Radiation Hydrodynamic Simulations Research Articles

Thomson scattering measurements of electron temperature and electron density in laser-driven Gd plasmas

In this study, we demonstrate the laser intensity scaling of electron temperature in Gd laser-produced plasmas (LPPs) through experiment and simulation. The spatial and temporal profiles of electron density ne and temperature Te in Gd LPPs were directly measured during a drive laser pulse duration of 7 ns at a laser wavelength of 1064 nm using collective Thomson scattering. We found that the measured maximum Te value in Gd LPPs increases with increasing laser intensity IL , with the dependence , in the IL range of 1010–. Radiation-hydrodynamic simulation code of the STAR-2D was performed under identical conditions to those used in the experiment, and extended further to higher laser intensities of up to ; the simulated Te was found to be in good agreement with the experimental data over the used IL range. The simulation indicates that higher overall maximum Te typically exists 50–70 above the target and modifies the dependence as . Experiments reveal that a laser intensity of is required to achieve the optimum condition ( 100 eV and ion charge states , at –) for efficient beyond extreme ultraviolet (EUV) light source. In addition, two spot sizes (150 and 250 in diameter) were used to study the effect of the spot size on the Te profile. Our experimental findings, supported by the simulations, show that larger spots can create uniform Te profiles, higher maximum Te , and larger high-Te regions in LPPs. The results and scaling laws presented in this study provide important information for developing more powerful and energy-efficient EUV light sources.

Radio Observations of Six Young Type Ia Supernovae

Type Ia supernovae (SNe Ia) are important cosmological tools, probes of binary star evolution, and contributors to cosmic metal enrichment; yet, a definitive understanding of the binary star systems that produce them remains elusive. Of particular interest is the identity of the mass-donor companion to the exploding carbon–oxygen white dwarf (CO WD). In this work, we present early-time (first observation within 10 days post-explosion) radio observations of six nearby (within 40 Mpc) SNe Ia taken by the Jansky Very Large Array, which are used to constrain the presence of synchrotron emission from the interaction between ejecta and circumstellar material (CSM). The two motivations for these early-time observations are: (1) to constrain the presence of low-density winds and (2) to provide an additional avenue of investigation for those SNe Ia observed to have early-time optical/UV excesses that may be due to CSM interaction. We detect no radio emission from any of our targets. Toward our first aim, these non-detections further increase the sample of SNe Ia that rule out winds from symbiotic binaries and strongly accreting white dwarfs. and discuss the dependence on underlying model assumptions and how our observations represent a large increase in the sample of SNe Ia with low-density wind constraints. For the second aim, we present a radiation hydrodynamics simulation to explore radio emission from an SN Ia interacting with a compact shell of CSM, and find that relativistic electrons cannot survive to produce radio emission despite the rapid expansion of the shocked shell after shock breakout. The effects of model assumptions are discussed for both the wind and compact shell conclusions.

The Planetary Accretion Shock. III. Smoothing-free 2.5D Simulations and Calculation of Hα Emission

Surveys have looked for Hα emission from accreting gas giants but found very few objects. Analyses of the detections and nondetections have assumed that the entire gas flow feeding the planet is in radial freefall. However, hydrodynamical simulations suggest that this is far from reality. We calculate the Hα emission from multidimensional accretion onto a gas giant, following the gas flow from Hill sphere scales down to the circumplanetary disk (CPD) and the planetary surface. We perform azimuthally symmetric radiation hydrodynamics simulations around the planet and use modern tabulated gas and dust opacities. Crucially, contrasting with most previous simulations, we do not smooth the gravitational potential but do follow the flow down to the planetary surface, where grid cells are 0.01 Jupiter radii small. We find that roughly only 1% of the net gas inflow into the Hill sphere directly reaches the planet. As expected for ballistic infall trajectories, most of the gas falls at too large a distance on the CPD to generate Hα. Including radiation transport removes the high-velocity subsurface flow previously seen in hydrodynamics-only simulations, so that only the free planet surface and the inner regions of the CPD emit substantial Hα. Unless magnetospheric accretion, which we neglect here, additionally produces Hα, the corresponding Hα production efficiency is much smaller than usually assumed, which needs to be taken into account when analyzing (non)detection statistics.

