arXiv subjects

Mark R. Krumholz

Publications and source records attributed to Mark R. Krumholz.

At least 127 records · Page 7Linked to original sources

arXiv2021

The physics of gas phase metallicity gradients in galaxies

We present a new model for the evolution of gas phase metallicity gradients in galaxies from first principles. We show that metallicity gradients depend on four ratios that collectively describe the metal equilibration timescale, production, transport, consumption, and loss. Our model finds that most galaxy metallicity gradients are in equilibrium at all redshifts. When normalized by metal diffusion, metallicity gradients are governed by the competition between radial advection, metal production, and accretion of metal-poor gas from the cosmic web. The model naturally explains the varying gradients measured in local spirals, local dwarfs, and high-redshift star-forming galaxies. We use the model to study the cosmic evolution of gradients across redshift, showing that the gradient in Milky Way-like galaxies has steepened over time, in good agreement with both observations and simulations. We also predict the evolution of metallicity gradients with redshift in galaxy samples constructed using both matched stellar masses and matched abundances. Our model shows that massive galaxies transition from the advection-dominated to the accretion-dominated regime from high to low redshifts, which mirrors the transition from gravity-driven to star formation feedback-driven turbulence. Lastly, we show that gradients in local ultraluminous infrared galaxies (major mergers) and inverted gradients seen both in the local and high-redshift galaxies may not be in equilibrium. In subsequent papers in this series, we show that the model also explains the observed relationship between galaxy mass and metallicity gradients, and between metallicity gradients and galaxy kinematics.

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arXiv2021

Magnetic field amplification in accretion discs around the first stars: implications for the primordial IMF

Magnetic fields play an important role in the dynamics of present-day molecular clouds. Recent work has shown that magnetic fields are equally important for primordial clouds, which form the first stars in the Universe. While the primordial magnetic field strength on cosmic scales is largely unconstrained, theoretical models strongly suggest that a weak seed field existed in the early Universe. We study how the amplification of such a weak field can influence the evolution of accretion discs around first stars, and thus affect the primordial initial mass function (IMF). We perform a suite of 3D ideal magneto-hydrodynamic (MHD) simulations with different initial field strengths and numerical resolutions. We find that, in simulations with sufficient spatial resolution to resolve the Jeans scale during the collapse, even initially weak magnetic fields grow exponentially to become dynamically important due to both the so-called 'small-scale turbulent dynamo' and the 'large-scale mean-field dynamo'. Capturing the small-scale dynamo action depends primarily on how well we resolve the Jeans length, while capturing the large-scale dynamo depends on the Jeans resolution as well as the maximum absolute resolution. Provided enough resolution, we find that fragmentation does not depend strongly on the initial field strength, because even weak fields grow to become strong. However, fragmentation in runs with magnetic fields differs significantly from those without magnetic fields. We conclude that the development of dynamically strong magnetic fields during the formation of the first stars is likely inevitable, and that these fields had a significant impact on the primordial IMF.

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arXiv2021

On the origin of nitrogen at low metallicity

Understanding the evolution of the N/O ratio in the interstellar medium (ISM) of galaxies is essential if we are to complete our picture of the chemical evolution of galaxies at high redshift, since most observational calibrations of O/H implicitly depend upon the intrinsic N/O ratio. The observed N/O ratio, however, shows large scatter at low O/H, and is strongly dependent on galactic environment. We show that several heretofore unexplained features of the N/O distribution at low O/H can be explained by the N seen in metal-poor galaxies being mostly primary nitrogen that is returned to the ISM via pre-supernova winds from rapidly rotating massive stars ($M \gtrsim 10$ M$_\odot$, $v/v_{\rm crit} \gtrsim 0.4$). This mechanism naturally produces the observed N/O plateau at low O/H. We show that the large scatter in N/O at low O/H also arises naturally from variations in star-formation efficiency. By contrast, models in which the N and O come primarily from supernovae provide a very poor fit to the observed abundance distribution. We propose that the peculiar abundance patterns we observe at low O/H are a signature that dwarf galaxies retain little of their SN ejecta, leaving them with abundance patterns typical of winds.

