulate or directly collapse to planetesimals.

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arXiv2018

The Maximum Stellar Surface Density Due to the Failure of Stellar Feedback

A maximum stellar surface density $Σ_{max} \sim 3 \times 10^5\,{\rm M_\odot\,pc^{-2}}$ is observed across all classes of dense stellar systems (e.g. star clusters, galactic nuclei, etc.), spanning $\sim 8$ orders of magnitude in mass. It has been proposed that this characteristic scale is set by some dynamical feedback mechanism preventing collapse beyond a certain surface density. However, simple analytic models and detailed simulations of star formation moderated by feedback from massive stars argue that feedback becomes {\it less} efficient at higher surface densities (with the star formation efficiency increasing as $\sim Σ/Σ_{crit}$). We therefore propose an alternative model wherein stellar feedback becomes ineffective at moderating star formation above some $Σ_{crit}$, so the supply of star-forming gas is rapidly converted to stars before the system can contract to higher surface density. We show that such a model -- with $Σ_{crit}$ taken directly from the theory -- naturally predicts the observed $Σ_{max}$. $Σ_{max}\sim 100Σ_{crit}$ because the gas consumption time is longer than the global freefall time even when feedback is ineffective. Moreover the predicted $Σ_{max}$ is robust to spatial scale and metallicity, and is preserved even if multiple episodes of star formation/gas inflow occur. In this context, the observed $Σ_{max}$ directly tells us where feedback fails.

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arXiv2018

When Feedback Fails: The Scaling and Saturation of Star Formation Efficiency

We present a suite of 3D multi-physics MHD simulations following star formation in isolated turbulent molecular gas disks ranging from 5 to 500 parsecs in radius. These simulations are designed to survey the range of surface densities between those typical of Milky Way GMCs ($\sim 10^2 M_\odot\,pc^{-2}}$) and extreme ULIRG environments ($\sim 10^2 M_\odot\,pc^{-2}}$) so as to map out the scaling of the cloud-scale star formation efficiency (SFE) between these two regimes. The simulations include prescriptions for supernova, stellar wind, and radiative feedback, which we find to be essential in determining both the instantaneous per-freefall ($ε_{ff}$) and integrated ($ε_{int}$) star formation efficiencies. In all simulations, the gas disks form stars until a critical stellar surface density has been reached and the remaining gas is blown out by stellar feedback. We find that surface density is a good predictor of $ε_{int}$, as suggested by analytic force balance arguments from previous works. SFE eventually saturates to $\sim 1$ at high surface density. We also find a proportional relationship between $ε_{ff}$ and $ε_{int}$, implying that star formation is feedback-moderated even over very short time-scales in isolated clouds. These results have implications for star formation in galactic disks, the nature and fate of nuclear starbursts, and the formation of bound star clusters. The scaling of $ε_{ff}$ with surface density is not consistent with the notion that $ε_{ff}$ is always $\sim 1\%$ on the scale of GMCs, but our predictions recover the $\sim 1\%$ value for GMC parameters similar to those found in sprial galaxies, including our own.

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arXiv2018

Simulating galaxies in the reionization era with FIRE-2: galaxy scaling relations, stellar mass functions, and luminosity functions

We present a suite of cosmological zoom-in simulations at z>5 from the Feedback In Realistic Environments project, spanning a halo mass range M_halo~10^8-10^12 M_sun at z=5. We predict the stellar mass-halo mass relation, stellar mass function, and luminosity function in several bands from z=5-12. The median stellar mass-halo mass relation does not evolve strongly at z=5-12. The faint-end slope of the luminosity function steepens with increasing redshift, as inherited from the halo mass function at these redshifts. Below z~6, the stellar mass function and ultraviolet (UV) luminosity function slightly flatten below M_star~10^4.5 M_sun (fainter than M_1500~-12), owing to the fact that star formation in low-mass halos is suppressed by the ionizing background by the end of reionization. Such flattening does not appear at higher redshifts. We provide redshift-dependent fitting functions for the SFR-M_halo, SFR-M_star, and broad-band magnitude-stellar mass relations. We derive the star formation rate density and stellar mass density at z=5-12 and show that the contribution from very faint galaxies becomes more important at z>8. Furthermore, we find that the decline in the z~6 UV luminosity function brighter than M_1500~-20 is largely due to dust attenuation. Approximately 37% (54%) of the UV luminosity from galaxies brighter than M_1500=-13 (-17) is obscured by dust at z~6. Our results broadly agree with current data and can be tested by future observations.