Spectral Observations and Modeling of a Solar White-light Flare Observed by CHASE

The heating mechanisms of solar white-light flares remain unclear. We present an X1.0 white-light flare on 2022 October 2 (SOL2022-10-02T20:25) observed by the Chinese Hα Solar Explorer that provides two-dimensional spectra in the visible light for the full solar disk with a seeing-free condition. The flare shows a prominent enhancement of ∼40% in the photospheric Fe i line at 6569.2 Å, and the nearby continuum also exhibits a maximum enhancement of ∼40%. For the continuum near the Fe i line at 6173 Å from the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory, it is enhanced up to ∼20%. At the white-light kernels, the Fe i line at 6569.2 Å has a symmetric Gaussian profile that is still in absorption and the Hα line at 6562.8 Å displays a very broad emission profile with a central reversal plus a red or blue asymmetry. The white-light kernels are cospatial with the microwave footpoint sources observed by the Expanded Owens Valley Solar Array and the time profile of the white-light emission matches that of the hard X-ray emission above 30 keV from the Gamma-ray Burst Monitor (GBM) on Fermi. These facts indicate that the white-light emission is qualitatively related to a nonthermal electron beam. We also perform a radiative hydrodynamic simulation with the electron-beam parameters constrained by the hard X-ray observations from Fermi/GBM. The result reveals that the white-light enhancement cannot be well explained by a pure electron-beam heating together with its induced radiative backwarming but may need additional heating sources such as Alfvén waves.

The formation of globular clusters with top-heavy initial mass functions

ABSTRACT We study the formation of globular clusters (GCs) in massive compact clouds with the low metallicity of Z = 10−3 Z⊙ by performing three-dimensional radiative-hydrodynamic simulations. Considering the uncertainty of the initial mass function (IMF) of stars formed in low-metallicity and high-density clouds, we investigate the impacts of the IMF on the cloud condition for the GC formation with the range of the power-law index of IMF as γ = 1−2.35. We find that the threshold surface density (Σthr) for the GC formation increases from 800 M⊙ pc−2 at γ = 2.35 to 1600 M⊙ pc−2 at γ = 1.5 in the cases of clouds with Mcl = 106 M⊙ because the emissivity of ionizing photons per stellar mass increases as γ decreases. For γ < 1.5, Σthr saturates with ∼2000 M⊙ pc−2 that is quite rare and observed only in local starburst galaxies due to e.g. merger processes. Thus, we suggest that formation sites of low-metallicity GCs could be limited only in the very high-surface density regions. We also find that Σthr can be modelled by a power-law function with the cloud mass (Mcl) and the emissivity of ionizing photons (s*) as $\propto M_{\rm cl}^{-1/5} s_{*}^{2/5}$. Based on the relation between the power-law slope of IMF and Σthr, future observations with e.g. the JWST can allow us to constrain the IMF of GCs.

Water-window x-ray emission from laser-produced Au plasma under optimal target thickness and focus conditions.

Optimal laser irradiation conditions for water-window (WW) x-ray emission (2.3-4.4nm) from an Au plasma are investigated to develop a laboratory-scale WW x-ray source. A minimum Au target thickness of 1 µm is obtained for a laser intensity of ∼10^{13} W/cm^{2} by observing the intensity drop in the WW spectra. Au targets produced by thermal evaporation are found to have a higher conversion efficiency than commercial foil targets for WW x-ray radiation. In addition, optimal laser spots for fixed laser energies (240 and 650mJ) are found for an Au target ∼1mm in front of the focal point, where suitable conditions for plasma temperature and plume volume coupling are achieved. The mechanism of the optimal target thickness and spot size can be well explained using a radiation hydrodynamic simulation code.

