Descendants of the first stars: the distinct chemical signature of second generation stars
Tilman Hartwig, Naoki Yoshida, Mattis Magg, Anna Frebel, Simon C. O. Glover, Facundo A. Gómez, Brendan Griffen, Miho N. Ishigaki, Alexander P. Ji, Ralf S. Klessen, Brian W. O'Shea, Nozomu Tominaga
MMon. Not. R. Astron. Soc. , 1–18 (2017) Printed 17 July 2018 (MN L A TEX style file v2.2)
Descendants of the first stars: the distinct chemicalsignature of second generation stars
Tilman Hartwig , , , (cid:63) , Naoki Yoshida , , Mattis Magg , Anna Frebel , Simon C.O. Glover , Facundo A. G´omez , , Brendan Griffen , Miho N. Ishigaki , AlexanderP. Ji , , Ralf S. Klessen , , Brian W. O’Shea , , , , Nozomu Tominaga , Department of Physics, School of Science, University of Tokyo, Bunkyo, Tokyo 113-0033, Japan Kavli IPMU (WPI), The University of Tokyo, Kashiwa, Chiba 277-8583, Japan Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7095, Institut d’Astrophysique de Paris, 75014 Paris, France CNRS, UMR 7095, Institut d’Astrophysique de Paris, 75014, Paris, France Universit¨at Heidelberg, Zentrum f¨ur Astronomie, ITA, Albert-Ueberle-Straße 2, 69120 Heidelberg, Germany Department of Physics and Kavli Institute for Astrophysics and Space Research, MIT, Cambridge, MA 02139, USA Instituto de Investigaci´on Multidisciplinar en Ciencia y Tecnolog´ıa, Universidad de La Serena, Ra´ul Bitr´an 1305, La Serena, Chile Departamento de F´ısica y Astronom´ıa, Universidad de La Serena, Av. Juan Cisternas 1200 N, La Serena, Chile The Observatories of the Carnegie Institution of Washington, 813 Santa Barbara St., Pasadena, CA 91101, USA Universit¨at Heidelberg, Interdisziplin¨ares Zentrum f¨ur Wissenschaftliches Rechnen, INF 205, 69120 Heidelberg, Germany Department of Computational Mathematics, Science and Engineering, Michigan State University, MI, 48823, USA Department of Physics and Astronomy, Michigan State University, MI, 48823, USA National Superconducting Cyclotron Laboratory, Michigan State University, MI, 48823, USA Joint Institute for Nuclear Astrophysics - Center for the Evolution of the Elements, USA Department of Physics, Faculty of Science and Engineering, Konan University, 8-9-1 Okamoto, Kobe, Hyogo 658-8501, Japan Hubble Fellow
17 July 2018
ABSTRACT
Extremely metal-poor (EMP) stars in the Milky Way (MW) allow us to infer theproperties of their progenitors by comparing their chemical composition to the metalyields of the first supernovae. This method is most powerful when applied to mono-enriched stars, i.e. stars that formed from gas that was enriched by only one previoussupernova. We present a novel diagnostic to identify this subclass of EMP stars. Wemodel the first generations of star formation semi-analytically, based on dark matterhalo merger trees that yield MW-like halos at the present day. Radiative and chemicalfeedback are included self-consistently and we trace all elements up to zinc. Mono-enriched stars account for only ∼
1% of second generation stars in our fiducial modeland we provide an analytical formula for this probability. We also present a novelanalytical diagnostic to identify mono-enriched stars, based on the metal yields of thefirst supernovae. This new diagnostic allows us to derive our main results indepen-dently from the specific assumptions made regarding Pop III star formation, and weapply it to a set of observed EMP stars to demonstrate its strengths and limitations.Our results may provide selection criteria for current and future surveys and thereforecontribute to a deeper understanding of EMP stars and their progenitors.
Key words: early Universe – stars: Pop III – Local Group – stars: abundances –methods: analytical
The first stars in the Universe (the so-called “Pop III” stars)are of fundamental importance for understanding galaxy for- (cid:63)
E-mail: [email protected] mation. They enriched the primordial interstellar medium(ISM) and intergalactic medium with heavy elements, theycontributed to the reionization of the Universe, and theyplayed a crucial role in the formation of the first supermas-sive black holes. Owing to the lack of efficient coolants inmetal-free gas, we expect the first stars to have a higher c (cid:13) a r X i v : . [ a s t r o - ph . GA ] J u l T. Hartwig et al. characteristic mass than is found for present-day star forma-tion. Direct observations of the first stars to test the theoriesof their formation are also lacking. Our knowledge aboutthe mass distribution of the first stars is thus mainly basedon theoretical models and simulations (Glover 2013; Greif2015). Another independent constraint is the absence of anylow-mass Pop III survivors in the Milky Way (MW), whichlimits the masses of the first stars to (cid:38) . (cid:12) (Bond 1981;Hartwig et al. 2015; Komiya et al. 2016; Ishiyama et al. 2016;Dutta et al. 2017; Magg et al. 2018).Stellar archaeology provides a powerful approach toconstrain the nature and properties of the first stars (Frebel& Norris 2015). Spectroscopic observations of extremelymetal-poor (EMP) stars in the MW enable measurementsof their chemical composition. The relative abundances ofthe different elements can then be compared with the theo-retically predicted yields of their putative progenitor super-novae (SNe). Several studies have successfully interpretedthe abundance signatures of individual EMP stars as thefingerprint of Pop III SNe, and obtained estimates for thestellar mass of the corresponding progenitor (Ishigaki et al.2014; Tominaga et al. 2014; Keller et al. 2014; Ji et al. 2015;Placco et al. 2015, 2016; Fraser et al. 2017; Chen et al. 2017;Ishigaki et al. 2018). However, a major assumption of thisreverse-engineering problem is that the EMP star carriesthe chemical imprint of only one SN. Accounting for metalcontributions from several SNe would require additional freeparameters, and consequently weakens the constraints dueto degeneracies between the individual yields.A key challenge of stellar archaeology is therefore toidentify mono-enriched second generation stars, as they aremost valuable for constraining the properties of the firststars. Here, we define “mono-enriched” second generationstars as stars that formed from gas that was enriched byexactly one Pop III SN. In contrast, we refer to stars thatcarry the combined chemical signature of more than one SNas “multi-enriched”.Metallicity alone is not a reliable tracer of the stellarpopulation because the metallicity of gas enriched by a sin-gle Pop III SN depends sensitively on the metal yield of theSN, which varies greatly, particularly for an element such asFe, and on the degree of metal mixing, which is poorly con-strained. For example, in our models, we find mono-enrichedsecond generation stars with metallicities [Fe / H] > − / H] ∼ −
3. The carbon-enhancement of most EMPstars has been claimed as an additional signature of secondgeneration stars, emerging from faint Pop III SNe (Beerset al. 1992; Aoki et al. 2007; Ishigaki et al. 2014; Sk´ulad´ottiret al. 2015). In this paper, we investigate further indicatorsand diagnostic to successfully identify mono-enriched sec-ond generation stars, based on their chemical abundance.This allows us to construct samples of stars that are mono-enriched based on our current understanding of Pop III SNe.A special subclass of second generation stars are thosethat form from gas that was enriched by a pair-instability su- Defined as [A / B] = log ( m A /m B ) − log ( m A , (cid:12) /m B , (cid:12) ),where m A and m B are the abundances of elements A and B and m A , (cid:12) and m B , (cid:12) are the solar abundances of these elements (As-plund et al. 2009). pernova (PISN). These very energetic explosions of massivemetal-poor stars are the final fates of non-rotating Pop IIIstars in the mass range 140 −
260 M (cid:12) (Rakavy & Shaviv 1967;Barkat et al. 1967; Fraley 1968; Bond et al. 1984; Fryer et al.2001). They eject more metals than core collapse SNe andcan therefore enrich the ISM of their host halo to highermetallicities, beyond [Fe/H] ∼ −
3. This makes it more dif-ficult to search for second generations stars that form fromthe debris of a PISN because the number of ordinary starsincreases with metallicity and the fraction of PISN-enrichedstars at [Fe/H] > − < − Cosmic structure formation proceeds hierarchically fromsmall matter overdensities in the early Universe via accre-tion and mergers. Hierarchical structure formation is domi-nated by dark matter, which accounts for most of the matterin the Universe. To model the baryonic physics of star andgalaxy formation, we can therefore decouple the formationand mergers of dark matter halos and the physics and stellarfeedback within them.Our semi-analytical approach is based on dark mat-ter merger trees that were separately generated from high-resolution N -body simulations. On top of this dark matterframework, we model star formation and the correspondingfeedback self-consistently with a set of analytical recipes. Forthis study, we use 30 MW-like merger trees from the Cater-pillar project (Griffen et al. 2016), which assumes the cur-rent dark energy plus cold dark matter (ΛCDM) paradigmwith cosmological parameters from (Planck Collaboration2014). The halos were selected based on three criteria toresemble the MW: virial masses in the range 0 . ×
