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Chemical evolution of neutron capture elements in our Galaxy and in the dwarf spheroidal galaxies of the Local Group

By adopting a chemical evolution model for the Milky Way already reproducing the evolution of several chemical elements, we compare our theoretical results with accurate and new stellar data of neutron capture elements and we are able to impose strong constraints on the nucleosynthesis of the studied elements. We can suggest the stellar sites of production for each element. In particular, the r-process component of each element (if any) is produced in the mass range from 10 to 30 Msun, whereas the s-process component arises from stars in the range from 1 to 3 Msun. Using the same chemical evolution model, extended to different galactocentric distances, we obtain results on the radial gradients of the Milky Way. We compare the results of the model not only for the neutron capture elements but also for alpha-elements and iron peak elements with new data of Cepheids stars. We give a possible explanation to the considerable scatter of neutron capture elements observed in low metallicity stars in the solar vicinity, compared to the small star to star scatter observed for the alpha-elements. In fact, we have developed a stochastic chemical evolution model, in which the main assumption is a random formation of new stars, subject to the condition that the cumulative mass distribution follows a given initial mass function. With our model we are able to reproduce the different features of neutron capture elements and alpha-elements. Finally, we test the prescriptions for neutron capture elements also for the dwarf spheroidal galaxies of the Local Group. We predict that the chemical evolution of these elements in dwarf spheroidal galaxies is different from the evolution in the solar vicinity and indicates that dwarf spheroidal galaxies (we see nowadays) cannot be the building blocks of our Galaxy.


Introduction
"It is written in the stars above" by Depeche Mode "The stars above us, govern our conditions" by William Shakespeare

The galactic chemical evolution
Galactic chemical evolution is the study of the evolution in time and in space of the abundances of the chemical elements in the interstellar gas in galaxies. This process is influenced by many parameters such as the initial conditions, the star formation and evolution, the nucleosynthesis and possible gas flows. So, in order to build a chemical evolution model one needs to specify the initial conditions, namely whether the system is closed or open and whether the gas is primordial (no metals 1 ) or already chemically enriched. Then, it is necessary to define the stellar birthrate function, which is generally expressed as the product of two independent functions, the star formation rate (SFR) and the initial mass function More recently, multi-slope expressions of the IMF have been adopted since they better describe the luminosity function of the main sequence stars in the solar vicinity (Scalo 1986(Scalo , 1998; Kroupa et al. 1993). Generally, the IMF is assumed to be constant in space and time, with some exceptions such as the one suggested by Larson (1998), which adopts a variable slope: x = 1. 35(1 + m/m 1 ) −1 (1.8) where m 1 is variable with time and associated to the Jeans mass, (the typical mass at which the internal pressure is no longer strong enough to prevent gravitational collapse). The effects of a variable IMF on the galactic disk properties have been studied by Chiappini et al. (2000), who concluded that only a very "ad hoc" variation of the IMF can reproduce the majority of observational constraints, thus favoring chemical evolution models with IMF constant in space and time.
The stellar yields are fundamental ingredients in galactic chemical evolution. In the past ten years a large number of calculations has become available for stars of all masses and metallicities. However, uncertainties in stellar yields are still present. This is due to uncertainties in the nuclear reaction rates, treatment of convection, mass cut, explosion energies,

The neutron capture elements
In this thesis we will mainly deal with very heavy elements. Early work by Gilroy et al. (1988) Ryan et al. 1996). A more extensive study by Burris et al. (2000) confirmed the very large star-to-star scatter in the early Galaxy, while studies of stars at higher metallicities, involving mostly disk stars (Edvardsson et al. 1993 To have an insight on this peculiar behaviour of these elements, it is important to understand how they are formed. The neutron capture is the main mechanism which forms elements heavier than iron; the other mechanism, the p-process, is required for proton-rich isotopes, whose abundances in the solar system is less than 1%. Two major neutron capture mechanisms are generally invoked: the slow process (s-process) and the rapid process(r-process), 7 where the slow and rapid are defined relative to the timescale of β-decay.

