Star Formation at Very Low Metallicity. IV. Fragmentation does not depend on metallicity fo

a rXiv:079.353v1[astro-ph]21Se p27Star Formation at Very Low Metallicity.IV.Fragmentation Does Not Depend on Metallicity for Cold Initial Conditions.Anne-Katharina Jappsen Canadian Institute for Theoretical Astrophysics,Toronto,ON,Canada jappsen@cita.utoronto.ca Ralf S.Klessen Institut f¨u r Theoretische Astrophysik,Zentrum f¨u r Astronomie der Universit¨a t Heidelberg,Heidelberg,Germany rklessen@ita.uni-heidelberg.de Simon C.O.Glover Astrophysikalisches Institut Potsdam,Potsdam,Germany sglover@aip.de and Mordecai-Mark Mac Low 1American Museum of Natural History,New York,NY,USA mordecai@ ABSTRACTPrimordial star formation appears to result in stars at least an order of mag-nitude more massive than modern star formation.Bromm et al.proposed that the transition from primordial to modern initial mass functions occurs due to the onset of effective metal line cooling at a metallicity Z/Z ⊙=10−3.5.However,their simulations neglected molecular cooling.We here perform simulations using the same initial conditions,but including molecular cooling,using both a simpli-fied chemical network that primarily follows molecular hydrogen formation,and amore complex network that also directly follows carbon monoxide and water.Wefind in both cases that molecular cooling allows roughly equivalent fragmentationto proceed even at zero metallicity for the Bromm et al.initial conditions.Theapparent transition just represents the point where metal line cooling becomesmore important than molecular cooling.In all cases,the fragments are mas-sive enough to be consistent with models of primordial stellar masses,suggestingthat the transition to the modern initial mass function may be determined byother physics such as dust formation.We conclude that such additional coolingmechanisms,combined with the exact initial conditions produced by cosmologi-cal collapse are likely more important than metal line cooling in determining theinitial mass function,and thus that there is unlikely to be a sharp transition inthe initial mass function at Z/Z⊙=10−3.5.Subject headings:stars:formation–stars:mass function–early universe–hydrodynamics–equation of state–methods:numerical1.IntroductionThe formation of stars is a key process in the early universe with far-reaching conse-quences for cosmic reionization and galaxy formation(Loeb&Barkana2001;Bromm&Larson2004;Glover2005).The physical processes that govern stellar birth in a metal-free or metal-poor environment,however,are still very poorly understood.Numerical simulations of thethermal and dynamical evolution of gas in primordial protogalactic halos indicate that themetal-freefirst stars,the so called Population III,are expected to be very massive,withmasses anywhere in the range20–2000M⊙(Abel,Bryan,&Norman2002;Bromm et al.2002;Yoshida et al.2006;O’Shea&Norman2007a).In contrast,the stellar mass spec-trum in the present-day universe is dominated by low-mass stars,peaking at around0.2M⊙(Scalo1998;Kroupa2002;Chabrier2003).This means that at some stage of cosmic evolutionthere must have been a transition from primordial,high-mass star formation to the“nor-mal”mode of star formation that dominates today.The discovery of extremely metal-poorsubgiant stars in the Galactic halo with masses below one solar mass(Christlieb et al.2002;Beers&Christlieb2005)indicates that this transition occurs at abundances considerablysmaller than the solar value.At the extreme end,these stars have iron abundances less than10−5times the solar value,and carbon or oxygen abundances that are still<∼10−3times the solar value.These stars are thus strongly iron deficient,which could be due to unusual abun-dance patterns produced by enrichment from pair-instability supernovae(Heger&Woosley2002)from Population III stars or due to mass transfer from a close binary companion(Ryan et al.2005;Komiya et al.2007).There are hints for an increasing binary fraction with decreasing metallicity for these stars(Lucatello et al.2005).In a seminal paper,Bromm et al.(2001)proposed that the transition from high to low mass star formation should occur at a critical metallicity Z crit≈10−3.5Z⊙as a result of atomicfine-structure line cooling from alpha elements,such as carbon and oxygen,becoming important.Their study is based on smoothed particle hydrodynamics simulations of the collapse of cold gas in a top-hat potential at densities below n=106cm−3.It assumes that cooling from molecular hydrogen is negligible in the density and metallicity range considered. The value of Z crit can also be estimated analytically by calculating the metallicity required to produce a cooling rate equal to the rate of adiabatic compression heating for given halo properties(Bromm&Loeb2003;Santoro&Shull2006;Frebel,Johnson&Bromm2007). However,Bromm et al.(2001)noted that neglecting H2cooling could significantly influence the resulting fragmentation pattern.Along the same lines,Omukai et al.