The formation of the [/Fe] radial gradients in the stars of elliptical galaxies
Aims:The scope of this paper is two-fold: i) to test and improve our previous models of an outside-in formation for the majority of ellipticals in the context of the SN-driven wind scenario, by means of a careful study of gas inflows/outflows; ii) to explain the observed slopes, either positive or negative, in the radial gradient of the mean stellar [/Fe], and their apparent lack of any correlation with all the other observables.
Methods:In order to pursue these goals we present a new class of hydrodynamical simulations for the formation of single elliptical galaxies in which we implement detailed prescriptions for the chemical evolution of H, He, O and Fe.
Results:We find that all the models which predict chemical properties (such as the central mass-weighted abundance ratios, the colours as well as the  gradient) within the observed ranges for a typical elliptical, also exhibit a variety of gradients in the  ratio, in agreement with the observations (namely positive, null or negative). All these models undergo an outside-in formation, in the sense that star formation stops earlier in the outermost than in the innermost regions, owing to the onset of a galactic wind. We find that the predicted variety of the gradients in the  ratio can be explained by physical processes, generally not taken into account in simple chemical evolution models, such as radial flows coupled with different initial conditions for the galactic proto-cloud. The typical  gradients predicted by our models have a slope of -0.3 dex per decade variation in radius, consistent with the mean values of several observational samples. However, we also find a quite extreme model in which this slope is -0.5 dex per decade, thus explaining some recent data on gradients in ellipticals.
Conclusions:We can safely conclude that the history of star formation is fundamental for the creation of abundance gradients in ellipticals but that radial flows with different velocity in conjunction with the duration and efficiency of star formation in different galactic regions are responsible for the gradients in the  ratios.
In this paper we exploit the radial variations in the chemical properties of the Composite Stellar Populations (CSPs) inhabiting elliptical galaxies in order to gain new insights into the mechanism of galaxy formation. Theoretical investigations show, in fact, that these properties strongly vary as a function of either the duration or the intensity of the star formation (SF), as well as of the infall history at each radius. A useful tool to understand the complex issue of galaxy formation is represented by the study of the radial gradients in either the mean abundance ratios or in the mean metallicity in the stars. There is general consensus on the fact that the observed increase of line-strength indices such as the and the (e.g. Carollo et al., 1993; Davies et al., 1993; Trager et al., 2000a) and the reddening of the colours (e.g. Peletier et al. 1990) toward the centre of elliptical galaxies, should be interpreted as an increase of the mean metallicity of the underlying stellar populations. In particular, the existence of possible trends of the gradient slopes with the galactic mass not only could favour a specific galaxy formation scenario, but also it might tell us about the degree of uniformity of this process. Davies et al. (1993) did not find any correlation linking the gradients to the mass or the of the galaxies, whereas Carollo et al. (1993) claimed a bimodal trend with mass, in which the gradient grows with mass below a certain galactic mass () and becomes flatter in more massive ellipticals. On the other hand, Gonzalez Gorgas (1996) found that the gradient correlates better with the central value of than with any other global parameter. Kobayashi & Arimoto (1999) analysed a compilation of data in the literature, finding that the metallicity gradients do not correlate with any physical property of the galaxies, including central indices and velocity dispersion, as well as mass and B magnitude. Finally, in a very recent paper, Ogando et al. (2005) claimed that the relation originally found by Carollo et al. (1993) for low mass ellipticals, might be extended to very massive spheroids (see also Forbes et al. 2005).
From a theoretical point of view, instead, dissipative collapse models (Larson 1974, Carlberg, 1984) predicted quite steep gradients which correlate with galactic mass. Mergers, on the other hand, are expected to dilute the gradients (Kobayashi, 2004). In the framework of chemical evolution models, Martinelli et al. (1998) suggested that gradients can arise as a consequence of a more prolonged SF, and thus stronger chemical enrichment, in the inner zones. In the galactic core, in fact, the potential well is deeper and the supernovae (SN) driven wind develops later relative to the most external regions (see also Carollo et al. 1993). Similar conclusions were found by Pipino & Matteucci (2004, PM04), with a more sophisticated model which takes also into accont the initial infall of gas plus a galactic wind triggered by SN activity. PM04 model predicts a logarithmic slope for indices such as which is very close to typical observed gradients, and, on the average, seems to be independent from the mass of the galaxies.
Gradients in abundance ratios such as the [/Fe] ratio are in principle very important, since we could use them as an clock for the duration of the SF process in that region (see Matteucci & Greggio 1986, Matteucci 1994). However, we will show that the estimate of the relative duration of the star formation process between two different galactic regions with similar mean [/Fe] ratios in their stars (, hereafter), is also affected by either the local SF efficiency or by (differential) metal-enhanced gas flows. This is one of the main novelties of our approach with respect to our previous work. A prediction made by the PM04 best model was that the galaxy should form outside-in with an increase in the  ratio as a function of the radius. To date, only a handful of observational works inferred the gradients in the  ratios from the indices such as and (Melhert et al. 2003, Annibali et al. 2006, Sanchez-Blazquez et al. 2007). These papers show that the slope in the  gradient can be either negative or positive, with a mean value close to zero, and that it does not correlate with galactic properties. In other words, they suggest that there is not a preferred mechanism for the formation of single galaxies, such as either an outside-in or an inside-out mechanism, at work. A drawback of these studies is that their samples are relatively small and the variations in the indices had been often evaluated either well inside one effective radius or by neglecting the galactic core, thus rendering the compilations of the slopes not homogeneous. On the other hand, a few recently observed single galaxies (NGC4697, Mendez et al., 2005; NGC821, Proctor et al., 2005, even though in the latter the authors use an empirical conversion in order to obtain [O/Fe]), seem to support PM04’s predictions, as shown by Pipino, Matteucci & Chiappini (2006, PMC06). PMC06 also stressed the fact the ellipticals are made of composite stellar populations (CSPs) with properties changing with radius; therefore, it cannot be taken for granted that the abundance pattern used to build theoretical SSP and to infer abundance ratios from the line indices really reflect the actual chemical composition of the stars (see also Serra & Trager 2006).
Finally, a limitation of the chemical evolution models is that gas flows cannot be treated with the same detail of a hydrodynamical model. This may affect not only the infall history or the development of the galactic wind, but also hampers an estimate of the role of possible internal flows on the build-up of the gradients.
