Supernova-Driven Outflows and Chemical Evolution of Dwarf Spheroidal Galaxies

Supernova-driven outflows and chemical evolution of dwarf spheroidal galaxies

Yong-Zhong Qiana,1 and G. J. Wasserburgb,1

aSchool of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455; and bThe Lunatic Asylum, Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125

Contributed by G. J. Wasserburg, January 30, 2012 (sent for review October 13, 2011)

We present a general phenomenological model for the metallicity Recent observations (14) give results for eight dSphs with rather distribution (MD) in terms of ½Fe∕H for dwarf spheroidal galaxies detailed structure of their MDs and provide a basis for exploring (dSphs). These galaxies appear to have stopped accreting gas from models of their chemical evolution (e.g., ref. 15). Among the key the intergalactic medium and are fossilized systems with their stars issues that we try to address are MDs exhibited by stellar popula- undergoing slow internal evolution. For a wide variety of infall his- tions of dSphs. Our general approach follows that of Lynden-Bell tories of unprocessed baryonic matter to feed star formation, most described in his incisive and excellent article on “theories” of the of the observed MDs can be well described by our model. The key chemical evolution of galaxies (16). It will be shown that the me- requirement is that the fraction of the gas mass lost by supernova- tallicity at which the MD peaks is directly related to the efficiency driven outflows is close to unity. This model also predicts a relation- of SN-driven outflows and that the MD of a dSph and the rela- ship between the total stellar mass and the mean metallicity for tionship between the stellar mass and the mean metallicity for dSphs in accord with properties of their dark matter halos. The these galaxies are direct consequences of the model. These results model further predicts as a natural consequence that the abun- are in strong support of those of ref. 17, where earlier and less dance ratios ½E∕Fe for elements such as O, Mg, and Si decrease for precise data on metallicities of dwarf galaxies were used to ad- stellar populations at the higher end of the ½Fe∕H range in a dSph. dress this problem. We show that, for infall rates far below the net rate of gas loss to In our approach, we consider evolution of Fe in a homoge-

star formation and outflows, the MD in our model is very sharply neous system of condensed gas governed by ASTRONOMY ∕ peaked at one ½Fe H value, similar to what is observed in most globular clusters. This result suggests that globular clusters may dM dM g g − ψ − [1] be end members of the same family as dSphs. ¼ ðtÞ FoutðtÞ; dt dt in n this paper, we show that supernova-driven gas outflows play a dMFe − MFeðtÞ ψ [2] Iprominent role in the chemical evolution of dwarf spheroidal ¼ PFeðtÞ ½ ðtÞþFoutðtÞ; galaxies (dSphs). In the framework of hierarchical structure for- dt MgðtÞ mation based on the cold dark matter cosmology, dwarf galaxies ∕ are the building blocks of large galaxies such as the Milky Way. In where MgðtÞ is the mass of gas in the system at time t, ðdMg dtÞin support of this picture, some recent observations showed that ele- is the infall rate of pristine gas, ψðtÞ is the star formation rate mental abundances in dSphs of the Local Group match those in (SFR), FoutðtÞ is the rate of gas outflow, MFeðtÞ is the mass of the Milky Way halo at low metallicities (e.g., refs. 1–5; see ref. 6 Fe in the gas, and PFeðtÞ is the net rate of Fe production by for a review of earlier works). It is expected that detailed studies all sources in the system. We assume that the SFR is proportional λ of chemical evolution of dwarf galaxies can shed important light to the mass of gas in the system with an astration rate constant , on the formation and evolution of the Milky Way in particular ψ λ [3] and large galaxies in general. Here we present an analysis of ðtÞ¼ MgðtÞ: the evolution of ½Fe∕H¼logðFe∕HÞ − logðFe∕HÞ⊙ focusing on ∕ 1 2 dSphs. The approach is a phenomenological one that takes into Given ðdMg dtÞin, FoutðtÞ, and PFeðtÞ, Eqs. and can be solved 0 0 0 0 account infall of gas into the dark matter halos associated with with the initial conditions Mgð Þ¼ and MFeð Þ¼ . these galaxies, star formation (SF) within the accumulated gas, The MD of a system measures the numbers of stars formed in and outflows driven by supernova (SN) explosions. The sources different metallicity intervals that survive until the present time. for production of Fe are core-collapse SNe (CCSNe) from pro- We use ½Fe∕H to measure metallicity. As the mass fraction of H genitors of 8–100 M⊙ and Type Ia SNe (SNe Ia) associated with changes very little over the history of the universe, we take ∕ stars of lower masses in binaries. Observations require that some ½Fe H¼log ZFe, where SNe Ia must form early along with CCSNe without a significant delay. It will be shown that there is a direct and simple connection ≡ MFeðtÞ [4] ZFeðtÞ ⊙ : between the metallicity distribution (MD) for a given dSph and X FeMgðtÞ λ ∕λ α λ two parameters Fe and . The ratio refers to the net rate Fe of ⊙ Fe production and the net rate λ of gas loss to SF and SN-driven Here X Fe is the mass fraction of Fe in the sun. We assume that outflows, and α indicates the promptness for reaching peak infall the initial mass function of SF does not change with time and is of rates. This model explicitly predicts the ratio of the stellar mass in the Salpeter form. Then the number of stars formed per unit mass the dSph to the total mass of the host dark matter halo. interval per unit time is related to the SFR as Phenomenological models for chemical evolution have a long

