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Eve C. Ostriker

Publications and source records attributed to Eve C. Ostriker.

At least 55 records · Page 3Linked to original sources

Effects of Magnetic Fields on Gas Dynamics and Star Formation in Nuclear Rings

Nuclear rings at the centers of barred galaxies are known to be strongly magnetized. To explore the effects of magnetic fields on star formation in these rings and nuclear gas flows, we run magnetohydrodynamic simulations in which there is a temporally-constant magnetized inflow to the ring, representing a bar-driven inflow. The mass inflow rate is $1\,M_\odot\,\mathrm{yr}^{-1}$, and we explore models with a range of field strength in the inflow. We adopt the TIGRESS framework developed by Kim & Ostriker to handle radiative heating and cooling, star formation, and resulting supernova (SN) feedback. We find that magnetic fields are efficiently amplified in the ring due to rotational shear and SN feedback. Within a few $100\,\mathrm{Myr}$, the turbulent component $B_\mathrm{trb}$ in the ring saturates at $\sim 35\,μ\mathrm{G}$ (in rough equipartition with the turbulent kinetic energy density), while the regular component $B_\mathrm{reg}$ exceeds $50\,μ\mathrm{G}$. Expanding superbubbles created by clustered SN explosions vertically drag predominantly-toroidal fields from near the midplane to produce poloidal fields in high-altitude regions. The growth of magnetic fields greatly suppresses star formation at late times. Simultaneously, strong magnetic tension in the ring drives radially inward accretion flows from the ring to form a circumnuclear disk in the central region; this feature is absent in the unmagnetized model.

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Star Formation Laws and Efficiencies across 80 Nearby Galaxies

We measure empirical relationships between the local star formation rate (SFR) and properties of the star-forming molecular gas on 1.5 kpc scales across 80 nearby galaxies. These relationships, commonly referred to as "star formation laws," aim at predicting the local SFR surface density from various combinations of molecular gas surface density, galactic orbital time, molecular cloud free-fall time, and the interstellar medium dynamical equilibrium pressure. Leveraging a multiwavelength database built for the PHANGS survey, we measure these quantities consistently across all galaxies and quantify systematic uncertainties stemming from choices of SFR calibrations and the CO-to-H$_2$ conversion factors. The star formation laws we examine show 0.3-0.4 dex of intrinsic scatter, among which the molecular Kennicutt-Schmidt relation shows a $\sim$10% larger scatter than the other three. The slope of this relation ranges $β\approx0.9{-}1.2$, implying that the molecular gas depletion time remains roughly constant across the environments probed in our sample. The other relations have shallower slopes ($β\approx0.6{-}1.0$), suggesting that the star formation efficiency (SFE) per orbital time, the SFE per free-fall time, and the pressure-to-SFR surface density ratio (i.e., the feedback yield) may vary systematically with local molecular gas and SFR surface densities. Last but not least, the shapes of the star formation laws depend sensitively on methodological choices. Different choices of SFR calibrations can introduce systematic uncertainties of at least 10-15% in the star formation law slopes and 0.15-0.25 dex in their normalization, while the CO-to-H$_2$ conversion factors can additionally produce uncertainties of 20-25% for the slope and 0.10-0.20 dex for the normalization.

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The structure and composition of multiphase galactic winds in a Large Magellanic Cloud mass simulated galaxy

We present the first results from a high resolution simulation with a focus on galactic wind driving for an isolated galaxy with a halo mass of $\sim 10^{11}$ M$_{\odot}$ (similar to the Large Magellanic Cloud) and a total gas mass of $\sim 6 \times 10^{8}$ M$_{\odot}$, resulting in $\sim 10^{8}$ gas cells at $\sim 4$ M$_{\odot}$ mass resolution. We adopt a resolved stellar feedback model with non-equilibrium cooling and heating, including photoelectric heating and photo-ionizing radiation, as well as supernovae (SNe), coupled to the second order meshless finite mass (MFM) method for hydrodynamics. These features make this the largest resolved-ISM galaxy model run to date. We find mean star formation rates around $0.05$ M$_{\odot}$ yr$^{-1}$ and evaluate typical time averaged loading factors for mass ($η_\mathrm{M}$ $\sim$ 1.0, in good agreement with recent observations) and energy ($η_\mathrm{E}$ $\sim$ 0.01). The bulk of the mass of the wind is transported by the warm ($T < 5 \times 10^5$K) phase, while there is a similar amount of energy transported in the warm and the hot phases ($T > 5 \times 10^5$K). We find an average opening angle of 30 degrees for the wind, decreasing with higher altitude above the midplane. The wind mass loading is decreasing (flat) for the warm (hot) phase as a function of the star formation surface rate density $Σ_{\rm SFR}$, while the energy loading shows inverted trends with $Σ_{\rm SFR}$, decreasing for the warm wind and increasing for the hot wind, although with very shallow slopes. These scalings are in good agreement with previous simulations of resolved wind driving in the multi-phase ISM.

