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Supernova-Driven Outflows and Chemical Evolution of Dwarf Spheroidal Galaxies

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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 Ã, 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 ÃMgðtÞ: 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 ÃMgðtÞ; d½Fe∕H d½Fe∕H∕dt R λ mmax ðtÞ −2.35 λ ⊙ [10] à 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 à 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 à 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.
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