history (e.g., ref. 7) and were applied to dSphs previously (e.g., Author contributions: Y.-Z.Q. and G.J.W. designed research, performed research, analyzed refs. 8–10). Dynamic models for dSphs including dark matter data, and wrote the paper. were also studied (e.g., refs. 11 and 12). The first effort was made The authors declare no conflict of interest. in ref. 13 to reconcile models of hierarchical structure formation 1To whom correspondence may be addressed. E-mail: [email protected] or gjw@ involving dark matter halos with the then-available luminosity- gps.caltech.edu. radius-metallicity relationships for dwarf galaxies. There it was This article contains supporting information online at www.pnas.org/lookup/suppl/ shown that SN-driven outflows could explain the observed trends. doi:10.1073/pnas.1201540109/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1201540109 PNAS Early Edition ∣ 1of6 Downloaded by guest on September 26, 2021 2 ψ −2.35 d N ðtÞ R m [5] between the birth and death of all SN progenitors, except when ¼ m −1.35 ; dmdt M⊙ u m dm the amount of gas in the system is so low that SNe Ia would ml dominate. where m is the stellar mass in units of M⊙ with ml and mu being Under our assumption, the total rate of CCSNe and SNe Ia, 0 1 the lower and upper limits, respectively. We take ml ¼ . and and hence P ðtÞ, are proportional to the SFR. We further as- 100 Fe mu ¼ . Assuming that ZFeðtÞ increases monotonically with sume that the rate of outflows driven by these SNe is also propor- time, we obtain the MD tional to the SFR. Specifically, we take R mmax ðtÞ 2 dN 0 1 ðd N∕dmdtÞdm ηλ [9] ¼ . FoutðtÞ¼ MgðtÞ; d½Fe∕H d½Fe∕H∕dt R λ mmax ðtÞ −2.35 λ ⊙ [10] 0R.1 m dm PFeðtÞ¼ FeX FeMgðtÞ; ¼ 100 −1.35 log e 0.1 m dm where η is a dimensionless constant that measures the efficiency M ðtÞ Z t × g Feð Þ [6] of the SN-driven outflows, and λ is a rate constant that is pro- ∕ ; Fe M⊙ dZFe dt λ portional to and the effective Fe yield of SNe. We take the infall rate to be where m ðtÞ is the maximum mass of those stars formed at time max t that survive until the present time (cf. ref. 18). There is little SF α dM ðλ tÞ in dSphs at the present time. We assume that SF ended at time t g λ in −λ [11] f ¼ inM0 Γ α 1 expð intÞ; in a system. Then the total number of stars in the system at the dt in ð þ Þ present time is λ Z Z where in is a rate constant, M0 is the total mass of gas infall over 2 0 ≤ ∞ Γ α 1 α −1 Γ tf mmax ðtÞ d N t < , and ð þ Þ with > is the function of argu- Ntot ¼ dmdt ment α 1. The above modified exponential form was specifi- 0 0.1 dmdt þ Z Z cally chosen to explore the role of the time dependence of the λ tf M ðtÞ mmax ðtÞ dmdt R g [7] infall rate in chemical evolution (cf. ref. 7). The infall rate peaks ¼ 100 −1.35 2.35 : 0 1 m dm 0 M⊙ 0.1 m 0 −1 α ≤ 0 α∕λ . at t ¼ for < , and the peak time increases to in for α > 0. The form with α > 0 allows a slow start of significant gas The integral involving m ðtÞ in Eqs. 6 and 7 increases only by max accumulation in the system. 