The aim of this paper is, therefore, manyfold:
i) to test the PM04 prediction of an outside-in formation for the majority of ellipticals in the context of the SN-driven wind scenario by means of a careful study of gas inflows/outflows;
ii) to improve the PM04 formulation by means of a detailed treatment of gas dynamics;
iii) to show how the observed variety of slopes in the  gradients in stars might be related to the different initial conditions and reconciled within a quasi-monolithic formation scenario.
In this sense we complete and supersede the work of Kobayashi (2004), who, with SPH models, studied only the metallicity gradients and found that nearly half of ellipticals have a pure monolithic origin, while the other half had undergone mergers during their life. In order to do that, we couple a simplified chemical evolution scheme with a hydrodynamical code (Bedogni & D’Ercole, 1986; Ciotti et al. 1991) presented in Section 2, whereas our model results will be discussed in Section 3, 4 and 5; we summarise our main conclusions in Section 6.
2 The model
We adopted a one-dimensional hydrodynamical model which follows the time evolution of the density of mass (), momentum () and internal energy () of a galaxy, under the assumption of spherical symmetry. In order to solve the equation of hydrodynamics with source term we made use of the code presented in Ciotti et al. (1991), which is an improved version of the Bedogni & D’Ercole (1986) Eulerian, second-order, upwind integration scheme (see their Appendix), to which we refer the reader for a thorough description of both the set of equations and their solutions. Here we report the gas-dynamical equations:
The parameter is the ratio of the specific heats, and are the gravitational acceleration and the fluid velocity, respectively. The source terms on the r.h.s. of equations (1)–(4) describe the injection of total mass and energy in the gas due to the mass return and energy input from the stars. is the sum of the specific mass return rates from low-mass stars and SNe of both Type II and Ia, respectively. is the injection energy per unit mass due to SN explosions (see Sec. 2.2). is the astration term due to SF. Finally, is the cooling rate per unit volume, where for the cooling law, , we adopt the Sutherland & Dopita (1993) curves. This treatment allows us to implement a self-consistent dependence of the cooling curve on the metallicity (Z) in the present code. We do not allow the gas temperature to drop below K. This assumption does not affect the conclusions.
represents the mass density of the element, and the specific mass return rate for the same element, with . Basically, eq. (2) represents a subsystem of four equations which follow the hydrodynamical evolution of four different ejected elements (namely H, He, O and Fe). We divide the grid in 550 zones 10 pc wide in the innermost regions, and then slightly increasing with a size ratio between adjiacent zones equal to 1.03. This choice allows us to properly sample the galaxies without wasting computational resources on the fraction of the simulated box at distances comparable to the galactic tidal radius (see Sec. 2.3 for its value). At the same time, however, the size of the simulated box is roughly a factor of 10 larger than the stellar tidal radius. This is necessary to avoid that possible perturbations at the boundary affect the galaxy and because we want to have a surrounding medium which acts as a gas reservoir for the models in which we start from an initial flat gas density distribution (see Sec. 2 for the model definition. We adopted a reflecting boundary condition in the center of the grid and allowed for an outflow condition in the outermost point.
At every point of the mesh we allow the SF to occur with the following rate:
where and are the local cooling and free-fall timescales, respectively, whereas is a suitable SF parameter which contains all the uncertainties on the timescales of the SF process that cannot be taken into account in the present modelling and its value is given a priori. In particular, we stress that the adopted parametrization of the SF process might appear simplistic, although it is a rather standard assumption in many galaxy formation simulation where the sub-grid physics cannot be properly modelled. A more detailed representation should at least discriminate between a cold molecular gas phase which is actually feeding the SF process, and the hot surrounding medium where the ejecta from SN are deposited. On the other hand, eq. 5 does not imply that the SF is occurring in the hot gas phase; in fact, we assume that a suitable fraction proportional to the average density in the gridpoint forms stars once it had cooled down. 111Note also that if , namely if the gas is cooling on a very long timescale.
gives the speed of the SF process, whereas the final efficiency, namely the fraction of gas which has eventually turned into stars, is an output of the model.
We assume that the stars do not move from the gridpoint in which they have been formed. We are aware that this can be a limitation of the model, but we prefer this solution than moving the stars in order to match some pre-defined luminosity profile (as done in, e.g., Friaca & Terlevich 1998), because this might artificially affect the resulting metallicity gradients. Moreover, we expect that the stars will spend most of their time close to their apocentre. In order to ensure that we match the observed mass-to-light ratio for the given potential well, we stop the SF in a given grid-point only if the mass density of low-mass stars created at that radius exceeds a given threshold profile. The adopted profile is a King distribution, with core radius of 370 pc and a central stellar mass density of . Integrating over the whole galactic volume, the above mentioned limiting profile yields a total stellar mass of . In the next Section we will show that this assumption does not flaw our simulated galaxies, because the occurrence of a galactic wind, which halts the SF process, coincides with or occurs even earlier than the time at which such a threshold profile is attained.
At the beginning the gas is subject only to the Dark Matter (DM) halo gravity and to its own self-gravity; once the SF begins, the gravitational potential due to the stellar component is self-consistently evaluated.
The DM potential has been evaluated by assuming a distribution inversely proportional to the square of the radius at large distances (see Silich & Tenorio-Tagle 1998). We classify each model according to the size of the DM halo (see next Section). The adopted core radii for the DM distribution, instead, are reported in Table 1.