The challenges of identifying Population III stars in the early Universe

ABSTRACT The recent launch of JWST has enabled the exciting prospect of detecting the first generation of metal-free, Population III (Pop. III) stars. Determining characteristics that robustly signify Pop. III stars against other possible contaminants represents a key challenge. To this end, we run high-resolution (sub-pc) cosmological radiation hydrodynamics simulations of the region around a dwarf galaxy at z ≥ 10 to predict the emission line signatures of the Pop. III/Pop. II transition. We show that the absence of metal emission lines is a poor diagnostic of Pop. III stars because metal-enriched galaxies can maintain low [O iii] 5007 Å that may be undetectable due to sensitivity limits. Combining spectral hardness probes (e.g. He ii 1640 Å/H α) with metallicity diagnostics is more likely to probe metal-free stars, although contamination from Wolf−Rayet stars, X-ray binaries, or black holes may be important. The hard emission from Pop. III galaxies fades fast due to the short stellar lifetimes of massive stars, which could further inhibit detection. Pop. III stars may be identifiable after they evolve off the main sequence due to the cooling radiation from nebular gas or a supernova remnant; however, these signatures are also short-lived (i.e. few Myr). Contaminants including flickering black holes might confuse this diagnostic. While JWST will provide a unique opportunity to spectroscopically probe the nature of the earliest galaxies, both the short time-scales associated with pristine systems and ambiguities in interpreting emission lines may hinder progress. Special care will be needed before claiming the discovery of systems with pure Pop. III stars.

Radiative hydrodynamical simulations of super-Eddington accretion flow in tidal disruption event: the accretion flow and wind

ABSTRACT One key question in tidal disruption events theory is how much of the fallback debris can be accreted to the black hole. Based on radiative hydrodynamic simulations, we study this issue for efficiently ‘circularized’ debris accretion flow. We find that for a black hole disrupting a solar-type star, $15{{\, \rm per\, cent}}$ of the debris can be accreted for a 107 M⊙ black hole. While for a 106 M⊙ black hole, the value is $43{{\, \rm per\, cent}}$. We find that wind can be launched in the super-Eddington accretion phase regardless of the black hole mass. The maximum velocity of the wind can reach 0.7c (with c being the speed of light). The kinetic power of wind is well above 1044 erg s−1. The results can be used to study the interaction of wind and the circumnuclear medium around quiescent supermassive black holes.

On the frequencies of circumbinary discs in protostellar systems

ABSTRACT We report the analysis of circumbinary (CB) discs formed in a radiation hydrodynamical simulation of star cluster formation. We consider both pure binary stars and pairs within triple and quadruple systems. The protostellar systems are all young (ages < 105 yrs). We find that the systems that host a CB disc have a median separation of ≈11 au, and the median characteristic radius of the discs is ≈64 au. We find that 89 per cent of pure binaries with semimajor axes a < 1 au have a CB disc, and the occurrence rate of CB discs is bimodal with log-separation in pure binaries with a second peak at a ≈ 50 au. Systems with a > 100 au almost never have a CB disc. The median size of a CB disc is between ≈5 and 6 a depending on the order of the system, with higher order systems having larger discs relative to binary separation. We find the underlying distributions of mutual inclination between CB discs and binary orbits from the simulation are in good agreement with those of observed CB discs.

SpK: A fast atomic and microphysics code for the high-energy-density regime

SpK is part of the numerical codebase at Imperial College London used to model high energy density physics (HEDP) experiments. SpK is an efficient atomic and microphysics code used to perform detailed configuration accounting calculations of electronic and ionic stage populations, opacities and emissivities for use in post-processing and radiation hydrodynamics simulations. This is done using screened hydrogenic atomic data supplemented by the NIST energy level database. An extended Saha model solves for chemical equilibrium with extensions for non-ideal physics, such as ionisation potential depression, and non thermal equilibrium corrections. A tree-heap (treap) data structure is used to store spectral data, such as opacity, which is dynamic thus allowing easy insertion of points around spectral lines without a-priori knowledge of the ion stage populations. Results from SpK are compared to other codes and descriptions of radiation transport solutions which use SpK data are given. The treap data structure and SpK’s computational efficiency allows inline post-processing of 3D hydrodynamics simulations with a dynamically evolving spectrum stored in a treap.

3D Radiation-hydrodynamic Simulations Resolving Interior of Rapidly Accreting Primordial Protostar

Direct collapse of supermassive stars is a possible pathway to form supermassive black hole seeds at high redshifts. Whereas previous three-dimensional (3D) simulations demonstrate that supermassive stars form via rapid mass accretion, those resolving the stellar interior have been limited. Here, we report 3D radiation-hydrodynamic (RHD) simulations following the evolution of rapidly accreting protostars resolving the stellar interior. We use an adaptive mesh refinement code with our newly developed RHD solver employing an explicit M1 closure method. We follow the early evolution until the stellar mass reaches ∼10 M ⊙ from two different initial configurations of spherical and turbulent clouds. We demonstrate that, in both cases, a swollen protostar whose radius is 100–1000 R ⊙ appears, as predicted by the stellar evolution calculations. Its effective temperature remains a few thousand Kelvin, and the radiative feedback by ionizing photons is too weak to disturb the accretion flow up to the epoch examined in this work. In the turbulent case, the protostar rotates rapidly at more than 0.4 times the Keplerian velocity owing to the angular momentum provided by the initial turbulence. The protostar approximates an oblate spheroid, and its equatorial radius is more than twice the polar radius. Our results suggest that we need to consider the rapid stellar rotation to elucidate the realistic 3D protostellar evolution in the supermassive star formation.