50 Myr at z <
6. This guarantees a high temporal resolu-tion to model accurately the radiative and chemical feedbackof Pop III stars. Our semi-analytical model of Pop III starformation is based on Hartwig et al. (2015) with improve-ments by Magg et al. (2016, 2018). For further details onthe model and a resolution study see Magg et al. (2018).In the early Universe the main components of primor-dial gas clouds are hydrogen and helium with H being themost efficient coolant under the conditions considered here. c (cid:13)000
6. This guarantees a high temporal resolu-tion to model accurately the radiative and chemical feedbackof Pop III stars. Our semi-analytical model of Pop III starformation is based on Hartwig et al. (2015) with improve-ments by Magg et al. (2016, 2018). For further details onthe model and a resolution study see Magg et al. (2018).In the early Universe the main components of primor-dial gas clouds are hydrogen and helium with H being themost efficient coolant under the conditions considered here. c (cid:13)000 , 1–18 ono-enriched second generation stars Once a pristine halo reaches the critical mass M crit = 3 . × M (cid:12) (cid:18) z (cid:19) − / , (1)cooling by molecular hydrogen is efficient enough to allowthe gas to collapse to protostellar densities and trigger starformation (Yoshida et al. 2003; Hummel et al. 2012). Massivestars forming in these halos produce large numbers of softultraviolet photons in the Lyman and Werner bands of H .These Lyman-Werner (LW) photons can readily escape fromlow-mass halos (Schauer et al. 2015) and so the onset of PopIII star formation is quickly followed by the growth of anextragalactic LW background. We model the effect of thisLW feedback as a uniform background that increases withtime according to F ( z ) = 4 π − ( z − / , (2)where F has the units 10 − erg s − cm − Hz − (Greif &Bromm 2006). Most halos are illuminated by a LW flux thatis within a factor of two of this mean value (Dijkstra et al.2008), which justifies this approximate treatment. LW pho-tons can photodissociate H and hence destroy the mostimportant coolant in the early Universe and consequentlyprevent star formation. In addition to the critical mass re-quired for primordial star formation (Eq. 1) we thereforecheck that the halo mass is above (O’Shea & Norman 2008) M LW = 5 × M (cid:12) + 3 . × M (cid:12) F . . (3)Baryonic streaming velocities might further alter this thresh-old and require a higher critical mass, but the relative impor-tance of this effect is still debated (Stacy et al. 2011; Greifet al. 2011; Naoz et al. 2013; Tanaka & Li 2014; Schaueret al. 2017; Hirano et al. 2017; Schauer et al., in prep.).Once we identify a halo in which Pop III stars canform, we assign individual metal-free stars to it by sam-pling stochastically from a logarithmically flat initial massfunction (IMF) until the total stellar mass is above M ∗ = η III Ω b Ω m M h , (4)where η III is the star formation efficiency (SFE) of Pop IIIstars and M h is the mass of the halo. The SFE and thelower and upper limit of the Pop III IMF are calibrated tomatch observational constraints (see Sec. 3.1). We assumethat star formation is instantaneous and model the ionizingfeedback on subsequent star formation. The emerging H ii regions around star-forming halos suppress star formationin their vicinity by photoionization heating and we allowstar formation in halos that are within the H ii region of aneighbouring halo only if T vir > K.Once a star explodes as an SN, we follow the expansionof its metal-enriched shell. For Pop III SNe we assume aconstant velocity of 10 km s − in the intergalactic medium(Smith et al. 2015) and for metals from SNe of later genera-tion stars we model their expansion as a momentum-drivensnowplough (see Magg et al. 2018 for details on the ionizingfeedback and external enrichment).When a halo has been enriched with metals, the sec-ond generation of stars form from this enriched interstellarmedium (e.g., Chiaki et al. 2016). We distinguish two dif-ferent enrichment channels: if the halos has been enrichedinternally by Pop III stars in the same halo, we delay the formation of second generation stars by the recovery time t recov = 100 Myr (Greif et al. 2010; Whalen et al. 2013;Smith et al. 2015; Jeon et al. 2014, 2017; Chiaki et al. 2018).If a previously pristine halo is externally enriched and has amass above M LW , Pop II star formation occurs one freefalltime after this enrichment with t ff = 72 Myr (cid:18) z (cid:19) − / , (5)where we assume an overdensity of 200 times the mean cos-mic density. In this paper, we refer to second generationstars as those that form after the first metal enrichment of ahalo. Due to the delay between the first enrichment and theonset of second generation star formation, the host galaxycan be enriched by multiple enrichment events or merge withan already enriched galaxy before the second generation ofstars forms.The main topic of this paper are the nature, chemicalcharacteristics, and unique signature of second generationstars. We assume that the composition of such a secondgeneration star is defined at the moment of its formationand does not change during the lifetime due to possible pol-lution by ISM accretion (Tanaka et al. 2017, see also Yoshii1981; Frebel et al. 2009; Komiya et al. 2010; Hattori et al.2014; Komiya et al. 2015; Johnson 2015; Shen et al. 2017).Whenever we refer to the chemical composition of secondgeneration stars, we implicitly refer to the chemical compo-sition of the ISM from which these second generation starsform. One novel feature of our semi-analytical model is the track-ing of chemical elements up to zinc. This enables us to cali-brate our model based on various observations and we obtaincrucial insight into the chemical enrichment history of theMW. In this section, we briefly summarize the main featuresof our model of chemical evolution.For Pop III stars, we use the tabulated metal yields asa function of progenitor mass by Nomoto et al. (2013). Thetheoretical uncertainty for the metal yields between differ-ent models (Heger & Woosley 2010; Limongi & Chieffi 2012)is of the order 0 . f dil of all hydrogen in the halo mixes withthe metals. This approach is consistent with more advancedtheoretical models (Starkenburg et al. 2013; Hirai & Saitoh2017; Chen et al. 2017; Sarmento et al. 2017, 2018) andwe draw the dilution factors from a log-normal distributionwith mean µ = 10 − . and width σ = 0 .
75 dex. More realis-tic hydrodynamical simulations of the mixing of the first SNyields have been performed self-consistently in 3D by othergroups (Greif et al. 2007; Wise & Abel 2008; Whalen et al.2008; Greif et al. 2010; Wise et al. 2012; Ritter et al. 2012;Vasiliev et al. 2012; Jeon et al. 2014; Safranek-Shrader et al. c (cid:13) , 1–18 T. Hartwig et al. −4−2 0 2 4 610 PISN [ C / F e ] Pop III progenitor mass [M ⊙ ]fiducial Pop III SNfaint Pop III SN, Chen+17faint Pop III SN, Ishigaki+14SN Type II, Z=0.001SN Type Ia Figure 1.
Carbon-to-iron ratio, [C/Fe], as a function of thePop III SN progenitor mass (solid, Nomoto et al. 2013). For com-parison, we also show the yields of Type Ia (short-dashed, Seiten-zahl et al. 2013) and Type II SNe (long-dashed, Nomoto et al.2013). The yields for individual faint SNe are based on Chen et al.(2017) and Ishigaki et al. (2014). PISNe with a progenitor mass of ∼
150 M (cid:12) yield a very high [C/Fe] (because they eject relativelylittle iron), but PISNe with a progenitor mass of ∼
250 M (cid:12) yielda very low, even significantly subsolar value of [C/Fe]. The ex-plosion energies of Type II SNe are assumed to be 10 erg. FaintSNe with lower explosion energies have generally higher [C/Fe]because more iron falls back onto the compact remnant. Z = 0 .
001 and average the contributionby SNe with different progenitor masses over a Salpeter IMFin the range 10 −
40 M (cid:12) .One important observed characteristic of extremelymetal-poor stars is their frequently high carbon-to-iron ra-tio, which we aim to reproduce in our model by includingfaint SNe. We illustrate the [C/Fe] ratio as a function ofPop III progenitor mass in Fig. 1 for different types of SNe.A faint SN refers to an explosion with a very small ejected Ni mass either due to a low explosion energy (Chen et al.2017) or large-scale mixing and fallback in aspherical explo-sions (Tominaga et al. 2007). To account for faint SNe, weinclude the corresponding yields by Ishigaki et al. (2014) in
Parameter Valuemass threshold for Pop III Eq. 1mass threshold with LW feedback Eq. 3Pop III SFE η III = 0 . η II = 0 . f faint = 40%metal fallback fraction f fallback = 20%metal ejection fraction f eject = 80%Pop III SN wind velocity v = 10 km/slower IMF limit M min = 3 M (cid:12) upper IMF limit M max = 150 M (cid:12) recovery time t recov = 100 Myrmean of dilution distribution µ = 10 − . width of dilution distribution σ = 0 .
75 dex
Table 1.
