Early-Age Evolution of the Milky Way Related by Extremely Metal-Poor Stars
aa r X i v : . [ a s t r o - ph . GA ] J a n Draft version October 25, 2018
Preprint typeset using L A TEX style emulateapj v. 12/14/05
EARLY-AGE EVOLUTION OF THE MILKY WAY RELATED BY EXTREMELY METAL-POOR STARS
Yutaka Komiya , Takuma Suda , Masayuki Y. Fujimoto Draft version October 25, 2018
ABSTRACTWe exploit the recent observations of extremely metal-poor (EMP) stars in the Galactic halo andinvestigate the constraints on the initial mass function (IMF) of the stellar population that left theselow-mass survivors of [Fe / H] . − . ≃ M ⊙ for the stars of EMP population and the overwhelmingcontribution of low-mass members of binaries to the EMP survivors are derived from the statisticsof carbon-enriched EMP stars with and without the enhancement of s-process elements (Komiyaet al. 2007, ApJ, 658, 367)). We first examine the analysis to confirm their results for variousassumptions on the mass-ratio distribution function of binary members. As compared with theuniform distribution they used, the increase or decrease function of the mass ratio gives a higher- orlower-mass IMF, and a lower-mass IMF results for the independent distribution with the both membersin the same IMF, but the derived ranges of typical mass differ less than by a factor of two and overlapfor the extreme cases. Furthermore, we prove that the same constraints are placed on the IMF fromthe surface density of EMP stars estimated from the surveys and the chemical evolution consistentwith the metal yields of theoretical supernova models. We then apply the derived high-mass IMF withthe binary contribution to show that the observed metallicity distribution function (MDF) of EMPstars can be reproduced not only for the shape but also for the number of EMP stars. In particular,the scarcity of stars below [Fe / H] ≃ − / H] < − Subject headings: stars: abundances — stars: carbon — Galaxy: halo — Galaxy: formation — stars:mass function INTRODUCTION
To reveal the nature of the extremely metal-poor(EMP) stars in the Galactic halo is the key to the under-standing of the formation process of the Galaxy as wellas of the mechanism of star formation in the primordialand very metal-poor gas clouds. Because of the verylow abundances of iron and other metals, these stars arethought to be survivors from the early days, and hence,are expected to carry the precious information about theearly Universe when they were born while they residein our nearby space. For a past decade, a lot of EMPstars have been discovered by HK survey (Beers et al.1992) and Hamburg/ESO (HES) survey (Christlieb et al.2001), which enables us to use halo EMP stars as a probeinto the early Universe. The number of known EMP starsexceeds several hundreds even if we limit the metallicityrange below [Fe / H] . − . / H] ≃ − ∼
160 stars have been reg-istered in the metallicity range of − . [Fe / H] . − Astronomical Institute, Tohoku University, Sendai, Miyagi980-8578, Japan Department of Cosmoscience, Hokkaido University, Sapporo,Hokkaido 060-0810, Japan stars of [Fe / H] < −
5, HE 0107-5240 ([Fe / H] = − . / H] = − .
4; Frebel et al. 2005), and one ultra metal-poor(UMP) star of − < [Fe / H] < −
4, HE 0557-4840([Fe / H] = − .
8; Norris et al. 2007). This has attractedwide interest, in particular, before the discovery of HE0557-4840 in-between metallicity of − < [Fe / H] < − h M i ≃ − M ⊙ .Salvadori et al. (2007) also take a similar approach to in-vestigate the chemical evolution of our Galaxy with themass outflow from mini-halos. In these former works,the low-mass star formation under the metal-deficientcondition is introduced in rather arbitrary ways, and theproper explanation is yet to be devised about the nature Komiya et al.and origin of HMP/UMP stars.One of the decisive ingredients in studying the struc-ture formation and chemical evolution of Galactic halois the initial mass function (IMF) of stars in the earlydays. Most of existent studies have assumed the IMFof EMP stars more or less similar to that of the metal-rich populations except for HMP and UMP stars. Fromthe observations, however, we know that the EMP starshave the distinctive feature that a fair proportion of themshow the surface carbon enhancement relative to iron,the proportion by far larger than the stars of youngerpopulations (Rossi et al. 1999). In addition, it is re-vealed that the carbon-enhanced extremely metal-poor(CEMP) stars are divided into two sub-groups, CEMP- s and CEMP-no s according to the presence and absenceof the enhancement of s -process elements (Ryan et al.2005; Aoki et al. 2007). This also forms a striking con-trast with the fact that their correspondences among theyounger populations, CH stars and Ba stars, are all ob-served to exhibit the enhancement of s-process elements.Since the EMP survivors are low-mass stars, the enrich-ment of these elements are expected only through themass transfer and/or the wind accretion from the AGBprimaries in the binaries. Assuming this binary scenarioand the same mechanism of carbon enhancement as thestars of younger populations, Lucatello et al. (2005) ar-gue an IMF with the typical mass of M md ∼ . M ⊙ for EMP stars from the surplus of CEMP- s stars. Previ-ously, Abia et al. (2001) have also asserted an IMF peak-ing in the intermediate-mass range of 4 − M ⊙ for popu-lation III stars from the consideration of Galactic chem-ical evolution with the CN enrichment among the EMPstars. Furthermore, an IMF with M md ∼ . − . M ⊙ has been is discussed for the old halo stars from theMACHO observation in relation to the prospect thatthe observed micro-lensing may be caused by an allegedpopulation of white dwarfs (Adams & Laughlin 1996;Chabrier et al. 1996).In order to use the carbon-enhancement to constrainthe IMF, we should properly take into account the evo-lutionary peculiarity of EMP stars. It is known that forthe stars of [Fe / H] . − .
5, there are two mechanismsof carbon enhancement, while only one mechanism forthe stars of younger populations, Pop. I and II, and also,that a different mode of s-process nucleosynthesis works(Fujimoto et al. 2000; Suda et al. 2004; Iwamoto et al.2004; Nishimura et al. 2008). Applying these theoreti-cal understandings to the binary scenario, Komiya et al.(2007, referred to as Paper I in the following) find thatthe IMF for EMP stars has to be high-mass with thetypical mass of M md ≃ M ⊙ to explain the observedstatistic features of both CEMP- s and CEMP-no s stars.In particular, as a consequence, it follows that the ma-jority of EMP stars, including CEMP stars, were bornas the low-mass members of binary systems with the pri-mary stars which have shed their envelope by mass lossto be white dwarfs and have exploded as supernovae.Tumlinson (2007a) discuss the binary scenario for HMPstars in the similar way.The purpose of this paper is twofold, first to demon-strate the robustness of the high-mass IMF derived inPaper I, and then to discuss the implications to the for-mation and early evolution of Galaxy. In the follow-ing, we make a distinction between the total assembly of EMP stars that were born in the early Galaxy, in-cluding massive stars which were already exploded assupernovae, and the low-mass EMP stars that are stillalive in the nuclear burning stages by calling the for-mer “EMP population” and the latter “EMP survivors”.In deriving the constraints on the IMF of stars for theEMP population, one has to make the assumptions onthe binary characteristics, among which the most crucialis the distribution function of mass ratio between the pri-mary and secondary stars in binaries. Paper I adopts aflat distribution for simplicity. It seems plausible fromthe observations of the stellar systems of younger pop-ulations (Duquennoy & Mayor 1991; Mayor et al. 1992),and yet, it is true that the mass-ratio distribution is yetto be properly established both observationally and the-oretically even for the binaries of younger populations.Several different mechanisms have been proposed for thebinary formation, such as the fragmentation during thecollapse and the capture of formed stars, and are thoughtto give different mass-ratio distributions (see also e.g.,Goodwin et al. 2007, and the references therein). Thedistribution may increase or decrease with the mass-ratio, or the two stars may form in the same IMF assuggested for the capture origin. In this paper, we exam-ine the dependence of the resultant IMF on the assumedmass-ratio distributions of various functional forms, in-cluding the independent coupling of the both stars in thesame IMF to demonstrate that the high-mass nature ofIMF of EMP population is essentially unaltered.The recent large-scaled surveys of EMP stars providethe additional information on the early history of Galac-tic halo. A fairly large number of known metal-poorstars (144 and 234 stars of [Fe / H] < − by the HK and HES surveys,respectively; Christlieb 2003; Beers & Christlieb 2005)allows to consider the total number of EMP survivors inthe Galactic halo. We demonstrate that the latter alsoplaces an independent constraint on the IMF of EMPpopulation in combination with the metal yields pro-duced by the EMP supernovae if the binary contributionis properly taken into account.We then apply the IMF, thud derived, to discuss thechemical evolution in which the stars of EMP popula-tion take part. It is shown that the resultant IMFs canreproduce the number and slope of observed metallic-ity distribution functions (MDF) for EMP stars, andalso, to give an explanation to the scarcity and origin ofHMP/UMP stars with the effects of hierarchical struc-ture formation process included. In this paper, we dealonly with the iron production by SNe since we are inter-ested in the MDF, and discuss the basic characteristicsof hierarchical structure formation by using simple an-alytic approximations. Tumlinson (2007b) studies thelow-mass star formation taking into account the contri-bution of binary stars, but his approach is different fromours in the uses the hypothesized IMF with the effect ofcosmic microwave background. In addition, he consid-ered only the CEMP- s stars, but not CEMP-no s stars orMDF.This paper is organized as follows. In §
2, we discussarly-Age evolution of Milky Way 3the constraints on the IMF of EMP population from thestatistics of CEMP stars and from the total number ofEMP survivors in our Galaxies. In §
3, we investigate themetallicity distribution of EMP stars in Galactic halowith the formation process of the Galaxy taken into ac-count. Then our conclusions follow with discussion of theorigin of observed MDF and also of HMP stars. In Ap-pendix, we re-discuss the relationship between the num-ber of EMP survivors, estimated from the surveys, andthe metal production by the EMP supernovae with thebinary contribution taken into account, to demonstratethat they entail the same IMFs as drawn independentlyfrom the statistics of CEMP stars. CONSTRAINTS ON IMF OF EMP STARS
In this section, we revisit the problem of constrain-ing the IMF for the stars of EMP population from theobservations of EMP survivors, studied in Paper I. Themethod is based on the analysis of statistics of CEMPstars in the framework of binary scenario, and hence, in-volves the assumptions of EMP binary systems. We startwith reviewing the method and assumptions used in Pa-per I in deriving the constraints on the IMF of EMPpopulation stars. We first investigate the dependenceof resultant IMF on these assumptions, in particular ofthe mass-ratio distribution of binary members. We thendiscuss the iron production by EMP population stars inrelation to the total number of EMP survivors, estimatedfrom the HK and HES surveys, to assess the constraintson the IMF through the chemical evolution of Galactichalo.