2.2 Chemical Evolution
We follow the chemical evolution of only four elements, namely H, He, O and Fe. This set of elements is good enough to characterize our simulated elliptical galaxyfrom the chemical evolution point of view. In fact, as shown by the time-delay model (Matteucci & Greggio, 1986, see also PMC06), the [/Fe] ratio is a powerful estimator of the duration of the SF. Moreover, both the predicted [Fe/H]-mass and [Z/H]-mass relationships in the stars can be tested against the observed Colour-Magnitude Relations (hereafter CMRs;e.g. Bower et al. 1992) and Mass-Metallicity relation (hereafter MMR; e.g. Carollo et al. 1993). In order to clarify this point, we recall that the O is the major contributor to the total metallicity, therefore its abundance is a good tracer of the metal abundance Z. However, we stress that we always refer to Z as the sum of the O and Fe mass abundances. On the other hand, the Fe abundance is probably the most commonly used probe of the metal content in stars, therefore it enables a quick comparison between our model predictions and the existing literature. We are aware that in the past literature the majority of the works used Mg as a proxy for the elements, as it can be easily observed in absorption in the optical bands giving rise to the well known and Lick indices. It is worth noticing, however, that the state-of-the-art SSPs libraries (Thomas et al. 2003, Hyun-Chul Lee & Worthey, 2006), are computed as functions of the total -enhancement and of the total metallicity. Moreover latest observational results (Mehlert et al. 2003, Annibali et al. 2006 and Sanchez-Blazquez et al. 2007), have been translated into theoretical ones by means of these SSPs; therefore the above authors provide us with radial gradients in [/Fe], instead of [Mg/Fe]. This is why in this paper we focus on the theoretical evolution of the elements, and the O is by far the most important. In any case, we will also present our predictions in the form of indices and show that we obtain reasonable values in agreement with observations. In fact, we will compare our results to recent observational data which have been transformed into abundance ratios by means of SSPs computed by assuming a global -enhancement. Finally, on the basis of nucleosynthesis calculations, we expect O and Mg to evolve in lockstep. This means that the [O/Fe]=[Mg/Fe]+const equation should hold (in the gas) during galactic evolution (see e.g. Fig. 1 of PM04); therefore the predicted slope of the [/Fe] gradient in the stars should not change if we adopt either O or Mg as a proxy for the s. There might be only an offset in the zero point of, at most, 0.1-0.2 dex which is within both the obseved scatter and the uncertainties of the calibration used to transform Lick indices into abundance ratios.
The nucleosynthetic products enter the mass conservation equations via several source terms, according to their stellar origin. A Salpeter (1955) initial mass function (IMF) constant in time in the range is assumed, since PM04 and PMC06 showed that the majority of the photochemical properties of an elliptical galaxy can be reproduced with this choice for the IMF. We adopted the yields from Iwamoto et al. (1999, and references therein) for both SNIa and SNII. The SNIa rate for a SSP formed at a given radius is calculated assuming the single degenerate scenario and the Matteucci & Recchi (2001) Delay Time Distribution (DTD). The convolution of this DTD with over the galactic volume, gives the total SNIa rate, according to the following equation (see Greggio 2005):
where is the fraction of binary systems which give rise to Type Ia SNe. Here we will assume it constant (see Matteucci et al. 2006 for a more detailed discussion). The time is the delay time defined in the range so that:
where is the minimum delay time for the occurrence of Type Ia SNe, in other words the time at which the first SNe Ia start occurring. We assume, for this new formulation of the SNIa rate that is the lifetime of a 8, while for , which is the maximum delay time, we assume the lifetime of a . The DTD gives the likelihood that at a given time a binary system will explode as a SNIa. Finally, is the number of stars per unit mass in a stellar generation and contains the IMF.
According to the adopted model progenitor and nucleosynthetic yields, each SNIa explosion releases erg of energy and of mass (out of which of O and of Fe, respectively). For the sake of simplicity, we assume that the progenitor of every SNII is a typical average (in the range ) massive star of , which pollutes the ISM with of ejecta during the explosion (out of which of O and of Fe, respectively). We recall that single low- and intermediate-mass stars do not contribute to the production of either Fe or O. We neglect the fact that they may lock some heavy elements present in the gas out of which they formed, and restore them on very long timescales; therefore single low- and intermediate-mass stars are only responsible for the ejection of H and He. Such a simplified scheme has been also tested with our chemical evolution code (PM04, their model IIb); it leads to relative changes smaller than the 10% in the predicted abundance ratios with respect to the ones predicted with the full solution of the chemical evolution equations.
These quantities, as well as the evolution of single low and intermediate mass stars, had been evaluated by adopting the stellar lifetimes given by Padovani & Matteucci (1993). The solar abundances are taken from Asplund et al. (2005).
We recall that in order to study the mean properties of the stellar component in ellipticals, we need average quantities related to the mean abundance pattern of the stars, which, in turn can allow a comparison with the observed integrated spectra. To this scope, we recall that, at a given radius, both real and model galaxies are made of a Composite Stellar Population (CSP), namely a mixture of several SSPs, differing in age and chemical composition according to the galactic chemical enrichment history, weighted with the SF rate. On the other hand, the line-strength indices are usually tabulated only for SSPs as functions of their age, metallicity and (possibly) -enhancement.
In particular we make use of the mass-weighted mean stellar metallicity as defined by Pagel & Patchett (1975, see also Matteucci 1994):
where is the total mass of stars ever born contributing to the light at the present time and Z is the metal abundance (by mass) in the gas out of which an amout of stars formed. In practice, we make use of the stellar mass distribution as a function of Z in order to derive the mean metallicity in stars.
One can further adapt eq. 8 in order to calculate the mean O/Fe ratio in stars. In this case, however, we make use of the stellar mass distribution as a function of O/Fe. Therefore we obtain:
where now in the abundance ratio characterising the gas out of which a mass of stars formed. This procedure will be repeated at each grid-point unless specified otherwise.
Then, we derive , taking the logarithm after the average evaluation (see Gibson, 1996). Similar equations hold for  and the global metallicity .
Another way to estimate the average composition of a CSP which is closer to the actual observational value is to use the V-luminosity weighted abundances (which will be denoted as ). Following Arimoto & Yoshii (1987), we have:
where is the number of stars binned in the interval centered around with V-band luminosity . Generally the mass averaged [Fe/H] and [Z/H] are slightly larger than the luminosity averaged ones, except for large galaxies (see Yoshii & Arimoto, 1987, Matteucci et al., 1998). However there might be differences between the two methods at large radii, as far as [Fe/H] and [Z/H] are concerned. In fact, the preliminary analysis of PMC06 showed that both distributions may be broad and asymmetric and their mean values can provide a poor estimate of the metallicity in complex systems with a chemical evolution history quite extended in time. On the other hand, PMC06 found the [Mg/Fe] distribution to be much more symmetric and narrow than the [Z/H] distribution. Therefore, we expect that at any radius and hence, we present mass-weighted values which are more representative of the physical processes acting inside the galaxy. After PMC06, we will present our results in terms of and , because the luminosity-weighted mean is much closer to the actual observations and might differ from the average on the mass, unless otherwise stated.
Finally, in order to convert the predicted abundances for a CSP into indices (especially in the case of short burst of SF), it is typically assumed that a SSP with a mean metallicity is representative of the whole galaxy. In other words, we use the predicted abundance ratios in stars for our CSPs to derive the line-strenght indices for our model galaxies by selecting a SSP with the same values for and from the compilation of Thomas, Maraston & Bender (2003, TMB03 hereafter).