Parameter values in our fiducial model. This set ofparameters best reproduces observations at [Fe/H] (cid:54) − our model and discuss the calibration of the fraction of faintSNe in Section 3.1. These models are all for faint SNe witha progenitor mass of 25 M (cid:12) , but can be taken as represen-tative for faint SNe occurring in the mass range 10 −
40 M (cid:12) . We use the observed fraction of carbon enhanced metal-poor(CEMP) stars and the distribution of EMP halo stars tocalibrate our model. However, our model is not intended toreproduce these functions over a broad metallicity range be-cause we focus on second generation stars. In general, metal-poor stars can form after any number of previous generationsof star formation, but each additional enrichment event re-sults in higher stellar metallicities. Therefore, we focus onthe stars with a metallicity of [Fe/H] (cid:54) − In this section, we present our fiducial parameters, motivatethat they are physically reasonable, and that they meet ad-ditional observational constraints. Throughout the paper,we restrict our analysis to the MW and satellites within R vir = 300 kpc from the MW centre at z = 0 (if not explic-itly stated otherwise).The main model parameters and their fiducial valuesare summarized in Table 1. The Pop III SFE is a crucialparameter for stellar archaeology since it defines the gasmass fraction that turns into stars and hence the averagenumber of Pop III SNe per minihalo. As well as calibratingit with stellar archaeology observations, we also enforce twoadditional constraints. We require that our choice of η III leads to an optical depth for the Thomson scattering of CMB c (cid:13)000
40 M (cid:12) . We use the observed fraction of carbon enhanced metal-poor(CEMP) stars and the distribution of EMP halo stars tocalibrate our model. However, our model is not intended toreproduce these functions over a broad metallicity range be-cause we focus on second generation stars. In general, metal-poor stars can form after any number of previous generationsof star formation, but each additional enrichment event re-sults in higher stellar metallicities. Therefore, we focus onthe stars with a metallicity of [Fe/H] (cid:54) − In this section, we present our fiducial parameters, motivatethat they are physically reasonable, and that they meet ad-ditional observational constraints. Throughout the paper,we restrict our analysis to the MW and satellites within R vir = 300 kpc from the MW centre at z = 0 (if not explic-itly stated otherwise).The main model parameters and their fiducial valuesare summarized in Table 1. The Pop III SFE is a crucialparameter for stellar archaeology since it defines the gasmass fraction that turns into stars and hence the averagenumber of Pop III SNe per minihalo. As well as calibratingit with stellar archaeology observations, we also enforce twoadditional constraints. We require that our choice of η III leads to an optical depth for the Thomson scattering of CMB c (cid:13)000 , 1–18 ono-enriched second generation stars M D F ( li n ) fducial modelYong+13Schörck+09 M D F ( l og ) [Fe/H] f r a c t i on C E M P - no s t a r s fducial modelPlacco+14 Figure 2.
Top: Fraction of CEMP-no stars ([C/Fe] > .
7) as afunction of [Fe/H] predicted by our model (orange) with the ob-served distribution (purple, Placco et al. 2014) shown for compari-son. Below: Predicted (orange) and observed (green: Sch¨orck et al.2009; purple: Yong et al. 2013) metallicity distribution functions,normalised to the number of stars below [Fe/H] (cid:54) − photons, τ = 0 . × M (cid:12) (the lowest massminihalo capable of forming Pop III stars at redshift z ∼ (cid:12) , in good agreement with the values of order100 − (cid:12) found in numerical simulations (Susa et al.2014; Hirano et al. 2014). These numerical results can beseen as a lower limit because most simulations focus on thefirst high redshift peaks but we also expect metal-free starformation at z <
10 in more massive halos. The fractions ofejected metals and metals that fall back onto the halo afteran SN are consistent with the results of Ritter et al. (2012).We show in Fig. 2 that we can reproduce the metallicitydistribution function (MDF) and the fraction of CEMP-nostars as a function of metallicity with this set of param-eters. CEMP-no stars are a subclass of CEMP stars with[Ba/Fe] (cid:54) .
0, i.e. with no enhancement in neutron captureelements. We limit this comparison to stars with [Fe/H] (cid:54) − M D F ( li n ) Yong+13 0.01 0.1 1 -6 -5.5 -5 -4.5 -4 -3.5 -3 -2.5 M D F ( l og ) [Fe/H] 0 0.2 0.4 0.6 0.8 1 fr ac ti on C E M P - no s t a r s µ = 10 -0.5 σ = 0.2dexM max =300M ⊙ f faint =20%only one treefiducial Placco+14 Figure 3.
Same as Fig. 2, but showing the effect of varying themodel parameters specified in the legend. The results of our fidu-cial model are shown in black. We also show the results from arealization based on only one tree (green), which highlights theexpected stochasticity of the distribution at low [Fe/H]. We notethat the yellow line in the middle and lower panels ( f faint = 20%)is identical with that in the fiducial model. our model. We used the fraction of CEMP-no stars with[C/Fe] > . (cid:54) − .
5, butdiscuss this effect separately below.Another important and poorly constrained parameter isthe fraction of faint SNe, which is assumed to have a directinfluence on the fraction of CEMP stars due to the high[C/Fe] yields of this type of SN. We find a best matchingvalue of f faint = 40%. Slightly higher values (Ji et al. 2015; deBennassuti et al. 2017) are also compatible within our errormargins. The fraction of CEMP stars is mainly controlledby the adopted model for mixing with the ISM and f faint . We now compare how different parameters affect the re-sults and demonstrate quantitatively that our fiducial setof parameters best reproduces the MDF and the fraction ofCEMP-no stars (Fig. 3). If we assume µ = 10 − . , i.e. thatmetals ejected by SNe mix with almost all available hydro-gen in a halo, we predict too few CEMP-no stars. If weassume that the distribution of dilution factors is too nar-row ( σ = 0 . ≈ −
3. The green line in this plot also demonstrates c (cid:13) , 1–18 T. Hartwig et al.
Parameter D MDF D CEMP
Σfiducial 0 .
08 0 .
07 0 . M min = 10 M (cid:12) .
09 0 .
11 0 . M max = 120 M (cid:12) .
13 0 .
11 0 . M max = 300 M (cid:12) .
24 0 .
07 0 . η III = 0 . .
14 0 .
05 0 . η III = 0 .
002 0 .
12 0 .
10 0 . η II = 0 .
02 0 .
12 0 .
06 0 . t recov = 10 Myr 0 .
13 0 .
05 0 . f faint = 0 . .
08 0 .
12 0 . f faint = 0 . .
08 0 .
08 0 . − .
14 0 .
18 0 . f eject = 0 . .
14 0 .
07 0 . µ = 10 − . .
07 0 .
15 0 . µ = 10 − . .
09 0 .
07 0 . σ = 0 . .
16 0 .
16 0 . Table 2.
Parameter study and KS-test values (Eq. 6). Our fidu-cial model yields the smallest maximum differences between thecumulative distributions of the observations and our model. How-ever, the only model that can be rejected based on this two-sampleKS test at the 95% level is the one with M max = 300 M (cid:12) (seetext). that our model for a single MW-like merger tree correctlyreproduces the sparsely sampled region at [Fe/H] (cid:54) − . D = max x (cid:54) − | F obs ( x ) − F model ( x ) | , (6)where F ( x ) is the cumulative distribution function and x =[Fe/H]. The resulting values for various models are sum-marized in Table 2. Our fiducial set of parameters mini-mizes the sum of D MDF and D CEMP . To reject the null-hypothesis that our model reproduces the observations at95% significance level, we determine the corresponding crit-ical distance to be D crit , MDF = 0 .
23 for the MDF and D crit , CEMP = 0 .
29 for the fraction of CEMP-no stars. Theonly parameter choice that can be excluded based on thisanalysis is M max (cid:62)
300 M (cid:12) as an upper limit for the Pop IIIIMF. Since we do not fully explore our 11D parameter space,we can only conclude that our fiducial parameters representa local optimum, while other parameter combinations mayyield a similar or even better fit to the observations. Unfor-tunately, this also illustrates the weak predictive power ofthis approach and we do not claim to constrain any of theparameters by fitting a model with 11 free parameters totwo observables. A full parameter space exploration couldbe performed by means of, e.g., Gaussian processes modelemulators (e.g. Bower et al. 2010; G´omez et al. 2012, 2014).Nonetheless, our set of initial parameters agrees with otherstudies and reproduces the main observations provided bystellar archaeology. Moreover, we will show later that ourmain conclusions can also be derived independently of thespecific cosmological model adopted.We also show the parameter dependence of the Pop IIIstar formation rate density (SFRd) in Fig. 4. It is calculatedwithin the co-moving volume of the MW and therefore rep- -6 -5 -4 -3 -2
0 5 10 15 20 25 30 P op III SF R d [ M ⊙ y r - c M p c - ] redshiftt recov =10Myr η III =0.002fiducial η III =0.0005Sarmento+17Johnson+13
Figure 4.
