Numerical simulation of the galactic chemical evolution: The revised solar abundance

Authors


Corresponding author. E-mail: sandeep@pu.ac.in

Abstract

We have developed a numerical code for galactic chemical evolution (GCE) of all the stable isotopes from hydrogen to gallium in accordance with the recently revised bulk solar photosphere abundances (Asplund et al. 2009) and the earlier deduced abundances (Anders and Grevesse 1989), herein referred as the pre-revised abundances. In contrast to solving the classical set of GCE equations, we have numerically simulated the evolution of the galaxy in the solar neighborhood in terms of evolution of the interstellar medium and numerous generations of stars. The evolution of the galaxy was simulated by adopting a two-stage accretion process. Numerous generations of stars were formed from the interstellar medium according to the adopted star formation rates and the stellar initial mass function (IMF). The simulated stars were evolved and their stellar nucleosynthetic contributions toward the inventories of the stable nuclides to the galaxy were accessed. Assessments were made regarding the type Ia, Ib/c, and II supernovae rates and the stellar remnants, e.g., white-dwarfs, neutron stars, and black holes. We have computed the age–metallicity relation and the solar abundances of the stable isotopes. A wide range of simulations were performed by parameterizing the choice of the accretion rate of the galaxy, the stellar initial mass function, the stellar evolution, and nucleosynthetic prescription to study their influence on galactic chemical evolution. The reduction in the observed solar metallicity results in significantly reduced supernova rate history of the galaxy, and a better match to the observed elemental evolutionary trend.

Introduction

Our galaxy consists of hundreds of billions of stars, interstellar gas, and dust. The primordial nucleosynthesis just after the big-bang produced the inventories of hydrogen, helium, and 7Li. Ever since the origin of the universe approximately 13.7 Ga ago, the process of stellar nucleosynthesis has resulted in the enrichment of elements ranging from carbon to uranium in the interstellar medium (Pagel 1997; Matteucci 2003). These elements are synthesized inside stars by a series of nuclear reactions that involve hydrostatic as well as explosive thermonuclear fusion reactions, slow and fast neutron capture processes, and nuclear statistical equilibrium processes. Galactic chemical evolution (GCE) essentially deals with understanding the gradual growth in the stable isotopic abundances over the galactic time scale of approximately 13 Ga due to the stellar nucleosynthetic contributions of numerous generations of stars (Matteucci and François 1989; Timmes et al. 1995; Chiappini et al. 1997, 2003a; Chang et al. 1999; Goswami and Prantzos 2000; Alibés et al. 2001; Cescutti et al. 2009; Huss et al. 2009; Kobayashi and Nomoto 2009; Matteucci et al. 2009; Chiappini 2011; Kobayashi and Nakasato 2011; Kobayashi et al. 2011). In the present work, we have developed numerical simulations of GCE for almost all the stable isotopes from 1H to 69Ga to understand the evolution of our galaxy in the solar neighborhood. The basic emphasis was to numerically simulate the evolution of the galaxy in terms of the formation and the evolution of numerous generations of stars in a realistic manner. The present adopted approach is unique and distinct from the traditional approach of solving the classical set of integro-differential equations (the nested differential equations that involve integration of the functions that have to be differentiated) dealing with GCE.

The significant growth in the field of stellar evolution and nucleosynthesis has been closely associated with the major refinements in GCE models. Several groups have been involved in developing GCE models (Matteucci and François 1989; Timmes et al. 1995; Chiappini et al. 1997, 2003a; Chang et al. 1999; Goswami and Prantzos 2000; Alibés et al. 2001; Fenner et al. 2003; François et al. 2004; Cescutti et al. 2006; Kobayashi et al. 2006; Matteucci et al. 2006; Cescutti et al. 2009; Kobayashi and Nomoto 2009; Matteucci et al. 2009; Chiappini 2011; Kobayashi and Nakasato 2011; Kobayashi et al. 2011). The essential aim of these studies on the local scale, i.e., in the annular ring of the galaxy associated with the solar neighborhood, is to develop numerical models to explain the age–metallicity relation; the solar abundances of the stable isotopes; the type Ia, Ib/c, and II supernova rates; the observed G-dwarf metallicity distribution; etc. (see, e.g., Matteucci and François 1989; Rana 1991; Chiappini et al. 1997, 2003a; Kobayashi et al. 2011). On the galactic scale, GCE models attempt to explain the observed elemental abundance gradient (e.g., Cescutti et al. 2006), and the formation of the galactic halo, the thin and the thick disks.

Apart from refinements in the theory of stellar evolution and nucleosynthesis, the observed elemental abundances of the various F, G, and K dwarf disk stars, red giants, planetary nebula, and open clusters by several groups have produced an enormous data set on the elemental abundance distribution of stars in the galaxy (Clegg et al. 1981; Gratton 1985; Laird 1985; Luck and Bond 1985; Gratton and Ortolani 1986; Tomkin et al. 1986; Carbon et al. 1987; François 1987, 1988; Gratton and Sneden 1988, 1991; Hartmann and Gehren 1988; Sneden and Crocker 1988; Magain 1989; Peterson et al. 1990; Zhao and Magain 1990; Meusinger et al. 1991; Sneden et al. 1991; Edvardsson et al. 1993; Nissen et al. 1994; McWilliam et al. 1995; Ryan et al. 1996; Israelian et al. 1998; Stephens 1999; Carretta et al. 2000; Chen et al. 2000; Mishenina et al. 2000; Prochaska et al. 2000; Rocha-Pinto et al. 2000; Bensby et al. 2003, 2005; Reddy et al. 2003; Carraro et al. 2004; Cayrel et al. 2004; Carney et al. 2005; Yong et al. 2005; Bensby and Feltzing 2006; Brewer and Carney 2006). The elemental abundance distribution of the disk stars is used to constrain GCE models.

One of the essential aims of developing a GCE model is to explain the bulk isotopic composition of the solar system (Matteucci and François 1989; Timmes et al. 1995; Alibés et al. 2001) at the time of its formation around 4.56 Ga ago (Sahijpal et al. 1998; Huss et al. 2009). The bulk isotopic abundance of the solar system is accessed on the basis of the spectral analysis of the solar photosphere and the composition of primitive meteorites (Anders and Grevesse 1989; Grevesse et al. 2007; Asplund et al. 2009). GCE models developed previously by other workers were based on the standard abundance distribution deduced by Anders and Grevesse (1989). Recently, there have been major revisions in the bulk solar elemental abundances estimated from the solar photospheric spectral analysis. These revisions are essentially due to the improvements associated with the analytical technique involving the conversion of the observed spectral information into elemental abundances (Grevesse et al. 2007; Asplund et al. 2009). This includes refinements in the atomic and molecular spectroscopy transition probabilities, the improvements associated with the modeling of the hydrodynamical solar atmosphere and the local thermodynamical equilibrium conditions in spectral line formation. The metallicity of the solar system, measured as Z, has gone down to approximately 0.014 (Asplund et al. 2009) from an earlier estimate of approximately 0.02 (Anders and Grevesse 1989). It should be mentioned that the revised abundances are not uniformly accepted as these abundances seem to be inconsistent with the helioseismology constraints on the standard solar model. However, some of the recent solar models that exclude the effects of rotation and magnetic field suggest better consistency (see e.g., Bi et al. 2011).

We have developed a GCE model in the present work by incorporating the latest understanding of the star formation rate (SFR), the stellar initial mass function (IMF), the updated stellar nucleosynthetic yields, and the stellar evolutionary theories for different stellar masses and metallicities. Our main objective was to develop numerical models for the galactic disk to explain the age–metallicity relation; the solar abundances of the stable isotopes; the type Ia, Ib/c, and II supernova rates; etc., on the basis of the earlier model (Anders and Grevesse 1989), herein referred to as the pre-revised bulk solar abundances and the recently revised estimated solar abundances (Lodders 2003, 2010; Grevesse et al. 2007; Asplund et al. 2009; Lodders et al. 2009) at the time of the formation of the solar system around 4.56 Ga ago. We performed a wide range of simulations by parameterizing the accretion rate of the galaxy, the stellar IMF, the stellar evolution, and nucleosynthetic yields to study their influence on GCE. As mentioned earlier, in contrast to solving the classical set of nonlinear-integro-differential equations associated with the GCE model (Matteucci and François 1989; Timmes et al. 1995; Pagel 1997), we have followed a unique and direct approach, whereby we actually made an attempt to simulate in a realistic manner the birth and evolution of successive generation of stars of different masses and metallicities during the approximately 13 Ga evolution of the galaxy. We followed the open chemical evolutionary model for the formation of the galaxy as suggested by Chiappini et al. (1997). The stars are formed in the simulation according to an adopted SFR throughout the evolution of the galaxy. We used the modified form of the Salpeter's IMF for the formation of stars of different masses (Alibés et al. 2001). We explored the parametric space associated with the SFR, the modified Salpeter's IMF, the supernovae rates, the accretion rate of the galaxy, and the stellar nucleosynthetic yields to obtain the solar metallicity of the galaxy at the time of the formation of the solar system around 4.56 Ga ago. The birth, evolution, and final evolved stages, e.g., supernova, red-giant stage, etc., of the ensemble of successive generation of stars were numerically simulated to develop a framework of a realistically evolving galaxy. The dependence of the life-time of a star on its zero age main sequence mass was considered. The stellar remnants, e.g., white-dwarf, neutron stars, black holes were appropriately accounted during the evolution of the galaxy. The preliminary results of this work were presented in a symposium (Sahijpal and Gupta 2010).

Methodology

As mentioned earlier, the conventional procedure for developing a GCE model involves solving the classical set of nonlinear-integro-differential equations (Matteucci and François 1989; Timmes et al. 1995; Chiappini et al. 1997, 2003a; Pagel 1997; Alibés et al. 2001; Matteucci 2003). Equation (1) represents one such nonlinear integro-differential equation that deals with the dynamical growth of the surface mass density (σi, M pc−2) of the ith isotopic species in the galaxy at a particular time, “t” (Matteucci and François 1989; Chiappini et al. 1997, 2003a; Pagel 1997; Matteucci 2003).

display math(1)

The first term in the equation deals with the nucleosynthetic contributions at the specific time, “t,” from all the stars in the mass (m) range extending over 0.1–100 M that had lived for their lifespan of τ(m). These stars eject their nucleosynthetic products into the interstellar medium at time, “t.” The stars will contribute to the bulk inventories of the isotope, “i,” according to the star formation history, expressed as the star formation rate, SFR(t−τ(m)), prevailing at their formation time, t−τ(m), and the IMF, φ(m). The second term in the equation deals with the reduction in the gas surface mass density associated with the isotopic species on account of star formation at the specific time, “t.” The third term deals with the accretion of infalling low metallicity intergalactic gases on the galaxy. To avoid the G-dwarf metallicity problem the accretion of the galaxy is considered to have extended over the galactic time scale (Matteucci and François 1989; Chiappini et al. 1997, 2003a; Matteucci 2003; Kobayashi et al. 2011). The stars in the mass range of 0.8–1.2 M are referred as the G type stars. Closed box GCE models (e.g., Pagel 1997) that deal with almost instantaneous accretion of the galaxy result in the predominant production of metal-poor G-type stars. However, the observed metallicity distribution of G-dwarf stars exhibit a paucity of metal-poor stars that is in contradiction with the predictions of the closed box GCE models. This is termed the G-dwarf metallicity problem. The gradual accretion of galaxy over galactic time scales provides a solution to the problem. The fourth term deals with the radioactive decay of the isotope in case it is radioactive, else this term contributes nothing.