2.3 Model description
The present work is aimed at understanding the origin of the radial gradients in the stars by means of models which have photochemical properties as well as radii comparable with those of typical massive ellipticals. Moreover, we would like to understand what causes the gradient slope to span the range of values dex per decade in radius. In order to do that, we will essentially vary the initial conditions by adopting reasonable hypotheses for the gas properties. A first classification of our set of models can be done according to their initial conditions (DM halo mass and available reservoir of gas):
Model M: a DM halo and of gas
Model L: a DM halo and of gas
These quantitites have been choosen in order to ensure a final ratio between the mass of baryons in stars and the mass of the DM halo around 0.1. Models by Matteucci (1992) and PM04 require such a ratio for ellipticals in order to develop a galactic wind. A more refined treatment of the link between baryons and DM is beyond the scope of this work, and a more robust study of the gradient creation in a cosmological motivated framework will be the topic of a forthcoming paper. The exact initial gas mass depends on the initial conditions and it is clear that gas can be accreted by the external environment. In particular, for each model we considered the following cases for the initial gas distribution:
isothermal density profile. In this case, the gas is assumed to start from an isothermal configuration of equilibrium within the galactic (i.e. considering both DM and gas) potential well. The actual initial temperature is lower than the virial temperature, in order to induce the gas to collapse. These initial conditions might not be justified by the current Cold DM paradigm for the formation of structures. However, we consider them very useful because they give the closest approximation of the typical initial conditions adopted by the chemical evolution models to which we will compare our results. The reader can visualise this model as an extreme case in which we let all the gas be accreted before the SF starts
constant density profile. In this case the gas has an initial value for the mass density which is constant with radius in the whole computational box (c.f.sec 2.1). The DM and, afterwards, the gas and stellar gravity will then create the conditions for a radial inflow to happen.This case might be more realistic than the former one, in the sense that the DM potential will “perturb” the gas which is uniformely distributed at the beginning of the simulation. At variance with the previous model, in this case we let the SF process start at the same time at which the gas accretion starts.
Table 1 summarises the main properties of each model that will be discussed in this paper, namely the core radius for both the DM and the gas profile, the SF parameter , the initial temperature and the SN efficiency respectively.
Concerning the class of models labelled a, we mainly vary the gas temperature and the parameter of star formation. We do not vary the gas mass (via the core gas density and radius) because we need that precise amount of gas in order to ensure that: i) enough stars can be created; ii) at the same time there is not too much gas left (we recall that present-day ellipticals are basically without gas). Also, the assumed profile guarantees the most of the gas is already within the final effective radius of the galaxy in a way which mimick the assumptions made in PM04 and PMC06.
For the class of models labelled b, instead, the initial gas density (as reported in Table 1 under the column pertaining ) can be a crucial parameter, as well as the gas temperature and . Here the values for is chosen in order to have the initial gas content in the whole grid not higher than the typical baryon fraction in high density environment (i.e. 1/5-1/10 as in galaxy cluster, e.g. McCarthy et al. 2007). In each case, the gas temperature ranges from K (cold-warm gas) to K (virialised haloes). We limit both the DM and the stellar profile to their tidal radii, chosen to be 66 kpc (both of them) in case M as well as 200 kpc and 100 kpc, respectively, in case L. These values are consistent with the radii of the X-ray haloes surrounding ellipticals of the same mass.
3 Results: a general overview
The main results of our models are presented in Table 2, where the final (i.e. after SF stops) values for the stellar core and effective radii, the time for the onset of the galactic wind in the central region (), the abundance ratios in the galactic center and the gradients in  and , are reported. In particular, we choose as the radius which contains 1/2 of the stellar mass of the galaxy and, therefore, it is directly comparable with the observed effective radius, whereas is the radius encompassing 1/10 of the galactic stellar mass. In most cases, this radius will correspond to , which is the typical size of the aperture used in many observational works to measure the abundances in the innermost part of ellipticals. We did not fix a priori, in order to have a more meaningful quantity, which may carry information on the actual simulated stellar profile. Finally, we did use the following notation for the metallicity gradients in stars ; a similar expression applies for both the  and the  ratios.
The slope is calculated by a linear regression between the core and the half-mass radius, unless otherwise stated. Clearly, deviations from linearity can affect the actual slope at intermediate radii. Before discussing in detail the galactic formation mechanism of our models, we must check whether they resemble typical ellipticals for a given mass. First of all, we have to ensure that the MMR is satisfied. The majority of our model galaxies exhibits a central mean values of  within the range inferred from integrated spectra, namely from -0.8 to 0.3 dex (Kobayashi & Arimoto 1999). On average, the more massive galaxies have a higher metal content than the lower mass ones. However, the small range in the final stellar masses as well as the limited number of cases presented here prevent us from considering our models as a complete subsample of typical ellipticals drawn according to some galactic mass function. Here we simply check whether our models fullfill the constraints set the MMR and the CMR for a galaxy of .
For instance, we applied the Jimenez et al. (1998) photometric code to both cases Ma1 and La (inside their effective radius), and found the results in good agreement with the classic Bower et al. (1992) CMRs. In fact, by assuming an age of 12.3 Gyr (which in a standard Lambda CDM cosmology means a formation redshift of 5), we have mag, U-V=1.35 mag, V-K=2.94 mag and J-K=0.97 mag for model Ma1, whereas for the case La we predict mag, U-V=1.28 mag, V-K=3.17 mag and J-K=1.06 mag. It can be shown that similar results apply to all the other cases, because their star formation histories as well as their mean metallicity are roughly the same. It is known, in fact, that broad-band colours can hardly discriminate the details of a SF episode if this burst occurred long ago in the past.
The models show an average  = 0.2 - 0.3 as requested by the observations (Worthey et al. 1992, Thomas et al. 2002, Nelan et al. 2005). In general, the predicted abundance ratios are consistent with the reported dex-wide observational scatter of the above mentioned articles, with the exception of a few cases which will be discussed in the following sections.
On the other hand, several models (not presented here) matching the chemical properties fail in fitting other observational constraints. As an example, here we report model Mb5, whose stellar core radius is by far too large to be taken into account in the remainder of the paper.
Model MaSN, instead, shows how a strong feedback from SN can suppress the SF process too early, as testified by the high predicted -enhancement in the galactic core. Also in this case the galaxy is too diffuse. It can be shown that in the range 0.1-0.2 does not lead to strong variations in the results. Therefore, we adopt = 0.1, in line with the calculations by Thornton et al. (1998).