Comparison of the SFRd for Pop III stars as a functionof redshift of our model (solid) to the rates by Johnson et al.(2013) and Sarmento et al. (2017) (dashed). The SFRd scalesroughly with the SFE, and in our fiducial model, we find a peakvalue of ∼ − M (cid:12) yr − cMpc − (co-moving Mpc) around z ≈
10. A shorter recovery time leads to a more efficient suppressionof Pop III SF at z (cid:38)
15 because Pop II stars can form earlier.The SFRd of our model is averaged over 30 MW-like trees. resents a cosmic overdensity. Our star formation rates areconsistent with those in Johnson et al. (2013), with the up-per limit advocated by Visbal et al. (2015), and with theThomson scattering optical depth measured by Planck Col-laboration (2016). Our results differ from Sarmento et al.(2017) because they allow Pop III star formation in slightlyenriched halos up to a metallicity of Z crit = 10 − Z (cid:12) , whichpermits more Pop III star-forming halos at z < The difference between internal and external enrichment isimportant because the timescales of the subsequent collapseand the overall enriching mass depend on the nature of theenrichment. As internal enrichment, we label the inevitablechemical enrichment of a halo after star formation. Externalenrichment occurs when the radius of a metal-enriched bub-ble is larger than the separation between the centres of twohalos (see Sec. 2.2), typically of the order 0 . −
10 kpc. Bothof these enriching events are passed through the merger treeso that a halo at z = 0 could have experienced several in-ternal and external enrichment events during its assemblyhistory. We investigate the relative contributions of inter-nal vs. external enrichment in Fig. 5. Internal enrichmentis dominant compared to external enrichment prior to theformation of second generation stars, as has also been shownby Griffen et al. (2018), Visbal et al. (2018), and Jeon et al.(2017). If halos are close enough for external enrichment,ionizing feedback is usually also strong enough to suppressstar formation, thereby preventing the formation of exter-nally enriched second generation stars. For this reason, vary-ing the recovery time makes little difference to the externalenrichment fractions. The metal contributions in Fig. 5 areaveraged and there are individual halos that are only en-riched externally by Pop III or Pop II stars, although theiroccurrence in number is small. We find that the outcome of c (cid:13)000
10 kpc. Bothof these enriching events are passed through the merger treeso that a halo at z = 0 could have experienced several in-ternal and external enrichment events during its assemblyhistory. We investigate the relative contributions of inter-nal vs. external enrichment in Fig. 5. Internal enrichmentis dominant compared to external enrichment prior to theformation of second generation stars, as has also been shownby Griffen et al. (2018), Visbal et al. (2018), and Jeon et al.(2017). If halos are close enough for external enrichment,ionizing feedback is usually also strong enough to suppressstar formation, thereby preventing the formation of exter-nally enriched second generation stars. For this reason, vary-ing the recovery time makes little difference to the externalenrichment fractions. The metal contributions in Fig. 5 areaveraged and there are individual halos that are only en-riched externally by Pop III or Pop II stars, although theiroccurrence in number is small. We find that the outcome of c (cid:13)000 , 1–18 ono-enriched second generation stars fr ac ti on [Fe/H]Pop III intPop III extPop II ext Figure 5.
Relative contribution to the metal enrichment of sec-ond generation stars via different enrichment channels (metalmass weighted). The three contributions sum up to 100%. In-ternal enrichment by Pop III stars dominates at all metallicitiesand external enrichment by Pop III stars accounts for ∼
10% ofthe enriching metals above [Fe/H]= −
4. External enrichment byPop II stars is always sub-dominant ( (cid:46) second generation star formation does not strongly dependon environmental effects, such as the clustering of halos. Wealso confirm in our semi-analytical model that the radialdistribution of halos hosting second generation stars followsthe radial distributions of all halos in the local volume at z = 0.In Fig. 6, we see the 3D distribution of halos in the lo-cal group at z = 0 for one exemplary MW-like merger tree.We find ∼
400 satellites with stellar masses above 1000 M (cid:12) .The observed number of MW satellites is around 50 (Drlica-Wagner et al. 2015), which seems to be in contradiction withour model and other DM simulations (the “missing satelliteproblem”, see Kauffmann et al. 1993; Klypin et al. 1999;Moore et al. 1999). However, this discrepancy can be solvedby correcting for the completeness bias of the surveys (Kimet al. 2017). We assign stellar masses at z = 0 via abundancematching based on the peak mass of each satellite during itsassembly history (Garrison-Kimmel et al. 2014). Note thatstellar masses below ∼ × M (cid:12) should be consideredas an extrapolation due to the incompleteness of their ob-servations for low mass satellites. Moreover, the scatter inthe relation between stellar and halo mass becomes moreimportant at lower masses (Garrison-Kimmel et al. 2017).Hydrodynamic simulations indicate that extrapolations tolow masses are reasonable (Munshi et al. 2017; Jeon et al.2017), but our stellar masses at z = 0 should be consideredas rough estimate for lower-mass satellites.For a direct comparison of the fractions of second gener-ation stars we assume for the mass of the stellar populationof the second generation an instantaneous starburst whichconverts 1% of the gas mass into stars. The resulting frac-tions as a function of the stellar mass can be seen in Fig. 7.During the assembly of the MW and its satellites, halos thathost second generation stars merge into larger systems andat z = 0 second generation stars can be found in satellitesof all masses. However, the relative contribution of second -300 -200 -100 0 100 200 300-300-200-100 0 100 200 300-300-200-100 0 100 200 300 log (M * /M ⊙ )kpc kpckpc 3 4 5 6 7 8 9 10 Figure 6.
Projection of all star-hosting halos at z = 0 within300 kpc of the MW main halo for one merger tree realization. Themain halo is indicated by the black asterisk and the satellites arecolour-coded by their stellar mass. -6 -5 -4 -3 -2 -1 fr ac ti on o f s t e ll a r m a ss M halo /M ⊙ Figure 7.
Fraction of all (purple), metal-poor ([Fe / H] < − z = 0. Second generation stars end upin satellites of all masses, but their fraction is much higher in lowmass halos. generation stars to the total stellar population depends onthe host mass, with less massive halos being more likely tohost a higher fraction of second generation stars. The MWat z = 0 consists of e.g. (cid:46) .
1% second generation stars, butonly ∼ − of all MW stars are metal-poor ([Fe/H] < − ∼ − are mono-enriched second generation stars. Ouranalysis shows that the stellar population in satellites with M h (cid:46) M (cid:12) originates dominantly from the second gen-eration of star formation. Although our model predicts afraction of close to 100% in this mass range, the actual frac- c (cid:13) , 1–18 T. Hartwig et al. N S N E SN =E b M h / M ⊙ z M crit M LW Figure 8.