6% when m ðtÞ increases from 0.8 to 100. Because stars with max With the above assumptions, Eqs. 1 and 2 become m ¼ 0.8 have a lifetime approximately equal to the age of the universe, we take m ðtÞ¼0.8 in both these equations to obtain max dM λ α the normalized MD g λ ð intÞ −λ − λ [12] ¼ inM0 Γ α 1 expð intÞ MgðtÞ; dt ð þ Þ 1 dN 1 M ðtÞ Z ðtÞ R g Fe [8] ¼ t : N d½Fe∕H log e f dZ ∕dt dMFe ⊙ tot 0 MgðtÞdt Fe ¼ λ X M ðtÞ − λM ðtÞ; [13] dt Fe Fe g Fe λ ≡ 1 η λ λ ≪ λ Model where ð þ Þ . Note that, for in , the solutions to the ∕ ≈ The key input for our model is the infall rate ðdMg dtÞin, the out- above equations approach a secular state for which MgðtÞ ∕ ∕λ ≈ λ ∕λ flow rate FoutðtÞ, and the net Fe production rate PFeðtÞ. The latter ðdMg dtÞin and ZFe Fe . This situation is analogous to two rates are closely related to the occurrences of CCSNe and the quasi-steady state for the metallicity of the interstellar med- SNe Ia in a system. CCSNe are associated with massive stars ium first proposed in ref. 24. For simplicity, we assume that λ (8 < m ≤ 100) that evolve rapidly. In contrast, SNe Ia require in and λ are so large that t ¼ ∞ can be used effectively in Eq. 8 for consideration of the evolution of binaries involving lower-mass f 0 ∞ 0 12 stars with longer lifetimes. In all our previous studies (e.g., the MD. As Mgð Þ¼Mgð Þ¼ , integrating Eq. over t gives λ∫ ∞ ref. 19), we considered that the evolution timescale for SNe Ia 0 MgðtÞdt ¼ M0. So the total gas mass used in SF is was ∼1 Gy using the lifetime of stars with m ∼ 2. This considera- λ ∫ ∞ ∕ 1 η 0 MgðtÞdt ¼ M0 ð þ Þ and the remainder of the gas infall tion meant that SNe Ia would not contribute to the Fe inventory is blown out as outflows. We assume that all outflows are lost during early epochs and was in accord with the general approach from the system into the broader intergalactic medium (IGM), used by other workers. The consequence of this assumption is that the MD for a system must have two peaks due to the assumed thus enriching the latter in metals. This approach gives a natural late onset of SNe Ia (e.g., ref. 19). However, of the eight dSphs cutoff to the chemical evolution of the system when the total mass studied in ref. 14, only a single peak is observed in the MD for of gas lost to SF and outflows is equal to the total mass of gas Fornax, Leo I, Leo II, Sextans, Draco, and Canes Venatici I, and infall (partial recycling of outflows would increase the degree there is only some indication for two peaks in the MD for Ursa of chemical enrichment but is ignored here for simplicity). At Minor and perhaps Sculptor. From this observation, we conclude the present time, the total mass of stars in the system can be es- that there must be a prompt component of SNe Ia that start in a timated as system on much shorter timescales than approximately 1 Gy. R Such a component is supported by both supernova surveys 0.8 −1.35 0 1 m dm M0 −2 Mh (e.g., ref. 20) and models that considered detailed evolution of M ¼ R . ¼ 9.65 × 10 ; [14] 100 m−1.35dm 1 þ η 1 þ η various binary configurations (e.g., refs. 21 and 22). The occur- 0.1 rences of “prompt” and “delayed” SNe Ia were investigated in Ω ∕Ω 0 17 ref. 23, where it was argued that both populations were present where we have used M0 ¼ð b mÞMh ¼ . Mh. Here Mh is at high redshift. The nature of these two classes of SNe Ia remains the total mass inside the dark matter halo hosting the system, Ω Ω unclear. For simplicity, we will lump the Fe production by these and b and m are the fractional contributions to the critical den- sources together with that by CCSNe and ignore the time delay sity of the universe from baryonic and all matter, respectively.