THE NEUTRON CAPTURE ELEMENTS
The s-process requires a relatively low neutron density and moves along the valley of β stability. The s-process feeds in particular the elements Sr-Y-Zr, Ba-La-Ce-Pr-Nd and Pb, the three major s-peaks. The reason for the existence of these peaks is that: the neutron capture process imposes certain features on the "spectrum" of the heavy element abundances. For certain neutron numbers N = 50, 82, 126 the neutron capture cross-sections are much smaller than for neighbouring neutron numbers. This means that once one of these "magic" numbers is reached, it becomes significantly less likely for the nucleus to capture more neutrons. These numbers represent a quantum mechanical effect of closed shells, in precisely the same way that closed electron shells produce high chemical stability in the noble gases. Therefore, if the neutron capture process operates in some environment for some finite length of time and then shuts off, we expect a fair number of nuclei to be "stuck" at these "magic" numbers. Elements that correspond to these "magic" numbers of neutrons will thus be especially abundant. The relevant properties necessary for describing the s-process chain are the neutron capture cross section and, in addition, the β-decay rates of those unstable isotopes, which are long-lived enough to allow neutron captures to compete with the β-decay. Käppeler  subtracting the predicted abundance of s-process fraction of each element to the solar system abundance of the same element. As we said, the classical analysis is a phenomenological model but nevertheless, more detailed considerations (see Käppeler et al. 1989) lead to a physical environment characteristic of helium shell-burning zones.
The site of production of the main component of the s-process, accounting for the s-process in the atomic mass number range 90 < A < 208, was shown to occur in the low-mass (1.5-3.0 M ⊙ ) asymptotic giant branch (AGB) stars during recurrent thermal pulses by stellar models (Gallino et al. 1998; Busso et al. 1999). A dredge down of protons can occur during these thermal pulses , as first suggested by Iben and Renzini (1982a, b). These protons move from the hydrogen-rich envelope into the helium zone in low mass AGB stars, due to the operation of a semiconvective mixing. Subsequently, these protons are captured by 12 C in the carbon- rich layer forming a 13 C pocket. This pocket is then engulfed by the growing convective region of the next pulse, releasing neutrons via the 13 C(α, n) 16 O reaction. The s-process mechanism operating in the AGB model depends on the initial stellar metallicity. In fact, although the 13 C pocket, which acts as a neutron producer with the reaction 13 C(α, n) 16 O, is of primary origin 3 in the work of Gallino et al. (1998) and Busso et al. (1999Busso et al. ( , 2001), the ensuing s-process production is dependent on the initial abundance of the Fe-group seeds, i.e. on the stellar metallicity. The neutron exposure (the neutron flux per nucleus seed) is roughly proportional to the number of available neutron sources (the 13 C nuclei) per seed (the iron nuclei), hence inversely proportional to the stellar metallicity. The strong s-component introduced by Clayton & Rassbach (1967) in order to reproduce more than 50% of solar 208 P b, is not necessary in the these stellar models, being naturally obtained in very metal poor stars by metallicity effects, where the iron seeds are very rare. Moreover this scenario provide to feed also chemical elements in the mass number ∼ 90 such Sr, Y and Zr. The 22 N e(α, n) 25 M g reaction plays also a marginal role in the production of neutron capture elements in low mass ABG stars. In fact, this reaction is triggered when the temperature T exceeds 3 · 10 8 K and in low mass stars the maximum temperature at the bottom of thermal pulses, although gradually increasing with the pulse number, barely reaches the mentioned value.
The weak s-component is responsible for the s-process nuclides up to A ≃ 90 and it is recognized as the result of neutron capture during the helium burning core of massive stars.
In this environment, the 22 N e(α, n) 25 M g neutron producer reaction can operate and first studies by Peters (1968) and Lamb et al. (1977) suggested that the lightest s-process nuclei can be provided by this source. Recent studies (Raiteri et al. 1992, Baraffe et al. 1992) discovered a decrease of the production at low metallicity. In fact, the elevated levels of nuclei from Ne to Ca in the He-burning core of a massive star prevent the neutrons to be captured by the relative rare iron seeds. For these reasons, the solar system contribution from weak s-process is less than 10% for Sr and negligible for Y and Zr (Travaglio et al. 2004).
The r-process takes place in extremely neutron-rich environments in which the mean time between two successive neutron captures is very short, compared with the time necessary for the β-decay. For some time (cfr. Truran et al. 1981, Mathews & Cowan 1990) the similar abundance distributions for Z>56 elements in metal deficient stars have been interpreted as evidence for a universal r-process abundance distribution in the early Galaxy. In particular, more recent observations (Sneden et al. , 1998(Sneden et al. , 2000Johnson & Bolte 2001) all seem to confirm this feature, that is generally referred as the "universality" of r-process. Hence, it is generally believed that, at least for Z>56 elements, the astrophysical site and associated yields of r-process nucleosynthesis are unique.
There are, however, some reasons to question the assumption of a universal r-process abundance curve. The material out of which these metal poor stars were formed is likely to have experienced the enrichment by only one or two supernovae before incorporation into the stars. Depending upon which particular progenitor supernova was involved, there might be substantially different abundance distribution curves for these stars, compared to the one represented in Solar-System material, which is a average on many episodes of chemical enrichment (Ishimaru & Wanajo 1999). Recent work by Otsuki et al. (2003) indicates that the coincidence of the observed abundance distribution for 56 < Z < 75 elements with the Solar r-process abundances does not necessarily mean that all r-process events occur in the same universal environment. Moreover, different abundance distributions for Z > 75 and Z < 56 elements are produced even when the universal 56 < Z < 75 abundances are reproduced.
Several scenarios have been proposed for the origin of r-process elements: neutrino winds in core-collapse supernovae (Woosley et al. 1994), the collapse of ONeMg cores resulting from stars with initial masses in the range 8-10M ⊙ (Wanajo et al. 2003) and neutron star mergers ), even if this last scenario seems to be ruled out from the recent work of Argast et al. (2004) at least as the major one responsible for r-process enrichment in our Galaxy. So if the r-process is generally accepted to take place in SNe II explosions (Hill et al. 2002; Cowan et al. 2002), no clear consensus has been achieved and r-process nucleosynthesis still remains uncertain. Theoretical predictions for r-process production still do not exist, with the exception of the results of Wanajo et al. (2003) and Woosley and Hoffmann (1992). However, the results of the model of Wanajo et al. cannot be used in galactic chemical evolution models because they do not take into account fallback (after the SN explosion some material can fall back onto central collapsing neutron star) and so the amount of neutron capture elements produced is probably too high (about 2 orders of magnitude higher than the chemical evolution predictions). Furthermore, Woosley and Hoffmann (1992) have given prescriptions for r-processes only until 107 Ru. In order to shed light on the nature (s-and/or r-processes) of heavy elements such as Ba and Eu one should examine the abundances of these elements in Galactic stars of all metallicities. These abundances can give us clues to interpret their nucleosynthetic origin when compared with detailed chemical evolution models. (1989) and Pagel & Tautvaisiene (1997) suggested that to reproduce the observed behaviour of Ba it is necessary to assume that at early stages Ba is produced as an r-process element by a not well identified range of massive stars. A similar conclusion was reached by Travaglio et al. (1999) who showed that the evolution of Ba cannot be explained by assuming that this element is only an s-process element mainly formed in stars with initial masses 2-4M ⊙ , but an r-process origin for it should be considered. In fact, in the hypothesis of a production of Ba only by s-process, a very late appearance of Ba should be expected, at variance with the observations indicating that Ba is already produced at [Fe/H]=-4.0. They suggested that low mass SNII (from 8 to 10M ⊙ ) could be responsible for the r-component of Ba