Top: Number of Pop III SNe per minihalo as a functionof redshift. Bottom: Halo masses at the moment of Pop III starformation. The solid line indicates the mean, the dark contoursthe 1 σ standard deviation, and the light contours the minimumand maximum values in this redshift bin. The increase of thenumber of SNe with decreasing redshift is related to the simul-taneous increase of the stellar mass that is available per Pop IIIstar-forming halo. In some rare cases at z >
15 there are mini-halos with only one SN, but generally we expect between 5 and20 SNe per minihalo. The dotted and dashed lines in the bottompanel illustrate the critical masses for Pop III star formation. Thedotted line in the top panel indicates the number of SNe requiredto expel all of the gas from the halo. Halos with more than thisnumber of SNe may be completely disrupted by Pop III SNe andhence may not form second generation stars. tion may be lower due to the scatter in the halo to stellarmass relation, which we do not take into account.These results are in agreement with previous modelsthat show that ultra-faint dwarf galaxies host ancient stel-lar populations and probe early cosmic star formation (Bul-lock et al. 2000; Salvadori & Ferrara 2009; Gao et al. 2010;Starkenburg et al. 2013; Weisz et al. 2014; Ji et al. 2016; Jeonet al. 2017; Griffen et al. 2018; Starkenburg et al. 2017). Thisis because ultra-faint dwarf galaxies with M h < × M (cid:12) formed (cid:38)
90% of their stellar mass prior to reionization(Jeon et al. 2017) and have an average iron abundance of[Fe/H] < − The chemical signature of second generation stars can beused to deduce the masses of their Pop III progenitors.For this purpose, we are especially interested in those caseswhere the ISM was enriched by exactly one previous Pop IIISN. However, in most minihalos we form Pop III stars insmall multiples (Turk et al. 2009; Stacy et al. 2010; Clarket al. 2011; Greif et al. 2011; Smith et al. 2011; Susa et al.2014; Hirano & Bromm 2017) and in Fig. 8, we show the av-erage number of SNe per minihalo. It is an increasing func- tion with decreasing redshift due to the increasing thresholdmass for Pop III star formation. At z (cid:38)
15 we expect fewerthan 10 SNe per halo and in individual cases there are haloswith just one Pop III SN. These are the cradles for mono-enriched second generation stars.Minihalos at high redshift have shallow potential wellsand SNe could unbind all the gas in the halo and hence pre-vent subsequent star formation. To derive the critical num-ber of SNe required to do this, we assume that an SN hason average an energy of 10 erg and that the halo has agravitational binding energy (Loeb 2010) of E b = 2 . × (cid:18) M h M (cid:12) (cid:19) / (cid:18) z (cid:19) erg . (7)Not all of the injected SN energy will effectively couple tothe gas and contribute to its ejection, as some will instead beradiated away. Also the low-density H ii region, which sur-rounds the first stars at the moment of their SN explosions,and the anisotropy of the ISM, which provides channels ofleast resistance for the energy to escape, reduce the efficiencyof SNe in ejecting gas from the galactic potential well. Toaccount for this effect, we assume that only 10% of the SNenergy couples efficiently to the gas (Kitayama & Yoshida2005; Whalen et al. 2008). This yields the number of SNeper halo that is required to unbind all gas as N SN = 62 (cid:18) M h M (cid:12) (cid:19) / (cid:18) z (cid:19) . (8)The black dashed line in the upper panel of Fig. 8 indi-cates that this critical value is above the average numberof SNe per halo. Nevertheless, some halos at every redshifthave values of N SN above this critical value, and may there-fore form fewer multi-enriched second generation stars thanour model assumes, because of the disruption of these halosby SN feedback. We note, however, that this is a simplis-tic order of magnitude estimate and more realistic modelsshow that gas fallback is also possible after several or moreenergetic SN explosions in a minihalo (Kitayama & Yoshida2005; Greif et al. 2010; Ritter et al. 2012; Chiaki et al. 2018).Therefore, we do not include this destructive effect of multi-ple SNe self-consistently in our model, but highlight possibleimplications in the discussion section.It is also interesting to examine whether the time be-tween two SNe is long enough for the gas to recollapse andform mono-enriched second generation stars before the sec-ond SN explodes. In Fig. 9 we show a histogram of the timesbetween the explosion of the first and the second SN in mini-halos. The average time between two SNe is much shorterthan our assumed recovery time for second generation starformation. Consequently, the presence of multiple SNe in oneminihalo indicates that most stars that form at the onset ofPop II star formation carry the imprint of several Pop IIISNe.We derive the probability that exactly one SN explodesin a minihalo, based on Poisson statistics. For a given Pop IIIIMF we calculate how much stellar mass we need on averageto form one SN. The mean number of SNe in a halo withstellar mass M ∗ is then given by λ = M ∗ M , (9)where M is the stellar mass to expect on average one SN. c (cid:13)000
15 we expect fewerthan 10 SNe per halo and in individual cases there are haloswith just one Pop III SN. These are the cradles for mono-enriched second generation stars.Minihalos at high redshift have shallow potential wellsand SNe could unbind all the gas in the halo and hence pre-vent subsequent star formation. To derive the critical num-ber of SNe required to do this, we assume that an SN hason average an energy of 10 erg and that the halo has agravitational binding energy (Loeb 2010) of E b = 2 . × (cid:18) M h M (cid:12) (cid:19) / (cid:18) z (cid:19) erg . (7)Not all of the injected SN energy will effectively couple tothe gas and contribute to its ejection, as some will instead beradiated away. Also the low-density H ii region, which sur-rounds the first stars at the moment of their SN explosions,and the anisotropy of the ISM, which provides channels ofleast resistance for the energy to escape, reduce the efficiencyof SNe in ejecting gas from the galactic potential well. Toaccount for this effect, we assume that only 10% of the SNenergy couples efficiently to the gas (Kitayama & Yoshida2005; Whalen et al. 2008). This yields the number of SNeper halo that is required to unbind all gas as N SN = 62 (cid:18) M h M (cid:12) (cid:19) / (cid:18) z (cid:19) . (8)The black dashed line in the upper panel of Fig. 8 indi-cates that this critical value is above the average numberof SNe per halo. Nevertheless, some halos at every redshifthave values of N SN above this critical value, and may there-fore form fewer multi-enriched second generation stars thanour model assumes, because of the disruption of these halosby SN feedback. We note, however, that this is a simplis-tic order of magnitude estimate and more realistic modelsshow that gas fallback is also possible after several or moreenergetic SN explosions in a minihalo (Kitayama & Yoshida2005; Greif et al. 2010; Ritter et al. 2012; Chiaki et al. 2018).Therefore, we do not include this destructive effect of multi-ple SNe self-consistently in our model, but highlight possibleimplications in the discussion section.It is also interesting to examine whether the time be-tween two SNe is long enough for the gas to recollapse andform mono-enriched second generation stars before the sec-ond SN explodes. In Fig. 9 we show a histogram of the timesbetween the explosion of the first and the second SN in mini-halos. The average time between two SNe is much shorterthan our assumed recovery time for second generation starformation. Consequently, the presence of multiple SNe in oneminihalo indicates that most stars that form at the onset ofPop II star formation carry the imprint of several Pop IIISNe.We derive the probability that exactly one SN explodesin a minihalo, based on Poisson statistics. For a given Pop IIIIMF we calculate how much stellar mass we need on averageto form one SN. The mean number of SNe in a halo withstellar mass M ∗ is then given by λ = M ∗ M , (9)where M is the stellar mass to expect on average one SN. c (cid:13)000 , 1–18 ono-enriched second generation stars −2 −1
0 5 10 15 20 a v e r a g e e v e n t s p e r M W time between first and second SN [Myr] Figure 9.
Histogram of the times between the explosion of thefirst and the second SN in minihalos per MW-like merger tree.Due to the very short lifetimes of massive stars, the second SN ex-plodes generally within less than 10 Myr after the first one (mindthe logarithmic y-axis). This is shorter than the typical recoverytime for second generation star formation ( ∼
100 Myr). The dom-inance of short times between SNe illustrates that there is gener-ally not enough time between two SN explosions to form secondgeneration stars. Instead, they form after most of the Pop III starsin the minihalo have exploded as SNe.
By applying Poisson statistics, we calculate the probabilityto have k SNe going off in one minihalo: p ( k ) = λ k k ! e − λ . (10)The probability to have one SN per halo is given by p (1)and the probability to have more than one SN per halo isgiven by 1 − p (0) − p (1). These probabilities can be seenas a function of the stellar mass in Fig. 10. This analyticalderivation is valid as long as the total stellar mass is higherthan the upper IMF limit because otherwise the entire IMFcannot be sampled. As we can see in the bottom panel, thiscriterion is almost always fulfilled in our fiducial model be-cause we form at least ∼
100 M (cid:12) of Pop III stars per halo(Eq. 4). Consequently, the probability to have only one SNeper minihalo is very low, of the order 1%. Instead, we expectsecond generation stars to form from gas that has been pre-viously enriched by several SNe. This analytical estimate isvery powerful and flexible because it predicts the probabilityof having more than one SN per minihalo for any possibleIMF or stellar mass. The chances to create mono-enrichedsecond generation stars are highest in the smallest miniha-los because the available gas mass to form stars is lower andhence it is more likely for these halos to host only one Pop IIIstar that explodes as an SN.
We aim to find the optimal diagnostic and selection criteriafor EMP stars that are promising mono-enriched candidatesgiven that only relatively few elements are observable inEMP stars with reasonable effort. We thus need to quantifythe likelihood for star-forming gas to have experienced onlyone prior enrichment event. We first use our semi-analyticalmodel to find which abundances are best suited for this pur-pose. Then, we present a novel diagnostic that is indepen-dent of any model for primordial star formation and onlydepends on the assumed SN yields. −3 −2 −1 p multi p mono p r ob a b ilit y simulationIMF 3−150IMF 1−100 IMF 3−300IMF 3−1000
10 100 1000 10000 c oun t Pop III stellar mass per minihalo [M ⊙ ] Figure 10.
Top: Probability to have exactly one SN (solid) ormore than one SN (dashed) per minihalo as a function of thestellar mass for different IMF ranges. The black lines correspondto the analytical prediction of our fiducial model and should becompared to the grey histogram, which is the average over all 30merger trees. Bottom: histogram of the stellar masses of Pop IIIstar-forming halos in one MW-like realization. Most Pop III starsform in minihalos with M ∗ (cid:46) (cid:12) but some form in atomiccooling halos with stellar masses up to M ∗ (cid:38) M (cid:12) . In thesemass ranges the probability to have exactly one SN in a randomlyselected minihalo is < In Fig. 11, we display as an example the distribution andprobability of finding mono-enriched second generationsstars, calculated for the [Mg/C] ratio. The mono-enrichedsecond generation stars populate specific regimes, differentfrom those of multi-enriched second generation stars. In gen-eral, the probability of mono-enrichment is a decreasingfunction of metallicity and we find even individual mono-enriched second generation stars with solar metallicities inour model. The abundance ratio [Mg/C] adds an additionalconstraint with the lowest probability for mono-enrichmentaround [Mg/C] ∼ . − . c (cid:13) , 1–18 T. Hartwig et al. −6 −4 −2 0 2[Fe/H]−2.0−1.5−1.0−0.50.00.51.01.5 [ M g / C ] multi-enrichedmono-enriched σ σ σ −6 −4 −2 0 [Fe/H] −2−1012 [ M g / C ] p m o n o [ % ] Figure 11.