2of6 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1201540109 Qian and Wasserburg Downloaded by guest on September 26, 2021 Exploration of Results from the Model A 2 The parameters governing the solutions to Eqs. 12 and 13 are α, λ λ λ in, , and Fe. We illustrate the dependences of the MD on these 1.5 parameters in the following subsections.

Dependence of the MD on λin∕λ with α ¼ 0. For α ¼ 0, the solutions 1 12 13 )dN/d[Fe/H] to Eqs. and are tot

λ (1/N 0.5 in −λ − −λ [15] MgðtÞ¼λ − λ M0½expð tÞ expð intÞ; in 0 λ λ -3 -2 -1 0 Fe in ⊙ M t X M0 exp −λ t [Fe/H] Feð Þ¼ λ − λ 2 Fe f ð in Þ ð in Þ 250 λ − λ − 1 −λ [16] B þ½ð in Þt expð tÞg; 200 from which we obtain λ λ − λ 150 Fe ð in Þt − 1 [17] ZFeðtÞ¼λ − λ 1 − − λ − λ ; exp t )dN/d[Fe/H] in ½ ð in Þ 100 tot (1/N 50 1 dN 1 λ λ f1 − exp½−ðλ − λÞtg2 ¼ in in N d½Fe∕H log e ðλ − λÞ2 expðλtÞ tot in 0 ðλ − λÞt − 1 þ exp½−ðλ − λÞt -1 -0.99 -0.98 -0.97 -0.96 -0.95 × in in [18] [Fe/H] 1 − 1 λ − λ − λ − λ : ½ þð in Þt exp½ ð in Þt

λ ∕λ 0 1 λ ∕λ α 0 ASTRONOMY Fig. 1. Example MDs for Fe ¼ . and various values of in with ¼ . λ ∕λ For λ ∕λ ¼ 1, the above two equations reduce to Note that in general the shape of the MD is determined by in and chan- in λ ∕λ ∕ A ging Fe only translates the MD along the ½Fe H axis. ( ) The solid, dashed, λ ∕λ 1∕2 ∞ λ dotted, and dot-dashed curves are the MDs for in ¼ , 1, 2, and , Fet [19] ∕ ZFeðtÞ¼ ; respectively. The vertical dashed line indicates the peak at ½Fe H¼ 2 λ ∕λ −1 B logð Fe Þ¼ for the dashed and dot-dashed curves. ( ) The solid curve λ ∕λ 0 1 is the MD for in ¼ . and the dashed line indicates the limiting value 2 ∕ λ ∕ λ − λ −0 954 1 dN ðλtÞ of ½Fe H¼log½ Fe ð inÞ ¼ . . Concentration of stars in an extremely ¼ expð−λtÞ: [20] ∕ N d½Fe∕H log e narrow range of ½Fe H as shown by the solid curve is typically not observed tot for dSphs but strongly resembles what is observed for most globular clusters. λ ∞ 0 In the limit in ¼ , which is equivalent to setting Mgð Þ¼M0 ∕ 0 0 metallicity range is typically not observed for dwarf galaxies and ðdMg dtÞin ¼ for t> , the results are but strongly resembles what is observed for most globular clus- λ [21] λ ∕λ 1∕2 ZFeðtÞ¼ Fet; ters. In general, for in significantly below , stars are con- ∕ λ ∕ λ − λ centrated immediately below ½Fe H¼log½ Fe ð inÞ.For 1 dN λt comparison, the solid curve in Fig. 1A shows that the MD for ¼ expð−λtÞ: [22] λ ∕λ 1∕2 N d½Fe∕H log e in ¼ first rises to a peak and then sharply drops to zero tot ∕ → λ ∕ λ − λ as ½Fe H log½ Fe ð inÞ. This general behavior also applies λ ∕λ 1∕2 In general, Eqs. 17 and 18 give the MD as a function of ½Fe∕H to in > but with a more extended tail at high metallicities λ ∕λ λ ∕λ λ ∕λ A in parametric form. This MD only depends on in and Fe , for a larger in (see Fig. 1 ). but not on the absolute values of these rates (this is true so long as λ ≫ 1 λ ≫ 1 8 15 λ ∕λ Dependence of the MD on α with λin ¼ λ. Based on the above dis- tf and intf ; see Eqs. and ). For a fixed in , chan- λ ∕λ ∕ cussion and the example MDs shown in Fig. 1, it seems reason- ging Fe only translates the MD along the ½Fe H axis. This λ ∕λ 1 λ λ result can be most easily seen in the special cases of in ¼ able to choose in ¼ and explore possible MDs for different λ ∞ −1 ∕ ∕ λ α λ ∕λ λ λ α −1 and in ¼ , where Ntot dN d½Fe H is a simple function of t values of and Fe .For in ¼ and > , the solutions ∕ λ λ ∕2λ 12 13 and ½Fe H differs from logð tÞ only by a shift of logð Fe Þ to Eqs. and are λ ∕λ and logð Fe Þ, respectively. The shape of the MD is determined λ αþ1 λ ∕λ A ð tÞ by in . This dependence can be seen from Fig. 1 , which uses M ðtÞ¼M0 expð−λtÞ; [23] λ ∕λ 0 1 λ ∕λ 1∕2 ∞ g Γðα þ 2Þ Fe ¼ . and shows that, as in increases from to , the position of the peak of the MD changes slightly [but staying ∕ λ ∕λ −1 λ λ αþ2 close to ½Fe H¼logð Fe Þ¼ , the exact peak position for Fe ⊙ ð tÞ λ ∕λ 1 λ ∞ M ðtÞ¼ X M0 expð−λtÞ; [24] in ¼ and in ¼ ] and the shape of the MD becomes broad- Fe λ Fe Γðα þ 3Þ er. Note the sharp cutoff of the MD to the right of the peak for λ ∕λ 1∕2 ∕ λ ∕ in ¼ with no stars formed above ½Fe H¼log½ Fe λ − λ −0 7 which give ð inÞ ¼ . . λ ∕λ 1∕2 The case of in < requires separate discussion. For illus- λ t λ ∕λ 0 1 λ ∕λ Z ðtÞ¼ Fe ; [25] tration, we again take Fe ¼ . and show the MD for in ¼ Fe α þ 2 0.1 in Fig. 1B. This MD has an extremely sharp peak with 90% of −1 154 ≤ ∕ −0 954 the stars having . ½Fe H < . . This result is simply 1 1 λ αþ2 a case close to secular equilibrium (see discussion below Eqs. 12 dN ð tÞ −λ [26] ∕ ¼ Γ α 2 expð tÞ: and 13; cf. ref. 24). Piling up of stars in an extremely narrow Ntot d½Fe H log e ð þ Þ

