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 [alpha/Fe], and their apparent lack of any correlation with all the other observables. 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. We find that all the models which predict chemical properties (such as the central mass-weighted abundance ratios, the colours as well as the [<Fe/H>] gradient) within the observed ranges for a typical elliptical, also exhibit a variety of gradients in the [<alpha/Fe>] 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. The typical [<Z/H>] 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. 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 [<alpha/Fe>] ratios.
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