(2005)argued, based on the results of detailed one-zone calculations,that molecular cooling would indeed dominate the cooling over many orders of magnitude in density.They argued that dust-induced cooling at densities above n≈1013cm−3may be a more viable mechanism to explain fragmentation in very low metallicity gas,a conjecture that has recently been confirmed by Clark et al.(2007).The effects of molecular cooling at densities up to n≈500cm−3have been discussed by Jappsen et al.(2007a,hereafter Paper II)in three-dimensional collapse simulations of warm ionized gas in protogalactic halos for a wide range of environmental conditions.This study used a time-dependent chemical network running alongside the hydrodynamic evolu-tion as described in Glover&Jappsen(2007,hereafter Paper I).The physical motivation was to investigate whether small protogalaxies that formed within the relic H ii regions left by neighboring Population III stars could form subsequent generations of stars themselves, or whether the elevated temperatures and fractional ionizations found in these regions sup-pressed star formation until larger protogalaxies formed.We found in Paper II that molecular hydrogen dominates the cooling of the gas for abundances up to at least10−2Z⊙.In addition, there was no evidence for fragmentation at densities below500cm−3.To understand whether this apparent discrepancy with the Bromm et al.(2001)re-sults comes from including molecular hydrogen cooling or from adopting vastly different initial conditions—here ionized halo gas in a dark matter halo with a NFW density profile (Navarro et al.1997),there cold atomic gas in a top-hat halo—we apply our time-dependent chemical network with molecular hydrogen cooling to Bromm et al.(2001)-type initial con-ditions.Our study is structured as follows.In the next section,§2,we give the details on the numerical set-up and the adopted initial conditions.In§3we report our results,and wesummarize in§4.2.Numerical Set-UpTo adequately describe star formation in the early universe,it is necessary to follow the cooling and possible fragmentation of the gas in the central regions of dark-matter halos over many orders of magnitude in density.Due to the nonlinear nature of theflow,it is not known in advance where and when gravitational collapse will occur,and consequently stars will form.To compute the time evolution of the system we therefore resort to smoothed particle hydrodynamics(SPH),which is a Lagrangian scheme to solve the equations of hydrodynamics (Benz1990;Monaghan1992,2005)coupled with the simple time-dependent chemical network discussed in Paper I,or with the more extended network discussed in Glover(2007,hereafter, Paper III).Thefluid is represented by an ensemble of particles,andflow quantities are obtained by averaging over an appropriate subset of SPH particles.The method is able to resolve high density contrasts as particles are free to move,and so the particle concentration increases naturally in high-density regions.We use a modified version of the parallel SPH code GADGET(Springel2005).Once the central parts of a collapsing fragment exceed a density n=105cm−3we introduce a“sink particle”(Bate et al.1995;Jappsen et al.2005), which is able to accrete gas from its surroundings while keeping track of the mass and linear and angular momentum of the infalling material.Replacing collapsing high-density cores by accreting sink particles allows us to follow the dynamic evolution of the system over many local free-fall timescales.2.1.Chemistry and CoolingWe present results from simulations performed using two different chemistry and cooling models.Thefirst model used in our simulations is the detailed model of low-metallicity gas chemistry outlined in Paper III.This model includes cooling both from atomicfine structures lines and also from the molecules H2,HD,CO,OH and H2O.It was designed to accurately follow the formation and destruction of these molecular species over a wide range of temperatures and densities.The other model used is the simpler one described in Paper I.This model follows the chemistry of just seven key coolants:H,H2,C+,C,O,Si,and Si+. It was designed to be used in conditions in which the majority of the gas phase metals are in atomic or ionized form,and so does not include cooling from species such as CO,OH or water,or any chemistry involving these species.In Paper I,we showed that the neglect of molecular species including metals can be justified in simulations of gravitationally collapsing protogalactic gas provided that n< 500/t char cm−3,where where t char is the characteristic physical timescale of interest,in Myr. At higher gas densities,molecule formation from metals may become more significant,al-though its ultimate importance depends on a number of factors,such as the temperature and metallicity of the gas,and the present(or absence)of an ultraviolet radiationfield.The second model is similar in design to that used in Bromm et al.