Qian and Wasserburg PNAS Early Edition ∣ 3of6 Downloaded by guest on September 26, 2021 ∕ λ ∕λ The above MD again has a peak at ½Fe H¼logð Fe Þ for all Careful inspection of these data shows that there is only some α −1 λ ∕λ > . Changing Fe only translates the MD, shifting the peak indication for two peaks in the MD for Ursa Minor and perhaps in particular, along the ½Fe∕H axis. The shape of the MD is de- Sculptor. An MD with two peaks would be typical if the turn-on α λ ∕λ termined by . For a specific Fe , the MD becomes narrower of SNe Ia were sudden and with a significant delay relative to while peaking at the same ½Fe∕H when α increases above −1.A CCSNe. It was the lack of such MDs from observations that positive α reflects slower infall at the start of the system, which led us to pursue a model where the net Fe production rate is with- suppresses SF at the early stages and reduces the extent of the out discontinuities. λ λ α low-metallicity tail of the MD. For α < 0, the initial infall rate We focus on models with in ¼ and different values of .In λ ∕λ α is enhanced and, consequently, more stars are formed at lower this case, a model MD is specified by Fe and , which deter- ∕ λ ∕λ metallicities. mine its peak position ½Fe H¼logð Fe Þ and its shape, respec- Note that, if we take the limit α ¼ −1, the MD given by Eqs. 25 tively. Positive values of α corresponding to slower initial infall 26 λ ∞ α 0 21 and coincides with that for in ¼ and ¼ (see Eqs. and give rise to narrower MDs, whereas negative values correspond- 22 λ ∕λ 0 1 α −1 ing to more rapid initial infall result in more extended MDs. As ). Thus, for Fe ¼ . ,as increases from to 0, the MD λ λ the MD is normalized, its peak height can be used to estimate α for in ¼ changes from the dot-dashed to the dashed curve shown in Fig. 1A while peaking at the same ½Fe∕H¼−1. More effectively. Using the position and the height of the peak for the λ λ α ≥−1 observed MD as guides, we fit a model MD for each of the eight example MDs for in ¼ and are shown in Fig. 2 where we compare them with observations of dSphs. dSphs reported in ref. 14. The observed and fitted MDs are shown as histograms and curves, respectively, in Fig. 2. The λ ∕λ α Comparison with Observations adopted values of Fe and are indicated for each galaxy. We now explore the implications of our model for observations of These values were not obtained from the best fits, but were simply dSphs. We first discuss the MDs using the high-quality dataset of picked to illustrate the overall adequacy of our model. Very good ref. 14 and then study the relationship between the mean metal- fits are obtained for Fornax and Leo I, which are the most massive ∕ licity h½Fe Hi and the stellar mass M of dSphs. of the eight dSphs. The fits for the least massive four, Draco, Sex- tans, Ursa Minor, and Canes Venatici I, are rather good, although MDs for dSphs. Important medium-resolution data on the MDs for the model MDs appear to underestimate their stellar populations dSphs were provided in ref. 14 and are summarized in Fig. 2. at the highest metallicities. The only exceptions are Leo II and