Method and basic assumptions
We give the outline of our method in studying thestatistics of CEMP stars and chemical evolution of Galac-tic halo with the discussion of the assumptions involved,and a brief summary of the observational facts that ourstudy rely on.
Statistics of CEMP stars
Our method is founded on the results of stellar evo-lution that the stars of [Fe / H] . − . < . M ⊙ undergo hydrogen mixing into the helium con-vection during the helium core or shell-flashes, differentlyfrom the stars of younger populations, Pop. I and II(Fujimoto et al. 1990, 2000). This triggers the helium-flash driven deep mixing (He-FDDM) to carry out car-bon to the surface(Hollowell et al. 1990). It is neces-sarily accompanied with the s-process nucleosynthesisin the helium convection as mixed protons are trans-formed into neutrons (Suda et al. 2004; Iwamoto et al.2004; Nishimura et al. 2008). For EMP stars, He-FDDMworks as the mechanism to enrich both carbon and s-process elements in their surface in addition to the thirddredge-up (TDU), the latter of which works in the starsof M > ∼ . M ⊙ in common with the stars of youngerpopulations.Consequently, the origins of two sub-groups of CEMPstars are identified with these two mechanisms. TheCEMP- s and CEMP-no s stars stem from the low-massmembers of EMP binaries with the primaries in the massranges of 0 . M ⊙ < M < . M ⊙ and 3 . M ⊙ ≤ M ≤ M up , respectively. Here M up is the upper limit to ini-tial mass of stars for the formation of white dwarfs. We take M up = 6 . M ⊙ (Cassisi & Castellani 1993, see alsoSiess 2007), which is also taken to be the lower masslimit to the stars that explode as supernova. This is thefundamental premise of our study. Among the CEMP-no s stars, there are the stars that show different charac-teristics such as CS22892-052 with a large enhancementof r-process elements. They may have different originsaccording to the scenarios such as proposed in connec-tion to supernovae yields (e.g., Tsujimoto & Shigeyama2001; Umeda & Nomoto 2002; Wanajo et al. 2006). Ac-cordingly, the CEMP-no s stars may be the admixtureof the stars of different origins. Since the ratio betweenthe CEMP- s and CEMP-no s stars depends strongly onthe IMF, as shown from Paper I, however, our resultswill not affected as long as the CEMP-no s stars containsthose with the AGB mass transfer origin that we pro-pose.For the formation of CEMP stars in the binary sys-tems, the initial separation, A , has to be large enoughto allow the primary stars to evolve through the AGBstage without suffering from the Roche lobe overflow,but small enough for the secondary stars to accretea sufficient mass of the wind to pollute their surfacewith the envelope matter processed and ejected by theAGB companion. The lower bound, A min ( m , m ),to the initial separation is estimated from the stellarradii of EMP stars taken from the evolutionary calcu-lation (Suda & Fujimoto 2007), where m and m arethe masses of primary and secondary stars. The AGBstar is assumed to eject the carbon enhanced matter of[C / H] = 0 with the wind velocity v wind = 20km s − untilit becomes a white dwarf, and we define CEMP stars as[C / Fe] ≤ .
5. The upper bound, A max ( m , m ), is esti-mated by the amount of accreted matter calculated byapplying the Bondi-Hoyle accretion rate, dm ( t ) dt = − G m ( t ) A ( t ) v rel ( t ) v rel ( t ) v wind × dm ( t ) dt , (1)in the spherically symmetric wind from the companion,and v rel is the relative velocity of the secondary star tothe wind. Accreted matter is mixed in surface convec-tion of depth 0 . M ⊙ and 0 . M ⊙ in mass for giantsand dwarfs, respectively. E.g., for the stellar metallic-ity [Fe / H] = − .
5, the mass of accreted matter has tobe larger than 3 . × − M ⊙ and 3 . × − M ⊙ , andhence, the upper bounds are ∼ ∼ ∼ . dex . This will not affectour results so much since the upper bound is itself suf-ficiently large to exceed the separation at the peak ofperiod distribution (see below). In addition, it has littleeffects on the giants which we mainly deal with in thefollowing because of by far deeper surface convection.If we specify the initial mass function, ξ b ( m ), andthe distributions of binary parameters, therefore, we canevaluate the frequency of CEMP- s and CEMP-no s stars,and through the comparison with the observations, we Komiya et al.may impose the constraints on the IMF and on the bi-nary parameters. The numbers of CEMP- s and CEMP-no s stars currently observable in flux-limited samples aregiven by ψ CEMP - s = f b Z . M ⊙ . M ⊙ dm N s ( L [ m ]) × Z . M ⊙ . M ⊙ dm ξ b ( m ) n ( q ) m Z A M ( m ,m ) A min ( m ,m ) f ( P ) dPda da (2) ψ CEMP - no s = f b Z . M ⊙ . M ⊙ dm N s ( L [ m ]) × Z M up . M ⊙ dm ξ b ( m ) n ( q ) m Z A M ( m ,m ) A min ( m ,m ) f ( P ) dPda da, (3)where f b is the binary fraction: n ( q ) is the distribution ofthe mass-ratio, q ≡ m /m , and f ( P ) is the distributionof the period of binaries: and N ( L ) is the probability ofthe stars in the Galactic halo with the luminosity L inthe survey volume of HES survey. Note that all of themstem from the low-mass members of binary since it is onlya very small fraction of stars that have experienced He-FDDM to develop CEMP- s characteristics in themselvesand now stay on AGB. Similarly the total number ofEMP survivors is given by ψ surv = Z . M ⊙ . M ⊙ dmN s ( L [ m ])[(1 − f b ) ξ s ( m ) + f b ξ b ( m )]+ f b Z . M ⊙ . M ⊙ dm N s ( L [ m ]) Z ∞ . M ⊙ dm ξ b ( m ) n ( q ) m , (4)with the contribution of the stars born as single underthe initial mass function ξ s ). The rest of the terms givethe number of EMP survivors formed as binary, ψ binary .The stellar luminosity and lifetime are taken from theevolution calculation of EMP stars by Suda & Fujimoto(2007). The AGB primaries dredge up to increase theirsurface helium abundances, and hence, may cause thesurface enrichment of helium to the companion stars inthe binaries at the same time with the carbon enrich-ment, though both suffering the dilution in the envelopeconvection. The surface enrichment increases the lumi-nosity during the RGB evolution to shorten their RGBlifetime of polluted EMP stars nearly in inverse propor-tion of the luminosity, but the survey volume increaseswith the luminosity. For flux limited sample, the ob-served number of EMP giants may rather increases withthe surface enrichment in proportion to a half power un-der a constant density distribution. On the other hand,the increase in the luminosity occurs only after the hy-drogen burning shell comes to take place in the shell towhich the pollutants are carried in by the surface con-vection, and hence, in the later stages for metal-poorerstars. Accordingly, the effect of helium enhancementwill little affect our results of constraints on the IMFsince HES survey is thought to reach far enough that thespatial distribution of halo stars decreases. Model Parameters
In this paper, we assume that the binary primarystars and single stars are born under the same IMF, i.e., ξ ( m ) = ξ b ( m ) = ξ s ( m ). For the form of IMF, we may well assume a lognormal function with the medium mass, M md , and the dispersion, ∆ M , as parameters ξ ( m ) ∝ m exp (cid:20) − (log m − log M md ) (cid:21) . (5)In addition, we assume the binary fraction f b = 0 . f b since not only the CEMP stars butalso most of the EMP survivors come from the sec-ondary companions of binaries unless M md < . M ⊙ ,as seen later. As for the binary period, we mayadopt the distribution derived for the nearby stars byDuquennoy & Mayor (1991), f ( P ) ∝ P exp " − (log P − . × . , (6)where P is the period in units of days. The binary frac-tions and period distributions of halo stars are observedto be not significantly different from those of nearby diskstars (Latham et al. 2002; Carney et al. 2003). Addi-tionally, it is shown in Paper I that this period distribu-tion is consistent with the observations of CEMP starsfor the periods of P .