To develop the GCE model, Equation (1) has to be solved for all the stable isotopes from hydrogen to gallium. This is the general approach that has been traditionally followed. Apart from this traditional approach, there are some very sophisticated chemodynamical models for GCE (see, e.g., Kobayashi and Nakasato 2011). In contrast to these approaches, we have adopted a direct approach of physically evolving the galaxy in a realistic manner without solving the classical set of nonlinear-integro-differential equations. In our numerical approach, we initiate the simulation with the commencement of the formation of the galaxy. The galaxy is considered to form by the accretion of the infalling intergalactic gases in a gradual manner as discussed elaborately in the following section. We produce stars in the simulation according to the SFR that essentially depends upon the prevailing surface gas density. These simulation stars evolve over their life-spans. At the time of their final evolution, these stars eject their matter back into the interstellar medium. The stellar debris are homogenized in the interstellar gas to further participate in the next generation of star formation, and hence, contribute to the further galactic chemical evolution. The stellar remnants from the evolved stars, i.e., the white dwarfs, neutron stars, and black holes, do not further contribute to the evolution of the galaxy in terms of isotopic evolution. We simulated the birth and evolution of successive generation of stars of different masses and metallicities during approximately 13 Ga evolution of the galaxy to re-create GCE. The isotopic inventories of all the isotopes from carbon to gallium apart from hydrogen and helium were gradually evolved by maintaining a proper account based on mass balance calculations of the processing of interstellar gas by various generations of stars.

We performed simulations by appropriate parameterization of the various aspects associated with the accretion growth of the galaxy, the SFR, the initial stellar mass function, and the stellar nucleosynthetic yields of various stars. The essential aim of all these simulations is to explain the standard isotopic abundance of the various isotopes (Lodders 2003, 2010; Asplund et al. 2009; Lodders et al. 2009) in the solar neighborhood at the time of formation of the solar system around 4.56 Ga ago. Furthermore, we deduce the differences in the evolutionary history of the galaxy in terms of the SFR and the supernovae rates based on the pre-revised and the revised bulk solar abundances.

The Solar Elemental Abundances

The bulk isotopic abundance of the solar system, considered as a standard reference for cosmic abundance, is accessed on the basis of a complex procedure involving the spectral analysis of the solar photosphere and the composition of primitive meteorites (Anders and Grevesse 1989; Lodders 2003, 2010; Grevesse et al. 2007; Asplund et al. 2009; Lodders et al. 2009). GCE models developed previously by other researchers were based on the standard abundance distribution deduced by Anders and Grevesse (1989). As mentioned earlier, there have been major revisions in the bulk solar elemental abundances. The metallicity of the solar system, measured as Z, has gone down to approximately 0.014 (Asplund et al. 2009) from an earlier estimate of approximately 0.02 (Anders and Grevesse 1989). The changes in the solar mass fraction of the various elements with respect to the pre-revised estimates (Lodders 2003, 2010; Asplund et al. 2009; Lodders et al. 2009) are graphically presented in Fig. 1. We obtained the revised (new) abundance mass fraction of the various elements using the data presented by Asplund et al. (2009). The updated solar abundances were appropriately modified from the present solar elemental abundances to the elemental abundances prevailing at the time of the formation of the solar system around 4.56 Ga ago by taking into account gravitational settling of elements in the Sun's photosphere (Grevesse et al. 2007; Asplund et al. 2009). The largest revisions have occurred in the case of nitrogen, oxygen, neon, and chlorine (Fig. 1).

Figure 1.

The change in the mass fraction of the various elements from H to Ga in the recently revised (new) solar photosphere abundance (Asplund et al. 2009) with respect to the pre-revised solar abundances (Anders and Grevesse, 1989). We assume that there is no change in the isotopic composition due to revision in the solar metallicity.

In the present work, we have performed simulations for GCE using the pre-revised (Anders and Grevesse 1989) as well as the revised (new) (Lodders 2003, 2010; Asplund et al. 2009; Lodders et al. 2009) bulk solar elemental abundances. The former set of simulations will help us to compare our results with those obtained by other researchers that are based on the solutions of the nonlinear integro-differential equations. As mentioned earlier, the previous researchers have developed GCE models for the pre-revised solar elemental abundances. Table 1 represents the various simulations along with the parameters. The simulations, P-Solar-A, -B, etc., and N-Solar-A, -I, -J, etc. represent the models based on the pre-revised and the revised (new) solar elemental abundances, respectively. The simulations have been short-named for easy reference apart from the detailed names, e.g., the simulation P-Solar-A and N-Solar-A are short-named as PSA and NSA, respectively. The simulations were run to understand GCE dependence on: (1) the stellar IMF, (2) the SFR, (3) the accretion scenario of the galaxy, (4) the assumed SNIa (type Ia) supernova rates, (5) the metallicity of the infalling matter on the galaxy in the form of mass accretion, and (6) the nucleosynthetic yields of SNII (type II) supernovae. Two parameters, “x” and “f,” were appropriately chosen to reproduce the required solar metallicity at the time of the formation of the solar system around 4.56 Ga ago. The parameter “x” refers to the power exponent of the stellar IMF, whereas the parameter “f” refers to the fraction of stars formed in the mass range 3–8 M and 11–16 M that will eventually evolve to SNIa in a binary system. These parameters are discussed elaborately in the following sections. The remaining parameters are considered as free parameters. The details of the Table 1 are subsequently discussed in the text. The first simulation listed in Table 1, PSA, was run to reproduce the pre-revised bulk solar isotopic abundances (Anders and Grevesse 1989) with the basic set of assumptions regarding the choice of the accretion of galaxy, the stellar IMF, the SFR, and the stellar nucleosynthetic prescription. This simulation is certainly not our best model as explained later on the basis of certain model predictions. We explored the parameter space on the basis of the primer model PSA by running ten additional simulations (Table 1) to identify the most critical parameter(s) that influence GCE. Some of the parametric choices eventually led to our best models, PSI and PSN, dealing with the pre-revised solar metallicity simulations. In the case of the revised solar metallicity models, we performed seven simulations with varied parameters (Table 1). All these simulations are able to explain most of the observations dealing with GCE.

Table 1. The characteristic set of simulations with varied parameters
ModelsaPower exponentcxParameter IMFcASNIa fractiondfSNIa FactoreηAccretionftT” (Gyr)AccretionftD” (Gyr)SFRgn(Z′) Metallicity of accreting matterhSupernova type II (30, 35, 40 M)iPredicted present SNII/SNIaPredicted [O/Fe] at [Fe/H] = −2
  1. a

    The models have been short-named for easy reference apart from detailed names, e.g., PSA is the short representation for P-Solar-A.

  2. b

    The bulk solar metallicity (Z) produced by the model at the time of the formation of the solar system around 4.56 Giga years ago. The parameters “x” and “f” were appropriately chosen to reproduce the required solar metallicity. The remaining simulation parameters are considered as free parameters.

  3. c

    The power exponent, “x,” and the constant, “A” in the initial mass function, φn(m) = A m−(1+x), in the mass range 11–100 M as in Equation (3).

  4. d

    The assumed fraction “f” of the stars formed in the mass range 3–8 M and 11–16 M that will eventually evolve to SNIa in a binary system. The remaining fraction “1 − f” will evolve to AGB stars and core collapse supernova in the mass ranges 3–8 M and 11–16 M, respectively.

  5. e

    The parameter associated with the life span of the binary system that will eventually explode as SNIa. In a binary system with a total mass “m,” the supernova SNIa occurs after a time span that corresponds to the age of the star with mass 1/η times the mass of the single star of equivalent mass “m.” However, in the case of simulations marked “Random,” at the time of the formation of the binary systems with a total mass “m” in the mass range 3–8 M, we randomly generated binary pairs with the help of random numbers that would eventually explode as SNIa subsequent to a time interval between approximately 5 Ga and the corresponding age of the star with mass 1/2 times the mass of the single star of equivalent mass “m” (see text for details).

  6. f

    The accretion time scale of the halo-thick disk (tT) and thin disk (tD) as defined in Equation (1) (Chiappini et al. 1997).

  7. g

    The star formation rate index, “n” as defined in Equations (4)-(6).

  8. h

    Assumed metallicity (Z′) of the accreting matter to the galaxy.

  9. i

    The core collapse supernova nucleosynthetic yields of the lowest remnant mass of the massive stars (≥25 M) models of Woosley and Weaver (1995) were considered in the simulation N-Solar-M (NSM). In the case of the remaining simulations, we averaged out the nucleosynthetic yields of the various models A, B (and C, if available) of the massive stars (≥25 M) of Woosley and Weaver (1995) to consider the contributions of the stars with distinct remnant masses produced at the time of supernova. The nucleosynthetic yields of the iron-peaked nuclides beyond iron were divided by a factor of 2 (Timmes et al. 1995) in several simulations. The models NSI and NSIm differ in the manner the extrapolation of stellar yields are performed for stars of masses >40 M (see text for details).

Z = 0.02 modelsb
P-Solar-A (PSA)1.355110.01002170.330.002All models180.10
P-Solar-B (PSB)1.348110.01012170.250.002All models180.10
P-Solar-C (PSC)1.365110.00902170.450.002All models200.09
P-Solar-D (PSD)1.362110.009720.570.330.002All models190.16
P-Solar-E (PSE)1.347110.010121.570.330.002All models180.05
P-Solar-F (PSF)1.374110.00852150.330.002All models200.09
P-Solar-G (PSG)1.341110.011021100.330.002All models170.10
P-Solar-H (PSH)1.354110.01303170.330.002All models100.10
P-Solar-I (PSI)1.358110.02052170.330.002All models, Iron-peak yields divided by 280.22
P-Solar-J (PSJ)1.354110.0099Random170.330.002All models170.10
P-Solar-N (PSN)0.91220.01682170.330.002All models70.32
Z = 0.014 modelsb
N-Solar-A (NSA)1.498110.03502170.330.0014All models30.55
N-Solar-I (NSI)1.498110.04002170.330.0014

All models, Iron-peak

yields divided by 2

30.63
N-Solar-Im (NSIm)1.510110.04202170.330.0014

All models, Iron-peak

yields divided by 2

30.63
N-Solar-J (NSJ)1.495110.0350Random170.330.0014All models40.56
N-Solar-K (NSK)1.511110.03402170.330.0020All models30.56
N-Solar-M (NSM)1.551110.02602170.330.0014Low remnant mass models40.38
N-Solar-N (NSN)1.05620.04202170.330.0014All models20.76

Accretion Growth of the Galaxy

We performed simulations with the gradual accretion of intergalactic gases on our galaxy. We followed the open chemical evolutionary model for the formation of the galaxy as suggested by Chiappini et al. (1997). The galaxy is considered to have formed by two main accretion episodes. The first episode involves the formation of the galactic halo and bulge. This is followed by a second episode that forms the thin-disk within a time scale, which is an increasing function of the galactocentric distance. As explained earlier, the G-dwarf metallicity distribution that is dominated by the paucity of metal-poor G-dwarf stars has been best explained by considering the infall of the extragalactic matter on the galactic disk over the extended time duration, and with a constant stellar IMF (Chiappini et al. 1997, 2003a; Matteucci 2003).

