Globular Cluster Formation at High Density: A model for Elemental Enrichment with Fast Recycling of Massive-Star Debris
aa r X i v : . [ a s t r o - ph . GA ] J a n Globular Cluster Formation at High Density: A model forElemental Enrichment with Fast Recycling of Massive-Star Debris
Bruce G. Elmegreen ABSTRACT
The self-enrichment of massive star clusters by p-processed elements is shownto increase significantly with increasing gas density as a result of enhanced starformation rates and stellar scatterings compared to the lifetime of a massive star.Considering the type of cloud core where a globular cluster might have formed,we follow the evolution and enrichment of the gas and the time dependenceof stellar mass. A key assumption is that interactions between massive starsare important at high density, including interactions between massive stars andmassive star binaries that can shred stellar envelopes. Massive-star interactionsshould also scatter low-mass stars out of the cluster. Reasonable agreementwith the observations is obtained for a cloud core mass of ∼ × M ⊙ and adensity of ∼ × cm − . The results depend primarily on a few dimensionlessparameters, including, most importantly, the ratio of the gas consumption timeto the lifetime of a massive star, which has to be low, ∼ Subject headings:
Galaxies: star clusters — Galaxies: star formation — globularclusters: general
1. Introduction
Most globular clusters (GCs) in the Milky Way have two populations of stars in ap-proximately equal proportion with a first generation (G1) relatively abundant in Oxygen IBM Research Division, T.J. Watson Research Center, 1101 Kitchawan Road, Yorktown Heights, NY10598; [email protected] Ne to Na along with the simultaneous destruc-tion of O at a temperature of 2 × K (Denissenkov & Denissenkova 1990; Decressin et al.2007a). Other elemental anti-correlations include Mg with Al (e.g., Carretta et al. 2014),explained by Langer et al. (1993) and others as a result of high temperature (
T > × K) proton-capture plus beta-decay that transforms Mg into Mg, Mg and Al. Also insome GCs, stellar Nitrogen anti-correlates with Carbon and Oxygen (Dickens et al. 1991),with a total C+N+O abundance that is about constant, suggesting a CNO cycle. Reviews ofthese abundance anomalies are in Gratton et al. (2004), Charbonnel (2005), Gratton et al.(2012), Renzini et al. (2015), and Bastian (2015).Light element anomalies from the CNO cycle and the NeNa- and MgAl-chains are pe-culiar to the GCs and are not in field or halo stars (Gratton et al. 2000; Charbonnel 2005;Prantzos & Charbonnel 2006). Because they involve high-temperature reactions and thestars in which they are observed today are too low in mass to have had such p-process re-actions themselves (Gratton et al. 2001), the Na-excess and Al-excess stars in current GCshad to form in gas that was pre-enriched with these elements (Cottrell & Da Costa 1981),most likely from the previous generation of stars as mentioned above. The debris from thisprevious generation may also have mixed with some left-over initial gas to explain an anticor-relation between Li, which is destroyed in stars, and Na, which is produced (Pasquini et al.2005; Bonifacio et al. 2007; Decressin et al. 2007a), and to explain the difference between thehigh enrichment in the nuclear burning regions of massive stars and the observed enrichmentin low mass stars (Decressin et al. 2007b, table 4).The first generation has been proposed to include normal massive stars (Cottrell & Da Costa1981), such as rapidly spinning massive stars (Prantzos & Charbonnel 2006), which, be-cause of their rotation, bring p-processed material from the H-burning zone into the en-velope and then shed it along the equator via centrifugal force and radiation pressure(Prantzos & Charbonnel 2006; Decressin et al. 2007b). Binary massive stars with Rochelobe overflow are also an option (de Mink et al. 2009). AGB stars (D’Ercole et al. 2008)shed processed gas at low speed too and may be a source of the anomalies, although thenear-constant C+N+O does not look like a byproduct of AGB stars, which make carbon dur-ing Helium burning (Charbonnel 2005); NGC 1851 may be an exception though (Yong et al.2014; Simpson et al. 2017). Renzini et al. (2015) discuss ways in which the AGB optionmight still be viable. Other models for the bimodality include proto-stellar disk accretion(Bastian et al. 2013a), supermassive stars (Denissenkov & Hartwick 2014; Denissenkov et al.2015), GC-merging in dwarf galaxy hosts (Bekki & Tsujimoto 2016), and AGB wind re-collection in the pressurized cavity around the GC (D’Ercole et al. 2016). 3 –A problem with these models is that the stellar debris is only a small fraction of thetotal stellar mass in a normal IMF. This implies that because the current G1 and G2 massesare comparable to each other, the mass in the original G1 population had to be at leastthe inverse of this fraction times the current G1 mass. For the model with rapidly rotatingmassive stars and a normal IMF, the original cluster had to be ∼