10 yr confirmed to date (see Fig. 3in Paper I).The mass ratio distribution is an essential factor indiscussing the evolution of binary systems, and yet, itis not well understood. The mass ratio distributionof metal-poor halo stars is investigated observationally(e.g., see Goldberg et al. 2003; Abt 2008), and yet, sub-ject to large uncertainties. Especially for the binary withintermediate-mass or massive primary stars, it is hard toknow the mass ratio distribution from the observations.Theoretically, neither the fragmentation of gas cloud northe accretion process onto proto-binaries are yet well un-derstood even for Population I stars (e.g. Bate & Bonnell1997; Ochi et al. 2005; Machida 2008). In order to testthe assumption on the mass-ratio distribution, we inves-tigate the constraints on the IMF for different mass-ratiodistributions. In Paper I, the simplest flat distribution isassumed in Paper I among the possible distributions. Inthis paper, we test some other assumptions and discussthe dependence of IMF parameters on the mass ratiodistributions, as stated in § χ ( m , m ), as the fraction of the binaries with a primaryand secondary star in the mass range of [ m , m + dm ]and [ m , m + dm ] ( m ≥ m ) and write it in the form; χ ( m , m ) dm dm = ξ ( m ) n ( q ) dqdm = ξ ( m ) n ( m , m ) /m dm dm . (7)Here the initial mass function, ξ , of the primary star isassumed to be the same as IMF of single stars: And n ( q )is the mass ratio distribution, for which we assume bothextremities of increase and decrease functional forms inaddition to the constant one, adopted in Paper I; n ( q ) = / (1 − . M ⊙ /m ) (Case A)2 q/ [1 − (0 . M ⊙ /m ) ] (Case B) q − / ln( m / . M ⊙ ) (Case C) . (8)Furthermore, we take up a different type of mass-ratiodistribution that the primary and secondary stars in-dependently obey the same IMF such as assumed byarly-Age evolution of Milky Way 5Lucatello et al. (2005). In this case, the coupling massdistribution function is given as a product of the sameIMF as; χ ( m , m ) dm dm = 2 ξ ( m ) ξ ( m ) dm dm (Case D) . (9)We shall refer to this distribution function as “inde-pendent” coupling. From the comparison with eq. (7),we may write the mass-ratio function in the form n ( m , m ) = 2 m ξ ( m ); it is should be noted, however,that the frequency of binaries with a primary star of mass m is not normalized and increases with m from zero to2, as given by the integral R . M ⊙ /m n ( m , m ) dq =2 R m . M ⊙ ξ ( m ) dm .With these specification and with the assumed mass-ratio distribution function, we may compute the fractionsof EMP survivors, ψ surv ( M md , ∆ M ) and of both cempsstars, ψ CEMP - s ( M md , ∆ M ) and ψ CEMP - no s ( M md , ∆ M ),and search the ranges of the IMF parameters, mediummass M md and dispersion ∆ M , that can reproduce thestatistics of CEMP stars consistent with observations Total iron yield of EMP supernovae
We can pose another constraint from the total ironyield, M Fe , of EMP population and the total number, N EMP , G , of the giant EMP survivors. For N EMP , G , es-timated from the results of existent surveys, the totalstellar mass, M EMP , of EMP population for an assumedIMF is given by, M EMP ( M md , ∆ M ) = m N EMP , G /f G , (10)where f G is the fraction of giant EMP survivors in allthe stellar systems, born as EMP population, and m isthe averaged mass of EMP population stars: f G = " ξ (0 . M ⊙ ) + f b Z . M ⊙ ξ ( m ) n (0 . M ⊙ /m ) dm m ∆ M G , (11) m = Z dm [ m ξ ( m ) + f b m Z m m n ( q ) dm ] . (12)The first terms of both equations denote the contribu-tions by the stars born as the single stars and as the pri-mary stars in the binaries and the second terms denotethe contributions by the stars born as the secondary starsin the binaries. The mass and mass range of EMP starsnow on the giant branch are taken to be M = 0 . M ⊙ and ∆ M G = 0 . M ⊙ , based on the stellar evolution cal-culation of stars with [Fe / H] = −
3, as in paper I.The massive stars of EMP population have explodedas supernovae to enrich the interstellar gas with metals.The amount of iron, M Fe , EMP , ejected by all the super-novae of EMP population of the total mass, M EMP , isgiven by M Fe , EMP = M EMP m f SN h Y Fe i = N EMP , G f SN f G h Y Fe i , (13)where f SN is the fraction of the stars that have explodedas supernovae and given by, f SN = Z M up dm ξ ( m )[1 + f b m Z m M up n ( q ) dm ] : (14)and h Y Fe i is the averaged iron yield per supernova, takento be h Y Fe i = 0 . M ⊙ in the following calculations. With these evaluations and the observed number ofEMP giants, we can give the total iron yield of starsof EMP population as a function of IMF parameters.The comparison with the total amount of iron estimatedfrom the chemical evolution of Galactic halo may imposeconstraint on the IMF parameters. Observational constraints
The first constraint is the number fraction of CEMP- s stars. The HK and HES observations tell that theCEMP stars with [C / Fe] > ∼
25% ofEMP stars (e.g., Beers 1999; Rossi et al. 1999; Christlieb2003). Cohen et al. (2005) suggest a slightly lower frac-tion of 14 . ±
4% with the errors in the abundanceanalysis taken into account while Lucatello et al. (2006)obtain a larger frequency of 21% ±
2% for the HERES(HES r-process enhanced star) survey sample, both for[Fe / H] < − .
0. It is claimed that the frequency of CEMPis higher at lower metallicity of [Fe / H] < − . s stars.In this paper, we adopt the observational constraint onthe fraction of the CEMP- s stars at 10 − . < ψ CEMP - s ( M md , ∆ M ) ψ surv ( M md , ∆ M ) < . . (15)The second constraint is the number ratio betweenCEMP-no s and CEMP- s stars. The observed frequencyof CEMP-no s to CEMP- s stars is ∼ / /
14 for [Fe / H] ≤ − .
5. Inaddition, EMP stars enriched with nitrogen are foundin number comparable with, or more than, CEMP-no s stars (“mixed” stars; Spite et al. 2005), whose origin canbe interpreted in terms of the same mechanism but withmore massive primary companions that experience thehot bottom burning (HBB) in the envelope of the AGB.Some other scenarios for CEMP stars have been proposedUmeda & Nomoto (2005); Meynet et al. (2006) but weassume all CEMP stars are formed in binaries with AGBin this paper. We adopt the observational constraint onthe relative frequency of CEMP-no s to CEMP- s stars at1 / −
1; 1 / < ψ CEMP - no s ( M md , ∆ M ) ψ CEMP - s ( M md , ∆ M ) < . (16)We note that the above two constraints are not depen-dent on the total mass nor on the spatial distributionof the stellar halo because they are concerned with therelative number ratios.The third constraint is the total iron yield from theEMP population. The HES survey obtained 234 starsof [Fe / H] < − S = 8225deg (Beers & Christlieb 2005).Taking the relative frequency between the giants anddwarfs (1 : 0 .
93) and the ratio of the stars of [Fe / H] < − / H] < − . σ EMP , G ≃
410 sr − . (17)We assume that all giant stars in the survey areas areobserved because of the fairly large limiting magnitude Komiya et al.of HES survey ( B = 17 . § N EMP , G = 5 . × in the Galaxy.On the other hand, the amount of iron necessaryto promote the chemical evolution of the whole gas inGalaxy of mass, M h = 10 M ⊙ , up to [Fe / H] = − . M Fe , halo = M h X Fe , ⊙ − . = 10 . M ⊙ , (18)and the supernovae of EMP population should have pro-vided this amount of iron unless there were other pop-ulation(s) of stars which made iron without producinglow-mass stars. Using eq. (13), this is transferred into aconstraint on the IMF as M Fe , EMP = 0 . M ⊙ × . × f SN ( M md , ∆ M ) f G ( M md , ∆ M ) ≃ . M ⊙ . (19)The estimated number of EMP survivors may be subjectto significant uncertainties. If we take into account theEMP stars in the outer halo and in the Galactic bulgethat HES survey cannot reach, N EMP , G can be larger,which demands a smaller amount of iron produced perEMP survivor, and hence, a smaller number of super-nova, leading to a lower-mass IMF. A lower-mass IMFalso results if the binary fraction is smaller and/or if thereis other source(s) of iron that does not accompany thelow-mass star formation. On the other hand, if a partof the supernovae ejecta is dispelled and lost from theGalaxy, it demands a larger amount of iron, M Fe , EMP ,and hence, a higher-mass IMF. Despite such uncertain-ties both of the observations and the theoretical assump-tions, the constraints on the IMF derived from eq. (19)are rather robust since the ratio, f SN /f G , is a rapidlyvarying function of IMF. Dependence on Mass-Ratio Distributions
For the four mass-ratio distributions, formulated in § M md and ∆ M ,the portion of stars that survive to date ( M ≤ . M ⊙ ),and then, the fractions of stars in these EMP survivorsthat evolved to CEMP- s and CEMP-no s stars accordingto the masses of primary stars and to the orbital separa-tions. Figure 1 compares the fractions of CEMP- s starsin the EMP survivors [ ψ CEMP - s /ψ surv ] and the ratios ofCEMP-no s to CEMP- s stars [ ψ CEMP - no s /ψ CEMP - s ], pre-dicted with use of these four different mass-ratio distri-butions, as a function of medium mass M md of IMFswith the dispersion of ∆ M = 0 .