The simplest GCE scenario involves the formation and evolution of the galaxy in the form of radial annular bins consisting of primordial gas from which the stars form according to the SFR and the IMF prescriptions with no loss or gain of the gas. This type of the scenario is referred as the closed box model. As mentioned earlier, the closed box models lead to the G-dwarf problem (Chiappini et al. 1997, 2003a; Matteucci 2003). The problem can be resolved by using the open models for the formation of the galaxy.

Several GCE models have explored different types of infall rate, but the exponentially decreasing infall rates for the formation of the galaxy are considered as the most successful (Prantzos and Aubert 1995; Timmes et al. 1995; Chiappini et al. 1997; Thomas et al. 1998). Larson (1972) suggested the ongoing infall of gases on the galactic disk based on the external high velocity clouds. Wakker et al. (1999) detected massive clouds falling on the disk with metallicity approximately 0.1 times the solar metallicity (Z). This observation provides a strong argument for the formation of the galaxy with the extragalactic origin of the cloud. In the present work, we form the galaxy in two episodes of exponentially decreasing infall. In the first episode, the halo and the thick disk formed in a short time scale of 1 Ga, and in the second episode, the thin disk began to form over a much longer time scale. The temporal evolution of the total surface mass density at a specific time, “t” at a distance of the solar neighborhood (approximately 8 kiloparsecs) from the galactic center is given by the Equation (2).

display math(2)
display math
display math

Here, σ(r, tG) and σT(r, tG) are the total local surface mass density and the thick disk local surface mass density, respectively, at the present time. The present observed estimates of approximately 54 and approximately 10 M pc−2, respectively, were taken for these surface mass densities (Chiappini et al. 1997, 2003a; Chang et al. 1999; Alibés et al. 2001; Matteucci 2003). The values of the other parameters are based on literature survey. The time scale, tG, was considered to be 13 Ga, whereas the time scales tT and tD associated with the accretion of the halo-thick disk and thin disk, respectively, were chosen as the simulation parameters as defined in Table 1. The time scale “tmax” of the maximum accretion of the thin disk was assumed to be identical to tT. On the basis of these accretion scenarios, we developed numerical codes to simulate the accretion of the galaxy along with the formation and evolution of several generations of the stars. Figure 2 graphically represents the deduced total (σTotal), stellar (σStar+remnant), and gas (σGas) surface mass densities of the solar neighborhood on account of the accretion history of the galaxy and stellar evolution in case of some representative set of simulations. The total surface density increases on account of accretion of new material to the galaxy. Its value at the present epoch matches with the present astronomically observed value of approximately 54 M pc−2 in all the simulations. The stellar surface density increases on account of star formation, at the cost of the gas surface.

Figure 2.

The deduced evolution of the total (σtotal), stellar (σStar+remnant), and gas (σgas) surface mass density of the solar neighborhood on account of the accretion history of the galaxy and stellar evolution. The deduced present gas surface mass density is within the observed values 7–13 M pc−2 for all the simulations. The stellar (σStar+remnant) surface mass density contributes to the remaining mass density out of the total surface mass density of approximately 54 M pc−2. The deduced evolution of the stellar remnant (σremnant) surface mass density (as white-dwarfs, neutron stars, and black holes) is presented in Fig. 7 for some specific simulations.

The accreting matter was assumed to have a metallicity (Z′) for the various simulations as mentioned in Table 1. As discussed earlier, to solve the G-dwarf problem, it is essential to accrete matter with approximately 0.1 times the final solar metallicity (Z′ = 0.1 × Z). However, in the case of the simulation N-Solar-K, we assumed Z′ = 0.002. This value is 0.1 times the pre-revised solar metallicity. We performed this simulation to understand the influence of the metallicity of the accreting matter on the GCE. In the remaining simulations, the steady-state infall of the intergalactic gases of metallicity approximately 0.1 times the solar metallicity was maintained throughout the accretion phase of the galaxy, initiating from the time (tZ) the solar neighborhood of the galaxy first acquired the identical value of the metallicity. Before this time (tZ), the metallicity of the infalling gas was considered to be equivalent to the prevailing metallicity of the evolving galaxy. Using an identical approach, we appropriately incorporated the isotopic abundance contributions of all the stable isotopes of the infalling gas. This procedure is analogous to the incorporation of the third term of Equation (1) in the conventional approach of solving GCE.

Temporal Resolution of the Simulations

We have considered a temporal step, Δt, of one million years for the simulations in the present work. The galactic evolution of approximately 13 Ga was essentially divided into equal time-steps of one million years each. At any specific time, “t,” ranging from 0 to 13 Ga of the galaxy evolution, the time-step (Δt) defines an interval after which assessments, followed by appropriate numerical modifications, are performed regarding:

  1. The accretion of the infalling intergalactic gas of appropriate metallicity (Z′) on the gradually evolving galaxy.
  2. The contributions in the form of stable isotopes and stellar remnants to the interstellar medium by all the stars that have ejected their nucleosynthetic debris at the specific time, “t.”
  3. The production of the next generation of stars based on the prevailing surface gas density, the SFR and the IMF at the specific time, “t.” We assumed a constant form of the IMF throughout the evolution of the galaxy.
  4. The entire inventories of all the stable isotopes, the stars of different masses and metallicities, the stellar remnants, and the supernovae events.

Since the most massive star in our simulations with the mass of 100 M lives with the least lifetime of approximately 3.5 million years, the assumed time-step (Δt) of one million years is well justified. It is 1/3.5 times the least temporal scale of any astrophysical process that can be simulated with our present numerical code. All the real numbers associated with lifetime of stars were rounded-off to integers. This introduces an insignificant error of <0.5 million years. All the calculations in the present work were performed with double precision. We performed detailed mass balance calculations to avoid any computation error within the stated precision.

The Initial Mass Function (IMF)

The IMF is essentially the stellar mass spectrum that deals with the number of stars born in a given mass interval at a specific region of a molecular cloud in the galaxy. The stars are formed in molecular clouds as a result of gravitational instabilities that can eventually produce thousands of stars. The astronomical observations indicate a power law distribution of stars that are formed together (Salpeter 1955; Pagel 1997; Matteucci 2003). These stars of different masses evolve differently over distinct time scales. They contribute distinct nucleosynthetic products to the interstellar medium on varied time scales. Thus, the IMF influences the relative and absolute elemental abundances of the elements. It determines the average stellar yields and remnants' masses for each generation of stars. The assumed form of the IMF is generally taken as a power law.

display math(3)

This differential function φn(m) defines the number of stars within a specific mass interval. The product of this function with the average mass of the stars in the specified mass range will define φ(m), the total stellar mass within the range. This is generally referred as the IMF. The integral, ∫φ(m) dm = ∫m φn(m) dm over the entire mass range will determine the total mass available for star formation. The IMF can have a single slope (Salpeter 1955) or multiple slopes (e.g., Kroupa 1998; Scalo 1998; Meyer et al. 2000) over the entire mass range of the stars. Salpeter (1955) defined the IMF with a slope, x = 1.35, for the entire mass range of the stars. Scalo (1998), Kroupa (1998) and Meyer et al. (2000) used a varied range of slopes.

In the present work, we have assumed a piecewise IMF with three distinct slopes in the power law mass distribution (Pagel 1997; Matteucci 2003). The values of the parameter, “A,” and the power exponent, “x,” used in the Equation (3) are presented in Table 2. The selection of these various constants and power exponents is certainly not unique. It is partially based on the data available in literature (Pagel 1997; Matteucci 2003), and our basic simulation requirement to produce the solar metallicity in the solar neighborhood at the time of the formation of the solar system around 4.56 Ga ago. In fact, the IMF, specifically the power exponent, “x,” in the mass range 11–100 M is one of our simulation's parameters that is required to reproduce the solar metallicity. The various choices of the power exponent are presented in Table 1 and Table 2.

Table 2. The values associated with the parameter, “A,” and the power exponent, “x.”
Mass rangeParameter APower exponent x
0.1–0.5 M110.0
0.5–1.0 M80.0
1.25–8.0 M111.7
11.0–100.0 M2, 110.9–1.55

In general, in the low mass stellar range 0.1–1 M, the slope of IMF is flat with x = 0.0 (Table 2). These low mass stars are the most abundant stars in the galaxy as the nature is more efficient in producing these stars. The IMF steepens to x = 1.7 for 1< m < 8 M, and for high mass stars (11–100 M), “x” was varied according to the simulation requirement (Table 1). The choice made for the power exponents is well within the range proposed by other researchers (Pagel 1997; Matteucci 2003). The coefficient “A” in the mass range 11–100 M was also considered as one of the free parameters (Table 1). We used a lower value of this constant in the mass range 11–100 M in two of our simulations to understand in a simplest manner the impact of the contributions of the massive stars on GCE. In the remaining simulations, an identical value was used for “A” in the mass ranges 1.25–8.0 M and 11–100 M. Apart from the variation in the two cases, we used a lower value of “A” in the mass range 0.5–1 M. This was done to reduce the stellar contributions in the mass range 0.1–1 M. We have used the discrete set of stellar masses in the mass range 1–100 M, whereas we covered the mass range 0.1–1 M by four distinct stellar masses, i.e., 0.1, 0.4, 0.8, and 1 M. The net stellar contributions in the mass range 0.1–1 M were reduced by lowering the value of the constant “A” in the mass range 0.5–1 M without changing the power exponent. Furthermore, it should be noted that in the present work the stellar nucleosynthetic contributions were considered for stars having mass ≥1.25 M. Stars in the mass range 0.1–1 M contribute only in terms of their formation and entrapment of galactic matter over galactic time scales.

In our simulations, we choose stars of a discrete set of masses. These stars of specific masses in the mass range 0.1–100 M were formed and evolved over numerous generations during the galactic evolution. The IMF for our selected set of stellar masses is graphically presented in Fig. 3 for two distinct set of simulations that eventually produced the pre-revised and the recently revised solar metallicity at the formation of the solar system around 4.56 Ga ago. In the low mass range, we selected the discrete set of masses, 0.1, 0.4, 0.8, and 1 M to appropriately include the most abundant stars in the distribution. The stars with mass less than 1 M live more than the age of the galaxy. The formation of these stars essentially leads to the exclusion of matter from further processing in the interstellar medium. Hence, these stars do not contribute to the GCE in an explicit manner. Nonetheless, a proper account of at least the formation of these stars is extremely important. We have taken care of this aspect. The fraction of the stellar mass that goes on in producing these stars is in accordance with Pagel (1997). The discrete integer values of the mass of the stars in the mass range 3–100 M were considered to include the entire broad IMF. We choose the stellar masses 1.25, 1.75, and 2.5 M as the nucleosynthetic yields of these masses are available for low mass asymptotic giant branch (AGB) stars of different metallicities (Karakas 2003, 2007, 2010; Karakas and Lattanzio 2003, 2007; Karakas et al. 2006).