Multiphase Gas Nature in the Sub-parsec Region of the Active Galactic Nuclei. I. Dynamical Structures of Dusty and Dust-free Outflow

We investigated dusty and dust-free gas dynamics for a radiation-driven sub-parsec-scale outflow in an active galactic nucleus (AGN) associated with a supermassive black hole 107 M ⊙ and bolometric luminosity 1044 erg s−1 based on the two-dimensional radiation-hydrodynamic simulations. A radiation-driven “lotus-like” multi-shell outflow is launched from the inner part (r ≲ 0.04 pc) of the geometrically thin disk, and it repeatedly and steadily produces shocks as mass accretion continues through the disk to the center. The shape of the dust sublimation radius is not spherical and depends on the angle (θ) from the disk plane, reflecting the nonspherical radiation field and nonuniform dust-free gas. Moreover, we found that the sublimation radius of θ ∼ 20°–60° varies on a timescale of several years. The “inflow-induced outflow” contributes to the obscuration of the nucleus in the sub-parsec region. The column density of the dust-free gas is N H ≳ 1022 cm−2 for r ≲ 0.04 pc. Gases near the disk plane (θ ≲ 30°) can be the origin of the Compton-thick component, which was suggested by the recent X-ray observations of AGNs. The dusty outflow from the sub-parsec region can be also a source of material for the radiation-driven fountain for a larger scale.

Simulating the diversity of shapes of the Lyman-α line

ABSTRACT The Ly α line is a powerful probe of distant galaxies, which contains information about inflowing/outflowing gas through which Ly α photons scatter. To develop our understanding of this probe, we post-process a zoom-in radiation-hydrodynamics simulation of a low-mass (Mstar ∼ 109 M⊙) galaxy to construct 22 500 mock spectra in 300 directions from z = 3 to 4. Remarkably, we show that one galaxy can reproduce the variety of a large sample of spectroscopically observed Ly α line profiles. While most mock spectra exhibit double-peak profiles with a dominant red peak, their shapes cover a large parameter space in terms of peak velocities, peak separation, and flux ratio. This diversity originates from radiative transfer effects at interstellar medium and circum-galactic medium (CGM) scales, and depends on galaxy inclination and evolutionary phase. Red-dominated lines preferentially arise in face-on directions during post-starburst outflows and are bright. Conversely, accretion phases usually yield symmetric double peaks in the edge-on direction and are fainter. While resonant scattering effects at <0.2 × Rvir are responsible for the broadening and velocity shift of the red peak, the extended CGM acts as a screen and impacts the observed peak separation. The ability of simulations to reproduce observed Ly α profiles and link their properties with galaxy physical parameters offers new perspectives to use Ly α to constrain the mechanisms that regulate galaxy formation and evolution. Notably, our study implies that deeper Ly α surveys may unveil a new population of blue-dominated lines tracing inflowing gas.

Observability of photoevaporation signatures in the dust continuum emission of transition discs

ABSTRACT Photoevaporative disc winds play a key role in our understanding of circumstellar disc evolution, especially in the final stages, and they might affect the planet formation process and the final location of planets. The study of transition discs (i.e. discs with a central dust cavity) is central for our understanding of the photoevaporation process and disc dispersal. However, we need to distinguish cavities created by photoevaporation from those created by giant planets. Theoretical models are necessary to identify possible observational signatures of the two different processes, and models to find the differences between the two processes are still lacking. In this paper, we study a sample of transition discs obtained from radiation–hydrodynamic simulations of internally photoevaporated discs, and focus on the dust dynamics relevant for current Atacama Large Millimetre Array observations. We then compared our results with gaps opened by a super-Earth/giant planets, finding that the photoevaporated cavity steepness depends mildly on gap size, and it is similar to that of a ${1}\, {\rm M_J}$ planet. However, the dust density drops less rapidly inside the photoevaporated cavity compared to the planetary case due to the less efficient dust filtering. This effect is visible in the resulting spectral index, which shows a larger spectral index at the cavity edge and a shallower increase inside it with respect to the planetary case. The combination of cavity steepness and spectral index might reveal the true nature of transition discs.