20 times the current G1mass (Decressin et al. 2007b), and for the AGB model, the original cluster had to be > M ⊙ and ages between 20 and 30 Myr are often clear of gas. This age isold enough for some supernovae to have occurred. Without gas, star formation stops and aprolonged epoch of secondary star formation that feeds on G1 debris does not occur.Secondary or delayed star formation as in the AGB model is also not observed in to-day’s massive clusters at the predicted age range of 10-1000 Myr (Bastian et al. 2013b;Cabrera-Ziri et al. 2014), although it was reported by Li et al. (2016) whose result was thenquestioned by Cabrera-Ziri et al. (2016). Neither do massive 100 Myr old clusters haveobvious gas from accumulated winds or secondary accretion (Bastian & Strader 2014).Here we investigate further the model by Prantzos & Charbonnel (2006) and others,where p-process contamination from massive stars is quickly injected into a star-formingcloud core and incorporated into other forming stars. We consider conditions appropriatefor the early Universe where the interstellar pressure was much higher than it is today. Thispressure follows from the observation that star formation was ∼
10 times more active perunit area than it is now (Genzel et al. 2010) as a result of a factor of ∼
10 higher gas columndensity (Tacconi et al. 2010; Daddi et al. 2010). Because interstellar pressure scales withthe square of the column density ( P ∼ G Σ ), the pressure was ∼ × larger when GCs 4 –formed than it is in typical star-forming regions today. Consequently, the core of the GC-forming cloud was likely to have a higher density and therefore a larger number of free-falltimes, such as 10 or more, before the first supernovae occurred. Star formation could haveoccurred so quickly and the energy dissipation rate at high density could have been so highthat feedback from massive stars had little effect on cloud dispersal before the supernovaera (e.g., Wunsch et al. 2015). Also at high density, massive stars experience a significantdrag force from gas and low-mass stars (Ostriker 1999), causing them to spiral into aneven more compact configuration. High stellar densities lead to the dispersal of extrusiondisks filled with stellar envelope material (Prantzos & Charbonnel 2006), and also to closeencounters between massive stars or between tight massive binaries and massive stars, whichcan shred the stellar envelopes (Gaburov et al. 2010). High stellar densities may also makesupermassive stars by coalescence (Ebisuzaki et al. 2001; Bally & Zinnecker 2005), and thesestars can produce the highest-temperature p-process elements (Denissenkov et al. 2015).A related point is that at high density, a significant fraction of early-forming low-massstars should have been ejected from the cloud core by interactions with massive binariesand massive multi-body collisional systems. This could occur long before “infant mortality,”when the final clearing of gas signals the end of the star formation process. Rapid and contin-uous stellar ejections by massive star interactions are well observed in numerical simulationsof cluster formation (Reipurth & Clarke 2001; Bate & Bonnell 2005; Fujii & Portegies Zwart2013). Because massive three-body collisions can also dump nearly a supernova’s worth ofkinetic energy into a region (Gaburov et al. 2010; Umbreit et al. 2008), the gas in the coreshould be continuously agitated. The corresponding changes in the gravitational potentialenergy of the gas should eject even more low-mass stars, in analogy to the proposed ejection ofstars and dark matter from the cores of young star-bursting dwarf galaxies (Governato et al.2012; El-Badry et al. 2016).Dense cores are likely to continue accreting from the cloud envelope for many coredynamical times. If the core plus envelope gas mixes with stellar debris and forms new stars,then a succession of stellar populations will occur with ever-increasing levels of p-processelements. By the time the first massive stars begin to supernova and clear away the gas,some 3-7 Myr after star formation begins (Heger et al. 2003), there should be a wide rangeof p-process elements in the stars that have formed. The model presented below shows thatthis range can reproduce the observations for certain values of dimensionless parameters.A key observation is that elemental enrichment in GCs seems to be discrete (Marino et al.2011; Carretta et al. 2012, 2014; Renzini et al. 2015). Discreteness requires burst-like con-tamination, and in the stellar interaction model here, that means intermittent events ofcatastrophic massive-star interactions. The interactions occur where the density is high- 5 –est, so either the stellar density in the cloud core varies episodically with a burst of stellarcollisions following each high-density phase, or there are several mass-segregated cores in aproto-GC (McMillan et al. 2007; Fujii & Portegies Zwart 2013) and each has its own burst ofintense interactions. Both situations are likely and could contribute to discrete populationsof stars. The temporal density variations would presumably follow from the time-changinggravitational potential in the cluster core, as stellar envelope mass is disbursed along withcloud mass through collisions, and then recollected in the core after a dynamical time becauseof self-gravity and background pressure.The following sections examine this model in more detail. Section 2.1 outlines the basicmodel of GC formation at high density, section 2.2 presents the equations that govern thismodel, giving some analytical solutions in section 2.3, and section 3 shows the results. Aconclusion that highlights the main assumptions of the model and their implications forhigh-density cluster formation is in Section 4.