Top: Mono-enriched second generation stars popu-late specific regions in this plot (green), compared to the dis-tribution of multi-enriched second generation stars, illustratedby the purple probability contours. Mono-enriched stars can befound at all metallicities up to almost solar, although most have[Fe / H] < −
2, and so the metallicity alone is not a reliable diag-nostic for whether the star is mono-enriched or multi-enriched.[Mg/C] further helps to quantify the likelihood of the gas be-ing enriched only once. Bottom: Probability of mono-enrichment, p = N mono / ( N multi + N mono ), for the same elemental ratios asin the top panel. There are regions of the parameter space inour model with a probability of almost 100% for finding secondgeneration stars that formed from gas that was enriched by onlyone previous SN. However, this probability does not reflect howmany stars in total are expected in these regions, as we can seeby comparing the two panels. ferent groups for Pop III SNe (Tominaga et al. 2007; Heger& Woosley 2010; Limongi & Chieffi 2012) and find a scat-ter between independent models of on average 0 . > . σ = 0 . −1.5−1−0.5 0 0.5 1−4.5 −4 −3.5 −3 −2.5 −2faint SN (12M ⊙ )CCSN (40M ⊙ ) CCSN+ faint SNchemical displacement:v=( ∆ [Mg/C], ∆ [Fe/H]) [ M g / C ] [Fe/H] Figure 12.
Illustration and definition of the chemical displace-ment for two example SNe. Combining the yields of two SNe withdifferent progenitor masses results in an effective displacement ofthe ISM metal abundances. We define the chemical displacementas the resulting vector field of this operation.
We do not account for observational or theoretical un-certainties in the top panel of Fig. 11. This is why the prob-ability map in the lower panel extends to regions that arenot sampled in the top panel. The two events at [Fe/H] < − ∼ . In this section, we propose a new, alternative method toidentify mono-enriched EMP stars based on their chemicalcomposition. This method is independent of the star forma-tion model, computationally efficient, and the qualitativeresults are insensitive to assumptions about the IMF or thefraction of faint SNe. We first introduce the underlying an-alytical arguments of this new diagnostic, compare it to theresults from our cosmological model, and finally apply it toobserved EMP stars.
Our new diagnostic is based on the chemical displacement,which is illustrated in Fig. 12. Commonly, the elementalabundances of observed EMP stars are plotted, but now wedirectly illustrate the SN yields and analyse how the chem-ical composition changes when we add the metal yields oftwo or more SNe. Each possible combination of SNe yieldsdefines two vectors which point to the resulting ISM abun-dance, as illustrated by the two arrows in this example. Theresulting vector field of the successive mixing of SN yieldsfrom different progenitor stars defines the chemical displace-ment, which we show in Fig. 13. This vector field of thechemical displacement reflects changes in the abundancesratios when more than one SN contributes to the metal en-richment. The local magnitude of this vector field quantifies c (cid:13)000
Our new diagnostic is based on the chemical displacement,which is illustrated in Fig. 12. Commonly, the elementalabundances of observed EMP stars are plotted, but now wedirectly illustrate the SN yields and analyse how the chem-ical composition changes when we add the metal yields oftwo or more SNe. Each possible combination of SNe yieldsdefines two vectors which point to the resulting ISM abun-dance, as illustrated by the two arrows in this example. Theresulting vector field of the successive mixing of SN yieldsfrom different progenitor stars defines the chemical displace-ment, which we show in Fig. 13. This vector field of thechemical displacement reflects changes in the abundancesratios when more than one SN contributes to the metal en-richment. The local magnitude of this vector field quantifies c (cid:13)000 , 1–18 ono-enriched second generation stars ● ●● ●●● ●●● ●●●● ●●●●●● ● ● ●●●●●●● ●●● ●●● ●●●● ●●●●●● ● ● ●●●●● ●● ●●● ●●● ●●●● ●●●●●● ● ● ●●●●●●● ●●● ●●● ●●●● ●●●●●● ● ● ●●●●●●● ●●● ●●● ●●●● ●●●●●● ● ● ●●●●● ●● ●●● ●●● ●●●● ●●●●●● ● ● ●●●●●●● ●●● ●●● ●●●● ●●●●●● ● ● ●●●●●●● ●●● ●●● ●●●● ●●●●●● ● ● ●●●●● ●● ●●● ●●● ●●●● ●●●●●● ● ● ●●●●●●● ●●● ●●● ●●●● ●●●●●● ● ● ●●●●●●● ●●● ●●● ●●●● ●●●●●● ● ● ●●●●●●● ●●● ●●● ●●●● ●●●●●● ● ● ●●●●● ●● ●●● ●●● ●●●● ●●●●●● ● ● ●●●●●●● ●●● ●●● ●●●● ●●●●●● ● ● ●●●●●●● ●●● ●●● ●●●● ●●●●●● ● ● ●●●●●●● ●●● ●●● ●●●● ●●●●●● ● ● ●●●●●●● ●●● ●●● ●●●● ●●●●●● ● ● ●●●●●●● ●●● ●●● ●●●● ●●●●●● ● ● ●●●●● ●● ●●● ●●● ●●●● ●●●●●● ● ● ●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ●●● ●●●● ●●●●●● ● ● ●●●●●●● ●●● ●●● ●●●● ●●●●●● ● ● ●●●●● ●● ●●● ●●● ●●●● ●●●●●● ● ● ●●●●●●● ●●● ●●● ●●●● ●●●●●● ● 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Illustration of the chemical displacement vector fieldof [Mg/C] and [Fe/H] for 25 SN progenitor masses, according toour fiducial IMF. The black points indicate the yields of singleSNe for different progenitor masses. The 25 dark grey points in-dicate the abundance ratios produced by combining the elementalyields of all possible combinations of two SNe from our set of 25.Similarly, the 25 light grey points represent the combined yieldsof three Pop III SNe. This plot illustrates how adding yields fromseveral SNe changes the typically expected elemental ratios. Forhydrogen, we assume a constant dilution mass of 7 × M (cid:12) ,which is the median hydrogen mass in our sample of halos thatare about to form second generation stars. The actual hydrogenmass and hence [Fe/H] might vary, but such an offset will notchange the qualitative results for [Mg/C]. The length of the vec-tors illustrates the local magnitude of the chemical displacementfield. The dynamical range of the vectors is decreased for betterillustration and their length is therefore not to scale. The color ofthe arrows is an additional qualitative guidance to illustrate themagnitude of the vector field. the tendency for enriched gas to be displaced from this re-gion (i.e. to change its [Mg/C] and [Fe/H] abundances) whenthe elements of an additional SN are added.To further quantify the chemical displacement, and themost promising elements for identifying mono-enriched sec-ond generation stars, we calculate the divergence of thechemical displacement field. The divergence describes theeffective outward flux of a vector field that is emanatingfrom a point. To guarantee numerical stability, we do notdifferentiate the resulting sparsely sampled vector field butapply Gauss’ theorem: for each point where a displacementvector starts, we add the length of this vector to the diver-gence of this point. Where a displacement vector ends, wesubtract the length of this vector from the divergence of thispoint.Regions in abundance space with a high negative di-vergence attract SN yield contributions from other regionsof the abundance space. Conversely, areas with a high posi-tive divergence represent regions for which mixing with theyields of a second SN shifts the elemental abundances outof this region.The information about the exact enrichment channelcannot be reconstructed uniquely for stars in areas with anegative divergence. Therefore, the divergence of the chem-ical displacement simultaneously quantifies the informationloss that occurs when combining several SN yields. A nega-tive divergence corresponds to a high degeneracy. −8 −6 −4 −2 0 [Fe/H] −3−2−1012 [ M g / C ] d i v ( c h e m i c a l d i s p l a c e m e n t )
150 100 50050100150
Figure 14.
Divergence of the chemical displacement, based onthe SN yields by Nomoto et al. (2013). Positive values indicatepromising regions to find mono-enriched second generation stars.Negative values represent attracting regions with a high chanceof degeneracy due to yields being higher overall. To find mono-enriched second generation stars, EMP stars with [Mg/C] < − . ∼ (cid:38) − To highlight the strengths and weaknesses of our new di-agnostic we compare it to the probabilities of a star beingmono-enriched, as derived from our cosmological model.The divergence map for [Mg/C] can be seen in Fig. 14.This divergence map should be compared to Fig. 11 to seethat we can reproduce the same trend with a more flexiblemethod, fewer assumptions regarding the details of Pop IIIstar formation, and with less computational time. Our newdiagnostic does not reproduce the high probability region at[Mg/C] (cid:38) . < − ∼ −
2) and pair-instability SNe ([Fe/H] ∼− . < [Mg/C] < .