In all the other cases, the dimension of the model galaxies (i.e. their effective radii) are consistent with the values reported for bright ellipticals (e.g. Graham et al. 1996).
We stress that here we are not interested in a further fine tuning of the input parameters in order to reproduce the typical average elliptical as in PM04. Our aim is, instead, to understand whether it is possible to explain the observed variety of  gradient slopes once all the above constraints have been satisfied. In order to do this we first examine the formation of the stellar component of a typical elliptical galaxy. Then we derive further constraints by comparing both the predicted abundance and line-strength indices gradients with observations. Finally, we study in great detail the role of several factors in shaping the  gradients.
Values predicted after the SF has finished.
3.1 The outside-in formation of a typical elliptical
3.1.1 The gas-dynamical evolution
In this section we focus on the formation mechanism of a single galaxy: the time evolution of its abundance gradients will be the subject of Sec. 4.1 . A clear example of a massive elliptical is given by model La (massive elliptical with the gas in intial equilibrium at K and =10), whose chemo-dynamical evolution is shown in Figs. 1,3-6. We will refer to this particular model as a reference case for characterizing the hydrodynamical behaviour of our models, as well as to derive general hints on both the development of the metallicity gradients and the SF process. We will also compare the results of models La with those of models Lb, being the main difference between the two models in the initial gas distribution. Fig. 1 shows the stellar and the gas density profiles (upper panels) as well as the gas velocity and the temperature profiles (lower panels) at different times (see captions and labels). It can be clearly seen that at times earlier than 300 Myr the gas is still accumulating in the central regions where the density increases by several orders of magnitude, with a uniform speed across the galaxy. The temperature drops due to cooling, and the SF can proceed at a very high rate (). In the first 100 Myr the outermost regions are built-up, whereas the galaxy is still forming stars inside its effective radius. For comparison, the thick solid line in the star density panel shows the adopted threshold (King profile). We show the evolution predicted by model Lb (similar to La, but with an initial accretion of gas) in Fig. 2. We notice, that, despite the different initial conditions, the evolution of all the physically interesting quantities follows the results obtained for model La.
After 400 Myr, the gas speed becomes positive (i.e. outflowing gas) at large radii, and at a 500 Myr almost the entire galaxy is experiencing a galactic wind. This model proves that a massive galaxy can undergo a galactic wind, which develops outside-in, thanks to the sole energy input from SNIa+II. The wind is supersonic for, at least, the first Gyr after ,which is the time of the onset of the galactic wind and depends on the model assumptions. At roughly 1.2 Gyr, the amount of gas left inside the galaxy is below 2% of the stellar mass. This gas is really hot (around 1 keV) and still flowing outside. Therefore, as anticipated also by our chemical evolution studies (Pipino et al. 2002, PM04, Pipino et al. 2005), a model with Salpeter IMF and a value for can mantain a strong galactic wind for several Gyr, thus contributing to the ejection of the chemical elements into the surrounding medium.
The fact that the galactic wind occurs before externally than internally is simply due to the fact that the work to extract the gas from the outskirts is smaller than the work to extract the gas from the center of the galaxy. Therefore, since the galactic wind occurs first in the outer regions the star formation rate stops first in these regions, for lack of gas. In the following we will refer to the outside-in scenario as to the fact that the SFR halts before outside than inside due to the progressive occurrence of the galactic wind from outside to inside.
3.1.2 Chemical abundances: from the gas to the stars
In Fig. 3 we show the temporal evolution of the elemental abundances in the gas for the entire galactic volume. As expected, the prompt release of O by SNII makes the [O/H] in the gas to rise very quickly, whereas the Fe enrichment is delayed. As a result, the [O/Fe] ratio spans nearly two orders of magnitude, reaching the typical value set by the SNIa yields after 500 Myr. We can derive much more information from Figs. 4, where the metallicity distribution of stars as a function of  and  are shown. In these figures we plot the distribution of stars formed out of gas with a given chemical pattern (i.e. a given  and ) as contours in the - plane. In particular, the contours connect regions of the plane with the same mass fraction of stars. Since we consider the stars born in different points of the grid, which may have undergone different chemical evolution histories, it is useful to focus on two different regions: one limited to (upper panel) and the other extending to (lower panel). It is reassuring that in both panels the overall trend of the  versus  in the stars agrees with the theoretical plot of [O/Fe] versus [Fe/H] in the gas expected from the time-delay model (Matteucci & Greggio 1986). For comparison, we plot the output of PM04’s best model with roughly the same stellar mass as a dot-dashed line in fig. 4. Both the early and final stages of the evolution coincide. An obvious difference is that the knee in the [O/Fe] vs [Fe/H] relation predicted by our model is much more evident than the one of PM04. The reason must be ascribed to the fact that here we adopt a fixed O/Fe ratio in the ejecta of SNII, whereas the stellar yields show that there is a small dependence on the progenitor mass (which is taken into account in detailed chemical evolution models as the PM04 one). Moreover, as we will show in Sec. 5.1, most of the metals locked-up in the stars of the galactic core were produced outside the core. In practice, we anticipate that the inner regions suffer a metal-rich initial infall (i.e. inflowing gas has a higher [Fe/H] abundance with respect to the gas already present and processed in the inner regions), therefore the number of stars formed at is very small compared to number of stars created at very high metallicities. This fast increase of the [Fe/H] ratio in the gas also makes the knee of the upper panel of Fig. 4 more evident than the one in the lower panel. 222The physical mechanisms which produce such a metal-enhanced internal gas flows, as well as their role in changing the [O/Fe] ratio in the gas, will be discussed in great detail in Sec. 5.
The above results have two implications: first, the fact that our implementation of the chemical elements in the hydrodynamical code does not produce spurious chemical effects and it has been done in the proper way. Second, and perhaps more important, it shows that a chemical evolution model gives accurate predictions on the behaviour of the mean values, even though it does not include the treatment of gas radial flows and it has a coarser spatial resolution. As expected from the preliminary analysis of PMC06, the innermost zone (Fig. 4, upper panel) exhibits less scatter. At larger radii, the distribution broadens and the asymmetry in the contours increases. This can be more clearly seen in the classical G-dwarf-like diagram of Fig. 5, where the number of stars per [Fe/H] bin only is shown. We can explain the smooth early rise in the [Fe/H]-distribution in the inner part (solid line) as the effect of the initially infalling gas, whereas the sharp truncation at high metallicities is the first direct evidence of a sudden and strong wind which stopped the star formation. The suggested outside-in formation process reflects in a more asymmetric shape of the G-dwarf diagram at larger radii (dashed line), where the galactic wind occurs earlier (i.e. closer to the peak of the star formation rate), with respect to the galactic centre. The broadening of the curves, instead, reflects the fact that the outer zone (extending to ) encloses several shells with different SF as well as gas dynamical histories.