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arXiv2021

Reconstructing three-dimensional densities from two-dimensional observations of molecular gas

Star formation has long been known to be an inefficient process, in the sense that only a small fraction $ε_{\rm ff}$ of the mass of any given gas cloud is converted to stars per cloud free-fall time. However, developing a successful theory of star formation will require measurements of both the mean value of $ε_{\rm ff}$ and its scatter from one molecular cloud to another. Because $ε_{\rm ff}$ is measured relative to the free-fall time, such measurements require accurate determinations of cloud volume densities. Efforts to measure the volume density from two-dimensional projected data, however, have thus far relied on treating molecular clouds as simple uniform spheres, while their real shapes are likely filamentary and their density distributions far from uniform. The resulting uncertainty in the true volume density is likely one of the major sources of error in observational estimates of $ε_{\rm ff}$. In this paper, we use a suite of simulations of turbulent, magnetized, radiative, self-gravitating star-forming clouds to examine whether it is possible to obtain more accurate volume density estimates and thereby reduce this error. We create mock observations from simulations, and show that current analysis methods relying on the spherical assumption likely yield ~ 0.26 dex underestimations and ~ 0.51 dex errors in volume density estimates, corresponding to a ~ 0.13 dex overestimation and a ~ 0.25 dex scatter in $ε_{\rm ff}$, comparable to the scatter in observed cloud samples. We build a predictive model that uses information accessible in two-dimensional measurements -- most significantly the Gini coefficient of the surface density distribution -- to estimate volume density with ~ 0.3 dex less scatter. We test our method on a recent observation of the Ophiuchus cloud, and show that it successfully reduces the $ε_{\rm ff}$ scatter.

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arXiv2021

Cosmic rays across the star-forming galaxy sequence. I: Cosmic ray pressures and calorimetry

In the Milky Way, cosmic rays (CRs) are dynamically important in the interstellar medium, contribute to hydrostatic balance, and may help regulate star formation. However, we know far less about the importance of CRs in galaxies whose gas content or star formation rate differ significantly from those of the Milky Way. Here we construct self-consistent models for hadronic CR transport, losses, and contribution to pressure balance as a function of galaxy properties, covering a broad range of parameters from dwarfs to extreme starbursts. While the CR energy density increases from $\sim 1$ eV cm$^{-3}$ to $\sim 1$ keV cm$^{-3}$ over the range from sub-Milky Way dwarfs to bright starbursts, strong hadronic losses render CRs increasingly unimportant dynamically as the star formation rate surface density increases. In Milky Way-like systems, CR pressure is typically comparable to turbulent gas and magnetic pressure at the galactic midplane, but the ratio of CR pressure to gas pressure drops to $\sim 10^{-3}$ in dense starbursts. Galaxies also become increasingly CR calorimetric and gamma-ray bright in this limit. The degree of calorimetry at fixed galaxy properties is sensitive to the assumed model for CR transport, and in particular to the time CRs spend interacting with neutral ISM, where they undergo strong streaming losses. We also find that in some regimes of parameter space hydrostatic equilibrium discs cannot exist, and in Paper II of this series we use this result to derive a critical surface in the plane of star formation surface density and gas surface density beyond which CRs may drive large-scale galactic winds.

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arXiv2020

Evolution of Stellar Feedback in HII Regions

Stellar feedback is needed to produce realistic giant molecular clouds (GMCs) and galaxies in simulations, but due to limited numerical resolution, feedback must be implemented using subgrid models. Observational work is an important means to test and anchor these models, but limited studies have assessed the relative dynamical role of multiple feedback modes, particularly at the earliest stages of expansion when HII regions are still deeply embedded. In this paper, we use multiwavelength (radio, infrared, and X-ray) data to measure the pressures associated with direct radiation ($P_{\rm dir}$), dust-processed radiation ($P_{\rm IR}$), photoionization heating ($P_{\rm HII}$), and shock-heating from stellar winds ($P_{\rm X}$) in a sample of 106 young, resolved HII regions with radii $\lesssim$0.5 pc to determine how stellar feedback drives their expansion. We find that the $P_{\rm IR}$ dominates in 84% of the regions and that the median $P_{\rm dir}$ and $P_{\rm HII}$ are smaller than the median $P_{\rm IR}$ by factors of $\approx 6$ and $\approx 9$, respectively. Based on the radial dependences of the pressure terms, we show that HII regions transition from $P_{\rm IR}$-dominated to $P_{\rm HII}$-dominated at radii of $\sim$3 pc. We find a median trapping factor of $f_{\rm trap} \sim$ 8 without any radial dependence for the sample, suggesting this value can be adopted in sub-grid feedback models. Moreover, we show that the total pressure is greater than the gravitational pressure in the majority of our sample, indicating that the feedback is sufficient to expel gas from the regions.