3D Radiation Hydrodynamic Simulations of Gravitational Instability in AGN Accretion Disks: Effects of Radiation Pressure

We perform 3D radiation hydrodynamic local shearing-box simulations to study the outcome of gravitational instability (GI) in optically thick active galactic nuclei (AGNs) accretion disks. GI develops when the Toomre parameter Q T ≲ 1, and may lead to turbulent heating that balances radiative cooling. However, when radiative cooling is too efficient, the disk may undergo runaway gravitational fragmentation. In the fully gas-pressure-dominated case, we confirm the classical result that such a thermal balance holds when the Shakura–Sunyaev viscosity parameter (α) due to the gravitationally driven turbulence is ≲0.2, corresponding to dimensionless cooling times Ωt cool ≳ 5. As the fraction of support by radiation pressure increases, the disk becomes more prone to fragmentation, with a reduced (increased) critical value of α (Ωt cool). The effect is already significant when the radiation pressure exceeds 10% of the gas pressure, while fully radiation-pressure-dominated disks fragment at t cool ≲ 50 Ω−1. The latter translates to a maximum turbulence level α ≲ 0.02, comparable to that generated by magnetorotational instability. Our results suggest that gravitationally unstable (Q T ∼ 1) outer regions of AGN disks with significant radiation pressure (likely for high/near-Eddington accretion rates) should always fragment into stars, and perhaps black holes.

Black hole mergers as tracers of spinning massive black hole and galaxy populations in the OBELISK simulation

Massive black hole (BH) mergers will be key targets of future gravitational wave and electromagnetic observational facilities. In order to constrain BH evolution with the information extracted from BH mergers, one must take into account the complex relationship between the population of merging BHs and the global BH population. We analysed the high-resolution cosmological radiation-hydrodynamics simulation OBELISK, run to redshift z = 3.5, to study the properties of the merging BH population, and its differences with the underlying global BH population in terms of BH and galaxy properties. In post-processing, we calculated dynamical delays between the merger in the simulation at the resolution limit and the actual coalescence well below the resolution scale. We find that merging BHs are hosted in relatively massive galaxies with stellar mass M* ≳ 109 M⊙. Given that galaxy mass is correlated with other BH and galaxy properties, BH mergers tend to also have a higher total BH mass and higher BH accretion rates than the global population of main BHs. These differences generally disappear if the merger population is compared with a BH population sampled with the same galaxy mass distribution as merger hosts. Galaxy mergers can temporarily boost the BH accretion rate and the host’s star formation rate, which can remain active at the BH merger if sub-resolution delays are not taken into account. When dynamical delays are taken into account, the burst has generally faded by the time the BHs merge. BH spins are followed self-consistently in the simulation under the effect of accretion and BH mergers. We find that merging BHs have higher spins than the global population, but similar or somewhat lower spins compared to a mass-matched sample. For our sample, mergers tend to decrease the spin of the final BH remnant.

Synthetic photometry for carbon-rich giants

Context. The properties and the evolution of asymptotic giant branch (AGB) stars are strongly influenced by their mass loss through a stellar wind. This, in turn, is believed to be caused by radiation pressure due to the absorption and scattering of the stellar radiation by the dust grains formed in the atmosphere. The optical properties of dust are often estimated using the small particle limit (SPL) approximation, and it has been used frequently in modelling AGB stellar winds when performing radiation-hydrodynamics (RHD) simulations. Aims. We aim to investigate the effects of replacing the SPL approximation by detailed Mie calculations of the size-dependent opacities for grains of amorphous carbon forming in C-rich AGB star atmospheres. Methods. We performed RHD simulations for a large grid of carbon star atmosphere+wind models with different effective temperatures, luminosities, stellar masses, carbon excesses, and pulsation properties. Also, a posteriori radiative transfer calculations for many radial structures (snapshots) of these models were done, resulting in spectra and filter magnitudes. Results. We find that, when giving up the SPL approximation, the wind models become more strongly variable and more dominated by gusts, although the average mass-loss rates and outflow speeds do not change significantly; the increased radiative pressure on the dust throughout its formation zone does, however, result in smaller grains and lower condensation fractions (and thus higher gas-to-dust ratios). The photometric K magnitudes are generally brighter, but at V the effects of using size-dependent dust opacities are more complex: brighter for low mass-loss rates and dimmer for massive stellar winds. Conclusions. Given the large effects on spectra and photometric properties, it is necessary to use the detailed dust optical data instead of the simple SPL approximation in stellar atmosphere+wind modelling where dust is formed.