S-and r-process elements in dwarf spheroidal galaxies
The proximity and the relative simplicity of the Local Group (LG) dwarf spheroidal (dSph) galaxies make these systems excellent laboratories to test assumptions regarding the nucleosynthesis of chemical elements and theories of galaxy evolution. The Local Group is the group of galaxies that includes the Milky Way. The group comprises about 40 galaxies, with its gravitational center located between the Milky Way and the Andromeda Galaxy (see  The Milky Way is one of three large galaxies belonging to the group of galaxies called the Local Group which also contains several dozen dwarf galaxies. Most of these galaxies are depicted on the map. In chapter 5, we adopt the nucleosynthesis prescriptions for Ba and Eu which are able to reproduce the most recent observed data for our Galaxy, as shown in chapter 2, and we compare the predictions of the models with observational data in 5 dSph galaxies. In this way, it is possible to verify if the assumptions made regarding the nucleosynthesis of Ba and Eu can also fit also the data of local dSph galaxies.
In chapter 6, we use the results of chapter 2 and chapter 5, to compare the predictions of the Milky Way to those of the dSph galaxies. We choose, as typical dwarf spheroidal galaxy, Sculptor. We do not show all the data for all the dwarf spheroidal galaxies because as will be shown in chapter 5, the star formation histories are different and also the chemical evolution is different among the dSph galxies, even if they share common behaviors.
Finally, in chapter 7, the main conclusions of our work are drawn.

Chapter 2
Chemical evolution of neutron capture elements in the solar vicinity ""Man," I cried, "how ignorant art thou in thy pride of wisdom!" by Mary Shelley

Barium and europium
We present the results of a chemical evolution model based on the original two-infall model of Chiappini et al. (1997)