Figure 3.

The assumed mass function φn(m) of star formation for the two set of simulations that eventually produced the pre-revised solar metallicity, i.e., Z = 0.02 (Anders and Grevesse, 1989), and the recently revised metallicity, i.e., Z = 0.014 (Asplund et al. 2009) at the birth of the solar system. The power index for the IMF in the mass range 11–100 M was used as one of the fitting parameters in the simulations. The differential function φn(m) defines the number of stars within a specific mass interval. The product of this function with the average mass of the stars in the specified mass range will define φ(m), the total stellar mass within the range. The integral, ∫φ(m) dm = ∫m φn(m) dm over the entire mass range will determine the total mass available for star formation.

At any specific time, “t,” in the simulation, we estimated the total amount of interstellar gas mass available for star formation depending upon the prevailing SFR in the solar neighborhood. This total mass was distributed among the stars of distinct masses in the mass range 0.1–100 M using the IMF. The final obtained mass distribution of the stars consisted of the integer numbers of stars corresponding to each mass. Proper precautions were made to convert the IMF distribution, represented as real numbers or fractions in Fig. 3, to the final integer number of stars produced at a specific time, “t.” The entire procedure for obtaining the mass distribution at time, “t,” essentially involves a constant multiplier of the parameter, “A,” in the IMF to eventually obtain the integer numbers of stars of different masses.

Star Formation Rate (SFR)

As mentioned previously, we produce stars at any specific time, “t,” depending upon the prevailing surface gas density. This is postulated based on the observations that indicate that the SFR depends upon the rate at which gas is settling over the disk of the galaxy (Talbot and Arnett 1975; Alibés et al. 2001; Matteucci 2003). Schmidt (1959) defined the simplest form of the SFR.

display math(4)

where σgas is the surface mass density of the interstellar gas, and the power exponent, m, may range from 1 to 2. Dopita and Ryder (1994) described SFR by the following equation.

display math(5)

where σ is the total surface mass density with n = 1/3 and m = 5/3.

In the present work, we have used the normalized SFR in the solar neighborhood (Alibés et al. 2001; Matteucci 2003).

display math(6)

The parameter “n” was varied to explore the dependence of GCE on the SFR (Table 1). The value of “m” was appropriately modified according to the normalization. The deduced SFR for the various representative set of simulations is presented in Fig. 4. The SFR of the galaxy initially increases essentially due the initial fast accretion episode of the galaxy that lasts for the initial 1 Ga (Chiappini et al. 1997; Matteucci 2003). The termination of the rapid accretion episode of the galaxy is marked by a kink in the behavior of the SFR history. The typical SFR maximizes at approximately 2 Ga and thereafter, it gradually decreases. The decrease is essentially due to the decline in the accretion of new material on the galaxy and the entrapment of interstellar gas (Fig. 2) in the long-lived, low, and intermediate mass stars (0.1–8 M). As mentioned earlier, according to the stellar IMF (Fig. 3), the low and intermediate mass stars are formed in abundance compared with massive stars (>11 M). Furthermore, the low and intermediate mass stars live long, in certain cases for more than the galactic time-spans, thereby entrapping the interstellar inventories for a considerably long time.

Figure 4.

The deduced star formation rate (SFR) history of the solar neighborhood for the various simulations. The present observed SFR is in the range 2–5 M pc−2 Gyr−1 (Guesten and Mezger 1982). The deduced time integral values of the SFRs for all the simulations are mentioned in Table 3.

In Fig. 5, we present the deduced star formation history of some representative set of stellar masses during the entire evolution of the galaxy in the solar neighborhood. The star formation history is governed by the IMF, with the low mass stars as the most dominant and the massive stars as the least dominant.

Figure 5.

The star formation history of the stars of various masses in the range 0.8–100 M in the solar neighborhood for the simulation P-Solar-A (PSA). At least one 100 M star was considered to form during the simulation time-step of 1 million years with the assumed IMF. This resulted in an increase in the anticipated area of the solar neighborhood. An annular ring of width approximately 1.9 kiloparsecs enclosed within 7.0–8.9 kiloparsecs was assumed to define the solar neighborhood contrary to the traditionally accepted area within 7.5–8.5 kiloparsecs.

The Solar Neighborhood

The conventional approach involved in solving the GCE model is to divide the galaxy into equal concentric annular rings of one kiloparsec width. The interstellar medium within these rings is considered to evolve independently as far as chemical evolution is concerned with no amount of mixing with the consecutive rings. The annular ring, centered at a distance of approximately 8 kiloparsecs, the distance of the Sun from the galactic center, is considered as the solar neighborhood. In the present approach, we made modifications in defining the width of the annular ring. As we are simulating the realistic formation and the evolution of stars in our simulations, it is essential to produce the entire mass range of the stars as defined by the IMF. According to our opted criteria, the 100 M stars are the least abundant stars in the galaxy due to the nature of the IMF. To produce at least one 100 M star during the time-step Δt of one million years within a single annular ring according to the SFR (see e.g., Fig. 5), we appropriately modified the width of the annular ring. However, this condition need not be satisfied throughout the evolution of the galaxy. It should be mentioned that we could achieve the imposed requirement of the formation of at least one 100 M star in the case of simulations that produced the pre-revised solar metallicity. However, in the case of the simulations with the revised solar metallicity, the condition could not be satisfied due to the steeper slope of the IMF (Table 1). The nucleosynthetic contributions of the higher end stellar masses are missed in these simulations. An annular ring of width approximately 1.9 kiloparsecs was used to define the solar neighborhood area at a distance of 7.0–8.9 kiloparsecs from the center of the galaxy in contrast to the traditionally accepted area enclosed within 7.5–8.5 kiloparsecs. A smaller annular ring would result in a significant reduction in the production of massive star with masses >50 M due to the steepness in the IMF. This would make it difficult to properly account for the contributions of massive stars to the GCE within a time-step of 1 million years chosen for our simulations. The entire numerical analysis associated with the GCE that includes star formation and evolution of numerous generations was performed for the entire annular ring as a single entity. We made the traditional assumption of uniformly mixing the nucleosynthetic yields from the stars instantly throughout the annular ring subsequent to the evolution of stars.

Lifetimes of the Stars

The stars of different masses and metallicities evolve differently over distinct life-spans. We have used the compilations of the lifetime data made available by Pagel (1997). The lifetime data based on this compilation were appropriately interpolated or extrapolated to generate the lifetime of stars over the entire range of masses and metallicities as required according to the simulation. Even though the massive stars in the mass range 11–100 M are formed at a lower rate compared with the low and intermediate mass stars of masses 0.1–8 M (Figs. 3 and 5) due to the IMF, the massive stars evolve rapidly, and hence dominate the galactic evolution of the stable nuclides.

Stellar Nucleosynthetic Contributions

Stars of different masses evolve differently, and hence, contribute distinctly toward the stable isotopic inventories of the galaxy (Pagel 1997; Matteucci 2003). Single stars in the mass range 11–100 M eventually evolve through core-collapse supernova (SNII and SNIb/c). Stars in the mass range 11–33 M evolve to core collapse supernova (SNII), whereas the more massive stars, 34–100 M evolve to Wolf-Rayet stars that eventually lead to core-collapse supernova (SNIb/c) (Sahijpal and Soni 2006; Huss et al. 2009; Sahijpal and Gupta 2009). A black hole or neutron star remnant is eventually produced at the end of evolution of these stars.

A low and intermediate mass single star of mass 0.1–8 M evolves through the AGB phase, eventually leading to a white-dwarf remnant and a planetary nebula. These stars gradually eject their matter into interstellar medium during their evolution. At the final stages, these stars experience high mass losses in the form of strong winds leading to planetary nebula formation. We have considered the nucleosynthetic contribution of low and intermediate mass stars as well as massive stars. It should be mentioned that the evolution of single stars in the mass range 8–11 M is uncertain with no available nucleosynthetic data. Hence, we have ignored this stellar mass range in our simulations. We have considered the nucleosynthetic contributions of supernova of type Ia (SNIa). The core collapse supernova (SNII and SNIb/c) and supernova of type Ia (SNIa) are the major sources of oxygen and iron in the galaxy, respectively (Timmes et al. 1995). We have not considered the contribution of novae in the present work as these stellar sources contribute insignificantly to GCE except for 15N and 17O (Romano and Matteucci 2003).

To develop the GCE model, it is extremely important to consider the nucleosynthetic yields of all the stellar masses present in the IMF with varied metallicities ranging from zero metallicity to slightly more than the solar metallicity. The evolution of the galaxy starts with approximately 0.74 (X), 0.26 (Y), and 0.0 (Z) mass fractions of hydrogen, helium, and heavy elements (defined as metallicity), respectively. The first generation of stars is formed with zero metallicity. These stars evolve and finally eject heavier elements in the interstellar medium. The next generations of stars are formed with comparatively higher metallicity compared with the previous generations. The process of the formation of successive generations of stars with higher metallicity continues until the present epoch. As the nucleosynthetic yields of a star depend upon its metallicity, it is essential to appropriately consider the metallicity-dependant nucleosynthetic data of stars of various masses.

Single Massive Stars (M ≥ 11 M)

Several groups have estimated the nucleosynthetic yields of core collapse supernova while dealing with the stellar evolution and nucleosynthetic details (Woosley and Weaver 1995; Limongi et al. 2000; Chieffi and Limongi 2004; Woosley and Heger 2007). Among the nucleosynthetic yields produced by these groups, Woosley and Weaver (1995) have produced the most comprehensive work on the stellar evolution and nucleosynthesis of stars of masses 11–40 M and metallicities Z = 0–0.02. We have used these data for the nucleosynthetic yields of 11–100 M over the wide range of metallicities. It should be noted that even though Woosley and Heger (2007) have incorporated the revised solar metallicity in their stellar models, the nucleosynthetic yields of only the solar metallicity stars have been published so far. In the absence of the stellar yields of low-metallicity stars, it would be difficult to use the stellar yields of the recent solar metallicity models proposed by Woosley and Heger (2007) in our GCE models in a self-consistent manner. Hence, the nucleosynthetic yield data obtained by Woosley and Weaver (1995) were appropriately interpolated or extrapolated to obtain the nucleosynthetic yields for the stars of all the masses in the range 11–100 M and the metallicities spanning over the range of zero to slightly more than the solar metallicity.