(2001);the major differences between their model and ours are that we do not make any a priori assumptions regarding chemical equilibrium,and,more significantly,the fact that we include the formation of H2 and cooling from H2line emission.The majority of our modelling was done using the more complex network.However, for the purposes of comparison,we also ran a few simulations using our simpler model. Comparison between the results of these two sets of simulations helps to highlight how accurate our chemical modelling needs to be in order to properly model the thermodynamics and dynamical evolution of the gas at the densities treated in this paper.Although our chemical models are capable of accounting for the effects of dust,cosmic rays,and photodissociation by an external radiationfield,we omit these complications in the present study.A full list of the chemical reactions included in the versions of the simple network (reactions1–47)and complex network used in this paper is given in Table2,along with the corresponding rate coefficients and references to the source of the data.Given these rate coefficients,calculation of the appropriate chemical rates is for the most part straightforward. One minor exception is the case of H2collisional dissociation,which has a rate coefficient that depends on both temperature and density.For the collisional dissociation of H2by H or H2(reactions9&10),we calculate the rate coefficients using a function of the formlog k i= n/n cr1+n/n cr log k i,l,(1)where k i,l and k i,h are the low density and high density limits for the collisional dissociation rate due to collisions with species i(which can be found in Table2),and where the critical density,n cr,is given by1n cr,H +x H2(reaction75),we use a slightly different expression,taken from Dove et al.(1987):log k He=log k He,h−(log k He,h−log k He,l)|∇·v|,(4) where n(M)is the local number density of species M.The Neufeld&Kaufman(1993) and Neufeld,Lepp&Melnick(1995)treatments assume that only collisions with H2are important in determining the CO or H2O rotational cooling rates,as is appropriate in a fully molecular gas.However,in metal-poor gas,the H2abundance will often be much smaller than the abundances of atomic hydrogen or atomic helium.We therefore use an effective number density n eff,given by(Pavlovski et al.2002;Smith&Rosen2003)n eff=n H2+√2.2.Initial ConditionsWe model the initial dark matter distribution,following Bromm et al.(2001),as anisolated and roughly spherical overdensity described by the top-hat approximation with alow level of small-scalefluctuation added.We consider a halo virializing at z vir=30,whichcorresponds to a roughly3σpeak with a total mass of2×106M⊙,corresponding to2×105M⊙in baryons.The overdense region has a proper radius of150pc.For more details on theset-up of the dark matter and the gas distribution see Bromm et al.(2001).We carry outfive numerical experiments with different metallicities ranging from Z=0to Z=0.1Z⊙.Aside from the difference in metallicity,these calculations have identicalinitial conditions.The simulations were initialized at a redshift z=100,with an initialgas temperature T gas,i=200K.We performed a number of low resolution simulations,withN DM=13400dark matter particles and N gas=65536gas particles.In these simulations,our numerical resolution was the same as in Bromm et al.(2001),i.e.M res=150M⊙.Wealso performed three high resolution simulations with N DM=107177and N gas=8×105;inthese simulations,the mass resolution was M res=12.5M⊙.In all of our simulations,boththe dark matter and the gas particles were endowed with the same Hubble expansion andwere set in rigid rotation with a spin parameter of0.05,just as in Bromm et al.(2001).Wesummarize the properties of all of the runs in Table1.3.ResultsIn each of our calculations the halo contracts rapidly and the gas density increases byseveral orders of magnitude.A disk-like structure builds up in the very center of each halowith a density of n≈103cm−3and a diameter of10to20pc,as illustrated in Figure1.Thedisk mass at the time illustrated is of order M disk∼2–3×104M⊙,and so it is well-resolved even in the low resolution simulations.The disk is supported by a combination of thermalpressure and rotation and its growth is fed by material falling in alongfilaments and sheets.This complex disk provides the background for the small-scale initial densityfluctuations togrow,some of which become gravitationally unstable and collapse to form stars.In contrastto the results of Bromm et al.