Chemical evolution model for the solar vicinity
We model the formation of the Galaxy assuming two main infall episodes: the first forms the halo and the thick disk, the second the thin disk. The timescale for the formation of the halo-thick disk is ∼ 1Gyr. The timescale for the thin disk is much longer, implying that the infalling gas forming the thin disk comes mainly from the intergalactic medium and not only from the halo (Chiappini et al. 1997). Moreover, the formation of the thin disk is assumed to be a function of the galactocentric distance, leading to an inside out scenario for the Galaxy disk build up (Matteucci & François 1989). In this chapter, all the results shown are for the assumed solar galactocentric distance: 8 kpc. The main characteristic of the two-infall model is an almost independent evolution between the halo and the thin disk (see also Pagel & Tautvaisiene 1995). A threshold in the star formation process (Kennicutt 1989, Martin & Kennicutt 2001) is also adopted. The model well reproduces an extended set of observational constraints both for the solar neighborhood and for the whole disc.
One of the most important observational constraints is represented by the various relations between the abundances of metals (C,N,O,α-elements, iron peak elements) as functions of the where X i (r, t) is the abundance by mass of the element i and Q mi indicates the fraction of mass restored by a star of mass m in the form of the element i, the so-called "production matrix" as originally defined by Talbot and Arnett (1973). We indicate with M L the lightest mass that contributes to the chemical enrichment and it is set at 0.8M ⊙ ; the upper mass limit, M U , is set at 100M ⊙ .
The star formation rate (SFR) ψ(r, t) is defined: ν is the efficiency of the star formation process and is set to be 2Gyr −1 for the Galactic halo (t < 1Gyr) and 1Gyr −1 for the disk (t ≥ 1Gyr). Σ(r, t) is the total surface mass density, Σ(r ⊙ , t) the total surface mass density at the solar position, G gas (r, t) the surface density normalized to the present time total surface mass density in the disk Σ D (r, t Gal ), where t Gal = 14Gyr is the age assumed for the Milky Way and r ⊙ = 8 kpc the solar galactocentric distance (Reid 1993 For φ, the initial mass function (IMF), we use the Scalo (1986) one, constant in time and space, while τ m is the evolutionary lifetime of stars as a function of their mass "m". The SNIa rate has been computed following Greggio & Renzini (1983a) and Matteucci & Greggio (1986) and it is expressed as: where M 2 is the mass of the secondary, M B is the total mass of the binary system, µ = IMF is represented by φ(M B ) and refers to the total mass of the binary system for the computation of the SNeIa rate, f (µ) is the distribution function for the mass fraction of the secondary: with γ = 2; A is the fraction of systems in the appropriate mass range that can give rise to SNeIa events. This quantity is fixed to 0.05 by reproducing the observed SNeIa rate at the present time (Cappellaro et al. 1999). Note that in the case of SNIa the"production matrix" is indicated by Q SN Ia mi because of its different nucleosynthesis contribution (for details see Matteucci and Greggio 1986  where X A i are the abundances of the infalling material, assumed to have a primordial chemical composition , t max = 1Gyr is the time for the maximum infall rate on the thin disk, τ H = 2.0Gyr is the time scale for the formation of the halo thick-disk and τ D is the timescale of the thin disk at the solar galactocentric distance (τ D = 7Gyr). The timescale τ D increases with the galactocentric distance as we will see in chapter 3. The coefficients a(r) and b(r) are constrained by the present day total surface mass density. In particular, b(r) is assumed to be different from zero for t > t max (see Chiappini et al. 2003, for details).

S-process
We have adopted the yields of Busso et al. (2001) in the mass range 1.5-3M ⊙ for the smain component. In this process, the dependence on the metallicity is very important. The s-process elements are made by accretion of neutrons on seed elements (in particular iron) already present in the star. Therefore, this Ba component behaves like a secondary element.
The neutron flux is due to the reaction 13 C(α, n) 16 O which can easily be activated at the low temperature of these stars (see Busso et al. 1999). The yields are shown in Table 2. 1 and Fig. 2.3 as functions of the initial metallicity of the stars. The theoretical results by Busso et al. (2001) suggest negligible europium production in the s-process and therefore we neglected this component in our work. We have added for models 1 and 2 (see Table 2.

R-process
The production of r-process elements is still a challenge for astrophysics and even for nuclear physics, due to the fact that the nuclear properties of thousands of nuclei located between the valley of β stability and the neutron drip line, necessary to correctly compute this process, are ignored. In our models we have tested 6 different nucleosynthesis prescriptions for the r-process Ba and Eu, as shown in Tables 2.2, 2.3 and 2. 4. Some of the prescriptions refer   Moreover, we considered another independent indication for the r-process production of barium; Mazzali and Chugai (1995) observed Ba lines in SN 1987A, which had a progenitor star of 20M ⊙ . These lines of Ba are well reproduced with a overabundance factor f = X obs /X i = 5 (typical metal abundance for LMC X i = (1/2.75)× solar). From this observational data we can derive a X new Ba ∼ 2 · 10 −8 for a 20 M ⊙ star, which is in agreement with our prescriptions.
For Eu we assumed that it is completely due to the r-process and that the yields originate from massive stars in the range 12-30M ⊙ in model 1 and 10-25M ⊙ in model 2, as shown in We have tested prescriptions for Ba and Eu both for a primary production and a secondary production (with a dependence on the metallicity). In the first case the main feature of the yields is a strong enhancement in the mass range 12 − 15M ⊙ (model 1) with no dependence on the metallicity and so the elements are considered as primary elements. In the case of metallicity dependence (model 2), the yield behavior is chosen to have a strong enhancement in the range of metallicity 5 · 10 −7 < Z < 1 · 10 −5 with almost constant yields for Eu and Ba in the whole mass range for a given metallicity.