2 2 2 A 7 B C M M 6 6 Fornax * = 1.9x10 sun Leo I M* = 4.6x10 Msun Leo II M* = 1.4x10 Msun λ λ λ λ λ λ 1.5 log( Fe/ ) = -0.95 1.5 log( Fe/ ) = -1.35 1.5 log( Fe/ ) = -1.5 λ λ α λ λ α in/ = 1, = 1 in/ = 1, = 1 solid: λ λ α 1 1 1 in/ = 1, = 1 )dN/d[Fe/H] )dN/d[Fe/H] )dN/d[Fe/H] tot tot tot dashed: (1/N (1/N (1/N λ λ α 0.5 0.5 0.5 in/ = 0.8, = 0

0 0 0 -4 -3 -2 -1 0 -4 -3 -2 -1 0 -4 -3 -2 -1 0 [Fe/H] [Fe/H] [Fe/H]

2 2 2 5 D 6 E 5 F Sextans M* = 8.5x10 Msun Sculptor M* = 1.2x10 Msun Draco M* = 9.1x10 Msun λ λ λ λ λ λ log( Fe/ ) = -1.85 1.5 log( Fe/ ) = -1.55 1.5 log( Fe/ ) = -1.85 1.5 λ λ α λ λ α λ λ α in/ = 1, = -0.55 in/ = 1, = -1 in/ = 1, = -0.4 1 1 1 )dN/d[Fe/H] )dN/d[Fe/H] )dN/d[Fe/H] tot tot tot (1/N 0.5 (1/N 0.5 (1/N 0.5

0 0 0 -4 -3 -2 -1 0 -4 -3 -2 -1 0 -4 -3 -2 -1 0 [Fe/H] [Fe/H] [Fe/H] 2 2

G 5 H 5 Ursa Minor M* = 5.6x10 Msun Canes Venatici I M* = 3.0x10 Msun λ λ λ λ 1.5 log( Fe/ ) = -2.1 1.5 log( Fe/ ) = -1.85 λ λ α λ λ α in/ = 1, = -0.3 in/ = 1, = -1 1 1 )dN/d[Fe/H] )dN/d[Fe/H] tot tot (1/N 0.5 (1/N 0.5

0 0 -4 -3 -2 -1 0 -4 -3 -2 -1 0 [Fe/H] [Fe/H]

Fig. 2. Comparison of model MDs with observations of dSphs. The data are taken from ref. 14 and shown as histograms with error bars. The model MDs C M assume the indicated parameters and are shown as curves. The dashed curve in provides a better fit to the data than the solid curve. Values of are taken from ref. 25 for A–G and from ref. 26 for H. Note that, if the baryonic matter is not always blown out of the dark matter halo, but returns to the gas mass in a dSph after some time, then the curves will have a leading-edge tail going to higher ½Fe∕H.