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Photochemistry and Heating/Cooling of the Multiphase Interstellar Medium with UV Radiative Transfer for Magnetohydrodynamic Simulations

We present an efficient heating/cooling method coupled with chemistry and ultraviolet (UV) radiative transfer, which can be applied to numerical simulations of the interstellar medium (ISM). We follow the time-dependent evolution of hydrogen species (H$_2$, H, H$^+$), assume carbon/oxygen species (C, C$^+$, CO, O, and O$^+$) are in formation-destruction balance given the non-steady hydrogen abundances, and include essential heating/cooling processes needed to capture thermodynamics of all ISM phases. UV radiation from discrete point sources and the diffuse background is followed through adaptive ray tracing and a six-ray approximation, respectively, allowing for H$_2$ self-shielding; cosmic ray (CR) heating and ionization are also included. To validate our methods and demonstrate their application for a range of density, metallicity, and radiation field, we conduct a series of tests, including the equilibrium curves of thermal pressure vs. density, the chemical and thermal structure in photo-dissociation regions, H I-to-H$_2$ transitions, and the expansion of H II regions and radiative supernova remnants. Careful treatment of photochemistry and CR ionization is essential for many aspects of ISM physics, including identifying the thermal pressure at which cold and warm neutral phases co-exist. We caution that many current heating and cooling treatments used in galaxy formation simulations do not reproduce the correct thermal pressure and ionization fraction in the neutral ISM. Our new model is implemented in the MHD code Athena and incorporated in the TIGRESS simulation framework, for use in studying the star-forming ISM in a wide range of environments.

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Pressure-Regulated, Feedback-Modulated Star Formation In Disk Galaxies

The star formation rate (SFR) in galactic disks depends on both the quantity of available interstellar medium (ISM) gas and its physical state. Conversely, the ISM's physical state depends on the SFR, because the "feedback" energy and momentum injected by recently-formed massive stars is crucial to offsetting losses from turbulent dissipation and radiative cooling. The ISM's physical state also responds to the gravitational field that confines it, with increased weight driving higher pressure. In a quasi-steady state, it is expected that the mean total pressure of different thermal phases will match each other, that the component pressures and total pressure will satisfy thermal and dynamical equilibrium requirements, and that the SFR will adjust as needed to provide the requisite stellar radiation and supernova feedback. The pressure-regulated, feedback-modulated (PRFM) theory of the star-forming ISM formalizes these ideas, leading to a prediction that the SFR per unit area, Sigma_SFR, will scale nearly linearly with ISM weight W. In terms of large-scale gas surface density Sigma, stellar plus dark matter density rho_sd, and effective ISM velocity dispersion sigma_eff, an observable weight estimator is W~P_DE=pi G Sigma^2/2+Sigma(2G rho_sd)^{1/2} sigma_eff, and this is predicted to match the total midplane pressure P_tot. Using a suite of multiphase magnetohydrodynamic simulations run with the TIGRESS computational framework, we test the principles of the PRFM model and calibrate the total feedback yield Upsilon_tot = P_tot/Sigma_SFR ~ 1000 km/s, as well as its components. We compare results from TIGRESS to theory, previous numerical simulations, and observations, finding excellent agreement.