Iron
For the nucleosynthesis prescriptions of Fe, we adopted those suggested in François et al.  To better fit the new data we have to extend the mass range for the production of the r-processed barium toward higher mass in order to reproduce [Ba/Fe] at lower metallicity.
In Fig. 2 Table 2. 5. For error bars we use the standard deviation (see Table 2  The trend of the data is followed well by both models from low metallicity to solar metallicity. In Table 2.8 we show the predicted solar abundances of Eu and Ba for all our models compared to the solar abundances by Grevesse & Sauval (1998). We also give the predicted s-process fraction in the barium solar abundance. The results of almost all our models are in  Table 2. 6. For error bars we use the standard deviation (see Table 2   in turn affects the Ba production. This fraction of s-process Ba is also lower than the results obtained by means of stellar evolution models (e.g. Arlandini et al. 1999), although different s-process Ba fractions are possible in these models. Mod As model 1 4 As model 1 As model 1 4. 48. · 10 −10 5 As model 1 As model 1 1.86 · 10 −10 6 As model For this reason we believe that the mechanism which produces the observational spread does not affect the ratio of these two elements. We propose that the explanation of the smaller spread in the ratio of [Ba/Eu] is that the site of production of these two elements is the same: the neutronized shell close to the mass cut in a SNII (see Woosley et al. 1994).
What changes could be the amount of the neutronized material that each massive star expels during the SNII explosion. The mass cut and also the ejected neutronized material are still uncertain quantities and usually they are considered as parameters in the nucleosynthesis  Table 2. 6. For error bars we use the standard deviation (see Table 2.6). Model 1: solid line, model 2: long dashed line (Models are described in Table 2.2).  Table 2. 7. For error bars we use the standard deviation (see Table 2.7). Model 1: solid line, model 2: long dashed line (Models are described in Table 2.2).

Upper and lower limit to the r-process production
The purpose of this section is to give upper and lower limits to the yields to reproduce the observed spread at low metallicities for Ba and Eu. An inhomogeneous model would provide better predictions about the dispersion in the [r-process/Fe] ratios if due to yield variations, but it is still useful to study the effect of the yield variations by means of our model.  Table   2.9, where are indicated the adopted yields and the factors by which they have been modified relative to Model 1. In Fig. 2. 11  We can deduce from these upper and lower limit models that the large observed spread could also be due to a different production of heavy elements among massive stars (> 15M ⊙ ). Table 2. 9. The stellar yields for model 1Max and 1Min for barium and europium in massive stars (r-process) in the case of a primary origin.

Model 1Max
Model 1Min This type of stars could produce different amounts of these elements independently of the mass. As we have introduced in the previous subsect. (2. 1.4), it is possible to link this fact with the problems of mass cut and the fallback during the explosion of a SNII. If these elements are produced in a shell close to the iron core of the star, differences in the explosion behaviour can give rise to a different quantity of r-process elements expelled by each star. In this way we are able to explain the presence of the spread for the heavy elements and the absence of the same spread, for example, in the α-elements. The α-elements are produced mostly during the hydrostatic burning of massive stars and then ejected by the explosion.
Another approach can be followed to derive upper and lower limits for the model by changing the yields as functions of metallicity. Model 2, which is the model with yields independent of the mass and depending only on metallicity, will be our test model. In particular, model 2 assumes for the massive stars different yields for Ba and Eu in three ranges of metallicity (see Table 2.2). The new prescriptions for both Ba and Eu are summarized in Table 2.10, where are indicated the new limits for the ranges of metallicities for the model 2min and 2Max. Note that for the model 2Max we assume only two regimes of production of r-process elements as a function of the metallicities and so we distinguish only two ranges of metallicities.
In Fig. 2. 15  Changing the central range of metallicity, in which there is an enhancement of the pro-

Lanthanum
In this Section we study the evolution of La. This element has a nucleosynthesis origin very similar to that of Ba. Therefore we will follow the same method of study that we adopted for Ba.

Observational data
We adopted the very recent results by François (2007  tion are summarized in Table 2.11, in which the mass fraction of newly produced La is given as function of the mass.

Results
We divide the [Fe/H] axis in several bins and we compute the mean and the standard deviations from the mean of the ratios [La/Fe] for all the data inside each bin. These results are shown in Table 2.12, where we also summarize the center and the dimension of each bin and the number of data points contained in each of them. In Fig.2.18 we show the predictions   Unfortunately, for this element there are not so many data. Only recently, even if its solar abundance is about a tenth of the Ba solar abundance, it has been measured because with only one stable isotope, La has more favorable atomic properties than Ba does, making the abundance analysis more straightforward. The predicted trend of La, with the new adopted yields well fits the observational data in the solar neighborhood.

Strontium, zirconium and yttrium
In this Section we use the same methodology used for heavy neutron capture elements to constrain the yields in massive stars for light neutron capture elements, Sr, Zr and Y. Light elements are produced also by weak s-process in massive stars (see 1.2). In this work we do not distinguish between r-process production and weak s-process production and we rather  Table 2. 12. As error bars we consider the standard deviation (see Table 2.12). The solid line is the result of our model for La (see Table 2

Nucleosynthesis prescriptions for Sr, Y and Zr
For the nucleosynthesis prescriptions of s-process we have adopted the yields by Travaglio et For the r-process, we give new prescriptions following the same method adopted for Ba in model 1: we assume an r-process contribution in massive stars (12 -30 M ⊙ ), besides the s-process contribution from low mass stars. The yields are summarized in Table 2.13, in which the mass fraction of newly produced of Sr, Y, Zr is given as function of the mass.