4of6 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1201540109 Qian and Wasserburg Downloaded by guest on September 26, 2021 λ ∕λ ∝ β Sculptor with intermediate M. Using the same Fe , we ob- where we have assumed rh M in the second step, which gives λ ∕λ 0 8 α h tained a better fit to the data for Leo II with in ¼ . and ¼ 0 C λ ∕λ 1 α 1 ∕ ∼− η ∼ 1 − β [28] (dashed curve in Fig. 2 ) than with in ¼ and ¼ (solid h½Fe Hi log ð Þ log Mh þ constant: curve). However, neither type of MD can provide a good fit to the data for Sculptor, which may represent an MD with two peaks. In addition, Eq. 14 gives There is perhaps a more clear indication for such an MD for Ursa Minor. We note that, if infall took a more complicated form than ∝ ∕η ∝ 2−β [29] M Mh Mh : Eq. 11, then the shape of the MD would change accordingly. It is possible that the broadened peak in the MD for Sculptor might be ∕ Thus, a slope of 2.5 for the relationship between logðM M⊙Þ explained by an increase in the infall rate after the assumed and h½Fe∕Hi requires ð2 − βÞ∕ð1 − βÞ¼2.5,orβ ¼ 1∕3, which smooth rate in Eq. 11 peaked. However, in no case can a signifi- is in agreement with the framework of hierarchical structure cant part of the enriched outflows be returned to the infalling formation (e.g., Eq. 24 in ref. 27). matter because this would produce a population of stars with high ½Fe∕H values, which is not observed. Conclusions In a previous study (14) with alternative modeling, detailed We have shown that the MDs of dSphs can be reasonably well multiparameter fitting was used. As can be seen from Fig. 2, a described quantitatively by a phenomenological model based good to excellent fit of the model to the data can be obtained on general considerations of gas infall into a dark matter halo, in our approach. The model thus appears to give a good descrip- astration in the condensed gas, and SN-driven gas outflows. tion of the MDs of dSphs and is easily understood in terms of The peak position of the MD is governed by the ratio of the rate physical processes. constants for net Fe production and net gas loss to astration and λ ∕λ λ ∕ 1 η λ outflows, Fe ¼ Fe ½ð þ Þ . The observations require high Relationship Between h½Fe∕Hi and M for dSphs. The observed MDs efficiency η ∼ 10–360 (see Table S1) for SN-driven outflows and λ ∕λ ∼ 0 01–0 1 of dSphs require that Fe . . (see Fig. 2). Because hence, massive gas loss from dark matter halos associated with λ ∕λ 1 η −1 λ SI Fe is approximately ð þ Þ (see derivation of Fe in dSphs. The model also directly relates the stellar mass M re- Text), this requires that η ∼ 10–100. Consequently, only a fraction maining in a dSph to its mean metallicity h½Fe∕Hi through −1 ð1 þ ηÞ ∼ 1–10% of the gas falling into the dark matter halo the efficiency of outflows governed by the mass Mh and the radius hosting a dSph is used in SF and the rest is blown out of the halo rh of the dark matter halo. The observed relationship between

∕ ∕ ASTRONOMY by SN-driven outflows. The MD of a dSph peaks at essentially its logðM M⊙Þ and h½Fe Hi has a slope of 2.5 over the wide range ∕ ≈ λ ∕λ ∼− η 4 8 × 103 ≤ ∕ ≤ 4 6 × 108 SI mean metallicity h½Fe Hi logð Fe Þ log . The lower of . M M⊙ . for dwarf galaxies (see h½Fe∕Hi a dSph has, the lower fraction of the infalling gas is Text ∝ 1∕3 ), indicating rh Mh in agreement with the framework stored in its stars. There is thus a direct relationship between of hierarchical structure formation. Our results confirm the pre- ∕ h½Fe Hi and the stellar mass M for dSphs. Using the data vious studies of refs. 17 and 25, and are in support of the early ∕ on h½Fe Hi in ref. 14 and those on M in refs. 25 and 26, we work of Dekel and Sulk (13) on the general model of dwarf galaxy show this relationship in Fig. 3. The dot-dashed line in Fig. 3 formation. has a slope of 2.5 and passes through the point defined by the λ ≪ λ Our model also demonstrates that, for slow infall rates in , average values of the data. It is nearly the same as the least- the resulting MD must be extremely sharply peaked (essentially square fit (solid line) and is in excellent agreement with the result concentrated at a single value of ½Fe∕H). Such MDs apply to all in ref. 25 (see SI Text). globular clusters with the exception of the most massive ones As discussed in ref. 17, the slope of 2.5 for the relationship (e.g., ref. 28), suggesting that globular clusters are the result of ∕ ∕ between logðM M⊙Þ and h½Fe Hi has a simple physical expla- the general process of astration governed by very slow infall rates nation. In general, this slope follows from the relationship be- compared to net gas loss rates. These slow infall rates further sug- tween the radius rh and the total mass Mh of the dark matter gest that globular clusters might be weak feeders during the in- halo hosting a dSph in addition to the dependences of M and homogeneous evolution inside a large system and that they are h½Fe∕Hi on η discussed above. The outflow efficiency η is inver- not responsible for significant depletion of the baryonic supply sely proportional to the depth of the gravitational potential well of that system. As such, the dark matter halo would be associated of the dark matter halo, with the entire (much larger) system, but not be specifically tied to the globular clusters. This behavior is in contrast to that of η ∝ ∕ ∝ β−1 [27] rh Mh Mh ; dSphs, which result from processing all the baryonic supply in their dark matter halos and are naturally associated with these 8 dark matter halos. Nonetheless, from the point of view of the analysis presented here, we consider globular clusters to be part of a 7 For family of early formed dwarf galaxies but with low effective infall Leo I rates compared to outflow rates. With regard to the diverse mor- ) Dra Leo II phological types of dwarf galaxies, we consider dSphs to represent sun 6 UMi Sex Scl M