0. These SN have highyields of Mg, C, and Fe and therefore dominate the metalmass budget over those of other SNe, after they were com-bined with the metal yields of a second or third SNe. Thisillustrates that it is generally difficult with our diagnosticto uniquely identify mono-enriched second generation starsthat have abundance ratios close to those produced by a SNwith a high mass of ejected metals.This implies an important consequence for EMP starsthat formed from the gas enriched by such a dominatingPop III progenitor. Since the dominating Pop III SN haslarge absolute metal yields, it can thus not be excluded thatanother progenitor SNe with a lower yield is “hidden” in theobserved stellar signature. We thus conclude that only EMPstars enriched by one (faint) SN with a small absolute metalyield can be clearly identified as mono-enriched stars.A direct comparison for [Mg/C] in one dimension isgiven in Fig. 15. The different units of the two diagnos- c (cid:13) , 1–18 T. Hartwig et al. -800-600-400-200 0 200 400 600 800 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2.5-2-1.5-1-0.50 d i v ( c h e m i ca l d i s p l ace m e n t ) l og ( p m ono ) [Mg/C]div(chem. displ.)p mono Figure 15.
Comparison of the divergence of the chemical dis-placement (purple, left y-axis) to the probability of being mono-enriched based on our cosmological model (green, right y-axis).Due to the different units, the two methods can only be com-pared at a qualitative level. The overall behaviour and predictivepower of the two diagnostics is generally the same: the maximumof the divergence of the chemical displacements corresponds to p mono (cid:38)
30% and a negative divergence to p mono (cid:46) tics allows only a qualitative comparison. Both methodsidentify the range below [Mg/C] (cid:46) − > (cid:46)
3% for mono-enrichment. This indicates an attractingregion with a high degeneracy between mono- and multi-enriched second generation stars.This comparison shows a qualitative agreement betweenour semi-analytical cosmological model and the new diag-nostic based on the divergence of the chemical displacement.We highlight again that this new diagnostic is cheaper, moreflexible and involves fewer free parameters than the full cos-mological model.
In the previously presented example, this diagnostic toolwas derived in 2D for two elemental abundance ratios butit can also be applied in higher-dimensional vector spaces ifinformation is available on additional abundance ratios orfor a single elemental ratio to obtain the trends with theseelements.We canonically expect to find mono-enriched secondgeneration stars at the lowest metallicities (Ryan et al.1996), as we show in Fig. 16. This distribution is affectedby the Pop III SFE and by the efficiency of metal mixing.Allowing the ejected metals to mix on average with a largerfraction of the gas in a halo ( µ = − .
0) shifts this distribu-tion to lower metallicities compared to the fiducial model.A lower SFE yields higher values for the fraction of mono-enriched second generation stars at all metallicities becausewe expect fewer Pop III SN to explode per halo. The fraction fr ac ti on o f m ono - e n r i c h e d s t a r s [Fe/H]fiducial, µ =-1.5 µ =-1.0 µ =-2.0 η III =3E-4
Figure 16.
Fraction of mono-enriched second generation stars asa function of the metallicity, based on our semi-analytical model.In our fiducial model, this fraction is 100% for [Fe/H] (cid:54) − − (cid:46) [Fe/H] (cid:46) −
4. There can also bemulti-enriched second generation stars at [Fe/H] (cid:46) −
6, althoughthe probability for this case is small. This distribution dependson the SFE, η III , and on the assumed fraction of hydrogen thatmixes with the ejected metals after an enrichment event, 10 µ . -1000-500 0 500 1000 -3 -2 -1 0 1 2 3 d i v ( c h e m i ca l d i s p l ace m e n t ) [A/B][C/Fe][Ca/Fe] [Mg/C][Al/Mg] [C/Cr][Sc/Mn] Figure 17.
Divergence of the chemical displacement for vari-ous elemental abundance ratios. For example, [Mg/C] < − (cid:38) < of mono-enriched stars increases with decreasing metallicity.Therefore, the [Fe/H] values on the abscissae of figures 11-14do not represent novel information as such.In a further step, we therefore calculate the 1D diver-gence of various elemental ratios as an additional diagnostic.The results are shown in Fig. 17. This not only highlights themost promising abundance ratios that should be used to find c (cid:13)000
Divergence of the chemical displacement for vari-ous elemental abundance ratios. For example, [Mg/C] < − (cid:38) < of mono-enriched stars increases with decreasing metallicity.Therefore, the [Fe/H] values on the abscissae of figures 11-14do not represent novel information as such.In a further step, we therefore calculate the 1D diver-gence of various elemental ratios as an additional diagnostic.The results are shown in Fig. 17. This not only highlights themost promising abundance ratios that should be used to find c (cid:13)000 , 1–18 ono-enriched second generation stars mono-enriched second generation stars, but it also allows usto compare different element diagnostics: the absolute valueof the divergence quantifies how strongly a certain region isgoing to be attracting or repulsing. Moreover, it is impor-tant to examine the size of the difference in the abundanceratios between regions of positive and negative divergence.If this difference is too small, as for [Al/Mg], uncertainties inboth the aluminium and magnesium yields weaken the pre-dictive power. A reliable diagnostic requires a peak of highdivergence that is significantly separated from regions withnegative divergence.Although [C/Fe] ∼ − . We apply our new diagnostic to a selection of observed starsfrom the JINAbase (Abohalima & Frebel 2017). We selectall stars with [Fe/H] < − . − − − . Our novel diagnostic based on the divergence of the chemi-cal displacement can be applied to assess the likelihood of astar to be mono-enriched. A representation of the divergencesuch as in Fig. 17 or Fig. 18 will be most useful to classifymetal-poor stars based on their measured abundances. How-ever, the divergence of the chemical displacement cannot bedirectly translated into a probability of mono-enrichment.It rather reveals regions with a positive divergence in themulti-dimensional space of stellar abundances, which aredominantly populated by mono-enriched stars. A negativedivergence is not a sufficient condition for multi-enrichment.A mono-enriched star can also be found in regions with anegative divergence, if it formed from gas enriched by an SNwith high metal yields. In a future project we will improvethis diagnostic and apply it to further EMP stars.Ji et al. (2015) also considered how the abundance ofsecond-generation stars would be affected by forming a smallmultiple of Pop III stars in minihalos. They focussed on twospecific scenarios of second-generation star formation: im-mediate gas recollapse in a minihalo and delayed formationin atomic cooling halos. These cases are applicable in theearliest stages of Pop III and Pop II star formation, butat later times global radiative feedback becomes important.Our model includes the effect of external radiation in a cos-mological context, extending its applicability to lower red-shifts. Ji et al. (2015) also focussed on specific element ratioswith critical ratios to investigate the carbon-enhanced andPISN signatures. Our new chemical divergence formalismgeneralizes their approach and allows more efficient search-ing of the ideal ratios in the full abundance space, indepen-dent of the specific assumptions of star formation.
In theoretical models of cosmic chemical evolution or of theformation of the first low-mass stars, the total metal contentis of fundamental importance. The metal content of a staris defined as the relative abundance of all elements heavierthan helium, relative to our Sun, which consists of ∼ (cid:12) = log ( M metals / (0 . M gas )). To connect thistotal metallicity to the observed abundances of individualelements, we show in Fig. 19 which element is a reliabletracer for the total metal content of a star.We find that [Ca/H] is on average about one dex aboveZ/Z (cid:12) for second generation stars, albeit with a large scatter.Iron and carbon abundances are more reliable tracers for thetotal metal content of a star and the usage of calcium canlead to severe misinterpretations: stars with an estimated[Ca/H] ≈ − (cid:12) < − ∼ . c (cid:13) , 1–18 T. Hartwig et al. −8 −6 −4 −2 0[Fe/H]−4−3−2−1012 [ M g / C ] d i v ( c h e m i c a l d i s p l a c e m e n t )
150 100 50050100150 −8 −6 −4 −2 0[Fe/H]−20246 [ C / N i ] d i v ( c h e m i c a l d i s p l a c e m e n t ) −1000−750−500−25002505007501000 −2 −1 0 1 2 3 4[N/Na]−2.0−1.5−1.0−0.50.00.51.0 [ A l / M g ] d i v ( c h e m i c a l d i s p l a c e m e n t ) −100−50050100 −4 −3 −2 −1 0 1 2[Al/Fe]−2.0−1.5−1.0−0.50.00.51.01.5 [ S c / M n ] d i v ( c h e m i c a l d i s p l a c e m e n t ) −200−1000100200 Figure 18.