Results
We    (1998), measured in meteorites and the resulting fraction produced by the main s-process in low mass stars, for our model. The Zr has a larger contribution by r-process than Sr and Y.
These have almost the same contribution by s-process. These results are quite different from other chemical evolution models (see Travaglio et al. 2004), and this is probably due to the different treatment of the coupling between halo phase and disc phase.
Sr is the element which presents the larger spread in the data at low metallicities, whereas   Table 2.14. As error bars we consider the standard deviation (see Table 2.14). The solid line is the result of our model for Sr (see Table 2. 13 Table 2. 15. As error bars we consider the standard deviation (see Table 2.15). The solid line is the result of our model for Y (see Table 2. 13 Table 2. 16. As error bars we consider the standard deviation (see Table 2.16). The solid line is the result of our model for Zr (see Table 2. 13), normalized to the solar abundance as measured by Asplund et al. (2005). and Zr show to well fit the trends of the observational data in the solar neighborhood over the whole range of metallicities.

Observational data
In this work we use the data by 4AL for all the studied elemental gradients. We use the data of Daflon & Cuhna (2004)

Chemical evolution model for the Milky Way
In our model, the Galaxy is assumed to have formed by means of two main infall episodes: the first forms the halo and the thick disk, the second the thin disk. The galactic thin disk is approximated by several independent rings, 2 kpc wide, without exchange of matter between them. We assume that the thin disk forms inside-out as first suggested by Matteucci A is also in good agreement with the abundance gradients traced by Cepheids up to ∼ 12 kpc, whereas for larger galactocentric distances this model tends to be systematically below observations. We will show the predictions of this model only for the α-elements. Model C shows a trend similar to model A, whereas model D tends to be below the observations for galactocentric distances greater than 10kpc, so we chose to not show their predictions. We recall that below a critical threshold for the gas surface density (7M ⊙ pc −2 for the thin disk and 4M ⊙ pc −2 for the halo phase) we assume no star formation. In Fig. ( only toward the end of the evolution. We note that at the solar galactocentric distance, which is assumed to be 8 kpc, the threshold also produces a hiatus between the halo phase and the thin disk phase.

Abundance gradients compared with the 4AL data
We used the model described in Sect. 3  for what concern elements as O and S, compared to the ones by Grevesse & Sauval (1998), whereas the differences are negligible for the neutron capture elements.

α-elements (O-Mg-Si-S-Ca)
We plot the results for these elements in  However, the predicted gradients for s-and r-process elements seem flatter than all the others. The reason for this is that they are produced in very restricted stellar mass ranges producing an increase of their abundances at low metallicities until they reach a constant value for [Fe/H] > −3.0 (see Fig. 2.18). The variations between gradients are small but they

Iron peak elements (Sc-Ti-Co-V-Fe-Ni-Zn-Cu-Mn-Cr)
The ten elements of the so-called iron peak are plotted in Figs. 3 Table 3.   shown by the results of the model normalized to the mean value of the bin centered at 8kpc.
A problem is present for Co, for which a too low abundance is predicted by the model at galactocentric distances > 12 kpc.

Neutron capture elements (Sr-Y-Zr-Ba-La-Eu)
As we have seen, neutron capture elements present a large spread at low metallicities. However, since the Cepheids are young metal-rich stars, this problem is not important. As shown in Fig. 3

Predicted abundance gradients compared with other sets of data
We compare the results of our models with different sets of observational data, as described in Sect. 2            The gradients generally tend to steepen with time (see Table 3.5). This behavior is expected in the model taking into account the inside-out formation of the disk, where the external disk regions are still forming now and the abundance gradient is still building up. In fact, at early epochs, the efficiency in the chemical enrichment of the inner Galactic regions is low (owing to the large amount of primordial infalling gas) leading to a flat initial abundance gradient.
Then, at late epochs, while the SFR is still much higher in the central than in the external regions, the infall of metal poor gas is stronger in the outer than in the inner regions, thus steepening the gradients. An important role in this scenario is also played by the gas density threshold. In fact, the gas density threshold, stopping the SFR, tends to slow the chemical enrichment in the outskirts which have a lower gas density compare to the central regions.  for all the considered elements, except Mg, are slightly above the predictions of our model.

The iron peak elements (Ti-Mn-Co-Ni-Fe)
We show the iron peak elements in Fig. 3. 11

The neutron capture elements (Zr-Ba-La-Eu)
We show the neutron capture elements in Fig. 3. 13

Observational data
In this section we consider also the α − elements and not only the r-s process elements.