/ isolated evolution without dynamic effects from mergers or tidal *

M CVn I interactions with nearby systems. Other morphological types may 5 log( Her result from such effects. Insofar as all dwarf galaxies are related Leo IV UMa I by the same general process as presented here, the observed on- 4 CVn II going astration in some dwarf irregular galaxies (e.g., ref. 18) UMa II Com poses a problem. These systems must be experiencing secondary 3 -2.5 -2 -1.5 -1 processes of gas accretion due to local infall or mergers. Their <[Fe/H]> MDs should have a second peak due to the late infall. M ∕M ∕ The analysis presented here assumes that there is not a discon- Fig. 3. The relationship between logð ⊙Þ and h½Fe Hi for dSphs. Values M tinuous onset of SNe Ia contributing Fe. Otherwise, the general of for filled and open diamonds are taken from refs. 25 and 26, respectively. All values of h½Fe∕Hi used here are taken from ref. 14. The solid line is outcome would be MDs with two peaks, which are at most only the least-square fit to the data and the dashed lines indicate the 1σ error in rarely observed. It is not appropriate to extend this analysis to M ∕M logð ⊙Þ. The dot-dashed line has a slope of 2.5 and is nearly the same as other metals than Fe without a more realistic treatment of the the solid line. relative contributions of CCSNe and SNe Ia that each produce

Qian and Wasserburg PNAS Early Edition ∣ 5of6 Downloaded by guest on September 26, 2021 very different yields of the other metals. We note here that, as the accord with the conclusions from previous studies to infer the rate of CCSNe decreases with decreasing gas mass, the contribu- halo masses from observations of dSphs (e.g., ref. 29). As a result tions to Fe from SNe Ia will become larger or dominant. The rea- of the massive gas loss, extensive enrichment of metals has oc- son is that previously formed stars of low to intermediate masses curred in the general hierarchical structures outside individual in binaries will continue to evolve and produce SNe Ia even after dSphs. This enrichment drastically alters the subsequent chemical the SFR decreases. Crudely speaking, the net Fe production rate evolution of the emerging larger galaxies as most baryonic matter appropriate for later times must be changed from Eq. 10 to must have passed through processing in dSphs. As more massive ∼ ⊙ λCC λIa − Δ λCC λIa PFeðtÞ X Fe½ Fe MgðtÞþ FeMgðt Þ, where Fe and Fe are dark matter halos are formed during hierarchical growth, the ef- the rate constants for CCSNe and SNe Ia, respectively, and Δ ficiency of outflows will decrease. Nevertheless, the general IGM is a typical delay between the birth and death of the progenitors must have been enriched by the net outflows from all dSphs. If the for SNe Ia. As SNe Ia do not produce, e.g., O, Mg, and Si, this will average IGM has ½Fe∕H ∼−3 and the outflows from dSphs have result in lower values of ½E∕Fe for these elements at higher ½Fe∕H ∼−1.5 on average, this would imply that approximately values of ½Fe∕H, which is in agreement with the general trend 3% of all baryonic matter was processed in dSphs. observed in dSphs (e.g., ref. 15). In conclusion, dSphs have evolved as a result of very massive ACKNOWLEDGMENTS. We greatly appreciate the thoughtful and helpful gas loss and this gas has gone into the medium outside the dark comments by D. Lynden-Bell and R. Blandford in their efforts to improve matter halos associated with these galaxies. The implication is our paper. We thank Evan Kirby for providing his data on the MDs of dSphs that these dark matter halos must have masses approximately in electronic form. This work was supported in part by Department of Energy 102–103 Grant DE-FG02-87ER40328 (to Y.-Z.Q.). G.J.W. acknowledges the National times greater than the present total mass in stars after Aeronautics and Space Administration’s Cosmochemistry Program for the gas loss and the cosmic ratio of baryonic to all matter are research support provided through J. Nuth at the Goddard Space Flight taken into account (see TableS1). This conclusion is in qualitative Center. He also appreciates the generosity of the Epsilon Foundation.

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