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Molecular Cloud Populations in the Context of Their Host Galaxy Environments: A Multiwavelength Perspective

We present a rich, multiwavelength, multiscale database built around the PHANGS-ALMA CO$\,$(2-1) survey and ancillary data. We use this database to present the distributions of molecular cloud populations and sub-galactic environments in 80 PHANGS galaxies, to characterize the relationship between population-averaged cloud properties and host galaxy properties, and to assess key timescales relevant to molecular cloud evolution and star formation. We show that PHANGS probes a wide range of kpc-scale gas, stellar, and star formation rate (SFR) surface densities, as well as orbital velocities and shear. The population-averaged cloud properties in each aperture correlate strongly with both local environmental properties and host galaxy global properties. Leveraging a variable selection analysis, we find that the kpc-scale surface densities of molecular gas and SFR tend to possess the most predictive power for the population-averaged cloud properties. Once their variations are controlled for, galaxy global properties contain little additional information, which implies that the apparent galaxy-to-galaxy variations in cloud populations are likely mediated by kpc-scale environmental conditions. We further estimate a suite of important timescales from our multiwavelength measurements. The cloud-scale free-fall time and turbulence crossing time are ${\sim}5{-}20$ Myr, comparable to previous cloud lifetime estimates. The timescales for orbital motion, shearing, and cloud-cloud collisions are longer, ${\sim}100$ Myr. The molecular gas depletion time is $1{-}3$ Gyr and shows weak to no correlations with the other timescales in our data. We publish our measurements online and expect them to have broad utility to future studies of molecular clouds and star formation.

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Slow Star Formation in the Milky Way: Theory Meets Observations

The observed star formation rate of the Milky Way can be explained by applying a metallicity-dependent factor to convert CO luminosity to molecular gas mass and a star formation efficiency per free-fall time that depends on the virial parameter of a molecular cloud. These procedures also predict the trend of star formation rate surface density with Galactocentric radius. The efficiency per free-fall time variation with virial parameter plays the major role in bringing theory into agreement with observations for the total star formation rate, while the metallicity dependence of the CO luminosity to mass conversion is most notable in the variation with Galactocentric radius. Application of these changes resolves a factor of over 100 discrepancy between observed and theoretical star formation rates that has been known for nearly 50 years.

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Cosmic-Ray Transport in Varying Galactic Environments

We study the propagation of mildly-relativistic cosmic rays (CRs) in multiphase interstellar medium environments with conditions typical of nearby disk galaxies. We employ the techniques developed in Armillotta+21 to post-process three high-resolution TIGRESS magnetohydrodynamic simulations modeling local patches of star-forming galactic disks. Together, the three simulations cover a wide range of gas surface density, gravitational potential, and star formation rate (SFR). Our prescription for CR propagation includes the effects of advection by the background gas, streaming along the magnetic field at the local ion Alfvén speed, and diffusion relative to the Alfvén waves, with the diffusion coefficient set by the balance between streaming-driven Alfvén wave excitation and damping mediated by local gas properties. We find that the combined transport processes are more effective in environments with higher SFR. These environments are characterized by higher-velocity hot outflows (created by clustered supernovae) that rapidly advect CRs away from the galactic plane. As a consequence, the ratio of midplane CR pressure to midplane gas pressures decreases with increasing SFR. We also use the post-processed simulations to make predictions regarding potential dynamical impacts of CRs. The relatively flat CR pressure profiles near the midplane argue that they would not provide significant support against gravity for most of the ISM mass. However, the CR pressure gradients are larger than the other pressure gradients in the extra-planar region (|z|>0.5 kpc), suggesting that CRs may affect the dynamics of galactic fountains and/or winds. The degree of this impact is expected to increase in environments with lower SFR.

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The Life and Times of Giant Molecular Clouds

Giant molecular clouds (GMCs) are the sites of star formation and stellar feedback in galaxies. Their properties set the initial conditions for star formation and their lifecycles determine how feedback regulates galaxy evolution. In recent years, the advent of high-resolution telescopes has enabled systematic GMC-scale studies of the molecular interstellar medium in nearby galaxies, now covering a wide range of physical conditions and allowing for the first studies of how GMC properties depend on galactic environment. These observational developments have been accompanied by numerical simulations of improving resolution that are increasingly accurately accounting for the effects of the galactic-scale environment on GMCs, while simultaneously improving the treatment of the small-scale processes of star-formation and stellar feedback within them. The combination of these recent developments has greatly improved our understanding of the formation, evolution, and destruction of GMCs. We review the current state of the field, highlight current open questions, and discuss promising avenues for future studies.