33, taken to be sameas the present-day IMF of Galactic spheroid compo-nent (Chabrier 2003). Figures 2 and 3 present the con-tour maps on the M md -∆ M diagram for the fractions ofCEMP- s stars and the ratio between the CEMP-no s andCEMP- s stars, respectively.Left top panels on these figures show the results forthe flat mass-ratio distribution of Case A, which repro-duces the results obtained in Paper I. In Fig. 1, theCEMP- s fraction peaks at M md = 4 . M ⊙ , slightly abovethe upper mass limit of the primary stars for CEMP- s . Note that when the secondary mass is specified, themass distribution of primary stars peaks at mass smallerthan M md for this mass-ratio function [ ∝ ξ ( m ) /m , see Fig. 12 in Paper I). Two ranges of M md , 0 . − . M ⊙ and 7 . − . M ⊙ (light shaded parts) gives the IMFscompatible with the observations, separated by the over-production of CEMP- s stars. The relative frequency ofCEMP-no s to CEMP- s stars is a steep increase func-tion of M md , and excludes the lower range of M md compatible with the CEMP- s fraction. The IMFs with M md = 4 . − . M ⊙ (dark shaded part) gives com-patible ratio with the observations This range of M md lies in the mass range of primary stars of CEMP-no s stars or even larger. Accordingly, the intersection ofthe light and dark shaded parts designates the ranges( M md = 7 . − . M ⊙ ) that can explain the both statis-tics of CEMP stars, and hence, high-mass IMFs resultsfor a dispersion ∆ M = 0 . M md − ∆ M diagram of Fig.2, the parame-ter space compatible with the observed CEMP- s frac-tion separates into two ranges for the dispersion smallerthan ∆ M ≃ .
43, converging to the narrow ranges around M md ≃ M ⊙ , respectively, as ∆ M decreases. Forlarger dispersion, on the other hand, it merges into onepart to cover wider range. As for the ratio between theCEMP-no s and CEMP- s stars, Fig. 3 shows that themedium mass compatible with observed ratio increaseswith the dispersion to cover wider range, from M md =3 . − . M ⊙ at ∆ M = 0 . M md = 12 . − M ⊙ at ∆ M = 0 .
54. Accordingly, for the IMFs that satisfy theboth statistical constraints, the medium mass increaseswith the dispersion from M md ≥ . M ⊙ for ∆ M = 0 . M md = 100 M ⊙ for ∆ M > . q of Case B (right top panel), the portion of bi-naries that have the secondary stars surviving to datedecreases with the mass of primary stars in propor-tion to ( m /m ) − , more steeply than in proportion to( m /m ) − for a flat mass-ratio distribution in Case A.Since the average mass of the primary stars is smallerfor a given EMP star, therefore, the fraction of CEMP- s stars is larger for a given M md , and the peak shifts tolarger M md , as compared with Case A. In Fig. 1, the M md of IMFs compatible with the observed fraction of CEMP- s stars separates into two ranges, as in Case A, but thein-between gap is larger; the higher mass range shifts up-ward in mass to greater extent ( M md = 14 . − M ⊙ )than the smaller mass range shifts downward ( M md =0 . − . M ⊙ ). This also causes a smaller ratio of CEMP-no s to CEMP- s stars for a given M md , and hence, theIMFs compatible with the observed ratio shift to a largermass of M md = 8 . − . M ⊙ , as compared with that forCase A. As a result, the IMFs consistent with the bothstatistics of CEMP stars turns out to be higher mass by afactor of ∼ M md = 14 . − . M ⊙ for∆ M = 0 . M md ,compatible with the observed fractions of CEMP- s star(shaded area), separates into two and the higher rangeshifts to larger mass for a given ∆ M . Similarly, in theFig. 3, the observed ratio of the CEMP-no s to CEMP- s stars also demands larger M md , and the range of M md ofIMFs compatible with the observation increases rapidlywith ∆ M to exceed 100 M ⊙ for ∆ M ≥ .
59. In order tosatisfy the both conditions of CEMP star observations,the IMFs fall in the range of higher medium mass and ina rather narrow range of dispersion, lying in the param-eter space of M md > . M ⊙ , larger by a factor of ∼ . f r a c t i on M md (M sun ) A (cid:13) f r a c t i on M md (M sun ) B f r a c t i on M md (M sun ) C f r a c t i on M md (M sun ) D Fig. 1.—
Constraints on the IMF of EMP population for four cases of assumption for mass-ratio distribution function.
Top left
CaseA: n ( q ) = const. , Top right
Case B. n ( q ) ∝ q , Bottom left
Case C. n ( q ) ∝ /q , and Bottom right
Case D. independent coupling. Thin andthick solid lines denote the fraction of CEMP- s in the EMP survivors as the function of the medium mass M md for a fixed dispersionof ∆ M = 0 .
33 for the EMP stars born as binaries ( ψ CEMP - s ( M md , . /ψ binary ( M md , . ψ CEMP - s ( M md , . /ψ surv ( M md , . s to CEMP- s stars ( ψ CEMP - no s ( M md , . /ψ CEMP - s ( M md , . s stars in the EMP survivors (10 − s to CEMP- s stars (1 / − than for Case A, and ∆ M > .
22 and of ∆ M = 0 . − . M md = 100 M ⊙ .For a mass ratio function decreasing with q of Case C,we see the opposite tendency of Case B (bottom leftpanels of Fig. 1 - 3). The portion of EMP bina-ries whose low-mass members survive to date dependsonly weakly on the primary mass ( ∝ log m ) so thatthe fraction of CEMP- s stars reduces because of largercontributions from the binaries with more massive pri-maries. As seen in Fig. 1, the fraction of CEMP- s starsin the total EMP survivors is well below the upper boundof the observations, and hence, the M md compatiblewith the observations merges into one narrower rangeof M md = 1 . − . M ⊙ for ∆ M = 0 .
33. The observedratio of CEMP-no s to CEMP- s stars can be reproducedalso by the IMFs with a smaller M md by a factor of ∼ M md = 3 . − . M ⊙ ). Accordingly,the M md for the IMFs consistent with the both CEMPstar statistics are smaller by a factor of 1 . − . M md = 3 . − . M ⊙ for∆ M = 0 . M md for theIMFs, compatible with the observed CEMP- s fractionvaries only little with ∆ M , and is restricted in the rangebetween M md = 1 . − M ⊙ , though it separates into twofor small ∆ M < .
31. As shown in Fig. 3 the dependenceof the ratio of CEMP-no s to CEMP- s stars on ∆ M is alsoweaker than for Case A. Consequently, the IMFs can re- produce the both CEMP star statistics with the mass assmall as M md = 3 . M ⊙ , smaller by a factor of ∼ . M md = 23 M ⊙ , regardless ofthe dispersion with a lower bound of ∆ M > . m , of primary stars, while the binary fre-quency itself increases with m . The former is similarlyto Case C, and then, the production of EMP survivorsfrom the binaries with massive primary poses a severeconstraint on the high-mass side of IMFs. On the otherhand, the latter favors the production of CEMP- s starsas compared with the low-mass binaries of m ≤ . M ⊙ .The both shift the IMFs, compatible with the observedfraction of CEMP- s stars, to smaller M md . In addi-tion, the single stars, born in the same number of bi-naries, contribute to significant fraction of EMP sur-vivors, increasing from 1 / / M md for M md < . M ⊙ since the low-mass binaries are countedas one object. As a result, the maximum fraction ofCEMP- s stars remains below the upper limit of the ob-served range, which makes the M md for the IMFs thatcan reproduce the observation lie in a single range withina relatively small upper bound. The observed ratio ofCEMP-no s to CEMP- s stars demands also lower-mass Komiya et al. (cid:131) ¢ (cid:13) M M (M ) md sun A (cid:131) ¢ (cid:13) M M (M ) md sun B (cid:131) ¢ M M (M ) md sun C (cid:131) ¢ (cid:13) M M (M ) md sun D Fig. 2.—
Dependence of the fraction of CEMP- s stars on the medium mass, M md , and dispersion, ∆ M of IMF of EMP population. Solidline denote the contour line of the fraction of CEMP stars among EMP sourvivors ( ψ CEMP - s ( M md , M md ) /ψ surv ( M md , ∆ M )). Attachednumerals designate the fractions. Shaded area denote the parameter ranges for the IMFs that can give rise to the observed fraction ofCEMP- s stars. IMFs, as for Case C. Accordingly, the IMFs that canreproduce the both CEMP star statistics fall in the nar-rowest range of M md = 2 . − . M ⊙ with rather smallupper mass limit, almost irrespective of the dispersion,on the M md − ∆ M diagram in Fig.3. The CEMP- s starfaction remains smaller than ∼
20% because of the con-tribution of the stars born as single.In conclusion, the statistics of CEMP stars demandthe IMFs for the EMP population, peaking at theintermediate-mass stars or the massive stars, by farhigher mass than those of Pop. I and II stars, irrespec-tively of the assumed mass-ratio distribution. The pres-ence of CEMP-no s stars in a significant number of theCEMP- s stars excludes the IMFs of small mass. Thederived mass range varies by a factor of ∼
2, from thehighest M md > . M ⊙ for the mass-ratio distribution ofincrease function of the mass ratio (Case B) to the lowest2 . < M md / M ⊙ < Constraints from Galactic Chemical Evolution