2. Star Formation in Stellar Debris2.1. Basic Model
The basic scale of GC formation considered here involves a molecular cloud core of mass ∼ × M ⊙ in a spherical region of radius ∼ ∼ M ⊙ or more, as observed for a massive denseregion in the Antenna galaxy (Herrera et al. 2012; Johnson et al. 2015). The core moleculardensity for the above numbers is 2 . × cm − and the free fall time (= (3 π/ Gρ ) . ) is0.03 Myr. The ratio of the core mass to the free fall time, 133 M ⊙ yr − , is a measure ofthe core accretion rate during core formation. If this core collapsed from the interstellar gasat the typical rate of σ /G for interstellar velocity dispersion σ ISM , then σ ISM = 82 kms − , which is not unusual for high-redshift disk galaxies (e.g., F¨orster Schreiber, et al. 2009).The accretion rate of low-density peripheral gas on to the core should be much less than theinitial core formation rate.The fiducial cloud core mass of ∼ × M ⊙ was chosen to produce a final GCmass of around 2 × M ⊙ , which is at the peak of the GC mass distribution function(Harris & Racine 1979). The first factor of ∼
10 in mass reduction follows from our modelincluding multiple generations of star formation in the core and mass loss from stellar ejec-tion, as required by the observed spread in p-processed elemental abundance. Another 6 –factor of ∼ . × M ⊙ pc − , is comparable tothe maximum for stellar systems (Hopkins et al. 2010; Walker et al. 2016). The core velocitydispersion is ∼
76 km s − , which is a factor of 2 higher than for massive clusters today, butnot unreasonable for a young galaxy where the gas turbulence speed is this high. Over time,the cluster should expand and the dispersion decrease (e.g. Gieles & Renaud 2016). Theoriginal cluster dispersion is high enough to make feedback-driven gas loss difficult beforethe supernova era (Matzner & Jumper 2015; Krause et al. 2016). We consider that becauseof this difficulty, the efficiency of star formation per unit free-fall time might be relativelyhigh, ∼ ∼
12 km s − internalvelocity dispersion and a much lower rotation speed (Walker et al. 2006), while WLM hasa 36 km s − rotation speed (Leaman et al. 2012). Stars that are ejected at a factor of 1.5to 2 times the escape speed of the GC could leave the galaxy. This possibility leads to theprediction that galaxies with slow internal motions should have a systematic depletion inhalo stars from the G1 population that escaped their GCs . A similar conclusion was reachedby Khalaj & Baumgardt (2016) based on stellar loss from GCs during gas expulsion.The IMF for all star formation is assumed to be fully populated and given by the log-normal distribution in Paresce & De Marchi (2000) for stellar mass 0 . M ⊙ < M < . M ⊙ ,with mass at the peak M C = 0 . M ⊙ and dispersion σ = 0 . M ⊙ , and by a powerlaw with the Salpeter slope − .
35 above 0 . M ⊙ (see also Prantzos & Charbonnel 2006).The upper limit to the stellar mass will be varied from M upper = 100 M ⊙ to 300 M ⊙ ,with the high value considered because of stellar coalescence. Note that a 320 M ⊙ starhas been suggested for the dense massive cluster R136 in the LMC (Crowther et al. 2010;Crowther et a. 2016), and stars more massive than 100 M ⊙ were found in the dense cluster inNGC 5253 (Smith et al. 2016). Stars with masses larger than 20 M ⊙ are assumed to make p- 7 –process elements (Decressin et al. 2007a) and mix them into their stellar envelopes, which aredefined to be all of the stellar mass outside of the He core, as given by Prantzos & Charbonnel(2006). For M upper = 100 M ⊙ , the fraction of the total stellar mass in the form of theseenvelopes is f env = 7 . M > M ⊙ (12.1%) and the average fraction of this stellar mass in the form of envelopes(65.1%). For M upper = 300 M ⊙ , f env = 9 .