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Effects of Varying Mass Inflows on Star Formation in Nuclear Rings of Barred Galaxies

Observations indicate that the star formation rate (SFR) of nuclear rings varies considerably with time and is sometimes asymmetric rather than being uniform across a ring. To understand what controls temporal and spatial distributions of ring star formation, we run semi-global, hydrodynamic simulations of nuclear rings subject to time-varying and/or asymmetric mass inflow rates. These controlled variations in the inflow lead to variations in the star formation, while the ring orbital period ($18\,{\rm Myr}$) and radius ($600\,{\rm pc}$) remain approximately constant. We find that both the mass inflow rate and supernova feedback affect the ring SFR. An oscillating inflow rate with period $Δτ_\text{in}$ and amplitude 20 causes large-amplitude (a factor of $\gtrsim 5$), quasi-periodic variations of the SFR, when $Δτ_\text{in} \gtrsim 50\,{\rm Myr}$. We find that the time-varying ISM weight and midplane pressure track each other closely, establishing an instantaneous vertical equilibrium. The measured time-varying depletion time is consistent with the prediction from self-regulation theory provided the time delay between star formation and supernova feedback is taken into account. The supernova feedback is responsible only for small-amplitude (a factor of $\sim 2$) fluctuations of the SFR with a timescale $\lesssim 40\,{\rm Myr}$. Asymmetry in the inflow rate does not necessarily lead to asymmetric star formation in nuclear rings. Only when the inflow rate from one dust lane is suddenly increased by a large factor, the rings undergo a transient period of lopsided star formation.

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Star Formation Regulation and Self-Pollution by Stellar Wind Feedback

Stellar winds contain enough energy to easily disrupt the parent cloud surrounding a nascent star cluster, and for this reason have been considered candidates for regulating star formation. However, direct observations suggest most wind power is lost, and Lancaster21a,b recently proposed that this is due to efficient mixing and cooling processes. Here, we simulate star formation with wind feedback in turbulent, self-gravitating clouds, extending our previous work. Our simulations cover clouds with initial surface density $10^2-10^4$ $M_{\odot} \, {\rm pc}^{-2}$, and show that star formation and residual gas dispersal is complete within 2 - 8 initial cloud free-fall times. The "Efficiently Cooled" model for stellar wind bubble evolution predicts enough energy is lost for the bubbles to become momentum-driven, we find this is satisfied in our simulations. We also find that wind energy losses from turbulent, radiative mixing layers dominate losses by "cloud leakage" over the timescales relevant for star formation. We show that the net star formation efficiency (SFE) in our simulations can be explained by theories that apply wind momentum to disperse cloud gas, allowing for highly inhomogeneous internal cloud structure. For very dense clouds, the SFE is similar to those observed in extreme star-forming environments. Finally, we find that, while self-pollution by wind material is insignificant in cloud conditions with moderate density (only $\lesssim 10^{-4}$ of the stellar mass originated in winds), our simulations with conditions more typical of a super star cluster have star particles that form with as much as 1\% of their mass in wind material.

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Cosmic-Ray Transport in Simulations of Star-forming Galactic Disks

Cosmic ray transport on galactic scales depends on the detailed properties of the magnetized, multiphase interstellar medium (ISM). In this work, we post-process a high-resolution TIGRESS magnetohydrodynamic simulation modeling a local galactic disk patch with a two-moment fluid algorithm for cosmic ray transport. We consider a variety of prescriptions for the cosmic rays, from a simple purely diffusive formalism with constant scattering coefficient, to a physically-motivated model in which the scattering coefficient is set by critical balance between streaming-driven Alfvén wave excitation and damping mediated by local gas properties. We separately focus on cosmic rays with kinetic energies of $\sim 1$ GeV (high-energy) and $\sim 30$~MeV (low-energy), respectively important for ISM dynamics and chemistry. We find that simultaneously accounting for advection, streaming, and diffusion of cosmic rays is crucial for properly modeling their transport. Advection dominates in the high-velocity, low-density, hot phase, while diffusion and streaming are more important in higher density, cooler phases. Our physically-motivated model shows that there is no single diffusivity for cosmic-ray transport: the scattering coefficient varies by four or more orders of magnitude, maximal at density $n_\mathrm{H} \sim 0.01\, \mathrm{cm}^{-3}$. Ion-neutral damping of Alfvén waves results in strong diffusion and nearly uniform cosmic ray pressure within most of the mass of the ISM. However, cosmic rays are trapped near the disk midplane by the higher scattering rate in the surrounding lower-density, higher-ionization gas. The transport of high-energy cosmic rays differs from that of low-energy cosmic rays, with less effective diffusion and greater energy losses for the latter.