In this section, we discuss that additional constraintscan be derived from the relationship between the totalnumber of EMP survivors and the iron yields from the EMP population.Figure 4 shows the contour maps of the total stellarmass, M EMP , of EMP population, in eq. (10), neces-sary to leave the observed number of EMP survivors onthe M md -∆ M diagram. The total stellar mass increasesfor higher mass IMFs to produce the given number oflow-mass survivors. For the flat mass-ratio distribution(Case A; left panel), the total stellar mass of EMP pop-ulation is essentially determined by M md in proportionto M and only weakly dependent on ∆ M for large M md , since almost all EMP survivors formed as sec-ondary (i.e. second term of eq. [11] is dominant) for IMFof log( M md / . M ⊙ ) ≫ ∆ M . Figure 5 shows the con-tours of the total iron mass, M Fe , EMP , produced by themassive stars of EMP population, in eq. (13). The ironproduction also increases with larger M md but is moresensitive to ∆ M since the dependence of the supernovaefraction differs across the border of M md ≃ M up ; theamount of produced iron increases (or decreases) with∆ M for given M md < M up (or M md > M up ), little de-pendent on ∆ M for M md ≃ M up .For the ”independent” coupling (Case D; left panel),the fractions of giant EMP survivor and supernovaamong EMP population stars are given by, f G = (1 + f b ) ξ (0 . M G (20) f SN = (1 + f b ) Z M up dm ξ ( m ) (21)The total mass and the amount of iron production ofEMP population are sensitive both to M md and ∆ M ,especially for small ∆ M and large M md in contrast toarly-Age evolution of Milky Way 9 (cid:131) ¢ (cid:13) M M (M ) md sun A (cid:131) ¢ (cid:13) M M (M ) md sun B (cid:131) ¢ M M (M ) md sun C (cid:131) ¢ (cid:13) M M (M ) md sun D Fig. 3.—
The contour line of the number ratio between CEMP- s stars and CEMP-no s stars( ψ CEMP - no s ( M md , ∆ M ) /ψ CEMP - s ( M md , ∆ M )) on the M md − ∆ M plane for cases A( Top left ), B(
Top right ), C(
Bottom left ) andD(
Bottom right ), respectively. Attached numerals designate the fractions. Light shaded area denote the constraint from observed fractionof CEMP- s stars as plotted in fig.2. Dark shaded area denote ranges for the IMFs that fulfill the both constraints for the fraction ofCEMP- s and the ratio between CEMP-no s and CEMP- s . ∆ M M md (M sun ) 10 A ∆ M M md (M sun ) 10 D Fig. 4.—
The contour of the total mass of EMP population stars, M EMP = 10 , and 10 M ⊙ on the diagram of parameters (mediummass M md and dispersion ∆ M ) in the log-normal form for the flat distribution (Case A: left pannel) of the mass ratio and independentcounling (Case D: right pannel). Shaded area denotes the parameter range consistent with the statistics of CEMP stars. with the other cases. For small ∆ M , therefore, the frac-tion of low-mass stars varies greatly with M md , and theboth contours of M EMP and M Fe ( M EMP ) converge to M md ≃ − M ⊙ . As ∆ M increases, the differencesfrom Case A diminish since the IMFs tend to extendinto the low-mass stars, and in particular, for ∆ M & . M md . M ⊙ , the contours in the both panels re-semble each other to run through the similar parameterspaces.From the comparison with the total amount ofiron M Fe , EMP , necessary for the chemical evolu-tion, in this diagram, the parameter space where M Fe , EMP ( M md , ∆ M ) ≫ M Fe , halo = 10 . M ⊙ is excludedby the overproduction of iron or by the underproduc-tion of EMP survivors. For the parameter space where M Fe , EMP ( M md , ∆ M ) ≪ M Fe , halo , on the other hand, thestars of EMP population can leave the number of EMPsurvivors currently observed but are short of iron pro-duction, so that the chemical evolution demands othersources of iron production without producing the low-mass stars that survive to date. For a flat mass-ratio dis-tribution, the IMFs that can satisfy the condition of ironproduction coincide the IMFs, derived above from thestatistics of CEMP stars (shaded area) in the parameter0 Komiya et al. ∆ M M md (M sun )012 10 A ∆ M M md (M sun )120 10 D Fig. 5.—
Constraints on the IMF of EMP population, derived from the number of EMP survivors and the iron production by the starsof EMP population on the M md − ∆ M plane. Solid lines denote, the loci of IMFs which can produce the amounts of metal production, M Fe , EMP = 10 , , and 10 M ⊙ . Dashed lines denote the contour of carbon production by AGB stars to the iron production, [C / Fe] = 2,1, and 0. range of M md ≃ − M ⊙ and ∆ M ≃ . − .
6. For the“independent” coupling, the parameter range of IMFsthat satisfy the condition of iron production also over-lap the shaded area of parameter range, derived abovefrom the statistics of CEMP stars, but with the mass M md ≃ . − . M ⊙ , slightly smaller than for Case A andonly for a small dispersion of ∆ M ≃ . − .
35. For larger∆ M , even the highest-mass IMFs of M md = 7 . M ⊙ re-sult to be slightly short of, or marginally sufficient at themost, iron production.For the two other mass-ratio distributions of n ∝ q (Case B) and n ∝ /q (Case C), the iron production, M Fe , EMP , with a given IMF results to be larger or smallerthan for Case A because of the difference in the numberof massive stars exploded as supernova per low-mass sur-vivor (e.g., by a factors of 1.38 and 0.47, respectively, pera star of m = 0 . M ⊙ and the IMF of M md = 10 M ⊙ and ∆ M = 0 . M is small and M md is in the rangeof intermediate- and low-masses, the intermediate-massstars much surpass the massive stars in number and ejectmore carbon than the latter eject iron. We compute theamount of carbon ejected by AGB stars by taking thecarbon abundance in the wind ejecta of AGB stars at[C / H] = 0, and the remnant mass at 1 M ⊙ . Contours of[C / Fe] = 2 , , M < . f SN /f g depends strongly onthe IMF. Distinctive Features of EMP Survivors
The different assumptions on the mass-ratio distribu-tions admit the parameter ranges of high-mass IMFs thatcan reproduce the statistics of CEMP stars and the chem-ical evolution, consistent with the existent observations.The predicted mass ranges differ by a factor of 2 or morebetween M md ≃ − M ⊙ . Although hardly distinguish-able from the observations discussed so far, they surelymake the differences in the properties of EMP survivors.arly-Age evolution of Milky Way 11 ξ ( m ) m(M sun ) singleCase ACase BCase C Fig. 6.—
The mass function of EMP survivors under the differentassumptions on the coupling mass distribution of binaries. Here theparameters of IMFs are taken to be M md = 10 M ⊙ and ∆ M = 0 . We discuss the imprints that the mass-ratio distributionshave left on the current EMP survivors and investigatethe possibility of discriminating the mass coupling of bi-nary systems in the EMP population, especially for thetwo distinct distributions of the flat mass-ratio distribu-tion and the “independent” coupling.Firstly, an obvious difference is the mass distributionfunction of EMP survivors. For a given IMF, ξ ( m ),the mass distribution, ξ EMP - surv ( m ), of EMP survivorsis given by; ξ EMP - surv ( m ) = (1 − f b ) ξ ( m ) + f b ξ ( m ) Z m n ( m /m ) /mdm + f b Z . M ⊙ ξ ( m ) n ( m/m ) /m dm . (22)Here a low-mass binary, whose components are both lessmassive than 0 . M ⊙ , is counted as one object with theprimary star. Figure 6 shows the mass distributions ofEMP survivors ( m ≤ . M ⊙ ) for different assumptionsof mass-ratio distributions Cases A-C. For these mass-ratio functions, the mass distribution of EMP survivorsis nearly proportional to the mass-ratio distribution n ( q )because almost all of them come from the secondarystars; the contribution from the primary components aredenoted by thin solid line, and the same contributioncomes from the stars born as single. For the “indepen-dent” coupling, in contrast, the ξ EMP - surv ( m ), has thesame form as the IMF and the number of EMP survivorsdecreases rapidly as the stellar mass decreases.Secondly, the fraction of double-lined binary and thecontribution of stars born as single among EMP sur-vivors may differ according to the mass-ratio distribu-tion. The EMP survivors born as binaries are dividedinto three categories according to the mass of the pri-mary stars: (1) the low-mass binaries with the primaryof mass m ≤ . M ⊙ , (2) the white-dwarf binaries ofprimary stars of mass between 0 . M ⊙ < m ≤ M up ,and (3) the supernova binaries of primary stars of mass m > M up . The fraction of low-mass binaries with theprimary stars of mass m ≤ . M ⊙ in the EMP survivorsof mass between m to m + dm is given by ϕ surv , LMB ( m ) = f b [ ξ ( m ) /m ] Z m n ( m /m ) dm /ξ EMP - surv ( m ) . (23)They can be detected as double-lined binary. For the flat mass-ratio distribution, this gives a significant frac-tion of ϕ surv , LMB (0 . M ⊙ ) = 7 .
3% for M md = 10 M ⊙ and∆ M = 0 .
4, and increases with ∆ M to 16% for ∆ M = 0 . M md to 18% for M md = 5 M ⊙ , re-spectively. We note that these values depend weakly on f b since most of the EMP survivors are from the bina-ries. The number of low-mass binary decreases rapidlyfor smaller masses while the number of EMP survivors,formed as the low-mass members of white dwarf binariesor supernova binaries, remains constant.For the “independent” coupling, the fraction of low-mass binaries in the EMP survivors reduces to; ϕ surv , LMB ( m ) = 2 f b Z m ξ ( m ) dm ," (1 + f b ) − f b Z . M ⊙ m ξ ( m ) dm , (24)which gives a much smaller fraction of ϕ surv , LMB (0 . M ⊙ ) = 1 .
6% for M md = 5 M ⊙ and∆ M = 0 . M md = 3 M ⊙ , and for larger dispersion,to 3.9 % and 9.5% at ∆ M = 0 . M md , as seen from Fig. 5(bottom panel). In this case, the proportion of the EMPsurvivors, born as single stars, is fairly large as given by ϕ surv , sing ( m ) ≃ (1 − f b ) , " (1 + f b ) − f b Z . M ⊙ m ξ ( m ) dm . (25)Consequently, nearly one third of EMP stars were bornas single stars, for f b = 0 .