3% (16.4% of the IMF is in
M > M ⊙ stars and56.8% of that mass is in envelopes). Also for these two upper masses, the fraction of thetotal stellar mass in long-lived, low-mass stars ( M < . M ⊙ ), is f LM = 31 .
2% and 29.7%,respectively. The formation of p-process elements and the delivery of these elements into thestellar envelopes and equatorial disks is assumed to proceed at a steady rate with an averagetimescale t evol = 3 Myr, the lifetime of a high-mass star.Regarding the contamination by p-processed elements, we note that the [O/Na] ratioin GCs varies from the large value of ∼ . ∼ − . − . ∼ ∼ . ∼ Consistent with these assumptions, we consider an initial cloud core of mass M gas ( t = 0)in which stars begin to form, and a continuous accretion of new cloud gas onto this core ata rate R acc . Stars are assumed to form in the core with a constant consumption time, t consume (equal to t ff divided by the efficiency per free fall time). The effect of varyingthe consumption time will be discussed below. The formation rates of low ( < . M ⊙ ),intermediate (0 . M ⊙ − M ⊙ ) and high ( > M ⊙ ) mass stars are given by the starformation rate in the core multiplied by the fractions of the IMF in these three mass intervals: dM star , LM /dt = f LM M gas ( t ) /t consume (1) dM star , IM /dt = f IM M gas ( t ) /t consume (2) dM star , HM /dt = f HM M gas ( t ) /t consume , (3)where f LM = 0 . f IM = 0 . f HM = 0 .
121 for an IMF with a most massive star of100 M ⊙ , and where f LM = 0 . f IM = 0 . f HM = 0 .
164 for an IMF with a mostmassive star of 300 M ⊙ (Sect. 2.1). The total mass formed in these stars is M star , LM ( t ) = Z t ˙ M star , LM dt, ... (4)and so on for the other mass ranges. Here we use the notation ˙ M = dM/dt . The total massfor all stars is M star = M star , LM + M star , IM + M star , HM . 9 –For the gas, we track the primordial and enriched gas masses separately. The primordialgas is a combination of what was originally in the cloud core plus what gets accreted afterstar formation begins, and it also includes the part of the stellar envelope debris that was notconverted into p-processed elements. We assume that massive stars make p-process elementscontinuously and that mixing from rapid rotation puts these elements into the stellar envelopscontinuously. Thus we conceptually divide the envelope mass into an unprocessed fraction atthe original primordial abundance, and a completely processed, or enriched, fraction at theabundance of fully processed material. The total envelope is a combination of these, givinga partially-processed elemental abundance that comes from the dilution of fully processedmaterial by the mass that is still in an unprocessed form.With these assumptions, the rate of change of the primordial (1st generation) gas massin the cloud core, M gas , , is the increase from stellar debris and envelope accretion minuswhat goes into stars,˙ M gas , ( t ) = Z t f debr ˙ M star , HM ( t ′ ) t evol ! (1 − f p ( t ′ )) (cid:20) − t − t ′ t evol (cid:21) dt ′ (5)+ R acc ( t ) − (1 − f p ( t )) ˙ M star ( t ) . The first term in the integral is from ejection of stellar debris (equatorial disks, collisionaldebris, Roche-lobe overflow). The term f debr is the average fraction of the mass of a high-mass star that is in the envelope and can be ejected. It equals 0 .
651 for M upper = 100 M ⊙ and 0 .