In practice, the adopted  and  are either the mass or the luminosity weighted values, taken from the distributions similar to the one of fig. 5 (but in linear scale) according to eq. 9 and 10 . They can be compared with SSP-equivalent values inferred from the observed spectra taken from the integrated light (see next Section). These quantities tell us that, models La and Lb exhibit a quite high  in the stars of the galactic core, although model Lb is in slightly better agreement with the observed central values of  (Carollo et al. 1993, Mehlert et al. 2003, Sanchez-Blazquez et al. 2006) than model La.
4 The formation of the abundance gradients
4.1 The temporal evolution of the gradients in the reference case
In this section we discuss the issue of radial gradients in the stellar abundance ratios. We concentrate on the actual gradients, namely on the ones whose properties can be measured by an observer. A snapshot of model La after 100 Myr, reveals gradients already in place with slopes (Fig. 7) and (luminosity-weighted, upper panel of Fig. 6). After the SF has been completed, we have and , respectively. Both values are consistent with the predictions by PM04. In the same time interval, and decrease by a factor of 3 and 1.5, respectively. The changes in these quantities are more evident if we look at other models such as Ma1, where the final and are smaller by a factor of 5 and 2 than the initial ones, respectively. In this case, however, the slope in the  changes more smoothly from -0.024 to 0.02, whereas the steepening in the Fe gradient (from 0.48 to -0.13) is more dramatic.
In conclusion, both models Ma1 and La experience an outside-in formation process, which creates the abundance gradients, within the observed range, although with different slopes. At this stage we can say that the galactic winds certainly play a role in the gradients build-up. The temporal evolution of the gradients for model La can be visualized in Fig. 6, where the mass-weighted values for the  are also displayed in the bottom panel. As expected from the analysis of PMC06, mass-weighted values might differ from luminosity-weighted quantities with increasing galactocentric radii, owing to the well-know strong metallicity dependence of the light in the optical bands. In this particular case, we predict either a quite flat gradient, when the mass-weighted values are taken into account. This happens because also at large radii there is a significant number of very metal-rich stars, even though the peak of the stellar metallicity distribution (see Fig. 5) occurs at a lower [Fe/H] with respect to the core. There are many concurring effects which generate this apparent dichotomy between peak values and averages. First of all, we remind the reader that the stellar metallicity distributions are generally asymmetric, thus the mathematical average does not coincide with the distribution’s mode (i.e. the peak value, see PMC06). Secondly, the integral in eq. 9 is performed by taking into account a linear sampling of star mass in Fe/H bins (instead of [Fe/H]). In other terms, is always higher than (see Gibson 1997). Therefore we stress that taking the observed (i.e. luminosity-weighted) gradients at their face values, it might not necessarily reflect the actual galaxy formation process. Moreover all these subtle differences in the choice of a SSP-equivalent value (either or or simply ) might lead to different final value for our gradients.
In order to guide the eye, in the upper panel of figure 6 the solid lines represent a linear regression fit of the mean (luminosity weighted) abundances, at each time, at the core and at the effective radius. With this example we want to give a warning: if an observer measures the abundance at both and and then tries to infer a metallicity gradient by a linear regression (i.e. a straight line of slope ), the difference between its findings and the actual behaviour of  versus the radius can be large.
By means of these models we have shown that a 10% SN efficiency, as adopted in purely chemical models (PM04, PMC06, Martinelli et al. 1998), is supported also by hydrodynamical models. In passing, we note that models with 100% SN efficiency (e.g. MaSN) undergo the galactic wind too early in their evolution, thus implying ithat their chemical properties are at variance with observations.
4.2 Gradients in Fe/H and total metallicity
We find a radially decreasing luminosity-weighted Fe abundance in all our models: spans the range -0.5 – -0.2 dex per decade in radius, with a mean value of -0.25, in good agreement with the analysis of Kobayashi & Arimoto (1999). Once transformed into observables by means of 12 Gyr old TMB03 SSPs, the predicted gradient slopes are mag per decade in radius, again in agreement with the typical mean values measured for ellipticals by several authors and confirming the PM04 best model predictions. We notice that for models such as Mb3 and Ma2, we obtain mag per decade in radius, possibly matching a few objects in the sample of Ogando et al. (2005, see also Baes et al. 2007). This conclusion is strenghtened by the fact that also the total metalliticy gradients, are similar among all the models, their slopes typically being dex per decade in radius, in agreement with the average value of the Annibali et al. (2006) sample, with the remarkable exception of model Ma2 (an average elliptical with the gas initially in equilibrium at K - as well as =10) whose slope of =-0.42 dex per decade in radius is close to the largest gradients observed in the galaxies in the sample of Ogando et al. (2005).
The build-up of such gradients can be explained to the non-negligible role of the galactic wind, which occurs later in the central regions, thus allowing a larger chemical enrichment with respect to the galactic outskirts. The predicted gradient slopes are independent from the choice of the intial setup given by either case a or b. We are conscious, however, that we relaxed the PM04 hypothesis of not-interacting shells; therefore, in the rest of the paper we will also highlight the role of the metal flows toward the center.
4.3 Gradients in O/Fe
Recent papers as Mehlert et al. (2003), Annibali et al. (2006) and Sanchez-Blazquez et al. (2007) have shown a complex observational situation relative to abundance gradients, especially the gradients of the [/Fe] ratio. A successful galactic model should be able to reproduce the [/Fe] radial stellar gradient, either if flat or negative, while keeping fixed all the other properties (including the  gradient). This is nearly impossible with standard chemical evolution codes, unless by using extreme assumptions which may worsen the fit of all the other observables.
The hydro-code presented in this paper helps us in tackling this issue. From the entries in Table 2, in fact, we notice the all the objects which present reasonable values for their chemical properties, including the  gradient, show a variety of gradients in the [/Fe] ratio, either positive or negative, and one model shows no gradient at all (Mb2, namely an average elliptical with the gas initially diffuse and cold - K - as well as =10).