5, which is much larger fractionthan in the case of the flat mass-ratio distribution.Thirdly, the fraction, ϕ surv , SNB , of supernova binarieswith the primary stars of mass m > M up also differsbetween the two mass-ratio distributions. For the flatmass-ratio distribution, almost all of the EMP survivorsbelong, or have been belonged, to the binary systems,and the fraction is given by ϕ surv , SNB ( m ) = f b Z M up n ( n/m ) ξ ( m ) /m dm /ξ EMP - surv ( m ) , (26)and amounts to ∼ ϕ surv , SNB ( m ) = 2 f b Z M up ξ ( m ) dm ," (1 + f b ) − f b Z . M ⊙ m ξ ( m ) dm , (27)and turns out to be ∼ / H] < −
3, for which the observa-tions with high-resolution spectroscopy may be re-garded as unbiased, there are two stars CS22876-032([Fe / H] ≃ − . V = 12 . P = 424 . m /m ≃ .
89; Thorburn & Beers 1993; Norris et al.2000; Gonz´alez et al. 2008) and CS 22873-139 ([Fe / H] ≃− . V = 13 . P = 19 .
165 d, m /m ≃ .
92; Preston1994, 2000; Spite et al. 2000) with the detailed analysesand one star HE 1353-2735 ([Fe / H] ≃ − . V = 14 . / H] < − g [ cm s − ] ≥ . & / ≃ . ϕ surv , LMB (0 . M ⊙ ) = 7 .
3% for thefrat mass-ratio distribution but to be a little larger, ormarginal, for the “independent” coupling. We note, how-ever, that only with two samples, the above fraction maybe subject to significant observational selection effects.These stars have to be concentrated near to the upper-end of main-sequence since they are found among thecandidates, selected from the flux limited surveys, andthe mass ratio has to be sufficiently large for the linesof two components to be observed. The actual fractionhas to be larger than observed if we take into accountthe detection probability due to the orbital phase andto the inclination angle, and the rather narrow range ofmass-ratios for the observed double-lined binaries. Onthe contrary, the larger survey volume by a factor upto 2 / for the double-lined binaries due to the sum ofluminosities may reduce the actual fraction. More obser-vations for the main-sequence EMP binaries and the biascorrections are necessary to discriminate the mass ratiodistributions.It may be more straightforward to compare our resultswith the mass distribution function of EMP survivors.From the existent observations, however, it is ratherhard to determine since the observed dwarfs are mostlyconcentrated near to the upper end of main sequence.An exception is a carbon dwarf G77-61 of [Fe / H] =4 .
03 (Plez & Cohen 2005) whose mass is inferred at0 . − . M ⊙ , but it was found among the proper-motion-parallax stars (Dahn et al. 1977), not from the surveys.We have to wait for the larger-scaled surveys in near fu-ture to reveal the distribution of EMP survivors of lowmasses. As for the supernovae binaries, they are ex-pected to be related to the large star-to-star variationsin the surface elemental abundances, in particular, withthose of r-process elements, ranging more than by twoorders of magnitude. It is necessary, however, to un-derstand the nature of interactions between the super-nova ejecta colliding at very high velocity and the near-by low-mass stars before the meaningful conclusions canbe drawn from the observations. METALLICITY DISTRIBUTION FUNCTION OF EMPSTARS
We have shown that the high-mass IMFs with the bi-nary provide a reasonable explanation of the observedproperties of EMP stars in the Galactic halo, revealedby the recent large-scaled HK and HES surveys. Inthis section we discuss the consequence of derived IMFon the metal enrichment history of Galactic halo up to[Fe / H] = − . M md = 10 M ⊙ and the disper-sion ∆ M = 0 . Simple Model of Chemical Evolution
Under the assumption that matter ejected from su-pernovae spreads homogeneously and is recycled instan-taneously, the iron abundance, X Fe , of our Galaxy of(baryonic) mass M h can simply be related to the cu-mulative number, N ( X Fe ), of the stars born before themetallicity reaches X Fe as; M h X Fe = h Y Fe i N ( X Fe ) f SN . (28)where h Y Fe i is the averaged iron yield per supernova and f SN is the fraction of EMP stars that have exploded as su-pernovae, defined in eq. (14). By differentiating it withrespect to [Fe / H] = log( X Fe /X Fe , ⊙ ), the number dis-tribution of EMP survivors is written as a function ofmetallicity in the form n ([Fe / H]) = dN ( X Fe ) d [Fe / H] = M h h Y Fe i f SN ln(10) X Fe ⊙ [Fe / H] . (29)This shows that the number distribution of EMP sur-vivors is simply proportional to the iron abundance apartfrom the variation of h Y Fe i through the IMF and the lat-ter is small enough to be neglected for M md . M ⊙ (see Fig. A1 in Appendix).Figure 7 depicts the number distribution of EMP sur-vivors and compares it with the observed MDF pro-vided by the HES survey (Beers et al. 2005). We as-sume stars of mass m > M up = 8 M ⊙ become typeII supernova and eject h Y Fe i = 0 . M ⊙ of iron. Inthis figure, the theoretical MDF, ν surv , is evaluatedunder the same flux-limited condition as the observedMDF is derived; ν surv ([Fe / H]) = n ([Fe / H]) f G × (40%) × (8225 degree / π sr) × .
93. Here the fraction of follow-up observation and the sky coverage are taken into ac-count: as for the contribution of TO stars, we take thesame ratio to the giants as in the observed sample underthe assumption that the giant survivors are all reachedin the survey area. Solid line shows the MDF for theIMF with M md = 10 M ⊙ and ∆ M = 0 . § − . [Fe / H] . − .
5, as expected from thediscussion in the previous section.In this figure, we also plot the MDF using the low-mass IMFs, the Salpeter’s power-law mass-function asarly-Age evolution of Milky Way 13 ν ( [ F e / H ] ) ∆ [ F e / H ] [Fe/H]M md =10, ∆ M =0.4SalpeterLucatello(2005) Fig. 7.—
Comparison of the theoretically predicted MDF fromthe IMF derived from the statistics of EMP stars (solid line) withthe observed MDF obtained by the HES survey (shaded columns;Beers et al. 2005). The number distribution, ν surv is computedwith the log-normal IMF of medium mass M md = 10 M ⊙ and dis-persion ∆ M = 0 . M md = 0 . M ⊙ and∆ M = 1 .
18, dotted line), derived by Lucatello et al. (2005) fromthe CEMP- s star statistics alone, which bring about the overpro-duction of low-mass survivors. observed among the present-day stellar populations andthat derived only from the statistics of CEMP- s starsby Lucatello et al. (2005, M md = 0 . M ⊙ and ∆ M =1 . / ln 10). They bring about the overproduction ofEMP survivors by a factor of more than a few hundredsnot only from the low-mass members of binaries but alsofrom the primary stars and the single stars; both theIMFs give the similar MDF since our flux-limited sam-ples are dominated by the giants and luminous dwarfs ofmass M ≃ . M ⊙ . This means that the EMP survivorsis by far a small population as compared with the stellarsystems of Pop. I and II, and it is only with the high-mass IMFs that can make the EMP population producesufficient amount of metals to enrich the early Galactichalo without leaving too many low-mass survivors nowobservable in Galactic halo.In addition, we see in this figure that the slope of ob-served MDF is consistent with the prediction from thesimple one-zone approximation at least for [Fe / H] > − / H] ≃ −
2, the observed MDF derived fromthe HK and HES surveys seems to be underestimatedsince those objects are out of the metallicity range soughtafter by the survey and subject to imperfect selection.
Effect of Hierarchical Galaxy Formation
The observed MDF of Galactic halo stars has a suddendrop at [Fe / H] . −
4, and only three stars are foundbelow it . We propose the mechanism responsible for thisdepression of low-metallicity stars from the considerationof the Galaxy formation process. There are two more stars with the iron abundances reportedbelow [Fe / H] < −
4; CD − ◦
245 with [Fe / H] = − . ± . / H] = − . ± . / H] = − . ± . − . ± .
15 (Norris et al. 2001) havebeen reported for the former, and hence, their abundances arecloser to the EMP stars of [Fe / H] & − In the current cold dark matter (CMD) model, galax-ies were formed hierarchically. They started from lowmass structures and grew in mass through merging andaccreting matter, finally to be large-scale structures likeour Galaxy. In the hierarchical structure formation sce-nario with ΛCDM cosmology, the typical mass of firststar forming halos is ∼ M ⊙ for the dark matter and ∼ × M ⊙ for the gas (e.g., see Tegmark et al. 1997;Spergel et al. 2007).In these first collapsed gas clouds, the first stars con-tain no pristine metals except for lithium. When the firststar explodes as supernova, it ejects ∼ . M ⊙ of iron,which enriches the gas cloud of mass ∼ × M ⊙ whereit was born up to the metallicity of [Fe / H] ∼ − . / H] ∼ − . M ⊙ . We may take the metallicity of thisearly Galactic halo to be [Fe / H] ≃ − / H] < −
4. The cu-mulative number of stars born before the early Galac-tic halo is enriched up to [Fe / H] = − N (10 − X Fe ⊙ ) = 7 × with taking into account the su-pernova fraction f SN . If the mini-halos of larger massesstand between the first collapsed halos and the Galac-tic halo, the dilution of iron with unpolluted primordialgas can give birth to the stars of smaller metallicity of[Fe / H] ≃ −
4, and then, the metallicity at the formationof Galactic halo can be larger to increase the cumulativenumber of stars in accordance (see below).We may estimate the fractions of both the first gener-ation stars without metals and the 2nd generation starsof the metallicity [Fe / H] ∼ − .