Way halo
We model the chemical evolution of the halo of the Milky Way for the duration of 1 Gyr. We consider that the halo has formed by means of the assembly of many independent regions each with a typical volume of 10 6 pc 3 . Each region does not interact with the others. Inside each region the mixing is assumed to be instantaneous. In each region we assume an infall episode with a timescale of 1Gyr and a threshold in the gas density for the star formation. We assume a timestep of 1 Myr. When the threshold density is reached, the mass of gas which is transformed at each timestep in to stars, M new stars , is assumed to be proportional to ρ 1.5 gas . We stop the star formation in each region when the total mass of the newly formed stars exceeds   1.5-3M ⊙ . We have assumed that Sr, Y, Zr Ba and La are produced as r-process elements by massive stars but only up to 30 M ⊙ and Eu is also considered to be a purely r-process element produced in the same range of masses.

The ratios of α-elements and neutron capture elements to Fe
We discuss here the results of our simulations compared to the observational data and to      In black line the predictions of the model, in red line the observational data by the observational errors. In the low metallicity range some discrepancies can be noticed.
For instance, the predicted [Si/Fe] distribution is slightly shifted toward higher metallicity ratios compared to observations. Therefore a more careful analysis of nucleosynthesis yields is required (for instance using metallicity dependent Si yields

The ratios [Ba/Eu] and [Ba/Y]
Our model predicts a spread in the abundances ratios if the ratios of the yields of the same elements are different as a function of the stellar mass. This is not the case for what concern the yields of neutron capture elements considered in our nucleosynthesis prescriptions. In  to 8M ⊙ (that we do not take in account in our nucleosynthesis), or by a self enrichment of the observed stars due to dredge-up or to a binary system with an AGB mass transfer to the presently observed companion star (see Aoki et al. 2006 A possible way to explain this spread could be that the r-process yields that we use in our model, are indicative of the mean contribution by r-process to the abundances of these elements. We recall from the introduction of this work, that these two elements are in different peaks both for what concern s-process (not so important at this stage), and for what concern r-process. It could be that, as introduced by Otsuki     is the baryonic initial mass of the galaxy, ν is the star-formation efficiency, w i is the wind efficiency, and n, t and d are the number, time of occurrence and duration of the SF episodes, respectively. to a misleading interpretation and to wrong conclusions regarding the processes and the site of production of the heavy elements analyzed. Therefore, we excluded these stars from our sample, whereas all the other stars were considered and included in the comparisons with the models predictions.

Chemical evolution models for the Local dSph galaxies
We use in this work the same chemical evolution model for dSph galaxies galaxies as described The main features of the model are: • one zone with instantaneous and complete mixing of gas inside this zone; • no instantaneous recycling approximation, i.e. the stellar lifetimes are taken into account; • the evolution of several chemical elements (H, D, He, C, N, O, Mg, Si, S, Ca, Fe, Ba and Eu) is followed in detail; In the scenario adopted in the previous works, the dSph galaxies form through a continuous and fast infall of pristine gas until a mass of ∼ 10 8 M ⊙ is accumulated. One crucial feature in the evolution of these galaxies is the occurrence of galactic winds, which develop when the thermal energy of the gas equates its binding energy (Matteucci & Tornambé 1987).
This quantity is strongly influenced by assumptions concerning the presence and distribution of dark matter (Matteucci 1992). A diffuse (R e /R d =0.1, where R e is the effective radius of the galaxy and R d is the radius of the dark matter core) but massive (M dark /M Lum = 10) dark halo has been assumed for each galaxy.
In table 5.1 we summarize the adopted parameters for the models of dSph galaxies.

Europium
The  The small differences in the SF and wind efficiencies do not affect strongly the predictions of the models. As one can see in Table 1   What should be highlighted is that the nucleosynthesis prescriptions adopted here allow the models to reproduce very well the data, supporting the assumption that Eu, also in dSph galaxies, is a pure r-process element synthesized in massive stars in the range M = 10−30M ⊙ , as it is in the Milky Way (see chapter 2 and Cescutti et al. 2006). Besides that, the low SF efficiencies and the high wind efficiencies are required also to explain the [Eu/Fe] observed pattern, especially the abrupt decrease of the data in some dSph galaxies.