A comparison between some of our models and data drawn from Annibali et al. (2006) paper (namely a subsample of only massive ellipticals with homogeneously measured gradients out to ) is made in fig. 8. Also the data for NGC 4697 are reported. The models predict a relationship between the abundance ratios and the radius which is not linear. This further complicates the comparison with observations and will be the subject of a future paper. As an example we only notice that the Annibali et al. (2006) sample is limited to ; therefore, it is not surprising that their mean slopes are smaller than expected if one takes into account the whole region . However the agreement with our model is very good, when considering the same galactic regions, see fig. 8
As expected from this comparison, the predicted values for span a range from -0.2 to + 0.3, which is similar to the observed one (e.g. Mehlert et al. 2003), whith an average gradient slope of -0.002 dex per decade in radius.
Remarkably, this occurs in spite of the fact that the galaxy formation process always proceeds outside-in.
No correlations between and other galactic properties are found, as expected from observations. We only notice that the galaxies showing the steepest (both positive and negative)  gradient slopes, have also a quite strong radial decrease in the  ratio, although a quantitative confirmation needs a sample statistically richer than ours. A correlation in this sense seems to emerge by the Annibali et al. (2006) data (Annibali, private communication).
For instance, in the case of model Ma2, we predict , but it has basically the same final stellar mass of both models La and Lb, being only more compact, and it shows average abundance ratios in stars matching the typical mean values observed for massive ellipticals. On the other hand, model Mb4 (as Ma1, the only difference is that the gas is diffuse at the beginning) has a gradient of and model Mb1 predicts . We stress again that all these models halt the SF at earlier than in the core.
5 Possible explanations for the observed variety of  radial gradient slopes
We have analysed the possible causes for the variety of the predicted gradients in the abundance ratios: metal-enhanced radial flows and variable timescale of SF with radius. Here we disentangle their different roles by studying the effects of the gas flows in determining the central values (i.e. the ), whereas the variation of the SF timescale along the radius will be mainly linked to the gradient slope. We stress that the results we present here and their interpretation is valid for the particular initial conditions that we explored. Therefore, the intial set-up of the simulations is a sufficient condition for such a variety of gradient to be created.
5.1 Radial gas flows
In order to understand the differences - both observed and predicted - in the [/Fe] gradients among ellipticals, we first study the gas composition in a sphere of radius at each time-step. In this way we can have insights on the role of the gas flows in the determination of .
Almost all the models predict that, after the first 100 Myr, a substantial fraction (i.e. 80-90%) of the metals present in the gas inside has an external (i.e. ) origin. This means that a non-negligible contribution to the gradients is due to the gas flows, as shown by the negative velocity field for in Figs. 1 and 2, and this is also expected in dissipative models such as the Larson (1974) and the Carlberg (1984) ones. This effect cannot be seen in standard chemical evolution models with non-interacting shells, where, at a fixed mass, the predicted is always smaller than in the models presented in this paper (e.g. see Table 5 of PM04).
In order to quantify the effects of the convolution of the SF with the gas flows, we make a step further and use eq. 9 in order to define, for a given chemical element, the mass-weighted ratio between the mass of this element produced in the galactic core and locked-up in stars to the amount produced in a more external region (and subsequently locked-up in stars inhabiting the core). In particular, for O we define the quantity as:
where, at variance with eq. 9, we now consider the distribution of stars as function of the ratio and extract its average. In this case is the mass of O produced in the external (i.e. outside ) part of the galaxies, sunk in the galactic centre because of the radial inflows and eventually locked-up in stars inhabiting the core 333We only subtract from the metal budget, a posteriori, those elements not produced in situ and followed in their evolution by means of a suitable tracer. On the other hand, O is the actual mass of oxygen out of which stars form inside a sphere of radius . A high efficiency of the radial flows in transferring O from the external regions of the galaxy will correspond to high values of the ratio . On the other hand, means that all the stars formed in the core incorporated only the O produced by the previous generations which populated the core. We also evaluate the same ratio in the case of the Fe, namely . Both the and the time evolutions for four selected models are shown in Table 3. These quantities give an estimate of the contribution of the metal rich radial flows to the build-up of the gradients. The last row of Table 3 shows the quantity , namely the expected central value of the  in the hypothetical case in which the metals produced outside do not flow into the core (to be compared to the entries of Table 2, 6th column).
Remarkably the 3/4 of the models have basically =0.3.
In model Ma1, the mild positive is not enhanced by the metal rich gas produced in the ouskirts and flowing toward the center because and evolve in lockstep because they are dominated by the external production of metals in the same way (see Table 3).
In other cases, such as the model Mb3 (average elliptical with the gas initially diffuse and hot - K - as well as =10), instead, we have . This model, in fact, starts from a uniform gas distribution, then most of the gas out of which the stars form, must have first sunk into the centre. As a consequence, the star formation rate peaks later with respect to model Ma1. On the other hand, the outermost regions halt their star formation process, and thus the O production, quite soon; therefore, most of the stars in the central regions preferentially lock the Fe which is coming from the SNIa exploding in the outskirst, rather than O. This explains the slightly low central , despite the high SF parameter (); in fact, is 0.18 at 100 Myr, 0 at 240 Myr and becomes negative soon after.
As anticipated above, due to the very fast star formation rate (with respect to the gas inflow rate) in model Lb we have the lowest and , therefore the gradient could reflect the real outside-in formation in a manner which resembles the multi-zone chemical evolution models with non-interacting shells. Nevertheless, also in this case, we have , which is the outcome of a differential inflow, as explained above for model Mb3.
In summary, radial flows may lower the core value of , (that we consider as the zero point of the gradient in [/Fe]) relative to the case with no radial flows. The reason for that is that -depleted material flows from the outermost into the innermost regions. Therefore, the variety of gradients (in particular positive, null or negative) depends on the efficiency of the -depleted gas to flow from the outside to the inside during the time of active star formation. In other words, it depends on the velocity of the inflowing gas. Clearly a larger or smaller parameter of SF can have a strong influence on this process. In order to help the visualization of such a complex process, we show Fig.9 where the solid line at the top represents a hypothetical pure outside-in model with non-interacting shells and =0.3. The gradient slope is chosen to be 0.15 dex per decade in radius. In this case the  gradient is set by the occurrence of the galactic wind, which happens earlier in the outermost regions. By no means such a model is real. It just helps in visualising the simplest scenario - which is quite a common assumption in the literature involving multi-zone chemical evolution modelling - and the differences introduced by taking into account radial flows and the local variation in the input star formation timescale. None of the models run correspond to this ideal case, therefore we cannot compare it with any of our predicted curves.