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arXiv2018

Resonant drag instability of grains streaming in fluids

We show that grains streaming through a fluid are generically unstable if their velocity, projected along some direction, matches the phase velocity of a fluid wave (linear oscillation). This can occur whenever grains stream faster than any fluid wave. The wave itself can be quite general--sound waves, magnetosonic waves, epicyclic oscillations, and Brunt-Väisälä oscillations each generate instabilities, for example. We derive a simple expression for the growth rates of these "resonant drag instabilities" (RDI). This expression (i) illustrates why such instabilities are so virulent and generic, and (ii) allows for simple analytic computation of RDI growth rates and properties for different fluids. As examples, we introduce several new instabilities, which could see application across a variety of physical systems from atmospheres to protoplanetary disks, the interstellar medium, and galactic outflows. The matrix-based resonance formalism we introduce can also be applied more generally in other (nonfluid) contexts, providing a simple means for calculating and understanding the stability properties of interacting systems.

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arXiv2018

Warm FIRE: Simulating Galaxy Formation with Resonant Sterile Neutrino Dark Matter

We study the impact of a warm dark matter (WDM) cosmology on dwarf galaxy formation through a suite of cosmological hydrodynamical zoom-in simulations of $M_{\rm halo} \approx10^{10}\,M_{\odot}$ dark matter halos as part of the Feedback in Realistic Environments (FIRE) project. A main focus of this paper is to evaluate the combined effects of dark matter physics and stellar feedback on the well-known small-scale issues found in cold dark matter (CDM) models. We find that the $z=0$ stellar mass of a galaxy is strongly correlated with the central density of its host dark matter halo at the time of formation, $z_{\rm f}$, in both CDM and WDM models. WDM halos follow the same $M_{\star}(z=0)-V_{\rm max}(z_{\rm f})$ relation as in CDM, but they form later, are less centrally dense, and therefore contain galaxies that are less massive than their CDM counterparts. As a result, the impact of baryonic effects on the central gravitational potential is typically diminished relative to CDM. However, the combination of delayed formation in WDM and energy input from stellar feedback results in dark matter profiles with lower overall densities. The WDM galaxies studied here have a wider diversity of star formation histories (SFHs) than the same systems simulated in CDM, and the two lowest $M_{\star}$ WDM galaxies form all of their stars at late times. The discovery of young ultra-faint dwarf galaxies with no ancient star formation -- which do not exist in our CDM simulations -- would therefore provide evidence in support of WDM.

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arXiv2018

Gas Kinematics in FIRE Simulated Galaxies Compared to Spatially Unresolved HI Observations

The shape of a galaxy's spatially unresolved, globally integrated 21-cm emission line depends on its internal gas kinematics: galaxies with rotation-supported gas disks produce double-horned profiles with steep wings, while galaxies with dispersion-supported gas produce Gaussian-like profiles with sloped wings. Using mock observations of simulated galaxies from the FIRE project, we show that one can therefore constrain a galaxy's gas kinematics from its unresolved 21-cm line profile. In particular, we find that the kurtosis of the 21-cm line increases with decreasing $V/σ$, and that this trend is robust across a wide range of masses, signal-to-noise ratios, and inclinations. We then quantify the shapes of 21-cm line profiles from a morphologically unbiased sample of $\sim$2000 low-redshift, HI-detected galaxies with $M_{\rm star} = 10^{7-11} M_{\odot}$ and compare to the simulated galaxies. At $M_{\rm star} \gtrsim 10^{10} M_{\odot}$, both the observed and simulated galaxies produce double-horned profiles with low kurtosis and steep wings, consistent with rotation-supported disks. Both the observed and simulated line profiles become more Gaussian-like (higher kurtosis and less-steep wings) at lower masses, indicating increased dispersion support. However, the simulated galaxies transition from rotation to dispersion support more strongly: at $M_{\rm star} = 10^{8-10}M_{\odot}$, most of the simulations produce more Gaussian-like profiles than typical observed galaxies with similar mass, indicating that gas in the low-mass simulated galaxies is, on average, overly dispersion-supported. Most of the lower-mass simulated galaxies also have somewhat lower gas fractions than the median of the observed population. The simulations nevertheless reproduce the observed line-width baryonic Tully-Fisher relation, which is insensitive to rotation vs. dispersion support.