568 for M upper = 300 M ⊙ (Prantzos & Charbonnel 2006). Division by t evol indicatesthat we assume this debris is ejected steadily over the evolution time of the massive star,nominally assumed to be t evol = 3 Myr. The quantity f p ( t ′ ) is the processed fraction inthe gas at time t ′ , and therefore also the processed fraction in stars that form at time t ′ ,assuming rapid mixing. Thus, 1 − f p ( t ′ ) is the unprocessed fraction of the mass of the star thatpreviously formed at t ′ . The last term, 1 − ([ t − t ′ ] /t evol ), tracks the remaining unprocessedfraction in the stellar envelope as the concentration of processed material increases linearlywith time. This linear increase assumes the p-process elements from the stellar core mixinto the stellar envelope at a steady rate. In addition to the integral that represents debrisoutput, the unprocessed gas mass also increases by accretion at the rate R acc . From theseadditions we subtract the unprocessed gas mass in the cloud core that goes into stars.The rate of change of processed gas mass in the cloud core, M gas , (2nd generation), isfrom the addition of stellar debris minus what goes into stars:˙ M gas , ( t ) = Z t f debr ˙ M star , HM ( t ′ ) t evol ! (1 − f p ( t ′ )) (cid:18) t − t ′ t evol (cid:19) dt ′ (6) 10 –+ Z t f debr ˙ M star , HM ( t ′ ) t evol ! f p ( t ′ ) dt ′ − f p ( t ) ˙ M star ( t ) . The first integral represents the originally unprocessed mass fraction in the star when itformed, (1 − f p ( t ′ )), that came out as debris and became more and more contaminated withtime (as measured by [ t − t ′ ] /t evol ), and the second integral represents the return of originallyprocessed mass (at fraction f p ) into the cloud core. Note that the sum of the processed andunprocessed gas mass rates from equations (5) and (6) equals R t [ f debr M HM ( t ′ ) /t evol ] dt ′ + R acc − dM star /dt , which is the total debris rate plus the accretion rate minus the star formationrate.Now we determine the masses of primordial and enriched gas in the star-forming cloudcore by integration, M gas , ( t ) = Z t ˙ M gas , ( t ) dt (7) M gas , ( t ) = Z t ˙ M gas , ( t ) dt, (8)we combine these to get the total gas mass, M gas = M gas , + M gas , , (9)and we determine the mass fractions of enriched gas used above. f p ( t ) = M gas , M gas . (10)The low mass stars do not contribute to the above equations except as a long-termsink for stellar mass. However, these stars are important for the observation of GCs today,and this is where stellar ejection and evaporation come in. We assume here that only lowand intermediate mass stars leave the cluster by these processes, and we trace only the lowmass stellar loss because the intermediate mass stars will have disappeared by now anyway,except as residual collapsed remnants. The high mass stars are assumed to segregate tothe center of the GC where essentially all of their p-processes elements are available for gascontamination, as written in the above equations. Thus we need to model the escape of lowmass stars.The discussion in Section 1 suggests that multi-star interactions and gas motions in theGC core occasionally accelerate low mass stars up to escape speed or beyond. Thus the rateof stellar ejection depends on the dynamical rate in the core, and this is directly proportionalto both the free-fall rate and the consumption rate in the basic model. This implies that 11 –there is an additional rate of change of the mass of low-mass stars, so equation (1) shouldbe revised to contain an additional term, dM star , LM /dt = f LM M gas ( t ) /t consume − f eject M star , LM /t consume (11)for f eject a number less than unity. Recall that t consume ∼ t ff , so the ejection time for f eject = 0 . The time dependence of the star formation rate and cloud mass have analytical solutionsthat are conveniently written using the above equations in dimensionless form. We normalizethe gas and stellar masses to the initial cloud core mass, M gas ( t = 0), and the time to theconsumption time, t consume , assumed to be constant. Normalized quantities are denoted witha tilde. Then the star formation rate is d ˜ M star d ˜ t = ˜ M gas . (12)The gas mass changes from the addition of stellar debris from high mass stars and cloudenvelope accretion and the subtraction of new stars: d ˜ M gas d ˜ t = f debr f HM ˜ t evol Z ˜ t ˜ M star ( t ′ ) d ˜ t dt ′ + ˜ R acc − d ˜ M star d ˜ t . (13)Here, ˜ R acc = R acc t consume /M gas ( t = 0).Equation (13) can be differentiated with respect to time and equation (12) can besubstituted to give a second order linear differential equation, d ˜ M gas d ˜ t + d ˜ M gas d ˜ t − γ ˜ M gas = 0 (14)where γ = f debr f HM t consume t evol (15)is a dimensionless measure of the relative rate of return of processed gas. For the numbersin the basic model, γ ∼ − . The general solution of equation (15) is the sum of ˜ M gas ∝ exp (cid:0) − α ˜ t (cid:1) with two values of α , α = 0 . (cid:0) ± [1 + 4 γ ] . (cid:1) . (16) 12 –Using equation (13) again to fit d ˜ M gas /d ˜ t at ˜ t = 0, and defining Γ = (1 + 4 γ ) . , we obtainthe solution for normalized gas mass,˜ M gas (˜ t ) = Γ + 0 . − ˜ R acc Γ ! e − . t + Γ − . R acc Γ ! e . − t . (17)The normalized stellar mass is the time integral over the normalized gas mass, or, ignoringstellar ejection,˜ M stars (˜ t ) = Γ + 0 . − ˜ R acc . ! (cid:16) − e − . t (cid:17) + Γ − . R acc . − ! (cid:16) e . − t − (cid:17) . (18)This solution consists of an exponentially decaying initial burst of star formation and gasdepletion with a timescale approximately equal to the consumption time (0 . ∼ . − ∼ γ <<
3. Results
Figure 1 shows numerical solutions to equations (1) to (10) with mass and time nor-malized as above. The three curves are for different normalized accretion rates, as indicatedby their colors and the labels for ˜ R acc . The solutions have the parameters discussed abovefor an IMF extending out to 300 M ⊙ , i.e., f HM = 0 .