5, respectively, assumingthat stars are born with an equal probability whether inthe gas clouds, polluted with metals, or in the primordialgas clouds. Here it is worth noting that some recentcomputations of star formation demonstrate that thelow-mass stars can be formed as the binary members evenout of metal-free gas (Clark et al. 2008; Machida et al.2008). Accumulated number, N PopIII , of Pop III stars,born of gas unpolluted by SN ejecta, when the averagemetallicity reaches X Fe , is given by N PopIII = M h M c f SN (cid:20) − exp (cid:18) − M c X Fe h Y Fe i (cid:19)(cid:21) , (30)where M c (= 2 × M ⊙ ) is the mass of gas in the firststar forming clouds. If we assume the same IMF andbinary parameters as in the stars of EMP population,then, we expect that the number of Pop III stars is N PopIII (10 − X Fe , ⊙ ) = 3 . × , (31)and the number of Pop III survivors is3 . × × Z . M ⊙ . M ⊙ dm (cid:2) ξ ( m ) + f b Z . M ⊙ n ( m/m ) ξ ( m ) dm m (cid:3) = 1 . × , (32)and similarly we have 5 . × and 2 . × of the 2ndgeneration stars and their survivors, formed before theaveraged metallicity of the Galaxy reaches [Fe / H] = − / H] < − / H] ∼ − ∼
23 Pop. III survivors in the existing flux-limited samples of HES surveys. It is true, however, thatthere is no star with zero metallicity among the starsdetected by the existent surveys. We may propose onescenario to explain this absence that Pop. III survivorsare no longer remain metal-free at present since their sur-face are polluted by the accretion of interstellar matter,enriched with metals ejected by the supernovae of thefirst and subsequent generations. With the surface pol-lution of [Fe / H] ∼ −
5, they are observed as HMP stars.We discuss about evolution of Pop. III stars with pollu-tion in § ∼ CONCLUSIONS AND DISCUSSION
We have studied the initial mass function (IMF) andthe low-mass star formation with the chemical evolutionof the Galactic halo population on the basis of the charac-teristics of extremely metal-poor (EMP) stars, revealedby the recent large-scaled HK and HES surveys; the ob-servational facts that we make use of are; (1) the over-abundance of carbon-enhanced EMP (CEMP) stars, (2)the relative frequencies of CEMP stars with and withoutthe enrichment of s-process elements, (3) the estimateof surface density or total number of EMP stars in theGalactic halo, and (4) the metallicity distribution func-tion (MDF). We take into account the contribution ofbinary stars properly, as expected from the younger pop-ulations. In Paper I, the high mass IMF peaking around ∼ M ⊙ is derived for the stars of EMP population andit is shown that the binary population plays a major rolein producing the low-mass stars that survive to date,but by using the flat mass-ratio distribution betweenthe component stars. In this paper, we examine theseproperties of the stars of EMP population and EMP sur-vivors for the different types of mass-ratio distributionsand investigate the constraints on the IMFs of the starsof EMP population and discuss the observational testsof discriminating them. The derived IMFs are applied tounderstand the characteristics of MDF and the nature ofEMP stars including HMP/UMP stars, provided by thesurveys.Our main conclusions are summarized as follows;(1) The statistics of CEMP stars are explained by thehigh-mass IMFs with the binaries of significant fraction. Predicted typical mass is significantly larger than Pop-ulation I or II stars, irrespective of the assumptions ofthe mass-ratio distribution. The mass-ratio distribu-tion with a preference for nearby equal masses demandsthe IMF with higher typical mass M md > M ⊙ andsmaller dispersion (∆ M < . M md ∼ M ⊙ irrespective of ∆ M .(2) High mass IMFs with M md ∼ − M ⊙ derivedfrom the statistics of CEMP stars agree with those de-rived from the low-mass star formation and the chemicalevolution of Galactic halo based on the number of gi-ant EMP sourvivors evaluated from the surveys. IMFswith M md ≫ M ⊙ are excluded by overproduction ofiron or underproduction of EMP survivors. IMFs with M md ≪ M ⊙ need other iron source(s) without produc-ing low-mass stars.(3) The mass-ratio distribution of binaries in the EMPpopulation can be discriminated by the imprints left onthe EMP survivors such as the mass function, the binaryfraction, and the fraction of stars influenced by the su-pernova explosion of primary stars. In particular, theflat mass-ratio distributions predict significant fraction( ∼ . f orM md = 10) of double-lined spectroscopic bi-naries while the mass-ratio distribution of “independent”coupling predict much lower fraction. Among the 39 un-evolved stars of [Fe / H] < −
3, studied spectroscopicallyto date, three double-lined binaries are found, but theremay be significant uncertainties and biases for the ex-istent surveys and future observations can discriminatethe distributions.(4) The observed MDF of EMP survivors is consequentupon the derived IMF with the contribution of the bina-ries. There is no indication of significant change in theIMFs between the metallicity of − . [Fe / H] . −
2. Thedepression of stars below [Fe / H] < − / H] . − .
5) with the detailed stellar param-eters amount to ∼
400 in number (SAGA Database;Suda et al. 2008), and allow us to discuss the averagedproperties as studied in this paper. The existent sur-veys reach sufficiently deep and the nominal depth inthe magnitude 12 < B < . d ≃
10 kpc or beyond for giants of L ≃ L ⊙ while d ≃ . − ρ ∼ r − ∼− . (Majewski 1993). Furthermore, be-cause of strong dependence of the number ratio betweenthe low-mass stars that survive to date to the massivestars that have exploded as supernovae on IMF, our dis-cussion in § / H] & −
2, Ryan & Norris (1991) report that themetallicity distribution extends continuously from thepeak at [Fe / H] ≃ − . / H] ≃ − M ∼ . M ⊙ in the diverse conditions includ-ing the Galactic spheroid population of the metallicity[Fe / H] ≃ − . − − . / H] ≃ − / H] ≃ −
4. The detailedchemical evolution with the merger history taken intoaccount is discussed in a subsequent paper (Komiya etal. 2008, in preparation), in the similar ways as doneby Tumlinson (2006) and by Salvadori et al. (2007), buttaking into account the high-mass IMF, derived above,and the contribution of binaries.
Origin of HMP/UMP Stars
We end by discussing the consequences of the presentstudy on the understanding of the origin of stars foundbelow the cut-off of MDF.In our model, the stars made after the first pollutionhave the metallicity [Fe / H] ≃ − . / H] ≃ − . −− / H] ≃ − / H] ≃ − ∼
100 times in mass on the giant branch. Thus,the Pop. III survivors have evolved to giants to be ob-served as HMP/UMP stars. As for a sub-dwarf HMPstar HE1327-2623, the dilution of the accreted iron groupelements has occurred in the envelope of primary star onthe AGB because of the low-mass nature of its primarystar ( M . . M ⊙ , Nishimura et al. 2008).Majority of the Pop. III survivors have also to be thesecondary members of binary systems similar to the EMPsurvivors if their IMF and binary parameters are similarto EMP stars. Then some of Pop. III stars becomecarbon-enriched HMP/UMP stars with [Fe / H] ∼ − . M ⊙ < m < . M ⊙ and 3 . M ⊙ . m < M up , theprimary star enhances the surface abundances of carbonand nitrogen though the He-FDDM and of carbon and/ornitrogen through TDU and hot bottom burning in the en-velope, respectively, which are transferred onto the sec-ondary stars through the wind accretion. It is to be notedthat the primary stars of m > M ⊙ have the accretedpollutants mixed inward into the whole hydrogen-richenvelope at the second dredge-up, and thereafter, evolvelike the stars with the pristine metals. At the same time,the accreted matter is diluted in the envelope and theiron abundance is reduced to [Fe / H] ∼ − ∼
35% of Pop. III starsbecome carbon-rich HMP/UMP stars under the same as-sumptions on the binary parameters as in Paper I. Thesurface abundances of main sequence stars can be smallerthan stated in Paper I, however, since the accreted mat-ter mixes down and are reduced by an order of mag-nitude if the diffusion and thermohaline mixing works(Weiss et al. 2000; Stancliffe et al. 2007).In Fig. 8, solid lines denote the expected MDF atthe present days with the surface pollution taken intoaccount. The basic form of observed MDF is repro-duced, i.e., the cutoff around [Fe / H] ∼ −
4, the scarcityof stars for the metallicity below it and the existenceof a few HMP/UMP stars. From the above estimates,there should be ∼
23 Pop. III stars in the existent flux-limited samples of HES surveys; about a half of themmay be discovered as giants with the surface metal pol-lution and one third as carbon stars. In actuality, onlythree HMP/UMP stars (two giants and one sub-dwarf)6 Komiya et al. ν (cid:13) ( [ F e / H ] ) (cid:13) ∆ (cid:13) [ F e / H ](cid:13) [Fe/H](cid:13) surface pollution1ststars 2ndstarsdilution Fig. 8.—
Schematic drawing of MDF of EMP and Pop. III sur-vivors, constructed based on the hierarchical scenario for structureformation. Primordial main sequence stars with Z = 0 pollutedthrough the after-birth accretion of the interstellar gas, enrichedwith iron ejected by supernovae of the first and subsequent genera-tions, upto [Fe / H] ∼ −
3, and pollutants is diluted to [Fe / H] ∼ − are found to date, all enriched with carbon, among 153stars of [Fe / H] < −
3, registered in SAGA database(Suda et al. 2008). Since such low metal abundances canbe discriminated only with high dispersion spectroscopy, ∼ . / H] < −
3. The observednumbers are significantly smaller than prediction fromour model. The above estimates are made, however, un-der the assumption that the Pop. III stars are formed in the same IMF as EMP stars and with the same binaryparameters. This may not be warranted and rather wemay take that this deficiency may suggest a still higher-mass IMF and/or less efficiency of binary formation forPop. III stars than the EMP stars.In the above discussion, we assume the closed boxchemistry in the collapsed object before merging. Itis shown that the hypernovae, exploded with a largeenergy of 10 erg, blow off the first collapsed objectsof mass M ≃ M ⊙ (Machida et al. 2005); if thefirst stars are sufficiently massive, the metal yields arespread into larger masses, and pollute the ambient gasbefore they collapse to form mini-haloes, as discussedby Salvadori et al. (2007). After that, the first stars inthe collapsed clouds are no longer metal-free. Never-theless, those stars which are formed before each col-lapsed clouds are polluted by their own supernova forma distinct class from those which suffer from the firstpollution. Further study is necessary to make clear thepresent appearance of the possible Pop III survivors andto settle the origin of HMP/UMP stars, in particular, fortiny amounts of iron-group metals and the overwhelmingcarbon-enhancement, shared by all these stars known todate.We benefit greatly from discussion with Dr. W. Aoki.This paper is supported in part by Grant-in-Aid for Sci-entific Research from Japan Society for the Promotion ofScience (grant 18104003 and 18072001). APPENDIX
LOW-MASS STAR FORMATION AND IRON PRODUCTION BY EMP POPULATION
One of the important findings of the recent large-scaled surveys is the scarcity of EMP stars in the Galactic Halo. TheHES survey gives the total number of EMP stars in our Galactic Halo at σ EMP ≃
796 sr − (giants of σ EMP , G ≃
412 sr − in eq. (17) plus turn-off stars σ EMP , TO ≃
384 sr − ) within the limiting magnitude B ≃ .