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arXiv2018

Formation of Globular Cluster Candidates in Merging Proto-galaxies at High Redshift: A View from the FIRE Cosmological Simulations

Using a state-of-the-art cosmological simulation of merging proto-galaxies at high redshift from the FIRE project, with explicit treatments of star formation and stellar feedback in the interstellar medium, we investigate the formation of star clusters and examine one of the formation hypothesis of present-day metal-poor globular clusters. We find that frequent mergers in high-redshift proto-galaxies could provide a fertile environment to produce long-lasting bound star clusters. The violent merger event disturbs the gravitational potential and pushes a large gas mass of ~> 1e5-6 Msun collectively to high density, at which point it rapidly turns into stars before stellar feedback can stop star formation. The high dynamic range of the reported simulation is critical in realizing such dense star-forming clouds with a small dynamical timescale, t_ff <~ 3 Myr, shorter than most stellar feedback timescales. Our simulation then allows us to trace how clusters could become virialized and tightly-bound to survive for up to ~420 Myr till the end of the simulation. Because the cluster's tightly-bound core was formed in one short burst, and the nearby older stars originally grouped with the cluster tend to be preferentially removed, at the end of the simulation the cluster has a small age spread.

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arXiv2018

The AGORA High-Resolution Galaxy Simulations Comparison Project. II: Isolated Disk Test

Using an isolated Milky Way-mass galaxy simulation, we compare results from 9 state-of-the-art gravito-hydrodynamics codes widely used in the numerical community. We utilize the infrastructure we have built for the AGORA High-resolution Galaxy Simulations Comparison Project. This includes the common disk initial conditions, common physics models (e.g., radiative cooling and UV background by the standardized package Grackle) and common analysis toolkit yt, all of which are publicly available. Subgrid physics models such as Jeans pressure floor, star formation, supernova feedback energy, and metal production are carefully constrained across code platforms. With numerical accuracy that resolves the disk scale height, we find that the codes overall agree well with one another in many dimensions including: gas and stellar surface densities, rotation curves, velocity dispersions, density and temperature distribution functions, disk vertical heights, stellar clumps, star formation rates, and Kennicutt-Schmidt relations. Quantities such as velocity dispersions are very robust (agreement within a few tens of percent at all radii) while measures like newly-formed stellar clump mass functions show more significant variation (difference by up to a factor of ~3). Systematic differences exist, for example, between mesh-based and particle-based codes in the low density region, and between more diffusive and less diffusive schemes in the high density tail of the density distribution. Yet intrinsic code differences are generally small compared to the variations in numerical implementations of the common subgrid physics such as supernova feedback. Our experiment reassures that, if adequately designed in accordance with our proposed common parameters, results of a modern high-resolution galaxy formation simulation are more sensitive to input physics than to intrinsic differences in numerical schemes.

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arXiv2018

Universal Scaling Relations in Scale-Free Structure Formation

A large number of astronomical phenomena exhibit remarkably similar scaling relations. The most well-known of these is the mass distribution $\mathrm{d} N/\mathrm{d} M\propto M^{-2}$ which (to first order) describes stars, protostellar cores, clumps, giant molecular clouds, star clusters and even dark matter halos. In this paper we propose that this ubiquity is not a coincidence and that it is the generic result of scale-free structure formation where the different scales are uncorrelated. We show that all such systems produce a mass function proportional to $M^{-2}$ and a column density distribution with a power law tail of $\mathrm{d} A/\mathrm{d} \lnΣ\proptoΣ^{-1}$. In the case where structure formation is controlled by gravity the two-point correlation becomes $ξ_{2D}\propto R^{-1}$. Furthermore, structures formed by such processes (e.g. young star clusters, DM halos) tend to a $ρ\propto R^{-3}$ density profile. We compare these predictions with observations, analytical fragmentation cascade models, semi-analytical models of gravito-turbulent fragmentation and detailed "full physics" hydrodynamical simulations. We find that these power-laws are good first order descriptions in all cases.