164 and f debr = 0 . t consume /t evol = 0 .
1, which together make γ = 0 . f eject = 0 .
4. The lowerleft panel shows the processed fraction in the cloud core, which is also the processed fractionin the stars that form at that time. The analytical solutions to the gas and stellar massesfit exactly on top of the numerical solutions and are not shown in the figure. However, thetop left panel has each exponential function from the analytical solution plotted separatelyas a dotted line; the total solution is the sum of these, as in equation (17).The lower right panel shows the distribution of the processed fraction, f p , among low-mass stars after star formation is assumed to stop, which is at the time t evol in these modelsto avoid supernova contamination. All three cases assume stellar ejection with f eject = 0 . f p can range up to 0.7 from the Li abundance (Decressin et al.2007b), and that is how far the blue histogram goes (which is for ˜ R acc = 0). Also fromobservations, the ratio of stellar mass in the second generation to total stellar mass rangesfrom ∼ . ∼ .
8. For the 3 histograms in Figure 1, the ratios of the masses in all but the 13 –lowest- f p interval (i.e., the G2 stars) to the sum of all the masses (the G1+G2 stars) is aboutthe same as the observations: 0.63 for the red curve ( ˜ R acc = 0 . R acc = 0 . R acc = 0). The processed fraction f p increaseswith time as more and more massive stellar envelopes with their ever-increasing p-processcontaminations disperse inside the cloud core (lower left panel). The maximum value of f p at the end of the star formation time (right-hand limit to the curves in the lower-left panel)increases with decreasing ˜ R acc because there is less dilution of the cloud core gas with pristineinfall from the cloud envelope.In equation (11), ejection operates on all low-mass stars regardless of when they form,but it tends to remove a higher proportion of the first stars that form, i.e., weighted towardthe G1 stars, than the later stars that form because the first stars have been exposed forthe longest time to the multi-star interactions and gas motions that cause ejection. The topright panel of Figure 1 shows that ejection with the assumed f eject = 0 . f p bin would be muchhigher compared to the others, although the f p range in the histogram would be the same.Related to this age-dependence of stellar ejection is a prediction from this model thatthe G1 stars should on average be less concentrated in the GC than the G2 stars. This isbecause the G1 stars have had more opportunities to absorb kinetic energy from multi-stellarinteractions in the core than the G2 stars, regardless of whether the stars escape the cluster.Such a central concentration of G2 stars is observed (Gratton et al. 2012).The left-hand panel of Figure 2 shows the effect of stellar ejection through the parameter f eject on the fraction of the mass in the G2 population. The blue curves assume the usual t consume /t evol = 0 . f eject in equal steps between 0.1 and 0.7. The dashedblue curve with f eject = 0 . f eject = 0 . t consume /t evol = 0 .
05 and 0.2,respectively, for the same value of γ = f debr f HM t consume /t evol (i.e., f debr is 2 × larger andsmaller when t evol is 2 × larger and smaller, respectively, to keep γ the same). There arethree important dependencies shown in this figure: (1) the G2 mass fraction is low for bothlow accretion and high accretion rates, with a peak at ˜ R acc ∼ .