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Influence of Ion-Neutral Damping on the Cosmic-Ray Streaming Instability: Magnetohydrodynamic Particle-in-cell Simulations

We explore the physics of the gyro-resonant cosmic ray streaming instability (CRSI) including the effects of ion-neutral (IN) damping. This is the main damping mechanism in (partially-ionized) atomic and molecular gas, which are the primary components of the interstellar medium (ISM) by mass. Limitation of CRSI by IN damping is important in setting the amplitude of Alfvén waves that scatter cosmic rays and control galactic-scale transport. Our study employs the MHD-PIC hybrid fluid-kinetic numerical technique to follow linear growth as well as post-linear and saturation phases. During the linear phase of the instability -- where simulations and analytical theory are in good agreement -- IN damping prevents wave growth at small and large wavelengths, with the unstable bandwidth lower for higher ion-neutral collision rate $ν_{\rm in}$. Purely MHD effects during the post-linear phase extend the wave spectrum towards larger $k$. In the saturated state, the cosmic ray distribution evolves toward greater isotropy (lower streaming velocity) by scattering off of Alvén waves excited by the instability. In the absence of low-$k$ waves, CRs with sufficiently high momentum are not isotropized. The maximum wave amplitude and rate of isotropization of the distribution function decreases at higher $ν_{\rm in}$. When the IN damping rate approaches the maximum growth rate of CSRI, wave growth and isotropization is suppressed. Implications of our results for CR transport in partially ionized ISM phases are discussed.

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PHANGS-ALMA: Arcsecond CO(2-1) Imaging of Nearby Star-Forming Galaxies

We present PHANGS-ALMA, the first survey to map CO J=2-1 line emission at ~1" ~ 100pc spatial resolution from a representative sample of 90 nearby (d<~20 Mpc) galaxies that lie on or near the z=0 "main sequence" of star-forming galaxies. CO line emission traces the bulk distribution of molecular gas, which is the cold, star-forming phase of the interstellar medium. At the resolution achieved by PHANGS-ALMA, each beam reaches the size of a typical individual giant molecular cloud (GMC), so that these data can be used to measure the demographics, life-cycle, and physical state of molecular clouds across the population of galaxies where the majority of stars form at z=0. This paper describes the scientific motivation and background for the survey, sample selection, global properties of the targets, ALMA observations, and characteristics of the delivered ALMA data and derived data products. As the ALMA sample serves as the parent sample for parallel surveys with VLT/MUSE, HST, AstroSat, VLA, and other facilities, we include a detailed discussion of the sample selection. We detail the estimation of galaxy mass, size, star formation rate, CO luminosity, and other properties, compare estimates using different systems and provide best-estimate integrated measurements for each target. We also report the design and execution of the ALMA observations, which combine a Cycle~5 Large Program, a series of smaller programs, and archival observations. Finally, we present the first 1" resolution atlas of CO emission from nearby galaxies and describe the properties and contents of the first PHANGS-ALMA public data release.

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Star Formation in Nuclear Rings with the TIGRESS Framework

Nuclear rings are sites of intense star formation at the centers of barred galaxies. To understand what determines the structure and star formation rate (SFR; $\dot{M}_{\rm SF}$) of nuclear rings, we run semi-global, hydrodynamic simulations of nuclear rings subject to constant mass inflow rates $\dot{M}_{\rm in}$. We adopt the TIGRESS framework of Kim \& Ostriker to handle radiative heating and cooling, star formation, and related supernova (SN) feedback. We find that the SN feedback is never strong enough to destroy the ring or quench star formation everywhere in the ring. Under the constant $\dot{M}_{\rm in}$, the ring star formation is very steady and persistent, with the SFR exhibiting only mild temporal fluctuations. The ring SFR is tightly correlated with the inflow rate as $\dot{M}_{\rm SF}\approx 0.8\dot{M}_{\rm in}$, for a range of $\dot{M}_{\rm in}=0.125-8\,M_\odot\,{\rm yr}^{-1}$. Within the ring, vertical dynamical equilibrium is maintained, with the midplane pressure (powered by SN feedback) balancing the weight of the overlying gas. The SFR surface density is correlated nearly linearly with the midplane pressure, as predicted by the pressure-regulated, feedback-modulated star formation theory. Based on our results, we argue that the ring SFR is causally controlled by $\dot{M}_\text{in}$, while the ring gas mass adapts to the SFR to maintain the vertical dynamical equilibrium under the gravitational field arising from both gas and stars.