5. Similarly, the HK surveygives σ EMP ≃
528 sr − within the limiting magnitude of B ≃ .
5; 114 stars of [Fe / H] < − and 4100 deg areas in the Northern and Southern Hemisphere (Beers & Christlieb 2005). Because of thesignificantly large areas covered by these surveys ( ∼
20% of all sky with the follow-up observations), we may placereliance on these results, granted that they may not be complete. This also constrains on the IMF of stellar populationthat promoted the chemical evolution, or more specifically, the formation of metals and the low-mass survivors. Inthe paper, we have discussed the chemical evolution starting with the statistics of CEMP stars. In this Appendix, weshow that the chemical evolution with the total number of EMP survivors provides more stringent constraints on theIMF of EMP population with the aid of the amount of ejecta from supernova models, independently of the statisticsof CEMP stars.Our basic premise is that the same stellar population is responsible both for the production of metals and of low-mass survivors. In discussing the low-mass survivors, it is indispensable to take into account the contribution from thebinaries. This is one of the major conclusions in Paper I. We assume that the stars are born not only as single starsbut also as the members of binaries in an equal number and with the primary stars in the same IMF as the singlestars. For a given IMF, then, the total number, N EMP , surv of EMP survivors, currently observed in the Galactic halo,is related to the cumulative number, N EMP , of stars of EMP population as; N EMP , surv = N EMP f surv = N EMP Z . M ⊙ dm [ ξ ( m ) + f b Z m . M ⊙ n ( m/m ) dm m ] , (A1)and hence, to the cumulative number of EMP supernovae as N EMP , SN = N EMP f SN = N EMP , surv ( f SN /f surv ). Thesesupernovae have to supply the amount of iron, M Fe , EMP in eq. (18), in order to enrich the gas in the Galaxy of mass M h with iron to promote the chemical evolution up to the metallicity [Fe / H] = − .
5. Then, we may derive the averagediron yield, h Y Fe i EMP , per supernova of EMP population, necessary to explain the chemical evolution of Galaxy, by therelation h Y Fe i EMP = M Fe , EMP /N EMP , SN for an assumed IMF with the mass-ratio distribution function.We show in Figure A1 the averaged yield, h Y Fe i EMP , as a function of M md for ∆ M = 0 .
4: upper panel for theobservations of EMP stars of different evolutionary stages from the HES survey and of the total EMP stars fromarly-Age evolution of Milky Way 17 < Y F e > M md UN02WW95+HW02HES(total)HES(giant)HES(dwarf)HK(total) 0.01 0.1 1 0.1 1 10 100 < Y F e > M md CaseACaseBCaseC
Fig. A1.—
The average iron yields per supernova, h Y Fe i EMP , demanded from the chemical evolution of Galactic halo that leaves theEMP survivors consistent with the observed flux-limited samples, and the theoretical iron yields, h Y Fe i SN , computed from the theoreticalsupernova models with use of ion yields by Umeda & Nomoto (2002), and by Woosely & Weaver (1995) and Heger & Woosley (2002) as afunction of M md with ∆ M = 0 .
4. Top panel compares h Y Fe i EMP computed for the different samples of EMP stars, i.e., giants, dwarfs andtotal stars from the HES survey and total stars from HK survey, with the flat mass ratio function, while the bottom panel compares theresults with the different mass-ratio distributions for the giant samples of the HES survey. the HK survey with use of the IMFs with the flat mass-ratio function, and lower panel for the different mass-ratiofunctions with use of the observation of EMP giants from the HES survey. In order to compare the stars of differentevolutionary stages, we include the effects of the limiting magnitude of the surveys by assuming the de Vaucouleursdensity distribution, ρ ∝ exp( − r / ), with the radial distance, r , from the Galactic center, the same as the stars in theGalactic halo and by assigning the luminosity of L = L ⊙ ( M/ M ⊙ ) . and 100 L ⊙ to dwarfs and giants, respectively.The amount of iron demanded by the chemical evolution turns out to be a steep decrease function of M md since inorder to leave a fixed number of low-mass survivors, the total number of stars of EMP populations, and hence, thesupernova fraction increase rapidly with M md in particular near M md ≃ M up . The necessary yields computed fromthe different samples in upper panel show a fairly good agreement with each other. The difference between the giantsand dwarfs for the HES samples is indicative of a relatively deficiency of dwarf stars compared with giants by a factorof ∼ . ∼ . ∼ q give a larger (smaller) number of supernovae to produce one EMP survivors; the difference of which increasesfor higher-mass IMFs.These iron yields necessary to promote the chemical evolution may be compared with the theoretical iron yieldspredicted from the supernova models. The IMF-weighted iron yields, h Y Fe i SN , per supernova is given by using the ironmass, Y Fe ( m ), ejected from a massive star of initial mass m as; h Y Fe i SN = R M up dm ξ ( m )[ Y Fe ( m ) + f b R M up /m Y Fe ( m ) n ( q ) dq ] R M up dm ξ ( m )[1 + f b R M up /m n ( q ) dq ] . (A2)The IMF averaged yield h Y Fe i SN is also shown in this figure, for which the theoretical yields are taken from themetal-deficient supernova models computed by Umeda & Nomoto (2002), and by Woosely & Weaver (1995) andHeger & Woosley (2002). It is a slowly increase function of M md for M md . M ⊙ with the increase in the fractionof more massive stars that ended as supernovae, while beyond it, the gradient grows steeper owing to the contributionof the electron pair-instability supernovae of M > M ⊙ .The averaged yields, demanded by the chemical evolution, and the theoretical IMF-weighted iron yields both meetwith each other near M md ≃ − M ⊙ and with the iron yield h Y Fe i ≃ . − . M ⊙ per supernova. As typicallyseen for the flat mass-ratio distribution, the parameter range coincides with that we have derived for the IMFs fromthe CEMP statistics in Figs. 1-3. For higher-mass IMFs, the EMP stellar population cannot produce the sufficientnumber of low-mass survivors by themselves, while for lower-mass IMFs, it results short of iron production. Thedifferences arising from the mass-ratio distributions seem discernible but not large enough to differentiate these mass-ratio distributions in view of the uncertainties of current observations. As compared with the flat distribution, themass-ratio distribution increasing (decreasing) with q demands smaller (larger) M md , the opposite tendency derivedfrom the CEMP statistics. These distributions prefer smaller (larger) number of EMP survivors, larger (smaller)fraction of CEMP- s stars and smaller (larger) ratio of CEMP-no s to CEMP- s stars. In principle, however, we candiscriminate the mass-ratio functions in the EMP binaries, including those destructed already by the evolution, withuse of the survey and observations of EMP stars in sufficiently large number and with sufficient accuracy, which waitsfor future works.8 Komiya et al.In summary, the observed surface density of EMP stars indicates the high-mass IMF for the stars of early stageof Galactic evolution independently of the CEMP star statistics, and also indifferently of the assumed mass ratiodistribution function. It is true that the current estimate of N EMP /, surv may be subject to significant errors, andyet, this result is robust because of a strong dependence of h Y Fe on M md , as seen from the figure, which gives δ log M md ≃ . N EMP surv . In the above discussion, we assume a single log-normal IMF with the binary fractionfor the stellar population. It is possible to assume the bi-modal IMF and to explain the production of iron and theformation of low-mass stars, separately, in terms of the combination of two stellar populations, one with a higher-massIMF responsible for the iron production and the other with a lower-mass IMF for the low-mass survivors, respectively.In the case of bi-modal IMFs, the constraints, derived here, place an upper mass limit to the IMF of lower-masspopulation and an lower mass limit to the IMF of higher-mass population. It is to be noted that the IMF with thebinary mass function of Cases A-C is regarded as a sort of bi-modal IMF with the primary plus single stars as thehigher-mass population and the secondary stars as the lower-mass population (see Fig. 12 in Paper I); the separationof two IMFs differs with the mass-ratio function and the relative contributions of two populations vary with the binaryfraction. 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