(2001),wefind that fragmentation of the disk occurs in allof our simulations,and that the degree of fragmentation is independent of the metallicity ofthe gas.This result can be understood as follows.Regardless of the metal content,H2line coolingmaintains the gas temperature at densities above103cm−3close to the CMB temperature(T CMB=105K at the redshift z=37.5considered in Figure2);even in the primordial case,the gas temperature is within a factor of two of this value.The resulting equation of stateis therefore approximately isothermal(at high densities)or softer than isothermal(at low densities).The thermal behavior of the gas is depicted in the left column of Figure2.Massive,isothermal disks are known to be prone to gravitational fragmentation(Bodenheimer 1995;Saigo et al.2000)and so it is unsurprising that our disks fragment.The physical conditions where the equation of state changes from softer than isothermalto isothermal or stiffer,i.e.from a regime where the gas cools with increasing density to one where the temperature remains constant or rises again,imprint a characteristic mass inthe fragmentation process(Jappsen et al.2005;Larson2005).Because this fragmentation occurs at approximately the same density and temperature in each simulation,the mass distribution of the fragments does not vary greatly from simulation to simulation.In eachof our low resolution simulations,between15and20fragments are formed,with massesin the range102M⊙≤M≤103M⊙.The resulting mass spectra are shown as the solid lines in the right-hand column of Figure2.For comparison,the thermal Jeans mass for gas with temperature T gas=T CMB is approximately2.3×103M⊙at n=103cm−3,the density above which H2line cooling maintains the gas as approximately isothermal,and 230M⊙at n=105cm−3,the density above which we form sinks.To quantify whetherthe mass spectra differ significantly,we can use the two-sided Kolmogorov-Smirnov test to determine for each of our low resolution,metal-enriched simulations the probability thatthe mass spectrum produced is drawn from the same underlying distribution as in the Z=0 simulation.Wefind probabilities of0.16,0.05,0.49and0.63for runs Z-4,Z-3,Z-2and Z-1 respectively,indicating that the differences between the fragment mass spectra are in generalnot statistically significant.The one exception is the Z=10−3Z⊙case,and even then the significance is marginal.In our high resolution simulations,wefind a greater degree of fragmentation and a broader fragment mass distribution.However,just as in our low resolution simulations,the mass distribution is dominated by fragments with masses M>100M⊙.Wefind veryfew objects with masses close to our resolution limit of12.5M⊙,and there is no indication from our current results that we are missing a substantial population of objects with masses below this limit.As before,we can compare the fragment mass distributions resulting fromour three high-resolution simulations with the K-S test.Wefind that the probabilities thatthe fragment mass distributions in runs Z-3-BIG and Z-1-BIG are drawn from the same underlying distribution as in run Z0-BIG are0.78and0.34respectively,indicating that again the differences are not statistically significant.One quantity does change significantly as the metallicity of the gas increases,the peak temperature reached during the evolution.In the zero-metallicity case,the peak temperatureT peak reached due to compressional heating during the initial contraction of the gas is∼103K.At higher metallicities,however,T peak gradually declines,until it hardly exceeds the CMB temperature in the run with Z=0.1Z⊙.This decrease in T peak reflects the growing importance of metal-line cooling in cold gas as the metallicity increases.From Figure2it is apparent that the metals begin to significantly affect T peak once the metallicity exceeds Z=10−3Z⊙.This is a lower threshold than found in Jappsen et al.(2007a),but the difference is easily understood as a consequence of the fact that in these simulations we start with cold gas,and that the H2cooling rate of the cold,low-density gas in these simulations is much lower than H2cooling rate in the warm,low-density gas studied in Jappsen et al.(2007a).Our threshold of Z∼10−3Z⊙is in reasonable agreement with the Bromm et al.(2001)threshold of Z∼10−3.5Z⊙;however,the key difference here is that at lower metallicities,the H2alone is sufficient to cool the gas,and the outcome of the fragmentation process does not appear to depend on whether the cooling comes from the H2 or from the metals.Finally,as noted above,we performed two runs using the simplified chemical model of Paper I(runs Z-3S and Z-1S)in order to investigate the importance of treating the high density metal chemistry accurately.The results of these runs are illustrated in Figure3, along with the results of the corresponding runs utilizing the full chemical network(runs Z-3and Z-1).Comparison of the phase diagrams of these runs shows very little difference in the outcome.This result is easy to understand if,as argued in Paper I,CO and H2O formation and cooling are unimportant at low densities.At high densities,we might naively expect to see a greater difference between the simulations,but in practice,all of the gas at those densities is located at or very close to the CMB temperature,and the identity of the coolant