Consistent wide-range equation of state of silicon by a unified first-principles method

Using a unified method of the extended first-principles molecular dynamics, we report a wide-range (density $\ensuremath{\rho}=2.329--18.632$ $\mathrm{g}/{\mathrm{cm}}^{3}$, temperature $T=1.00\ifmmode\times\else\texttimes\fi{}{10}^{4}--1.29\ifmmode\times\else\texttimes\fi{}{10}^{8}$ K) equation of state (EOS) and principal Hugoniot of silicon, which agree well with experimental results, and then evaluate the precision of the results of first-principles calculations and a universal database. Moreover, we investigate the effects of finite-temperature exchange-correlation functionals and emphasize their importance and necessity in the warm dense matter regime, although it vanishes in the nondegenerate and fully degenerate limits. Finally, we show the ionic radial distribution function and density of electronic states to study the evolution of warm dense Si towards the classic plasma limit along its principal Hugoniot. The established standard theoretical EOS table of Si provides a benchmark for the development of high-precision first-principles methods and may improve radiation-hydrodynamics simulations of inertial confinement fusion and deepen the understanding of high-energy-density physics.

Emerging planetary nebulae within 3D spiral patterns

ABSTRACT We present the first 3D radiation-hydrodynamic simulations of the formation of planetary nebulae (PNe) emerging from 3D spiral patterns. We use the guacho code to create 3D spiral structures as a consequence of the distortions on the geometry of the intrinsically isotropic wind of an asymptotic giant branch (AGB) star produced by a companion star in a circular orbit. We found that the orbital period of the binary producing the 3D spiral pattern has consequences on the formation and shaping of the PN itself. Stellar systems with longer period create less entwined 3D spirals, producing PNe with rounder inner cavities, and prevent the expansion of jet towards the polar directions. The spiral fitting procedure used in the literature to predict the binary’s orbital period may be misleading in the case of proto-PNe and PNe as spiral patterns are diluted by their own thermal expansion down to the average AGB density profile within a few hundred years and are further disrupted by the action of jets. By adopting a phase of jet ejections between the AGB and post-AGB stages, we are able to recover the morphologies of proto-PNe and PNe that exhibit ring-like structures in their haloes.

Star cluster formation and survival in the first galaxies

ABSTRACT Using radiation-hydrodynamic cosmological simulations, we present a detailed (0.1 pc resolution), physically motivated portrait of a typical-mass dwarf galaxy before the epoch of reionization, resolving the formation, and evolution of star clusters into individual 10 M⊙ star particles. In the rest-frame ultraviolet, the galaxy has an irregular morphology with no bulge or disc, dominated by light emitted from numerous, compact, and gravitationally-bound star clusters. This is especially interesting in light of recent James Webb Space Telescope observations that − aided by the magnifying power of gravitational lenses – have imaged, at parsec-scale resolution, individual young star clusters forming in similar galaxies at z> 6. Because of their low metallicities and high temperatures, star-forming gas clouds in this galaxy have densities ∼100 times higher than typical giant molecular clouds; hence, their expected star formation efficiencies (SFEs) are high enough (around 10 − 70 per cent) to produce a sizeable population of potential globular cluster progenitors, but typically smaller (a few 100 − 2 ×104M⊙, half-mass radii of up to 3 pc) and of lower metallicities (10−3.5– 10−2.5 Z⊙). The initial mass function of the star-forming clouds is log-normal, whereas the bound star cluster mass function is a power-law with a slope that depends mainly on SFE but also on the temporal proximity to a major starburst. We find slopes between −0.5 and −2.5 depending on the assumed sub-grid SFE. Star formation is self-regulated on galactic scales; however, the multimodal metallicity distribution of the star clusters and the fraction of stars locked into surviving bound star clusters depends on SFE.