03; (2) the G2 mass fractionincreases with higher ejection parameter, and (3) the G2 mass fraction depends strongly onthe ratio of the gas consumption time to the stellar evolution time.The first result arises because high ˜ R acc causes severe dilution of the p-processed materialfor later stars that form, and because low ˜ R acc causes most of the stars to form quickly anddeplete the gas before p-processed elements have time to get into the cloud core. The secondresult is a consequence of ejecting more early-forming stars than late-forming stars because 14 –of the greater exposure of early-forming stars to time-changing gravitational forces.The third result indicates the importance of the dimensionless parameter t consume /t evol .A low value means there are more dynamical times available for stellar ejection before thesupernova era begins (at t = t evol ), so the first-forming stars become much more depletedcompared to the last-forming stars. This increases M G2 /M stars , as shown on the left of Figure2. It also means there are more massive stars at early times compared to late times, becauseof the more rapid drop in the star formation rate from faster gas consumption at small t consume . These greater numbers of early massive stars send relatively more p-processedmatter into the smaller amount of remaining gas mass in the cloud core, increasing themaximum f p . These trends with lower t consume /t evol occur when the density of the cloudcore increases, because that lowers the free fall time and, correspondingly, t consume , at a fixedstellar evolution time. Thus we have the important result that p-process contamination islarger and involves a higher fraction of remaining long-lived stars in a globular cluster whenthe initial cluster gas density is higher. This may be the critical clue to the distinct originof old globular clusters.Figure 2 suggests that realistic results require a fairly high ejection parameter f eject , onthe order of several tenths, which corresponds to a mean time before ejection of low massstars equal to several tens of dynamical times (= t consume / ( t ff f eject )). Fujii & Portegies Zwart(2013) follow the ejection of stars of various masses from the collapsed core of a clusterowing to binary star interactions. The number of ejected high mass stars per cluster is fairlylow, although it can account well enough for runaway OB stars in the field. The number ofejected low mass stars, which are of greatest interest in the present context, is much higher,following the stellar IMF (see their figure 1). They did not do a simulation that is exactlywhat we need, which would involve a lowest stellar mass less than 0 . M ⊙ and a clustermore massive than 10 M ⊙ . In their most relevant case, which had the lowest mass stars,the fraction of stars ejected was 0.4%. That is only for the binary star mechanism, however.Here we envision a much more dynamic environment in which gaseous mass motions frommulti-star interactions and winds stochastically change the whole cluster core potential, as insimulations of dwarf galaxy nuclei (El-Badry et al. 2016). That second process, in additionto the possible presence of several dense cores in a proto-GC, each of which has one or moreimportant massive binaries for stellar ejection, could possibly bring the ejection fraction perdynamical time up to the required value.The dependence of relative G2 mass and processed fraction, f p , on t consume is shownin Figure 3, where we assume fixed f debr , f HM , f eject , and t evol and vary only t consume . Inthis case the input parameters are dimensional, with t consume up to ∼ . R acc = 0(blue curves) and 4 × M ⊙ Myr − (red curves). There are sharp decreases in the relative 15 –mass of the second generation (left panel) and the maximum processed fraction (right panel)as the consumption time increases, which corresponds to a decrease in cloud core density.Because the bulk peculiarity of old GCs is in these two quantities, it seems evident that the primary distinction between clusters that formed in the early universe and most of themain-disk clusters forming today is cloud-core density .
4. Conclusions
We present a model of star formation in massive dense clusters that may be relevantto the formation of old globular clusters. This model has several key features that shouldcause a cluster to form stars with a wide range of p-processed elements, in agreement withobservations. These features are: • A 10 − M ⊙ cloud core with a density of ∼ cm − , giving a free fall time of ∼ .