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Efficiently Cooled Stellar Wind Bubbles in Turbulent Clouds I. Fractal Theory and Application to Star-Forming Clouds

Winds from massive stars have velocities of 1000 km/s or more, and produce hot, high pressure gas when they shock. We develop a theory for the evolution of bubbles driven by the collective winds from star clusters early in their lifetimes, which involves interaction with the turbulent, dense interstellar medium of the surrounding natal molecular cloud. A key feature is the fractal nature of the hot bubble's surface. The large area of this interface with surrounding denser gas strongly enhances energy losses from the hot interior, enabled by turbulent mixing and subsequent cooling at temperatures T = 10^4-10^5 K where radiation is maximally efficient. Due to the extreme cooling, the bubble radius scales differently (R ~ t^1/2) from the classical Weaver77 solution, and has expansion velocity and momentum lower by factors of 10-10^2 at given R, with pressure lower by factors of 10^2 - 10^3. Our theory explains the weak X-ray emission and low shell expansion velocities of observed sources. We discuss further implications of our theory for observations of the hot bubbles and cooled expanding shells created by stellar winds, and for predictions of feedback-regulated star formation in a range of environments. In a companion paper, we validate our theory with a suite of hydrodynamic simulations.

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Efficiently Cooled Stellar Wind Bubbles in Turbulent Clouds II. Validation of Theory with Hydrodynamic Simulations

In a companion paper, we develop a theory for the evolution of stellar wind driven bubbles in dense, turbulent clouds. This theory proposes that turbulent mixing at a fractal bubble-shell interface leads to highly efficient cooling, in which the vast majority of the input wind energy is radiated away. This energy loss renders the majority of the bubble evolution momentum-driven rather than energy-driven, with expansion velocities and pressures orders of magnitude lower than in the classical Weaver77 solution. In this paper, we validate our theory with three-dimensional, hydrodynamic simulations. We show that extreme cooling is not only possible, but is generic to star formation in turbulent clouds over more than three orders of magnitude in density. We quantify the few free parameters in our theory, and show that the momentum exceeds the wind input rate by only a factor ~ 1.2-4. We verify that the bubble/cloud interface is a fractal with dimension ~ 2.5-2.7. The measured turbulent amplitude (v_t ~ 200-400 km/s) in the hot gas near the interface is shown to be consistent with theoretical requirements for turbulent diffusion to efficiently mix and radiate away most of the wind energy. The fraction of energy remaining after cooling is only 1-Θ~ 0.1-0.01, decreasing with time, explaining observations that indicate low hot-gas content and weak dynamical effects of stellar winds.

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Cosmic Ray Transport in the Ionized and Neutral ISM: MHD-PIC Simulations and Effective Fluid Treatments

Cosmic rays (CRs) have critical impacts in the multiphase interstellar medium (ISM), driving dynamical motions in low-density plasma and modifying the ionization state, temperature, and chemical composition of higher-density atomic and molecular gas. We present a study of CR propagation between the ionized ISM and a neutral cloud. Using one-dimensional magnetohydrodynamic particle-in-cell simulations which include ion-neutral drag to damp Alfv$\acute{\text{e}}$n waves in the cloud, we self-consistently evolve the kinetic physics of CRs and fluid dynamics of the multiphase gas. By introducing the cloud in our periodic domain, our simulations break translational symmetry and allow the emergence of spatial structure in the CR distribution function. A negative spatial gradient forms across the fully-ionized ISM region while a positive gradient forms across the neutral cloud. We connect our results with CR hydrodynamics formulations by computing the wave-particle scattering rates as predicted by quasilinear, fluid, and Fokker-Planck theory. For momenta where the mean free path is short relative to the box size, we find excellent agreement among all scattering rates. By exploring different cloud sizes and ion-neutral collision rates, we show that our results are robust. Our work provides a first-principles verification of CR hydrodynamics when particles stream down their pressure gradient, and opens a pathway toward comprehensive calibrations of transport coefficients from self-generated Alfv$\acute{\text{e}}$n wave scattering with CRs.

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