Woosley and Weaver (1995) have considered distinct models, namely, models A, B (and C, in case available) corresponding to different remnant masses of a star of mass ≥25 M. The remnants in the form of neutron stars and black holes are produced at the time of supernova. These remnants do not participate further in GCE. For a star of specific mass, the size of the remnant mass decreases from the model A to C, thereby increasing the supernova nucleosynthetic yields of the heavier elements, specifically, the iron-peaked nuclei beyond iron. As there are uncertainties associated with the precise determination of the remnant masses in supernova explosion of stars of masses ≥25 M, we averaged out the nucleosynthetic yields of all the stable nuclides for the various models of distinct remnant masses in all our simulations except for the simulation NSM (Table 1). In the case of the simulation NSM, we only considered the models with the lowest remnant masses that have the highest nucleosynthetic yields of the iron-peaked nuclei. This simulation was attempted to understand the dependence of GCE on the remnant masses of the massive stars. Furthermore, we performed the simulations PSI, NSI, and NSIm (Table 1) by dividing the nucleosynthetic yields of the iron-peaked nuclei beyond iron by a factor of 2 (Timmes et al. 1995) to appropriately consider the relative yields of SNII and SNIa. The difference between the NSI and NSIm simulations is in the manner in which the extrapolation is performed in the nucleosynthetic yields of the stars of masses > 40 M (see e.g., François et al. 2004). In the former case, the nucleosynthetic yields of the stars of masses 35 and 40 M were used for the linear extrapolation, whereas in the latter case, we used the yields of the stellar masses 25 and 40 M for linear extrapolation. The former scenario can result in a lower, and hence, underestimated nucleosynthetic yields of the major isotopes, e.g., the yield of 24Mg (Woosley and Weaver 1995) is lower for a 40 M star than for a 35 M star. As the stellar yields of at least the major stable isotopes are higher for the 40 M models than the more accurate stellar yields of 25 M, the latter approach would yield comparatively more accurate extrapolated yields for the stars of masses >40 M. It should be mentioned that the stellar nucleosynthetic yields of the stars of masses >40 M would remain uncertain with any adopted criteria of extrapolation. The simulations NSI and NSIm would be helpful in understanding the influence of the criteria adopted for interpolation and/or extrapolation of the nucleosynthetic yields in cases where the stellar yields are not available. It should be mentioned that we have not performed rigorous analysis of the stellar yields of SNII as carried out by François et al. (2004).

Low and Intermediate Mass Stars

The nucleosynthetic yields of the low and intermediate mass AGB stars in the mass range 1.25–8 M have been calculated at different metallicities (Karakas 2003, 2007, 2010; Karakas and Lattanzio 2003, 2007; Karakas et al. 2006). These nucleosynthetic yields are more reliable than the yields based on the extrapolation of synthetic models developed by Forestini and Charbonnel (1997). These authors have also otherwise produced comprehensive nucleosynthesis yields. The low and intermediate mass stars contribute significantly to the stable nuclei up to silicon (Sahijpal and Soni 2006). On the basis of available nucleosynthetic yields at distinct metallicities, we appropriately interpolated or extrapolated to obtain the nucleosynthetic yields for the stars of all the metallicities spanning over the range of zero metallicity to slightly more than the solar metallicity. It should be mentioned that Cristallo et al. (2011) have recently published the nucleosynthetic yields of low-mass (1.5–3 M) AGB stars of different metallicities. A comparative analysis of the nucleosynthetic yields of a 3 M AGB star with a metallicity of 0.02 obtained by Cristallo et al. (2011) and Karakas (2010) exhibits reasonable agreement expect for 20Ne, 24Mg, and 28Si. These differences have been attributed to either net AGB production or destruction in comparison with initial abundances of these isotopes (Cristallo et al. 2011).

Type 1a Supernovae

Apart from the single evolving stars, we have also considered the nucleosynthetic contributions from the binary systems that are assumed to eventually evolve to supernova of type Ia (SNIa). The primary star (the heavier star) of the binary system evolves earlier and eventually leaves a white-dwarf core with the expulsion of planetary nebula in case its mass is less than 8 M. The white-dwarf can accrete mass from the companion secondary star at the time of its evolution through AGB stage. This can trigger the type Ia supernova if the mass of the accreting white-dwarf exceeds the Chandrasekhar mass limit. The evolution of the secondary star away from the main sequence decides the time frame of SNIa (Matteucci 2003; Matteucci et al. 2009). Hence, the mass of the secondary star essentially decides the time of SNIa.

The type Ia supernovae are considered to be the major contributors of the Fe-group nuclei in the galaxy. The explosive nucleosynthesis of some of the Chandrasekhar mass SNIa models with slow deflagrations and high central densities of an accreting white dwarf at thermal nuclear runway produces a wide range of neutron-rich iron group nuclei (Iwamoto et al. 1999). Travaglio et al. (2004) have indicated a dependence of the SNIa evolution on the dimensionality of the hydrodynamic models (Sahijpal and Soni 2006). In the present work, we have used the nucleosynthetic yields proposed by Iwamoto et al. (1999) for solar metallicity SNIa. We randomly triggered SNIa for these various models at different instances on the basis of a random number generator with equal weightage assigned to each model. This is essentially equivalent to averaging out the nucleosynthetic yields of the various models over the galactic time scales. The use of the solar metallicity model remains one of the short-comings of the present work. Metallicity-dependent SNIa nucleosynthetic yields are not available in literature.

The traditional procedure associated with the incorporation of SNIa into GCE models involves solving integro-differential equations dealing with the evolution of binary systems (Chiappini et al. 1997, 2003a; Pagel 1997; Matteucci 2003; François et al. 2004; Kobayashi and Nomoto 2009; Matteucci et al. 2009). The approach involves generating a fraction of the stars in the intermediate mass range 3–8 M and 11–16 M that will evolve through diverse binary systems with a wide range of life spans. The primary stars among the binary systems eventually lead to SNIa. This is a parametric approach with the primary aim of eventually explaining the present SNII/SNIa rates and the age–metallicity relation. We followed an alternative method that is quite simple in terms of the numerical operation in the presently adopted approach of evolving an ensemble of stars in a discrete manner. During the formation of the stars in the mass range 3–8 M and 11–16 M (Matteucci 2003), we assumed that a fraction (f) of these stars will eventually evolve to SNIa in a binary system. The remaining fraction (1 − f) of the stars will evolve to AGB stars and core collapse supernova (SNII and SNIb/c) in the mass range 3–8 M and 11–16 M, respectively. The fraction (f) was considered as a parameter (Table 1) to obtain the solar abundance of iron in the solar neighborhood at the time of the formation of the solar system around 4.56 Ga ago.

We followed two distinct approaches to estimate the time scale of the SNIa explosion after the formation of the binary systems. In the first approach, we averaged out the ages of the binary system following a simplified procedure, whereas in the second approach, we formed binary systems with ages that were randomly generated within a viable range in a manner to simulate the evolution of the binary systems in a realistic manner. Corresponding to a specific mass, e.g., the mass “m,” in the mass range, 3–8 M and 11–16 M, the binary system will share the mass with different secondary to primary star mass ratios. In the first approach based on average ages of the binary systems, we assumed that in the binary system of total mass “m,” the SNIa supernova occurs after a time span that corresponds to the age of the star with mass 1/η times the mass of the single star of equivalent mass “m.” η was considered as a free parameter in our simulations (Table 1). Its value was taken to be either 2 or 3. As mentioned earlier, the mass of the secondary star in the binary system decides the time of SNIa (Matteucci 2003). Hence, the value of η would be ≥ 2 as the secondary/primary star mass fraction would be ≤1. The mass of the secondary star was also accounted properly in the simulations.

In the second approach, at the time of the formation of a binary system of total mass “m” in the mass range 3–8 M, we randomly generated the age of the binary pair with the help of a random number in the time interval between 5 Ga (the age of a approximately 1.25 M single star) and the corresponding age of the star with mass 1/2 times the mass of the single star of equivalent mass “m.” For example, in the case of formation of a 8 M binary system, we randomly selected the age of this system to be between 5 Ga and the age of a single star of mass 4 M. This would cover the mass range of approximately 1.25–4 M for the secondary star in the binary system. The primary star explodes as SNIa subsequent to this temporal scale. The simulations based on this approach are marked as “random” in Table 1. In the mass range 11–16 M for the binary systems, we used the identical approach as used in the first method based on the averaged-out ages of the binary system with η = 2. The reason for this choice is based on the limited choice of the mass range of single individual stars. For example, a 16 M binary system could be produced essentially by a pair of 8 M single stars. Similarly, a 11 M binary system would consist of a secondary star in the mass range approximately 3–5 M. This can be approximated by an average choice of 5 M based on η = 2.

Predicted Supernova Rates and the Stellar Remnants

The deduced supernovae rates of the various types, i.e., SNIa, SNII, and SNIb/c, for the six distinct set of simulations are presented in Fig. 6. The presently predicted SNII/SNIa supernovae rate ratios are also included in Table 1 for all the simulations. This value has to be compared with the presently observed value that is in the range of 3–8 (Tammann et al. 1994). The deduced temporal evolution of the stellar remnant density in the form of white-dwarf and supernova remnants, i.e., neutron stars and black hole is presented in Fig. 7 for some selected simulations. The remnant density gradually increases due to the evolution of numerous generations of stars. The remnants do not further participate explicitly in the GCE. The time-integrated values of the star formation and supernovae explosion history over the last 13 Ga for all the simulations are presented in Table 3.

Table 3. The integrated history of the star formation and supernova explosion during 13 Gyr of the galactic chemical evolution in the solar neighborhood
ModelsStar formation M pc−2SNII pc−2SNIb/c pc−2SNIa pc−2
P-Solar-A (PSA)60.8070.2270.0420.011
N-Solar-A (NSA)59.5250.1530.0220.038
P-Solar-B (PSB)59.8830.2280.0430.010
P-Solar-C (PSC)62.2450.2270.0420.010
P-Solar-D (PSD)61.4960.2260.0420.010
P-Solar-E (PSE)60.0780.2230.0430.010
P-Solar-F (PSF)62.1780.2220.0400.009
P-Solar-G (PSG)59.5990.2300.0440.011
P-Solar-H (PSH)60.8080.2270.0420.011
P-Solar-I (PSI)60.7670.2240.0420.022
N-Solar-I (NSI)59.5220.1520.0220.044
N-Solar-Im (NSIm)59.4460.1470.0210.046
P-Solar-J (PSJ)60.8180.2280.0420.011
N-Solar-J (NSJ)59.5490.1540.0230.039
P-Solar-N (PSN)60.2840.1570.0490.017
N-Solar-N (NSN)59.1380.1050.0260.045
N-Solar-K (NSK)59.3980.1470.0210.037
N-Solar-M (NSM)59.1540.1330.0170.028
Figure 6.