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arXiv2018

The radial acceleration relation is a natural consequence of the baryonic Tully-Fisher relation

Galaxies covering several orders of magnitude in stellar mass and a variety of Hubble types have been shown to follow the "Radial Acceleration Relation" (RAR), a relationship between $g_{\rm obs}$, the observed circular acceleration of the galaxy, and $g_{\rm bar}$, the acceleration due to the total baryonic mass of the galaxy. For accelerations above $10^{10}~{\rm m \, s}^{-2}$, $g_{\rm obs}$ traces $g_{\rm bar}$, asymptoting to the 1:1 line. Below this scale, there is a break in the relation such that $\rm g_{\rm obs} \sim g_{\rm bar}^{1/2}$. We show that the RAR slope, scatter and the acceleration scale are all natural consequences of the well-known baryonic Tully-Fisher relation (BTFR). We further demonstrate that galaxies with a variety of baryonic and dark matter (DM) profiles and a wide range of dark halo and galaxy properties (well beyond those expected in CDM) lie on the RAR if we simply require that their rotation curves satisfy the BTFR. We explore conditions needed to break this degeneracy: sub-kpc resolved rotation curves inside of "cored" DM-dominated profiles and/or outside $\gg 100\,$kpc could lie on BTFR but deviate in the RAR, providing new constraints on DM.

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arXiv2018

On the deuterium abundance and the importance of stellar mass loss in the interstellar and intergalactic medium

We quantify the gas-phase abundance of deuterium and fractional contribution of stellar mass loss to the gas in cosmological zoom-in simulations from the Feedback In Realistic Environments project. At low metallicity, our simulations confirm that the deuterium abundance is very close to the primordial value. The chemical evolution of the deuterium abundance that we derive here agrees quantitatively with analytical chemical evolution models. We furthermore find that the relation between the deuterium and oxygen abundance exhibits very little scatter. We compare our simulations to existing high-redshift observations in order to determine a primordial deuterium fraction of 2.549 +/- 0.033 x 10^-5 and stress that future observations at higher metallicity can also be used to constrain this value. At fixed metallicity, the deuterium fraction decreases slightly with decreasing redshift, due to the increased importance of mass loss from intermediate-mass stars. We find that the evolution of the average deuterium fraction in a galaxy correlates with its star formation history. Our simulations are consistent with observations of the Milky Way's interstellar medium: the deuterium fraction at the solar circle is 85-92 per cent of the primordial deuterium fraction. We use our simulations to make predictions for future observations. In particular, the deuterium abundance is lower at smaller galactocentric radii and in higher mass galaxies, showing that stellar mass loss is more important for fuelling star formation in these regimes (and can even dominate). Gas accreting onto galaxies has a deuterium fraction above that of the galaxies' interstellar medium, but below the primordial fraction, because it is a mix of gas accreting from the intergalactic medium and gas previously ejected or stripped from galaxies.

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arXiv2018

Simulating galaxies in the reionization era with FIRE-2: morphologies and sizes

We study the morphologies and sizes of galaxies at z>5 using high-resolution cosmological zoom-in simulations from the Feedback In Realistic Environments project. The galaxies show a variety of morphologies, from compact to clumpy to irregular. The simulated galaxies have more extended morphologies and larger sizes when measured using rest-frame optical B-band light than rest-frame UV light; sizes measured from stellar mass surface density are even larger. The UV morphologies are usually dominated by several small, bright young stellar clumps that are not always associated with significant stellar mass. The B-band light traces stellar mass better than the UV, but it can also be biased by the bright clumps. At all redshifts, galaxy size correlates with stellar mass/luminosity with large scatter. The half-light radii range from 0.01 to 0.2 arcsec (0.05-1 kpc physical) at fixed magnitude. At z>5, the size of galaxies at fixed stellar mass/luminosity evolves as (1+z)^{-m}, with m~1-2. For galaxies less massive than M_star~10^8 M_sun, the ratio of the half-mass radius to the halo virial radius is ~10% and does not evolve significantly at z=5-10; this ratio is typically 1-5% for more massive galaxies. A galaxy's "observed" size decreases dramatically at shallower surface brightness limits. This effect may account for the extremely small sizes of z>5 galaxies measured in the Hubble Frontier Fields. We provide predictions for the cumulative light distribution as a function of surface brightness for typical galaxies at z=6.

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