Maps of the divergence of the chemical displacement for different elemental ratios overplotted with a sample of EMP starsfrom the JINAbase (Abohalima & Frebel 2017). Upper limits on the measured abundances are illustrated as arrows. This representationallows us to infer the trends of the stars being mono- or multi-enriched. Stars in regions with a high positive divergence (red) are likelyto be mono-enriched, whereas a high negative divergence (blue) indicates a possible degeneracy of elemental yields and therefore a highprobability of being multi-enriched. These divergence maps are based on the SN yields by Nomoto et al. (2013). find that carbon at Z/Z (cid:12) < − ∼ ∼ . ∼ < − .
2, a survey would reject stars with[Ca/H] > − . − . < [Ca / H] < − . < −
3. In particular, PISNe with a progenitor mass around ∼
150 M (cid:12) eject material with high [Ca/Fe] yields (Karlsson et al. 2008)and so second generation stars enriched primarily by thesePISNe will have high [Ca/Fe]. Our estimate can be used asan approximation for the completeness of surveys, althoughwe note that our simulated sample might not be completein this calcium range since we do not include enrichment bylater generations of star formation. For an assumed relationof [Ca/H]=[Fe/H]+0 .
4, we still find ∼
11% of EMP stars inthe corresponding range − . < [Ca / H] < − . . (12) Our diagnostic and predictions based on the divergence ofthe chemical displacement are only as good as the underlyingmodels for the SN nucleosynthetic yields. We use the tabu-lated SN yields as a function of the Pop III progenitor massby Nomoto et al. (2013) with additional models for faint SNeby Ishigaki et al. (2014). In a future study we will improveour model by including the metal contributions from otherenrichment channels, such as neutron star mergers, hyper-novae, AGB stars, and Type Ia SNe. Moreover we will assessthe sensitivity of our model to the assumed Pop III SN yield c (cid:13)000
11% of EMP stars inthe corresponding range − . < [Ca / H] < − . . (12) Our diagnostic and predictions based on the divergence ofthe chemical displacement are only as good as the underlyingmodels for the SN nucleosynthetic yields. We use the tabu-lated SN yields as a function of the Pop III progenitor massby Nomoto et al. (2013) with additional models for faint SNeby Ishigaki et al. (2014). In a future study we will improveour model by including the metal contributions from otherenrichment channels, such as neutron star mergers, hyper-novae, AGB stars, and Type Ia SNe. Moreover we will assessthe sensitivity of our model to the assumed Pop III SN yield c (cid:13)000 , 1–18 ono-enriched second generation stars [ X / H ] [Ca/H][C/H][Fe/H] Figure 19.
Individual elemental abundances as a function of thetotal metal content of second generation stars in our model. Thethree coloured lines indicate the binned medians of these distri-butions. Iron and carbon are better tracers of the total metallicityof second generation stars than calcium. Whereas the [Fe/H] dis-tribution lies closer to the diagonal (black dashed) and that for[C/H] slightly above, the mostly likely value for [Ca/H] is onedex above the corresponding Z/Z (cid:12) . For better illustration, weonly plot a small subset of all second generation stars. [ C a / H ] [Ca/H]=[Fe/H]+0.212% Figure 20.
Calcium against iron abundance for second genera-tion stars. The dark region represents the fraction of EMP stars([Fe/H] < −
3) that a survey would miss if it only selects starswith [Ca/H] < − . < − . models and derive a diagnostic based on elements that areleast sensitive to the underlying model assumptions.We include Pop III star formation as a sub-grid modelbased on the random sampling of individual stars from agiven IMF. However, UV feedback by the primary formedmassive star in a minihalo might prevent the formation offurther massive stars (Susa et al. 2014; Hosokawa et al.2016). Such a suppression of further Pop III stars with highermasses might result in a steeper slope of the IMF at highermasses. Moreover, we have no information on the exact po-sition of the first stars in a minihalo. Therefore we cannottake into account the effect of SNe that explode off-centre in the halo and have different metal ejection fractions, mixingefficiencies, or recovery times for the ISM.Throughout the paper we do not track individual secondgeneration stars. We rather follow their formation events andassume that such a burst of star formation creates a chem-ically homogeneous population of second generation stars.Therefore we cannot make reliable predictions about theabsolute number of second generation stars in our model.Moreover, the number of stars per halo might differ, depend-ing on the environment and the available gas mass. Largersystems, which are more likely to experience multiple previ-ous SNe, will also host more second generation stars. This isan additional bias that reduces the relative number of mono-enriched second generation stars, which tend to form in lessmassive systems. EMP stars in the MW provide a unique way to probe themass distribution of the first stars. They carry the charac-teristic chemical fingerprint of the SN that enriched the gasfrom which they formed. A comparison of their observedchemical abundances with models of Pop III SNe allows usto determine the Pop III progenitor masses of the SNe. Tofully exploit this method and avoid degeneracies in the fit-ting of the SN yields, it has to be applied to mono-enrichedsecond generation stars.In this paper, we have presented a novel diagnostic toidentify this precious subclass of mono-enriched stars. Wemodel the first generations of star formation with a semi-analytical model, based on dark matter merger trees fromthe Caterpillar simulations (Griffen et al. 2016). We findthat the Pop III star formation efficiency, the primordialIMF, the mixing efficiency of metals with the ISM, and thefraction of faint SNe are the main parameters to calibrateour model. The MDF and fraction of CEMP stars as ob-servational constraints are best reproduced by our fiducialmodel with a logarithmically flat IMF in the mass range3 −
150 M (cid:12) . With a two-sample KS test we can exclude aPop III IMF that extends up to M max = 300 M (cid:12) at the95% level. In our model, PISNe from stars with masses of (cid:38)
200 M (cid:12) fail to reproduce the MDF at [Fe/H] (cid:54) − ∼
1% of secondgeneration stars in our fiducial model. This fraction is astrong function of the primordial SFE and we provide ananalytical formula to independently calculate this fractionfor different model assumptions (Eq. 10). Dwarf satelliteshave the highest stellar fraction of mono-enriched secondgeneration stars because they formed the majority of theirstellar population early on. Satellites with M h < M (cid:12) host 10 − M h (cid:46) M (cid:12) contain only second generation stars, someof them only mono-enriched second generation stars. Thespecific numbers have to be treated with caution, since theyare affected by uncertainties in the abundance matching.We have also presented a novel analytical diagnostic toidentify mono-enriched stars, based on the divergence of thechemical displacement. This new diagnostic allows to derivethe likelihood of mono-enrichment independently from mostparameters that govern the first billion years. The fraction of c (cid:13) , 1–18 T. Hartwig et al. mono-enriched second generation stars is 100% for [Fe/H] (cid:54) − − (cid:46) [Fe/H] (cid:46) −
4. We alsopresent additional elemental ratios that are reliable tracersfor mono-enrichment, such as [Mg/C] < −
1, [Sc/Mn] < . > .
5, or [Ca/Fe] > Acknowledgements
We thank the reviewer for constructive suggestions and care-ful reading of the manuscript. The authors would like tothank Ken’ichi Nomoto, Gen Chiaki, Hajime Susa, KazuOmukai, and Wako Aoki for valuable discussions and help-ful contributions. TH is a JSPS International Research Fel-low. The authors were supported by the European ResearchCouncil under the European Community’s Seventh Frame-work Programme (FP7/2007 - 2013) via the ERC AdvancedGrant ‘STARLIGHT: Formation of the First Stars underthe project number 339177 (RSK) and via the ERC Grant‘BLACK’ under the project number 614199 (TH). SCOGand RSK also acknowledge funding from the DeutscheForschungsgemeinschaft via SFB 881 ‘The Milky Way Sys-tem’ (subprojects B1, B2, and B8) and SPP 1573 ‘Physicsof the Interstellar Medium’ (grant numbers KL 1358/18.1,KL 1358/19.2 and GL 668/2-1). AF is supported by NSFCAREER grant AST-1255160. This research has been sup-ported by the Canon Research Foundation. This researchis further supported in part by the National Science Foun-dation (NSF; USA) under grant No. PHY-1430152 (JINACenter for the Evolution of the Elements). APJ is sup-ported by NASA through Hubble Fellowship grant HST-HF2-51393.001 awarded by the Space Telescope Science In-stitute, which is operated by the Association of Universi-ties for Research in Astronomy, Inc., for NASA, under con- tract NAS5-26555. BWO was supported by the NationalAeronautics and Space Administration (NASA) throughgrant NNX15AP39G and Hubble Theory Grant HST-AR-13261.01-A, and by the NSF through grant AST-1514700.Computational support for the Caterpillar simulations wasprovided by XSEDE through the grants (TG-AST120022,TG-AST110038). Computations were carried out on thecompute cluster of the Astrophysics Division which was builtwith support from the Kavli Investment Fund, administeredby the MIT Kavli Institute for Astrophysics and Space Re-search. This work was supported by World Premier Interna-tional Research Center Initiative (WPI Initiative), MEXT,Japan.
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