The deduced supernova rates in the solar neighborhood for the various simulations. The 11–33 M stars were evolved eventually to SNII supernova. The 34–100 M stars were evolved to SNIb/c, whereas the SNIa resulted from the detonation of the primary star of a binary star. The broad band associated with the SNIa rates is due to the variations of the SNIa rates over the consecutive simulation time-step of 1 million years. Time averaging over several million years will reduce the band to a smooth curve. It was generally observed that in all the simulations, the upper limit of the broad band represented the general trend in the SNIa rates. The deduced time integral values of the supernovae rates for all the simulations are mentioned in Table 3. Matteucci and François (1989) estimated the present contribution of 20% for SNIa. Matteucci and Greggio (1986) deduced the values of approximately 1.5 × 10−5 pc−2 Myr−1 and 1 × 10−5 pc−2 Myr−1 for SNII and SNI, respectively. Chiappini et al. (1997) estimated a value of approximately 3 for the SNII/SNIa. Alibés et al. (2001) estimated the present SNII+SNIb/c and SNIa rates to be approximately 1.4 × 10−5 pc−2 Myr−1 and 3 × 10−6 pc−2 Myr−1, respectively. Matteucci (2003) estimated the values to be approximately 5.7 × 10−6 pc−2 Myr−1 and 1.2 × 10−6 pc−2 Myr−1 for SNII and SNIa, respectively. The observed present SNII/SNIa estimate for the galaxy is in the range of 3–8 (Tammann et al. 1994).

Figure 7.

The deduced evolution of the total remnant (σtotal remnant), white-dwarf (σWhite Dwarf), and supernova remnant (σsupernova remnant) surface mass density in the solar neighborhood from the stellar evolution in several simulations.

Predicted Age–Metallicity Relation and the Bulk Isotopic Composition of the Solar System

The predicted age–metallicity relation in the solar neighborhood is presented in Fig. 8. The metallicity is essentially the sum total of the mass fraction of all the elements except for hydrogen and helium. The temporal evolution of the iron abundance [Fe/H] of the solar neighborhood is presented in Fig. 9 for different models. Except for the difference in the initial trends in the iron abundance evolution, the two distinct set of simulations yield identical results when normalized to the pre-revised (Anders and Grevesse 1989) and the recently revised solar abundances (Asplund et al. 2009).

Figure 8.

The deduced temporal evolution of the metallicity of the solar neighborhood as measured in terms of Z, the total mass fraction of all the elements heavier than helium. Various simulations were run to reproduce the pre-revised solar abundance, i.e., Z = 0.02 (Anders and Grevesse 1989) and the recently revised abundances, i.e., Z = 0.014 (Asplund et al. 2009) at the birth of the solar system.

Figure 9.

The age–metallicity relation measured in terms of [Fe/H] (=Log[(Fe/H)Galaxy/(Fe/H)Solar]) in the solar neighborhood for the various simulations along with the observed data (Meusinger et al. 1991; Edvardsson et al. 1993; Rocha-Pinto et al. 2000). The pre-revised solar abundance (Anders and Grevesse 1989) and the recently revised abundances (Asplund et al. 2009) at the birth of the solar system were used in the two distinct set of simulations. It should be noted that the observed data do not explicitly represent the actual deduced spectral [Fe/H] abundances of the stars. The data have been binned on the temporal axis, with a bin size corresponding to 0.2 dex in terms of the metallicity for the defined age–metallicity relation (see e.g., Edvardsson et al. 1993). The observed [Fe/H] data have been averaged for various stars within a specific bin.

The estimated bulk abundance of the isotopes from 1H - 69Ga, at the time of formation of the solar system around 4.56 Ga ago (Sahijpal et al. 1998; Huss et al. 2009) in the solar neighborhood is presented in Fig. 10. Depending upon the simulation (Table 1), the estimated isotopic yields are normalized with respect to the pre-revised bulk solar isotopic abundance (Anders and Grevesse 1989), or the recently revised (new) bulk solar isotopic abundances (Lodders 2003, 2010; Asplund et al. 2009; Lodders et al. 2009). Due to the numerous uncertainties associated with star formation IMF, SFR, the stellar evolution, and nucleosynthesis, any departure within a factor of 2 is considered to be tolerable from the ideal normalized value of 1 (Fig. 10) (Matteucci and François 1989; Timmes et al. 1995; Chiappini et al. 1997, 2003a; Pagel 1997; Alibés et al. 2001; Kobayashi et al. 2011).

Figure 10.

The estimated galactic chemical evolution yield of the various stable nuclides in the solar neighborhood at the time of the formation of the solar system around 4.56 Ga ago (Sahijpal et al. 1998; Huss et al. 2009) for the various simulations. The isotopic yields are normalized with respect to either the pre-revised solar abundance (Anders and Grevesse 1989), or the recently revised (new) abundances (Asplund et al. 2009). The solid line represents the best match, whereas the deduced data enclosed within the two dashed lines represents a spread by a factor of 2 over the best match. This spread is considered tolerable, given the numerous uncertainties associated with the processes involved in stellar and galactic evolutions.

Predicted Relative Abundance Evolution of the Major Elements

As mentioned earlier, the bulk abundances of oxygen and iron in the galaxy are essentially produced by SNII+SNIb/c and SNIa supernova, respectively. As the galaxy evolves, the contributions from the SNIa gradually increase initially at a lower rate compared with the SNII+SNIa contributions (Fig. 6). This results in the reduction in the oxygen to iron ratio with the gradual evolution of the galaxy. The evolution of the oxygen to iron ratio in the solar neighborhood is presented in Fig. 11 for several selected models. The predicted [O/Fe] value at [Fe/H] = −2 is given in Table 1 for all the simulations to compare various models. We have also further selected additional major elements to understand their GCE in Fig. 11. The observational data are also presented to make comparison with our model predictions.

Figure 11.

The normalized elemental abundance evolution of [x/Fe] (=Log[(x/Fe)Galaxy/(x/Fe)Solar]) as a function of [Fe/H] for the various simulations. Here, “x” represents various major elements. The normalization has been performed with respect to (x/Fe)solar. The observation data points (Clegg et al. 1981; Gratton 1985; Laird 1985; Luck and Bond 1985; Gratton and Ortolani 1986; Tomkin et al. 1986; Carbon et al. 1987; François 1987, 1988; Gratton and Sneden 1988, 1991; Hartmann and Gehren 1988; Sneden and Crocker 1988; Barbuy and Erdelyi-Mendes 1989; Magain 1989; Peterson et al. 1990; Zhao and Magain 1990; Meusinger et al. 1991; Sneden et al. 1991; Edvardsson et al. 1993; Nissen et al. 1994; McWilliam et al. 1995; Ryan et al. 1996; Israelian et al. 1998; Boesgaard et al. 1999; Stephens 1999; Carretta et al. 2000; Chen et al. 2000; Mishenina et al. 2000; Rocha-Pinto et al. 2000) are plotted along with the predicted trends in the models.

Finally, the G-dwarf metallicity distribution of the revised solar metallicity model NSI is presented in Fig. 12. The identical distribution will also hold in the case of the remaining revised solar metallicity models as the general evolutionary trend in the [Fe/H] is almost identical in all these simulations (Fig. 9b). Due to the discrete nature of the stellar IMF adopted in the present approach, we choose the stellar masses 0.8, 1, and 1.25 M to make the assessment of G-dwarf metallicity distribution. The deduced distribution was compared with the observational data of Jørgensen (2000). Even though we do not achieve the perfect match, specifically around [Fe/H] = −2, the deduced distribution seems to explain the basic observational trend. The maxima of the predicted distribution seems to be offset toward [Fe/H] approximately 0 instead of the observational maxima in the range −0.2 to 0. This also seems to be reflected, to an extent, in the [Fe/H] evolutionary trend (Fig. 9b) that shows slight departure from the observational trend for [Fe/H] > 0. It should be mentioned that the observational evidence for an age–metallicity relation in the galaxy is not unambiguous. It has been argued that the observational data exhibit an overall scatter at any given time that is at least as large as any change over the history of the galaxy (see e.g., Feltzing et al. 2001; Nordström et al. 2004). One of the reasons for the observed predicted departure, at least in the case of the G-dwarf metallicity distribution (Fig. 12), could be due to the use of the discrete IMF, specifically, in the mass range 0.8–1.2 M (Fig. 2). We might not have appropriately accounted the number distribution of the stars in this mass range that evolve over the galactic time scales.

Figure 12.

The predicted G-dwarf metallicity distribution for one of the best simulations, N-Solar-I, is represented by thick line. The observational data (Jørgensen 2000), represented by histogram, are included for comparison. The 0.8, 1, and 1.25 M stars were included in the assessment of the G-dwarf metallicity distribution.

Discussions

An attempt has been made to numerically simulate the evolution of the galaxy in the solar neighborhood in a realistic manner by evolving an ensemble of numerous generations of stars. The essential aim was to numerically simulate the historic evolution of star formation, supernovae along with the bulk isotopic evolution of the isotopes from 1H - 69Ga. One of the main objectives was to explain the pre-revised bulk solar isotopic abundance (Anders and Grevesse 1989), and the recently revised bulk solar isotopic abundances (Lodders 2003, 2010; Asplund et al. 2009; Lodders et al. 2009). Several simulations were run to understand the dependence of GCE on the choice of various simulation parameters. These parameters deal with the assumed accretion rate of the galaxy, the SFR, the stellar IMF, the supernovae rates, the stellar evolution, and nucleosynthetic prescriptions.

Pre-Revised Solar Metallicity Models

As mentioned earlier, the basic model PSA (Table 1) was run to reproduce the pre-revised bulk solar isotopic abundances (Anders and Grevesse 1989). A basic set of assumptions were made in this simulation regarding the parametric choice. The accretion of the galaxy was performed according to the formulation used by Chiappini et al. (1997) and Alibés et al. (2001) with the time scales for the accretion of the halo-thick disk and thin disk to be 1 and 7 Ga, respectively (Table 1). The SFR parameter, “n” (Table 1) was adopted from the work by Alibés et al. (2001). This is reflected in the predicted SFR (Fig. 4a). The stellar IMF parameter, “A” (Tables 1 and 2) in the mass interval 11–100 M was adopted to be identical to the choice made for the mass interval 1.25–8.0 M. In terms of the stellar nucleosynthetic contributions, the stars in the mass range 11–100 M essentially contribute through SNII, and to an extent SNIa. The stars in the mass range 1.25–8.0 M contribute through AGB stars and SNIa. Hence, the parameter, “A” controls the relative contributions of AGB stars, SNII, and SNIa yields. In the majority of the simulations, we made an identical choice of this parameter in the two mass ranges. In terms of the contributions of ≥30 M SNII models (Woosley and Weaver 1995), we used all the proposed models of stars corresponding to distinct remnant masses (Table 1). The choice of Z′ was made identical to the suggestions made by several workers. Even though the PSA model is able to explain the trend in the age–metallicity relation (Fig. 8a), the temporal evolution of [Fe/H] (Fig. 9a) and the bulk solar isotopic composition of majority of the elements (Fig. 10a), the model is unable to reproduce the observed supernovae rates (Fig. 6a; Table 1) and the evolutionary trends in the major elements (Fig. 11), specifically the dependence of [O/Fe] on [Fe/H] with respect to the observed data (Fig. 11c). The predicted present SNII/SNIa rate is significantly higher than the observed value of 3–8 (Tammann et al. 1994). Hence, this simulation is certainly not our best model to reproduce GCE.

To reproduce all the essential features of GCE, we explored the parametric space by running several simulations with the pre-revised solar metallicity model (Table 1). The essential aim was to ascertain the most critical parameter(s) that can reproduce majority of the observed features of GCE.