or coolants responsible for keeping it there does not appear to be of great importance.That said,our choice of chemical model does have an impact on the fragment mass distribution,as illustrated in the right-hand column of Figure3.In runs Z-3S and Z-1S,we obtain fragments that are systematically smaller than in the corresponding runs Z-3and Z-1.Further investigation suggests that this difference is a result of the fact that the fragments form at a slightly later time in the runs performed using the simplified chemical model than in the runs performed using the full model,possibly indicating that the timing of the fragmentation is sensitive to the cooling time of the gas.4.Summary and ConclusionThe current study shows that gas in simulations of the type considered here,with low initial temperature,moderate initial rotation,and a top-hat dark-matter overdensity,willreadily fragment into multiple objects,regardless of metallicity,provided that enough H2 is present to cool the gas.Rotation leads to the build-up of massive disk-like structures in these simulations which allow smaller-scalefluctuations to grow and become gravitationally unstable to form protostars.The resulting mass spectrum of fragments peaks at a few hundred solar masses,roughly corresponding to the thermal Jeans mass in the disk-like structure.Our major conclusion is that the metallicity threshold at Z=10−3.5Z⊙reported by Bromm et al.(2001)does not represent a critical metallicity above which gas fragments and below which it does not.Rather,this threshold simply marks the point at which metal-line cooling becomes more important than H2cooling at the gas densities and temperatures relevant for fragmentation in the disk.This metallicity threshold only represents a criti-cal metallicity for fragmentation in these simulations if H2formation is strongly suppressed or not considered(e.g.Bromm et al.2001).This scenario may be relevant if the extra-galactic UV background is strong(Haiman et al.2000),although recent work suggests that even in this case,suppression of H2formation is less important than previously supposed (Wise&Abel2007;O’Shea&Norman2007b).We alsofind no indication that the mass spectrum of fragments formed in these simu-lations depends on the metallicity.In our low-resolution simulations,many of the fragments are only marginally resolved,but we obtain very similar results from our high-resolution simulations,in which the fragments are well-resolved,suggesting that this lack of influence is real.We further note that simulations performed by Smith&Sigurdsson(2007)that follow the evolution of cold,metal-enriched gas from cosmological initial conditions until collapse dofind tentative evidence for a metallicity threshold at around Z=10−3Z⊙that must be surmounted before fragmentation will occur.On the other hand,the simulations presented in Paper V that follow the collapse of hot,initially ionized gas into isolated NFW halosfind no evidence for any fragmentation at or below this metallicity.We conclude that the question of whether there is a critical metallicity below which the formation of low-mass stars is impossible remains unresolved.Our current results show no evidence for such a critical metallicity:fragmentation occurs even in our Z=0simulation, due only to efficient H2cooling,and the mass spectrum of the fragments formed does not appear to differ significantly from those obtained from our higher metallicity simulations. However,the degree of fragmentation occurring in these simulations appears to be a con-sequence of the initial conditions chosen:simulations utilizing substantially different initial conditionsfind very different results(Jappsen et al.2007b;Smith&Sigurdsson2007).To make further progress in understanding the role(if any)that metal-line cooling plays inpromoting fragmentation,we need to develop a much better understanding of how metals are initially dispersed into the high-redshift interstellar and intergalactic medium,in order to be able to select the most appropriate initial conditions for our simulations.Our results call attention to an alternative scenario for the transition to the present-day low-mass IMF.Omukai et al.(2005)stress the importance of dust-induced cooling at high densities which may become important at metallicities as low as Z≈10−6–10−5Z⊙depending on the adopted dust model(Schneider et al.2006).Recent numerical simulations by Clark et al.(2007)support this point of view and predict the existence of low-mass stars and even brown dwarfs with metal abundances of∼10−5the solar value(see also Tsuribe&Omukai2006).We acknowledge useful discussions with Tom Abel,Volker Bromm,Paul C.Clark, Kazuyuki Omukai,and Naoki Yoshida.This research was supported in part by the US National Science Foundation under Grants No.PHY05-51164and AST03-07854,and by the Emmy Noether Program of the German Science Foundation under Grant No.KL1358/1. 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