Barium
The    Therefore, in the light of what is said above, can we explain also the differences between the predicted [Ba/Fe] in dSph galaxies and in the Milky Way? Again, the SF efficiency is the major responsible parameter for this difference. In the Milky Way model the SF efficiency is much larger (10 -100 times) than the ones adopted for the dSph galaxies of the sample analyzed here. In the low efficiency regime, the contribution from LMS appears at lower metallicities than in the high SF regime, exactly for the same reason discussed for the [α/Fe] ratios. As a consequence, we predict a longer plateau for the [Ba/Fe] ratio in dSph galaxies than in the solar neighborhood and starting at lower metallicities. This prediction should in the future be confirmed or rejected by more data at low metallicities in dSph galaxies.   The observed trend is very well reproduced by the predicted curves in all four galaxies, especially in the case of Carina and Sculptor (Figures 5. 9 and 5.11, respectively). The abundance pattern of these two galaxies not only exhibits the "plateau" at low metallicities, but also the sudden observed increase of [Ba/Eu] after the onset of the wind, suggesting that the adopted nucleosynthesis prescriptions for both Ba and Eu are appropriate and that the scenario described by the chemical evolution models is suitable to explain the evolution of these galaxies. In the case of Draco and Ursa Minor (Figures 5. 10 and 5.12, respectively), there are no stars with metallicities larger than [Fe/H] ∼ -1.7 dex, the one characteristic for the onset of the galactic wind. Therefore, one cannot verify if this scenario (after the occurrence of the wind) holds also for these systems. However, the "plateau" is very well reproduced,  In Figure 5. 13  In this chapter we use the results of the previous chapters, in particular chapter 2 and chapter 5, to compare the results of the Milky Way to those of the dwarf spheroidal galaxies.
In particular, we choose Sculptor as typical dwarf spheroidal galaxy. We also include Y and La in our discussion, besides Eu and Ba discussed in the previous chapter. We show how in these two different systems, the Milky Way and Sculptor, the results of the chemical evolution predicts different abundance ratios and these results are confirmed by the observational data.
We concentrate on the ratios of neutron capture elements Y-Ba-La-Eu, which are the neutron capture elements measured in Sculptor.

Ratios of neutron capture elements over iron
In Fig. 6 The steepness of the decrease depends on the s-process contribution to each neutron capture elements. The s-process enrichment due to low-mass stars is delayed with respect to the galactic wind time for Sculptor; as a result, the elements with a contribution by the s-process show a smaller decrease of their ratios relative to iron.
In fact, the predictions of the model for the ratio of [Y/Fe], and yttrium has a contribution of s-process , show a smaller decrease than the results for the ratio of europium over iron (see Europium has a negligible production by s-process and the model predicts a very steep decrease for its ratio over iron in Sculptor and this well fits the observational data for this galaxy whereas the data for the Milky Way do not show this so steep decrease at this stage. For barium and lanthanum the results are similar to those of yttrium; this is in agreement with the data of Sculptor (see Fig. 6.3, 6.4), even if in these cases there are not significant differences with the data of the Milky Way.
The two different models, which share the same nucleosynthesis, give different results for the two systems due to the different conditions in which the chemical evolution take place, as the presence of a galactic wind in Sculptor, the different star formation history and the different efficiency for star formation (for others details see chapter 2 and 5     corresponds to different cosmic epochs in the two systems. 6        The comparison between model predictions and observational data showed that our model well reproduces the gradients of almost all the elements that we analyzed. Since abundance gradients can impose strong constraints both on the mechanism of galaxy formation, in particular of the galactic disk, and on the nucleosynthesis prescriptions, we can conclude that: •

Dwarf spheroidal galaxies
In the last part of this work, we have applied the nucleosynthesis prescriptions for the neutron capture elements, derived in chapter 2 for the Milky Way, to five dSph galaxies, in order to verify the assumptions regarding the production of these elements. We have implemented these new nucleosynthesis prescriptions in a chemical evolution model, which was able to reproduce several observational constraints of the Local Group dSph galaxies (such as [α/Fe], present day gas mass, estimated final total mass, metallicity distribution). The • The comparison between Sculptor and the Milky Way permits us to show the different predicted and observed values in these two systems. Although more data for the dSph galaxies are necessary, we can interpret these different ratios at a given metallicity as due to the much less efficient star formation that we need to adopt to reproduce the abundances of dSph galaxies, compared to the Milky Way. In the star formation regime adopted for the dSph galaxies, in fact, the metallicity increases more slowly and the different contributions for the enrichment of neutron capture elements in the ISM appear at lower metallicities than in the Milky Way.
• The dSph galaxies cannot be the building blocks of the Milky Way galaxy, being their chemical evolution and chemical composition very different. This cannot rule out that clumps of gas, progenitors of dSph galaxies in a isolate evolution, merged to our Galaxy at a very early stage; on the other hand, interactions between the Milky Way and dSph galaxies do exist as shown by the galaxy Sagittarius, which is gradually being disrupted by the tidal interaction with our Galaxy, as suggested also by the same chemical signature of Sagittarius and the globular cluster Palomar 12, which is nowadays in the halo of the Galaxy (Sbordone et al. 2006).
• We underline that only more data, possibly extended toward larger metallicities, and a robust statistical basis can confirm definitively our results.  Table 2. 5. For error bars we use the standard deviation (see  Table 2. 6. For error bars we use the standard deviation (see Table 2 Table 2.  Table 2. 7. For error bars we use the standard deviation (see Table   2.7). Model 1: solid line, model 2: long dashed line (Models are described in and for the other observational data (see Sect. 2 in the text, the filled circles).