Variations in the SFR parameter, “n,” were considered in the simulations PSB and PSC (Table 1). The increase (decrease) in this parameter results in increase (decrease) in the deduced density of the stars and remnants with respect to the simulation PSA (Fig. 2b; Table 3). This is essentially due to the increase (decrease) in the SFRs (Fig. 4a), specifically during 1–6 Ga. The deduced trend in the age–metallicity relation (Fig. 8a), the temporal evolution of [Fe/H] (Fig. 9a), and the predicted present SNII/SNIa ratio (Table 1; Fig. 6a) were observed to be almost identical to the simulation PSA in both the simulations. The predicted [O/Fe] at [Fe/H] = −2 was also found to be almost identical in the three scenarios. In general, we observed no major improvement over the PSA model even by varying the value of “n” over the range of 0.25–0.45. Nonetheless, it should be mentioned that an increase in the value of “n” results in the reduction in [O/Fe] at [Fe/H] = −2, and, an increase in the present predicted SNII/SNIa. Hence, a lower value of “n” infers better match of the GCE model predictions with the observational data. The dependence of GCE on SFR has been comprehensively studied by Chiappini et al. (1997). This work invokes high SFRs during the accretion of halo-thick disk of the galaxy. Even though we adopted the formulation of Alibés et al. (2001) for SFR, the variation in the simulation parameter “n” was performed to explore the high SFRs as worked out by Chiappini et al. (1997).

Variations in the accretion time scale of the halo-thick disk of the galaxy were attempted in the simulations PSD and PSE by varying “tT” over the range of 0.5–1.5 Ga. The change in “tT” results in distinct evolutionary history of the galaxy in terms of the total surface mass density (Fig. 2a). The reduction in “tT” results in higher stellar and remnant surface mass density. The maxima of the SFR is acquired around 1, 2, and 3 Ga in the case of the simulations, PSD, PSA, and PSE, respectively (Fig. 4a). Rapid early growth in terms of the metallicity (Fig. 8a) and [Fe/H] (Fig. 9a) occurs in the simulation PSD compared with the simulation PSE, with the simulation PSA exhibiting the intermediate behavior. The simulation PSD predicts higher [O/Fe] at [Fe/H] = −2 than the simulation PSE (Table 1). There is not much difference in the SNII/SNIa for the simulations (Table 1). Hence, the rapid accretion scenario of the halo-thick disk seems to be a viable approach to successfully deduce galactic chemical evolution.

Furthermore, we studied the variations in the accretion time scale of the thin disk in the simulations PSF and PSG by varying “tD” over the range of 5–10 Gyr (Table 1). The change in “tD” results in distinct evolutionary histories of the galaxy (Fig. 2a). A higher stellar and remnant surface mass density is due to the reduction in “tD.” The simulation PSF exhibits a high maxima in terms of SFR at approximately 2 Ga (Fig. 4a). The rate subsequently falls below the SFR for the simulation PSG beyond approximately 7 Ga. The age–metallicity (Fig. 8a) and [Fe/H] (Fig. 9a) early growths occur rapidly in the simulation PSG compared with the simulation PSF. The simulation PSF infers higher SNII/SNIa and lower [O/Fe] at [Fe/H] = −2, than the simulation PSG (Table 1).

The simulation PSH was run with a value of 3 for the SNIa parameter, “η” compared to the value of 2 in case of the simulation PSA. As discussed earlier, the increase in η would result in an increase in the assumed average age of the binary system (the secondary star) before the primary star explodes as SNIa (Table 1). This would result in relatively delayed contributions of SNIa in the case of simulation PSH compared with PSA. We noticed insignificant differences in the simulations PSH and PSA in terms of the fitting parameters, “x” and “f ” (Table 1), the age–metallicity relation (Fig. 8a), the predicted [Fe/H] evolution (Fig. 9a), and the integrated supernovae count (Table 3). However, the present predicted value of SNII/SNIa in the former case is significantly lower than the latter scenario, thereby indicating the delayed contributions from SNIa. One of the shortcomings of the present work is the use of metallicity-independent SNIa nucleosynthetic yields (see e.g., Kobayashi and Nomoto 2009). This results in an identical enrichment of the galaxy from SNIa in terms of nucleosynthetic component over the various epochs. The simulation PSJ also yielded almost identical results (Table 1). As discussed earlier, this simulation was run by appropriately selecting ages of the secondary stars in a random manner, but within a viable range.

The simulation PSI was performed by systematically reducing the yields of the iron-peaked nuclei beyond iron for ≥30 M stars by a factor of 2 (Table 1). This was motivated by the work of Timmes et al. (1995) that was subsequently followed by other workers (see e.g., Alibés et al. 2001). However, there are certain differences in the present approach. We have systematically reduced the yields of all the iron-peaked nuclei beyond iron for ≥30 M stars. We observed major differences in this simulation compared with the model PSA in terms of the predicted present SNII/SNIa (Table 1), and the evolutionary trends of the major elements (Fig. 11) that are also reflected in the last column of Table 1. The predicted present SNII/SNIa is in the upper limit of the observed range of 3–8 (Fig. 6b).

Another simulation, PSN, even performs comparatively better in terms of the predicted present SNII/SNIa (Table 1; Fig. 6c) and the evolutionary trends of some of the major elements, specifically oxygen (Fig. 11). However, it still falls short of explaining the observational trend (Fig. 11c). This simulation differs from PSA due to the reduction in the incorporation of the stellar yields from 11–100 M by reducing the constant, “A” associated with the stellar IMF. However, it should be noted that the deduced IMF is less steep compared with other simulations (Table 1). This simulation results in a reduced SFR compared with the other simulations, e.g., PSI and PSJ (Fig. 4b; Table 3).

In general, the pre-revised solar abundance simulations are able to explain most of the essential observed features of the galaxy in the cases where we choose the reduced yields of the iron-peaked nuclei from SNII (the simulation PSI) and systematically reduce the nucleosynthetic contributions of the stars in the mass 11–100 M (the simulation PSN).

Revised Solar Metallicity Models

We performed seven simulations with varied parameters in the case of the revised solar metallicity model (Table 1). On the basis of our experience from the pre-revised solar metallicity models, we emphasized here the major dependence of the simulations on the choice of the stellar nucleosynthetic prescription. The inferences drawn from the dependence of the pre-revised solar abundance models on the accretion criteria of the galaxy and the SFR can be generalized to the revised solar metallicity models.

All the revised solar metallicity models are able to explain most of the observations dealing with GCE, specifically, the deduced present SNII/SNI values that are well within the observed value of 3–8 (Tables 1 and 3; Figs. 6d–f) (Tammann et al. 1994). The evolutionary trends in the deduced metallicity (Fig. 8b) and the [Fe/H] (Fig. 9b) are almost identical in all the simulations. The evolution trends in some of the major elements (Fig. 11), specifically, the oxygen to iron abundance ratio evolution (Fig. 11c) is consistent with the astronomically observed values.

In contrast to the pre-revised solar metallicity models, the revised solar metallicity models are able to explain almost all the observational trends even with the choice of the simulation parameter, “A” associated with the stellar IMF (Tables 1 and 2). An identical value of “A” in the mass range, 1.25–8.0 M and 11.0–100.0 M, is able to explain GCE. However, the revised stellar metallicity models result in steeper slopes of the IMF in the mass range 11–100 M compared with the pre-revised solar metallicity models. Among the revised solar metallicity models, the simulation NSN results in a reduced SFR compared with the other simulations, e.g., NSI and NSJ (Fig. 4b, Table 3).

Except for the differences in the major elemental evolutionary trends (Fig. 11), the revised solar metallicity models do not exhibit any other major differences based on the choice of the stellar nucleosynthetic prescription as inferred from the NSI, NSIm, and NSM models that are discussed earlier. Nonetheless, the choice of SNII models that incorporate the averaged nucleosynthetic yields of the various models with different remnant masses of the stars ≥25 M (Woosley and Weaver 1995) seems to be more favorable. Furthermore, the reduction in the SNII yields of the iron-peaked nuclei beyond iron by a factor of 2 as in the models NSI and NSIm is found to be encouraging. We observed noticeable changes in the major elemental evolutionary trends (Fig. 11) in the case of NSI and NSIm models. As discussed earlier, the two models differ in the extrapolation criteria for SNII nucleosynthetic yields of stars >40 M. François et al. (2004) have performed rigorous analysis of the role of SNII stellar yields on galactic chemical evolution. It has been found that the stellar yields have to be appropriately modulated to explain the elemental evolutionary trends in a better manner.

The deduced supernovae rates (Fig. 6; Table 3) are in better agreement with the observed rates for the revised solar metallicity models compared with the pre-revised metallicity models. We also make comparison of our present supernovae rate predictions with the other theoretical works. Matteucci and François (1989) estimated around 20% SNIa contributions based on their chemical evolution model. Matteucci and Greggio (1986) deduced the values of approximately 1.5 × 10−5 pc−2 Myr−1 and 1 × 10−5 pc−2 Myr−1 for SNII and SNI, respectively. Chiappini et al. (1997) estimated a value of approximately 3 for the present SNII/SNIa. Alibés et al. (2001) estimated the present SNII+SNIb/c and SNIa rates to be approximately 1.4 × 10-5 pc−2 Myr−1 and 3 × 10-6 pc−2 Myr−1, respectively. Matteucci (2003) estimated these values to be approximately 5.7 × 10-6 pc−2 Myr−1 and 1.2 × 10-6 pc−2 Myr−1 for SNII and SNIa, respectively. In general, there is a range of estimated values for the predicted present supernovae rates. Our results are in general agreement with these works in the case of all the revised solar metallicity models, and in some simulations, e.g., PSI and PSN, corresponding to pre-revised solar metallicity models. However, we do not observe high SNII rates in the initial evolution of the galaxy as predicted by Chiappini et al. (1997) and Matteucci (2003). These higher deduced SNII rates are essentially due to the high SFRs used by these workers in the initial stage of galactic evolution.

We have time-integrated the supernovae rates over the entire evolution of the galaxy in Table 3. The revised solar metallicity simulations in general predict high SNIa rates and lower SNII+SNIb/c rates compared with the pre-revised solar metallicity models (Table 3, Fig. 6). As mentioned earlier, SNIa and SNII+SNIb/c are the major contributors of iron and oxygen, respectively. The reduction in the solar metallicity from 0.02 to 0.014 on account of metallicity revision results in the lower SNII+SNIb/c rates as oxygen is the predominant element that decides the metallicity. The lowering of the SNII+SNIb/c rates results in low iron abundances that need to be appropriately compensated by higher SNIa rates. This necessitates the increase in the fraction “f ” of the stars that evolve into the binary systems (Table 1). In general, we observe a significant reduction in the supernovae rates, specifically the SNII rates of the galaxy due to the reduction in the assumed solar metallicity (Table 3). However, it should be noted that there is not much significant reduction in the integrated star formation record (Table 3). This would imply that the relative production of low and intermediate mass stars compared with massive star (≥11M) increases in the revised solar metallicity models with respect to the pre-revised solar metallicity models. This is also inferred from the steeper IMF in the former case (Table 1). However, the absolute production of the low and intermediate mass stars is almost unaltered in the two set of models. The evolutionary trends in the stellar remnant surface mass density demonstrate the contributions of low and intermediate mass stars with respect to massive stars in the two set of simulations (Fig. 7). The simulations PSA and PSN have higher SNII+SNIb/c rates compared with the simulations NSA and NSI. This results in higher supernova remnant densities in the former case (Fig. 7). In general, the white-dwarf densities dominate the supernova remnant densities in all the simulations due to the stellar IMF.

