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SBU PHY 521 - Galactic Chemical Evolution
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1Galactic Chemical EvolutionIngredients The modeling of galactic chemical evolution requires four in-gredients:• Initial conditions: is the chemical composition of the initial gasprimordial or pre-enriched? Is the system closed or open (withinfall and outflow)?• The stellar birthrate function B(m, t), which is the rate at whichgas is turned into stars of a given mass. It is often expressed asthe product of the star formation rate (SFR) and the initial massfunction (IMF). There are reasons for b elieving that the IMF isindependent of time, in which case we can writeB (m, t) = SFR × IMF = ψ (t) φ (m) .• Stellar properties: stellar lifetimes τm, remnant masses wmandstellar yields Yi(m), or how elements are produced in stars andrestored into the ISM.• Gas flows, i.e., infall, outflow or ra dial flows.Some stellar properties The lifetime of a star of mass m (in M⊙) withsolar metallicity (Z⊙) is approximatelyτm≃ 11.3m−3+ 0.06m−.75+ 0.0012 Gyr.Stars of lower metallicity live shorter lives if their initial masses are lessthan about 10 M⊙; longer otherwise.Figure 1: Left: Lifetimes of stars of Z⊙. Blue curve is afit. Right: Ratio of stellar lifetimes Z⊙/20:Z⊙. (From Genevastellar models).2For low- and intermediate-mass stars, the remnant is a white dwar f withwm≃ 0.08m + 0.47; for stars 9 < m < 25 the remnant is likely a neutronstar with wm≃ 1.35; for larger masses the remnant appears to be a blackhole with wm≃ 0.24m − 4.Observational constraints• The age-metallicity t−Z relation, or AMR. This has been obtainedby combining metallicity measurements with stellar ages derivedfrom theoretical isochrones. The result can be expressed in anumber of ways, given the errors in the observations:log10t1 Gy=0.93 + 1.3 [Fe/H] − 0.04 [Fe/H]2[Fe/H] =0.68 −11.2 Gyt + 8 Gy,(1)where [Fe/H] ≡ log10(Z/Z⊙). The first form is more useful whenthe initial metallicity is very small, but it does not have an ef-fective upper limit. As t → ∞, we expect that gas will tend tobe continuously exhausted which raises Z to a terminal value, asin the second for m. The second form has Z(t → ∞)/Z⊙≈ 4.8,Z(t1)/Z⊙≈ 1.32 at the present, and Z(0)/Z⊙≈ 0.19 at t = 0.Interestingly, when one plots the second form in a linear-linearplot, one finds that it is well-approximated by a linear relationZ = Z (0) + [Z (t1) − Z (0)]tt1, (2)where t1≈ 12 Gy is the time since disk formation.• The present-time surface gas density σg= 13 ± 3 M⊙pc−2• The present-time sruface star density σ∗= 43 ± 5 M⊙pc−2• The present-time SFR ψ0= 2 − 5 M⊙pc−2Gyr−1• The present-time infall rate 0.3 − 1.5 M⊙pc−2Gyr−1• The present-time mass function• Solar abundances• Observed [Xi/Fe] vs. [Fe/H] relations• Observed G-dwarf metallicity distribution• Average SNIa (30 ± 20 yr−1) and SNI I (120 ± 80 yr−1) rates3Chemical evolution equationsTheoretically, we can attempt to calculate the evolution of Z in the solarneighborhood. Defining the birthrate B of stars as the mass of stars bornper unit time, the total stellar birthrate is:ψ (t) =Z∞mLmB (m, t) dm.mLis the lower mass limit for stars, usually taken to be 0.1 M⊙. If theinitial mass function (IMF) is constant in time it is φ(m) = B(m, t)/ψ(t)and is normalizedZ∞mLmφ (m) dm = 1.The total mass M and the mass of gas mgare usually defined in terms of amass per unit area integrated vertically through the galactic disc. The g asmass changes because of star formation (ψ), gas loss from stars Rψ, andinflow (f) and outflow (o):dmgdt= −ψ (t) + E (t) + f (t) − o (t) .E is the rate of mass ejection. If τmis the lifetime o f a star with mass m,which sheds at death all but a r emnant mass wm, and if m(t) is the massfor which τm= t, we haveE (t) =Z∞m(t)(m − wm) φ (m) ψ (t − τm) dmFor a specific chemical element,dmgXidt= −ψXi+ Ei(t) + Xi,ff (t) − Xi,oo (t) .Usually, Xi,o= Xi, but if hot supernova ejecta (rich in metals) leaves thesystem, then Xi,o> Xi. The rate of ejection of element i isEi(t) =Z∞m(t)Yi(m) φ (m) ψ (t − τm) dmwhere Yiis the stellar yield o f element i.4Derivation of the IMFFigure 2: The present-day mass function in differential formdN(m)/dm. Left: The evolution of the present-day mass func-tion. At t = 0 it is equal to the IM F φ(m). The final one(t = 13 Gyr) is the presently observed one. Right: The present-day mass function compared with data from Scalo.The present-day mass function N(m), for stars with initial masses 0.1 <m < 1 which have lifetimes τm> tG= 14 Gyr (the Hubble time),N (m) =ZtG0φ (m) ψ (t) dt.If the IMF is constant in time,N (m) = φ (m) < ψ > tG,where < ψ > represents the average SFR in the past. For stars with initialmasses 2 < m < ∞, τm<< tG,N (m) =ZtGtG−τmφ (m) ψ (t) dt ≃ φ (m) ψ0τm,where ψ0= ψ(tG) and we assumed ψ(t) didn’t change during the intervaltG− τmand tG. For 1 < m < 2 we ca nnot derive the IMF, but can usecontinuity to bridge the gap.5Figure 3: Top: Initial mass f unctions (φ(m)). Red curve isφSalpeter. Lower: Ratio of mass functions to φSalpeter.Based on stars in the solar neighborhood and accounting for variousbiases but not stellar multiplicity, Salpeter f oundφSalpeter=dNdm= Am−1−x, m > 0.1where x = 1.35. The normalization of φ, using mL= 0.1 M⊙, impliesthat A = 0.35 · 10−0.35≃ 0.156. Mor e recent studies by Scalo; Chabrier;Kroupa; and Reid and Gizis find fewer stars in the low-mass range. This isillustrated in Fig. 3.One can define the return mass fraction R(t) to beR (t) =Z∞m(t)(m − wm) φ (m) dm;it is the fraction of the mass of a stellar generation that returns to the ISM.For the IMF’s in the figure, RSalpeter(τ1) = 0.278, RKroupa(τ1) = 0.285 andRChabrier(τ1) = 0.34. For comparison, RSalpeter(τ9) and RSalpeter(τ25) are0.17 and 0.12, respectively.6Recipes for the SFRFigure 4: Left: Average surface density of star form ationrate (ψ) as a function of average gas (HI + H2) surface density(Σgas). The solid curve is a power law with exponent 1.4.(Right:) Average surface density of star formation rate as afunction o f Σgas/τdyn. The solid curve is a power law withexponent of 1.0.The SFR is probably the most uncertain quantity in ga lactic chemicalevolution models. Most observational information, such as from luminousstar counting, flux measurements in r ecombination lines from gas ionizedby OB stars, the UV flux of OB stars, far-infrared emission of dust


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SBU PHY 521 - Galactic Chemical Evolution

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