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arXiv2020

Pre-supernova feedback mechanisms drive the destruction of molecular clouds in nearby star-forming disc galaxies

It is a major open question which physical processes stop the accretion of gas onto giant molecular clouds (GMCs) and limit the efficiency at which gas is converted into stars within these GMCs. While feedback from supernova explosions has been the popular feedback mechanism included in simulations of galaxy formation and evolution, `early' feedback mechanisms such as stellar winds, photoionisation and radiation pressure are expected to play an important role in dispersing the gas after the onset of star formation. These feedback processes typically take place on small scales ($\sim 10-100$ pc) and their effects have therefore been difficult to constrain in environments other than the Milky Way. We apply a novel statistical method to $\sim 1$" resolution maps of CO and Ha emission across a sample of nine nearby disc galaxies, in order to measure the time over which GMCs are dispersed by feedback from young, high-mass stars, as a function of the galactic environment. We find that GMCs are typically dispersed within $\sim$ 3 Myr after the emergence of unembedded high-mass stars, showing no significant trend with galactocentric radius. Comparison with analytical predictions demonstrates that, independently of the environment, early feedback mechanisms (particularly photoionisation and stellar winds) play a crucial role in dispersing GMCs and limiting their star formation efficiency in nearby galaxies. Finally, we show that the efficiency at which the energy injected by these early feedback mechanisms couples with the parent GMC is relatively low (a few tens of per cent), such that the vast majority of momentum and energy emitted by the young stellar populations escapes the parent GMC.

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arXiv2020

An uncertainty principle for star formation -- III. The characteristic emission time-scales of star formation rate tracers

We recently presented a new statistical method to constrain the physics of star formation and feedback on the cloud scale by reconstructing the underlying evolutionary timeline. However, by itself this new method only recovers the relative durations of different evolutionary phases. To enable observational applications, it therefore requires knowledge of an absolute 'reference time-scale' to convert relative time-scales into absolute values. The logical choice for this reference time-scale is the duration over which the star formation rate (SFR) tracer is visible because it can be characterised using stellar population synthesis (SPS) models. In this paper, we calibrate this reference time-scale using synthetic emission maps of several SFR tracers, generated by combining the output from a hydrodynamical disc galaxy simulation with the SPS model SLUG2. We apply our statistical method to obtain self-consistent measurements of each tracer's reference time-scale. These include H$α$ and 12 ultraviolet (UV) filters (from GALEX, Swift, and HST), which cover a wavelength range 150-350 nm. At solar metallicity, the measured reference time-scales of H$α$ are ${4.32^{+0.09}_{-0.23}}$ Myr with continuum subtraction, and 6-16 Myr without, where the time-scale increases with filter width. For the UV filters we find 17-33 Myr, nearly monotonically increasing with wavelength. The characteristic time-scale decreases towards higher metallicities, as well as to lower star formation rate surface densities, owing to stellar initial mass function sampling effects. We provide fitting functions for the reference time-scale as a function of metallicity, filter width, or wavelength, to enable observational applications of our statistical method across a wide variety of galaxies.

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arXiv2020

Which feedback mechanisms dominate in the high-pressure environment of the Central Molecular Zone?

Supernovae (SNe) dominate the energy and momentum budget of stellar feedback, but the efficiency with which they couple to the interstellar medium (ISM) depends strongly on how effectively early, pre-SN feedback clears dense gas from star-forming regions. There are observational constraints on the magnitudes and timescales of early stellar feedback in low ISM pressure environments, yet no such constraints exist for more cosmologically typical high ISM pressure environments. In this paper, we determine the mechanisms dominating the expansion of HII regions as a function of size-scale and evolutionary time within the high-pressure ($P/k_\rm{B}$~$10^{7-8}$K cm$^{-3}$) environment in the inner 100pc of the Milky Way. We calculate the thermal pressure from the warm ionised ($P_\rm{HII}$; 10$^{4}$K) gas, direct radiation pressure ($P_\rm{dir}$), and dust processed radiation pressure ($P_\rm{IR}$). We find that (1) $P_\rm{dir}$ dominates the expansion on small scales and at early times (0.01-0.1pc; $<$0.1Myr); (2) the expansion is driven by $P_\rm{HII}$ on large scales at later evolutionary stages ($>0.1$pc; $>1$Myr); (3) during the first ~1Myr of growth, but not thereafter, either $P_{\rm IR}$ or stellar wind pressure likely make a comparable contribution. Despite the high confining pressure of the environment, natal star-forming gas is efficiently cleared to radii of several pc within ~2Myr, i.e. before the first SNe explode. This `pre-processing' means that subsequent SNe will explode into low density gas, so their energy and momentum will efficiently couple to the ISM. We find the HII regions expand to a radius of 3pc, at which point they have internal pressures equal with the surrounding external pressure. A comparison with HII regions in lower pressure environments shows that the maximum size of all HII regions is set by pressure equilibrium with the ambient ISM.