Bulk Solar Isotopic Composition and the Major Elemental Isotopic Trends

In the case of the simulations PSA, PSI, PSJ and PSN with the pre-revised solar abundance, the bulk solar isotopic abundance of most of the isotopes except for the heavier iron-peaked nuclei can be explained within a factor of 2 uncertainty (Figs. 10a and 10b). The deficiency of 15N is essentially due to the absence of the nova nucleosynthetic contribution in our simulations. Novae are considered a dominant source of 15N. The nucleosynthetic source(s) of the neutron-rich isotopes of the intermediate nuclei around calcium and titanium, e.g., 48Ca, is (are) still uncertain (Sahijpal et al. 2000). Rare types of SNIa are generally considered as their sources. The deficiency in 48Ca is essentially due to the uncertainties in the stellar sources(s) of this nuclide. The higher yields of the heavier iron-peaked nuclides are essentially due to the higher deduced SNII+SNIb/c supernova rates for these simulations. Massive stars are the major contributors of these nuclides. The situation improves in the case of the simulations with the revised solar metallicity models. In the case of simulation NSJ, NSM, and NSN, the yields of the heavier iron-peaked are well within the tolerable band (Fig. 10c). This is essentially due to the reduction in the SNII+SNIb/c supernova rates for these simulations (Table 3). However, this results in the reduction in the GCE yields of the intermediate nuclides in the neighborhood of calcium. The chlorine GCE yields of the revised solar metallicity model are significantly lower than the simulations with the pre-revised solar metallicity model. This is due to the significant enhancement in the revised bulk solar estimates of chlorine (Fig. 1). It should be noted that in case there are further revisions in the bulk solar estimates, the normalized abundances of the elements like chlorine that do not significantly contribute to the metallicity could be appropriately altered without any significant change in the entire GCE model. The observed differences among the various models are consistent with those anticipated with the simulation criteria (Table 1). The simulation PSI predicts lower yields of the iron-peaked nuclei beyond iron compared with PSA (Fig. 10a). The simulation PSN results in lower yields of the intermediate nuclei compared with PSJ due to the lower contributions of SNII compared with SNIa, low and intermediate mass stars (Fig. 10b). The identical trend is also observed in the simulation NSN with respect to NSJ (Fig. 10c). The simulation NSM differs from NSJ in the choice of the SNII nucleosynthetic models with low remnant masses for the stars ≥25 M (Table 1).

It should be noted that even in the case of the simulations with the revised bulk solar abundances, the majority of the isotopic species fall within the tolerable band (Fig. 10c). This should be appreciated especially in the context of the nucleosynthetic yields of the core-collapse supernova (Timmes et al. 1995; Woosley and Weaver 1995), SNIa (Iwamoto et al. 1999), and the AGB stars (Karakas 2003, 2007; Karakas and Lattanzio 2003, 2007; Karakas et al. 2006) that have been obtained on the basis of the pre-revised bulk solar abundances. Within the anticipated prevailing uncertainties associated with the stellar nucleosynthetic yields, and to an extent the assumptions associated with the various processes of the evolution of the galaxy, we have produced the revised bulk solar isotopic composition.

The inferred evolutionary trends in the major elements are presented in Fig. 11 along with the astronomically observed abundances. Except in few cases, the deduced trends in the revised solar metallicity models explain the observational data in a better manner compared with the pre-revised solar metallicity models.

The deduced evolutionary trend in [C/Fe] explains the observational data in almost all the simulations (Fig. 11a). The horizontal trend in the [C/Fe] results due to the fact that carbon and iron are produced by entirely distinct processes (see e.g., Chiappini et al. 1997; Alibés et al. 2001; Matteucci 2003; Cescutti et al. 2009). The deduced trends are almost identical to the trends obtained by other groups except for the prominent hump around [Fe/H] approximately −1. The hump is due to the significant contributions of carbon from the low mass stars (<2 M) that start contributing at a late stage. The subsequent contributions of iron from SNIa at a comparatively latter time significantly reduce the [C/Fe] values. The distinct simulations exhibit the dependence of nucleosynthetic contributions on chemical evolution, e.g., the simulation NSN exhibits the most prominent hump due to the choice of the stellar IMF with the dominance of low and intermediate mass stars (Table 1).

The observational scatter in the case of [N/Fe] vs. [Fe/H] is extremely large, and the two set of simulations, i.e., the pre-revised and the revised solar metallicity models exhibit trends that indicate an initial increase in the [N/Fe] (Fig. 11b). The increase is considered due to the increase in the nitrogen yield contributions from low and intermediate mass stars (Chiappini et al. 1997; Alibés et al. 2001; Matteucci 2003). The further decline is associated with the contributions of iron from SNIa at a latter time. The various models, NSA, NSI, NSIm, NSM, and NSN, that differ in the choice of SNII nucleosynthetic yields and IMF exhibit wide-ranging spreads. Our deduced trends in the case of pre-revised solar metallicity models are a better match for the trends observed by other workers. However, the evolutionary trends in the revised solar metallicity models could indicate that during the evolution of the galaxy, the [N/Fe] value increases and decreases around the average solar nitrogen/iron value. The large scatter in the observational data seems to be consistent with this scenario.

The bulk abundances of oxygen and iron in the galaxy are essentially produced by SNII and SNIa+SNII, respectively. As the galaxy evolves, the contributions from the SNIa increase initially at a lower rate compared with the SNII+SNIa contributions. This results in the reduction in the oxygen to iron ratio with the gradual evolution of the galaxy (Fig. 11c).

Based on the nucleosynthesis theories, the elemental evolutionary trends in the α-nuclei (Mg, Si, S, Ca, Ti) are expected to be identical to that of oxygen (Chiappini et al. 1997; Alibés et al. 2001; Matteucci 2003; François et al. 2004; Kobayashi et al. 2011). The initial contributions of these elements by massive stars (SNII) result in the initial higher abundance of these elements with respect to iron. The subsequent contribution of iron by SNIa results in the reduction in these ratios. The observational data for the various α-nuclei (Mg, Si, S, Ca) can be explained by our revised solar metallicity models (Fig. 11). However, the predicted trend in [Ti/Fe] does not follow this expectation (Alibés et al. 2001; Matteucci 2003; François et al. 2004). As mentioned earlier, to reproduce the observed astronomical data, the stellar yields of the intermediate elements have to be appropriately modified (François et al. 2004). Furthermore, as discussed earlier, the PSN model performs better among the pre-revised solar models in terms of its predictions (Fig. 11).

The evolutionary trends around the constant solar value are expected in [Ni/Fe] and [Zn/Fe] (Chiappini et al. 1997; Alibés et al. 2001; Matteucci 2003; François et al. 2004). This is supported by the observational data (Fig. 11). However, our predicted trends in [Ni/Fe] (Fig. 11i) at [Fe/H] = −3 initiate with [Ni/Fe] approximately −0.5, and gradually evolve to solar value. An identical behavior has been observed earlier (François et al. 2004). It can be appropriately accommodated by increasing the relative yields of SNII. The [Zn/Fe] predicted behavior is also different from the observed trend, specifically for the pre-revised solar metallicity models (Fig. 11j). However, the trends in the simulations NSI and NSIm differ by an amount of [Zn/Fe] <−0.4 from the observational data average and the solar value.

Summary and Conclusions

We propose an alternative approach to understand GCE. Numerical simulations have been performed for GCE on the basis of the evolution of an ensemble of numerous generations of stars in a realistic manner. The approach is unique and distinct from the traditional formulation that involves solving the integro-differential equations associated with GCE. A comprehensive analysis has been performed to parametrically understand the dependence of GCE on the accretion rate of the galaxy, the SFR, the stellar IMF, the supernovae rates, and the adopted SNII stellar nucleosynthetic yields. Besides understanding the age–metallicity relation, the G-dwarf metallicity distribution, and the elemental evolutionary trends, the main objective was to reproduce the recently revised bulk solar photosphere abundance (Lodders 2003, 2010; Asplund et al. 2009; Lodders et al. 2009), and the pre-revised bulk solar abundances (Anders and Grevesse 1989).

The simulations corresponding to the pre-revised solar metallicity (Anders and Grevesse 1989) model were performed with a dual purpose. One of the reasons was to make comparison of our predictions with the results based on the solution of integro-differential equations obtained by other researchers. The second reason was to explore the parametric space initiating with a basic model. Our parametric analysis eventually resulted in the simulations with a suitable set of parameters related to the evolution of the galaxy. The results from these specific simulations are indicative of the general features that have been observed in literature, specifically those related with the accretion scenario of the galaxy, the SFR, the supernovae rates, and the SNII stellar nucleosynthetic prescription.

On the basis of the experience acquired from the pre-revised solar metallicity model, the simulations dealing with the revised (Asplund et al. 2009) solar metallicity model were performed with the limited choice of parameters essentially dealing with the SNII stellar nucleosynthetic yields and the stellar IMF. Almost all of the essential features of GCE could be explained on the basis of these simulations. There were, however, some noticeable discrepancies in the observed elemental evolution trends, and the G-dwarf metallicity distribution at high metallicity. One of the reasons for these discrepancies could be the assumed discrete nature of the stellar IMF that limits the complete survey of the stars in the entire mass range. Nonetheless, in general, the proposed alternative approach for GCE seems to be consistent with the traditional methodology. Some of the major inferences drawn from the present work are:

  1. The recent revision of the solar metallicity results in the reduction of the deduced violent history of the galaxy in terms of the supernovae rates. This follows in spite of the insignificant change in the deduced SFR, specifically, the low and the intermediate mass stars. This is primarily due to the change in the stellar IMF that results in steep slope in the mass range 11–100 M.
  2. The revised bulk solar isotopic composition could be explained within the tolerable limits of a factor of 2. This follows in spite of the fact that the presently employed stellar nucleosynthetic yields have been estimated on the basis of pre-revised solar isotopic compositions. Significant departures are, however, found in some of the elements, e.g., chlorine, due to major revisions in their solar abundance.
  3. The present SNII stellar nucleosynthetic yields for at least iron are high by a factor of 2 compared with the predicted values. This observation has been followed throughout the literature. In general, the supernovae yields of the iron-peaked nuclides need to revised.

Acknowledgments

We are extremely grateful to our reviewers Drs. A. Davis, L. Nittler, and A. J. T. Jull for their numerous suggestions that led to significant improvement of this manuscript. This work is supported by PLANEX (ISRO) research project.

Editorial Handling

Dr. Marc Caffee

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