03 Myr. The corresponding virial velocity is ∼
80 km s − , which is comparable tothe turbulent speed in an L* galaxy at the same redshift where the GC forms. Such acore is tightly bound and should form stars with an elevated star formation efficiencyper unit free fall time, assumed here to be ∼ • Interactions between massive stars at close range, including massive binary star inter-actions with single stars and massive stellar mergers, which lead to the swelling anddispersal of massive stellar envelopes and the dispersal of extruded equatorial disksaround rapidly rotating massive stars. The dispersed envelope gas carries p-processedelements into the cloud core where it gets incorporated into new stars. • Rapid stellar ejection driven by the time-changing gravitational potential of multi-starinteractions and pressurized gas motions. The ejection rate is assumed to scale withthe dynamical rate in the cluster. We point out that stars ejected from a cluster witha terminal velocity equal to only several tenths of the cloud core virial speed can movethrough the host galaxy at greater than the galaxy’s escape speed and thereby leavethe host altogether. Such stellar loss may solve the problem of missing halo stars inFornax and WLM, which have very low escape speeds. • Star formation in the cloud core using gas that is a combination of original core gas,stellar debris that becomes more and more contaminated by p-processed elements over 16 –time, and newly accreted gas with the original abundances. Because the consumptiontime at high density is much less than the lifetime of a massive star, many generations ofstars form before supernovae finally clear the gas away. We assume that each generationhas a complete and normal IMF. We track the high mass stars for the production ofp-processed elements, and the low mass stars for comparison to GCs today.As a result of these assumptions, we produce GC stellar populations with approximatelythe observed range, mass, and radial distributions of p-process contamination. We find thatthe mass fraction of the final GC in the form of contaminated stars (i.e., the “2nd gener-ation” fraction), and the maximum amount of p-process contamination in the late-formingstars, both increase strongly with cloud core density. This result suggests that the primarydifference between old GCs with their significant amounts of p-process contamination andyoung super-massive star clusters that show little evidence for self-enrichment is that thecluster-forming clouds at high redshift have higher densities. This is to be expected be-cause high redshift galaxies have larger gas fractions, higher gas surface densities and fasterturbulent speeds than all but the most active galaxies today. As a result, the pressures inhigh-redshift galaxies are high, and the densities in the star-forming cloud cores should behigh too.This paper benefited from interesting discussions with Dr. Stefanie Walch at an earlystage of this investigation, and from comments by the referee.
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This preprint was prepared with the AAS L A TEX macros v5.2.
21 – consume M ga s / M c l oud ( t = ) R acc = 0.030.0030.0 0 2 4 6 8 100.010.11 t/t consume M s t a r s , L M / M c l oud ( t = ) f p M s t a r , L M consume f p Fig. 1.— Models for the formation of globular clusters with three different normalizedaccretion rates onto the cloud core. The top-left panel shows the relative core gas mass asa function of time, in units of the gas consumption time. The dotted lines are the separateexponential solutions from equation (17), whose sum equals the total solution. The coremass decreases faster when there is no accretion. The top right panel shows the mass inlow mass stars for these three cases (designated by the corresponding colors), with dashedlines representing the case when there is no stellar ejection from the cluster, and solid linesshowing the case with ejection using an efficiency of f eject = 0 .
4. The maximum time on theabscissa corresponds to the lifetime of a massive star, since we assume t consume /t evol = 0 . t/t consume = 10. Thereis a continuum of f p because stars form continuously as the core gas is contaminated. Weconsider the first generation stars to be those in the lowest bin. The numbers of these starsis relatively large in all cases, but the cumulative number in bins of higher f p can be larger. 22 – –3 –2 –1 acc M G / M s t a r s f eject = 0.70.60.50.4 –3 –2 –1 acc M a x i m u m f p t consume /t evol = 0.1= 0.05 ; = 0.2 Fig. 2.— (Left) The fraction of low mass stars in the “2nd generation,” defined to be thosein all but the lowest bin of f p from figure 1. This fraction is shown as a function of thenormalized accretion rate onto the cloud core, and for different values of the efficiency ofstellar ejection from the cluster, f eject . The G2 fraction is low at low ˜ R acc because thenstar formation ends quickly and there is too little time for p-processed elements to getinto the gas for subsequent generations of star formation. This fraction is low also at high˜ R acc because then the core is highly diluted with accreted pristine gas. It peaks at around˜ R acc ∼ .
3, which is when the accretion rate is 0.3 times the initial core mass dividedby the gas consumption time. Higher f eject produces higher G2 fractions because then ahigher proportion of early-forming low-mass stars, which have near-G1 abundances, havebeen ejected. (Right) The maximum value of the processed fraction, f p , is shown versus thenormalized accretion rate for three different cases of the relative consumption time. The lowand high cases are also shown in the left-hand panel using the same colors. The curves inthe right-hand panel are independent of f eject , which only affects the proportion of stars atvarious f p , but not the maximum in f p . The result indicates that G2 stars reach a higherp-processed fraction if the accretion rate is low, because then the dilution of stellar debrisby pristine gas is smaller. 23 – consume (Myr) M G / M s t a r s f eject = 0.2 0.8 0.05 0.1 0.2 0.4 100.20.40.60.81 t consume (Myr) M a x i m u m f p R acc =4x10 M O Myr –1 R acc =0.=0.