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arXiv2020

Formation and evolution of the local interstellar environment: combined constraints from nucleosynthetic and X-ray data

Several observations suggest that the Solar system has been located in a region affected by massive stellar feedback for at least a few Myr; these include detection of live $^{60}\text{Fe}$ in deep-sea archives and Antarctic snow, the broad angular distribution of $^{26}\text{Al}$ around the Galactic plane seen in all-sky $γ$-ray maps, and the all-sky soft X-ray background. However, our position inside the Galactic disc makes it difficult to fully characterise this environment, and our limited time baseline provides no information about its formation history or relation to large-scale Galactic dynamics. We explore these questions by using an $N$-body+hydrodynamics simulation of a Milky-Way-like galaxy to identify stars on Sun-like orbits whose environments would produce conditions consistent with those we observe. We find that such stars are uncommon but not exceptionally rare. These stars are found predominantly near the edges of spiral arms, and lie inside kpc-scale bubbles that are created by multiple generations of star formation in the arm. We investigate the stars' trajectories and find that the duration of the stay in the bubble ranges from 20 Myr to 90 Myr. The duration is governed by the crossing time of stars across the spiral arm. This is generally shorter than the bubble lifetime, which is $\sim 100$ Myr as a result of the continuous gas supply provided by the arm environment.

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arXiv2020

An uncertainty principle for star formation -- V. The influence of dust extinction on star formation rate tracer lifetimes and the inferred molecular cloud lifecycle

Recent observational studies aiming to quantify the molecular cloud lifecycle require the use of known 'reference time-scales' to turn the relative durations of different phases of the star formation process into absolute time-scales. We previously constrained the characteristic emission time-scales of different star formation rate (SFR) tracers, as a function of the SFR surface density and metallicity. However, we omitted the effects of dust extinction. Here, we extend our suite of SFR tracer emission time-scales by accounting for extinction, using synthetic emission maps of a high-resolution hydrodynamical simulation of an isolated, Milky-Way-like disc galaxy. The stellar feedback included in the simulation is inefficient compared to observations, implying that it represents a limiting case in which the duration of embedded star formation (and the corresponding effect of extinction) is overestimated. Across our experiments, we find that extinction mostly decreases the SFR tracer emission time-scale, changing the time-scales by factors of 0.04-1.74, depending on the gas column density. UV filters are more strongly affected than H$α$ filters. We provide the limiting correction factors as a function of the gas column density and flux sensitivity limit for a wide variety of SFR tracers. Applying these factors to observational characterisations of the molecular cloud lifecycle produces changes that broadly fall within the quoted uncertainties, except at high kpc-scale gas surface densities ($Σ_{\rm g}\gtrsim20~{\mathrm{M_{\odot}\,pc^{-2}}}$). Under those conditions, correcting for extinction may decrease the measured molecular cloud lifetimes and feedback time-scales, which further strengthens previous conclusions that molecular clouds live for a dynamical time and are dispersed by early, pre-supernova feedback.

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arXiv2020

Dynamics of small grains in transitional discs

Transitional discs have central regions characterised by significant depletion of both dust and gas compared to younger, optically-thick discs. However, gas and dust are not depleted by equal amounts: gas surface densities are typically reduced by factors of $\sim 100$, but small dust grains are sometimes depleted by far larger factors, to the point of being undetectable. While this extreme dust depletion is often attributed to planet formation, in this paper we show that another physical mechanism is possible: expulsion of grains from the disc by radiation pressure. We explore this mechanism using 2D simulations of dust dynamics, simultaneously solving the equation of radiative transfer with the evolution equations for dust diffusion and advection under the combined effects of stellar radiation and hydrodynamic interaction with a turbulent, accreting background gas disc. We show that, in transition discs that are depleted in both gas and dust fraction by factors of $\sim 100-1000$ compared to minimum mass Solar nebular values, and where the ratio of accretion rate to stellar luminosity is low ($\dot{M}/L \lesssim 10^{-10}$ $M_\odot$ yr$^{-1}$ $L_\odot^{-1}$), radiative clearing of any remaining $\sim 0.5$ $μ$m and larger grains is both rapid and inevitable. The process is size-dependent, with smaller grains removed fastest and larger ones persisting for longer times. Our proposed mechanism thus naturally explains the extreme depletion of small grains commonly-found in transition discs. We further suggest that the dependence of this mechanism on grain size and optical properties may explain some of the unusual grain properties recently discovered in a number of transition discs. The simulation code we develop is freely available.