In order to take into account the role of the gas flows we then correct the predicted gradient (solid line in the middle), thus obtaining something similar to the predictions by models La (dashed line) and Lb. This mechanism helps also in explaining the value for  predicted by other models, such as Mb3 (dotted line).
5.2 The role of the star formation timescale at different radii
Let us examine now the effect of varying the SF timescale. The analysis of the mass-weighted abundance ratio in the inner zone is not enough to explain the gradients. In fact we have studied only the build-up of the zero-point value, taken as the quantity . Even in the simplistic assumption in which the gradient can be well represented by a straight line we need another quantity in order to fix the slope steepness. We chose to study the radial variation of and , because another important difference with respect to PM04 and PMC06 is that here
We find that in a model such as Lb, which closely follows the PM04 best model. On the other hand, models with either a zero or a negative have for most of their evolution, thus favouring a higher [O/Fe] ratio in the stars belonging to the inner regions. Other test cases, not presented here, show us that if we run models with even higher values for the star formation parameter (i.e. ,) the strong feedback by SNe halts the gas flows; therefore the supply of baryons for the SF in the galactic center is strongly reduced and the outcome is a too diffuse galaxy. A similar result can be obtained by increasing , as shown by model MaSN.
The radial variation of means that the effect of the outside-in formation could be balanced by the interplay between local differences in the SF timescale and differential gas flows. Therefore the combined effect of gas flows plus a strong variation in the star formation timescale along the radius, make the hypothetical outside-in model gradient change slope (line labelled as fake inside-out model in Fig.9), thus matching the average trend predicted by model Mb3 (dotted).
In general, seems to fluctuate around a null value and to be a result of the interplay of many hydrodynamical factors, which render it more sensitive to the initial conditions of the gas rather than an indicator of the chemical enrichment process. Possible connections between the above mentioned trends and the other galactic properties will be investigated in a future paper and will help in sheding more light on this subject.
In the end we notice that the gradients in and may be affected by the particular formation history of one model only in their zero point, whereas their slopes are shaped by the strong role of the gas flows (both the final values of and are always larger than the 50%) and by the fact that the SF proceeds always outside-in.
Another important point is that the differences among the values of in the models presented in this paper are typically around a factor of 2 (if they are not presented in logarythmic units), values which are probably comparable to all the uncertainties involved in the measurements of the gradients as well as uncertainties related to the transformation from indices to abundances of such (see PMC06). This fact calls for newer, larger as well as homogeneous samples of gradients observed in ellipticals and extended to one effective radius. Only then, in fact, it will be possible to discriminate among the particular models presented in this work.
In this paper we have studied the formation and evolution of ellipticals by means of hydrodynamical models in which we implement detailed prescriptions for the chemical evolution of H, He, O and Fe, thus presenting a quite detailed treatment of both the chemical and the gas-dynamical evolution of elliptical galaxies. Within this framework we are able to relax the assumption of non-interacting shells which hampers many chemical evolution codes in the modelling of the gas flows, thus allowing us to perform a detailed study of the build-up of the metallicity gradients in stars. We suggest an outside-in formation for the majority of ellipticals in the context of the SN-driven wind scenario, thus confirming previous results of chemical evolution models, but we also show the necessity of taking into account in detail gas inflows/outflows.
Here we summarise our main results.
We find in the range -0.5 – -0.2 dex per decade in radius and -0.3 dex per decade in radius, in agreement with the observations (e.g. Kobayashi & Arimoto, 1999).
These gradients in the abundances, once transformed into predictions of the line-strenght indices, lead to mag per decade in radius, again in agreement with the typical mean values measured for ellipticals and confirming the PM04 best model predictions. We also find that some models predict a steeper gradient, and this seems to be in agreement with the observed gradients of a few massive objects in the Ogando et al. (2005) sample.
The build-up of the gradients is very fast and we predict negligible evolution after the first 0.5 - 1 Gyr.
suggested by PMC06.
We also find the actual (i.e. mass-averaged) metallicity gradients can be flatter than the luminosity-weighted ones (i.e. the observed ones).
The main novelty of this work is that we address the issue of the observed scatter in the radial gradient of the mean stellar [/Fe] ratio and its apparent lack of any correlation with all the other observables. By analysing typical massive ellipticals, we find that all the models which predict chemical properties, including the  and the global metallicity gradients, within the observed ranges, show a variety of gradients in the [/Fe] ratio, either positive, negative or null.
We explain this finding with the fact that the suggested outside-in mechanism for the formation of the ellipticals is not the only process responsible for the formation of abundance gradients. In particular, other processes should be considered such as the interplay between local differences in the SF timescale and gas flows. In particular, our models suggest the gradient in the [/Fe] ratio to be be related to the interplay between the velocity of the -enhanced radial flows, moving from the outer to the inner galactic regions, and the intensity and therefore duration of the SF formation process at any radius. In other words, if the flow velocity is fast relative to the star formation, the stars still forming at inner radii have time to form out of -enhanced gas coming from the outermost regions, thus flattening and even reversing the sign of the [/Fe] gradient.
In particular, we have shown that we do not need the merger events invoked in order to have a shallow  gradient.
Moreover, the predicted age gradients are very small, being typically a few Myrs per decade in radius, in agreement with Sanchez-Blazquez et al. (2007). This means that the estimate of the relative duration of the SF process between two different galactic regions by measuring the  is not a robust method. In other cases in which, instead, the age gradients are stronger we expect a much more evident radial variations in the , as those outlined in PMC06.
According to our fiducial cases, up to the 90% of the metals locked in stars in the galactic center could have been synthesized at larger radii.
We stress that the new class of models presented here make several and new predictions on both the shape and the fast evolution of the metallicity gradients which are left unconstrained by the lack of observations. What makes galaxies start from quasi-monolithic conditions is still to be understood. The quest for an explanation of such behaviours will be a challenging field of research if future observational campaigns will confirm the steep positive found in, e.g., NGC 4697 (see fig. 8), thus, further validate a particular type of models. These observables will be the testbench for our suggested galaxy formation scenario to be tested by future observations.
We acknowledge useful discussions with F.Annibali, L.Ciotti.
Then we warmly thank F. Calura, C. Chiappini, S.Recchi and P. Sanchez-Blazquez for a careful reading of the paper and many enlightening comments. The work was supported by the Italian Ministry for the University and the Research (MIUR) under COFIN03 prot. 2003028039.
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