– 1 –1. Chemical Evolution1.1. Analytical Chemical Evolution ModelsThe data we hope to reproduce:(1) the gas content of a stellar system as a function of time(2) the overall metallicity in the gas as a function of time(3) the metallicity distribution o f the stars a s a function of time(4) the detailed abundances of various elements as a function of time in the gas(5) the total mass of the system as a function of time(6) the fraction of mass locked up in stellar remnants (neutron stars, white dwarfs) as afunction of time(7) the number of type of SN, neutron stars, novae, X-ray binaries, etc as a function of timeObservational data for most of these issues exists for the various components of theMilky Way galaxy, less detailed data exists for other nearby galaxies within the LocalGroup. Beyond the Local Group, the data is quite limited with r ega rd to these issues.1.2. Simple Homogenous ModelIn a simple homogenous model, one assumes uniform mixing within the gas over theentire systmem, homogenous star formation, and no infall or outflow of gas from the system(i.e. a closed system). A power law initial mass function (IMF) φ(m) which is constant overtime and space, is often assumed; the Salp eter value is φ(m)dm ∝ m−2.35dm. More recentwo r k by Scalo (1986, Fund.Cosmic Phys., 11, 1) and Kroupa (2001, MNRAS, 322, 231)– 2 –suggests that the Salpeter IMF is too steep to fit the observations for low mass stars below0.5M⊙, and that a flatter slope is required, reaching −0.3 ± 0.7 for 0.01 < M/M⊙< 0.08.Note that the transformation from luminosity to mass (the IMF is a function of stellar mass)is not as well determined for the lowest mass stars). In their most recent work, Weidner,Kroupa & Bonell (2010, MNRAS, 401, 275) explore the statistical issues associated withmassive stars forming in clusters. How does the maximum mass of a star formed in a clusterdepend on the mass of the cluster gas ? Obviously a star with mass greater than this valuecannot form in such a cluster. Integrating over many clusters with an appropriate clustermass distribution, they then derive what they call the integrated galactic mass function,and which is somewhat steeper at the highest masses than a standard power law IMF.Instanteous recycling, that stars die and release their metals very quickly after theirbirth, is also assumed. The sp ecific simplifying assumption o ften made is that stars withM < 1M⊙live fo rever, while stars of higher mass die instantly. Since most metals comefrom fairly massive stars with lifetimes short compared to the age of the Galaxy, this isnot an unreasonable assumption for an intial pedagogic model. Another assumption whichmust be made is that the fraction of mass from each stellar generation which remains lockedup in long-lived remnants or in stars that do not evolve during t he entire timescale of thecalculation (i.e. the age of the Galaxy), α, is constant. This is equivalent to a constant IMFin practice.In this case, following Francesca Matteucci (The Chemical Evolution of the Galaxy,Kluwer Academic Publishers, 2 003) the equation for the metallicity as a function of timecan be solved analytically. The varia bles are µ, the fraction of the to t al mass which is ingas that can form stars, p the nuclear reactions yield, and Z the fraction of metals in thegas (by mass). The subscript 0 denotes inital values when star formation first started in thesystem. Msis the total mass in stars (both living and dead remnants) at the time t, Mgis– 3 –the fraction of the total mass in the form of gas at time t. ψ(t) is the star formation rate,usually taken to be a function of σgas/τdyn(the Schmidt-Kennicutt law), where σgasis thesurface density of gas and τdynis the local dynamical timescale for collapse of a gas cloud.The current value of ψ in the local disk is ∼ 4M⊙pc−2/Gyr.The relevant equations were first laid out by Beatrice Tinsley following earlier work byM. Schmidt (1959, 1963), and are given in many articles.Conservatio n of mass and of heavy element abundance,Mgas+ Ms= Mtot, Ms= (1 − µ)Mtot, Z = MZ/Mgas.The initial conditions are: Mgas(t = 0) = Mtotand Z(t = 0) = 0.The gas evolves according todMgasdt= − ψ(t) + E(t) (eq. 1)where E(t) is the ra te at which dying stars restore their gas to t he ISM, partially enrichedby nuclear r eactions in their cores. MR(m) is the permanently locked up remnant mass fora star of mass m, and τ(m) is the lifetime of a star of mass m. Denoting m(t) as the massof a star born at t = 0 and dying at time t, we get:E(t) =Z∞m(t)(m − MR)ψ(t − τm)φ(m)dmThe metallicity evolves through the addition of newly synt hesized material fr om stars,namely:– 4 –d(ZMgas)dt= Mtotd(Zµ)dt= − Zψ(t) + EZ(t) (eq. 2)where the second term represent the addition of material to the ISM f rom dying stars. Thisin turn has two components, the first is the pristine material from the outer layers of starsthat was returned to the ISM without a lteratio n, and the second is the processed materialwith newly formed and ejected metals, as indicated below in eq. 2a.EZ(t) =Z∞m(t)[(m − MR)Z(t − τm) + mpZm]ψ(t − τm)φ(m)dm (eq. 2a)Define R as the total mass restored in pristine condition (i.e. without any products o fnuclear reactions mixed in) to the ISM by each generation of stars. This is the gas lost bywinds from the stellar surface. We assume that all stars with masses less than 1M⊙alwaysbecome remnants and never return any mass to the ISM; this defines the lower limit on theintegral. Then R is:R =Z∞1(m − MR)ψ(m)dm,We also need the yield for nucleosynthesis, yZfor the stellar generation, which is the ratiobetween the total mass of the isotope (element) i newly formed and ejected into the ISMfrom all stars with M > 1M⊙, assumed to die immediately after being formed, and theamount of mass locked up in low mass stars and remnants,yZ=11 − RZ∞1m pZmφ(m)dm,where pZmis the mass fraction for a star of mass m of heavy elements freshly producedwithin the star and then ejected into the ISM upon its death. We denote the yield yZas– 5 –the effective yield yZef fwhich describes the stellar system assuming that the Simple Modelis adequate. If yZef f> yZthen the system has attained a higher metallicity for a g iven gasfraction µ than the Simple Model can produce. If this were the case, then some assumptionin the Simple Model must be incorrect.Substituting eq. 2 into eq. 1 and assuming instantaneous recycling, which