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arXiv2020

Understanding biases in measurements of molecular cloud kinematics using line emission

Molecular line observations using a variety of tracers are often used to investigate the kinematic structure of molecular clouds. However, measurements of cloud velocity dispersions with different lines, even in the same region, often yield inconsistent results. The reasons for this disagreement are not entirely clear since molecular line observations are subject to a number of biases. In this paper, we untangle and investigate various factors that drive linewidth measurement biases by constructing synthetic position-position-velocity cubes for a variety of tracers from a suite of self-gravitating magnetohydrodynamic simulations of molecular clouds. We compare linewidths derived from synthetic observations of these data cubes to the true values in the simulations. We find that differences in linewidth as measured by different tracers are driven by a combination of density-dependent excitation, whereby tracers that are sensitive to higher densities sample smaller regions with smaller velocity dispersions, opacity broadening, especially for highly optically thick tracers such as CO, and finite resolution and sensitivity, which suppress the wings of emission lines. We find that, at fixed signal to noise ratio, three commonly-used tracers, the J=4->3 line of CO, the J=1->0 line of C^{18}O and the (1,1) inversion transition of NH_3, generally offer the best compromise between these competing biases, and produce estimates of the velocity dispersion that reflect the true kinematics of a molecular cloud to an accuracy of about 10% regardless of the cloud magnetic field strengths, evolutionary state, or orientations of the line of sight relative to the magnetic field.

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arXiv2020

Distribution and kinematics of 26Al in the Galactic disc

26Al is a short-lived radioactive isotope thought to be injected into the interstellar medium (ISM) by massive stellar winds and supernovae. However, all-sky maps of 26Al emission show a distribution with a much larger scale height and faster rotation speed than either massive stars or the cold ISM. We investigate the origin of this discrepancy using an N-body+hydrodynamics simulation of a Milky-Way-like galaxy, self-consistently including self-gravity, star formation, stellar feedback, and 26Al production. We find no evidence that the Milky Way's spiral structure explains the 26Al anomaly. Stars and the 26Al bubbles they produce form along spiral arms, but, because our simulation produces material arms that arise spontaneously rather than propagating arms forced by an external potential, star formation occurs at arm centres rather than leading edges. As a result, we find a scale height and rotation speed for 26Al similar to that of the cold ISM. However, we also show that a synthetic 26Al emission map produced for a possible Solar position at the edge of a large 26Al bubble recovers many of the major qualitative features of the observed 26Al sky. This suggests that the observed anomalous 26Al distribution is the product of foreground emission from the 26Al produced by a nearby, recent supernova.

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arXiv2020

The importance of magnetic fields for the initial mass function of the first stars

Magnetic fields play an important role for the formation of stars in both local and high-redshift galaxies. Recent studies of dynamo amplification in the first dark matter haloes suggest that significant magnetic fields were likely present during the formation of the first stars in the Universe at redshifts of 15 and above. In this work, we study how these magnetic fields potentially impact the initial mass function (IMF) of the first stars. We perform 200 high-resolution, three-dimensional (3D), magneto-hydrodynamic (MHD) simulations of the collapse of primordial clouds with different initial turbulent magnetic field strengths as predicted from turbulent dynamo theory in the early Universe, forming more than 1100 first stars in total. We detect a strong statistical signature of suppressed fragmentation in the presence of strong magnetic fields, leading to a dramatic reduction in the number of first stars with masses low enough that they might be expected to survive to the present day. Additionally, strong fields shift the transition point where stars go from being mostly single to mostly multiple to higher masses. However, irrespective of the field strength, individual simulations are highly chaotic, show different levels of fragmentation and clustering, and the outcome depends on the exact realisation of the turbulence in the primordial clouds. While these are still idealised simulations that do not start from cosmological initial conditions, our work shows that magnetic fields play a key role for the primordial IMF, potentially even more so than for the present-day IMF.

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arXiv2020

Cosmic rays across the star-forming galaxy sequence. II: Stability limits and the onset of cosmic ray-driven outflows

Cosmic rays (CRs) are a plausible mechanism for launching winds of cool material from the discs of star-forming galaxies. However, there is no consensus on what types of galaxies likely host CR-driven winds, or what role these winds might play in regulating galaxies' star formation rates. Using a detailed treatment of the transport and losses of hadronic CRs developed in the previous paper in this series, here we develop a semi-analytic model that allows us to assess the viability of using CRs to launch cool winds from galactic discs. In particular, we determine the critical CR fluxes -- and corresponding star formation rate surface densities -- above which hydrostatic equilibrium within a given galaxy is precluded because CRs drive the gas off in a wind or otherwise render it unstable. We show that, for star-forming galaxies with lower gas surface densities typical of the Galaxy and local dwarfs, the locus of this CR stability curve patrols the high side of the observed distribution of galaxies in the Kennicutt-Schmidt parameter space of star formation rate versus gas surface density. However, hadronic losses render CRs unable to drive winds in galaxies with higher surface densities. Our results show that quiescent, low surface density galaxies like the Milky Way are poised on the cusp of instability, such that small changes to ISM parameters can lead to the launching of CR-driven outflows, and we suggest that, as a result, CR feedback sets an ultimate limit to the star formation efficiency of most modern galaxies.

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arXiv2020

The Role of Outflows, Radiation Pressure, and Magnetic Fields in Massive Star Formation

Stellar feedback in the form of radiation pressure and magnetically-driven collimated outflows may limit the maximum mass that a star can achieve and affect the star-formation efficiency of massive pre-stellar cores. Here we present a series of 3D adaptive mesh refinement radiation-magnetohydrodynamic simulations of the collapse of initially turbulent, massive pre-stellar cores. Our simulations include radiative feedback from both the direct stellar and dust-reprocessed radiation fields, and collimated outflow feedback from the accreting stars. We find that protostellar outflows punches holes in the dusty circumstellar gas along the star's polar directions, thereby increasing the size of optically thin regions through which radiation can escape. Precession of the outflows as the star's spin axis changes due to the turbulent accretion flow further broadens the outflow, and causes more material to be entrained. Additionally, the presence of magnetic fields in the entrained material leads to broader entrained outflows that escape the core. We compare the injected and entrained outflow properties and find that the entrained outflow mass is a factor of $\sim$3 larger than the injected mass and the momentum and energy contained in the entrained material are $\sim$25% and $\sim$5% of the injected momentum and energy, respectively. As a result, we find that, when one includes both outflows and radiation pressure, the former are a much more effective and important feedback mechanism, even for massive stars with significant radiative outputs.

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arXiv2020

The SAMI Galaxy Survey: Gas velocity dispersions in low-$z$ star-forming galaxies and the drivers of turbulence

We infer the intrinsic ionised gas kinematics for 383 star-forming galaxies across a range of integrated star-formation rates (SFR $\in [10^{-3}, 10^2]$ M$_\odot$ yr$^{-1}$) at $z \lesssim 0.1$ using a consistent 3D forward-modelling technique. The total sample is a combination of galaxies from the SAMI Galaxy Survey and DYNAMO survey. For typical low-$z$ galaxies taken from the SAMI Galaxy Survey, we find the vertical velocity dispersion ($σ_{v, z}$) to be positively correlated with measures of star-formation rate, stellar mass, HI gas mass, and rotational velocity. The greatest correlation is with star-formation rate surface density ($Σ_\text{SFR}$). Using the total sample, we find $σ_{v, z}$ increases slowly as a function of integrated star-formation rate in the range SFR $\in$ [$10^{-3}$, 1] M$_\odot$ yr$^{-1}$ from $17\pm3$ km s$^{-1}$ to $24\pm5$ km s$^{-1}$ followed by a steeper increase up to $σ_{v, z}$ $\sim 80$ km s$^{-1}$ for SFR $\gtrsim 1$ M$_\odot$ yr$^{-1}$. This is consistent with recent theoretical models that suggest a $σ_{v, z}$ floor driven by star-formation feedback processes with an upturn in $σ_{v, z}$ at higher SFR driven by gravitational transport of gas through the disc.

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