The G-dwarf problem in the solar neighbourhood: a statistical approach within a inhomogeneous, simple model

In dealing with the G-dwarf problem in the solar neighbourhood, simple comoving models of chemical evolution are generalized on two respects, assuming that star formation occurs: (i) within discrete regions instead of a continuous interstellar medium, and (ii) via stochastic processes instead of a uniform generation, with the additional hypotheses: (iii) mass conservation, (iv) instantaneous recycling, and (v) instantaneous mixing within the system at the end of a fixed time step, in connection with identical regions. In the special case of the expected evolution, when the number of star-forming regions equals the expected value of the related, binomial distribution, the gas mass fraction shows an exponential time dependence, the oxygen mass fraction a linear time dependence, and the star formation rate (related to the system) a linear dependence on the gas mass fraction, i.e. the gas density, whatever the star formation rate (related to a region) may be, provided the corresponding parameters are independent of evolution. In addition, it is shown that the dependence of oxygen gas mass fraction on gas mass fraction remains unchanged with respect to a simple model. Two main classes of theoretical age-metallicity relation (TAMR) are analysed and discussed, according if the contribution to an assigned metallicity bin comes from long-lived stars which are born during all the, or only a fraction of, time steps. It is found that, under some simplifying assumptions, the theoretical, differential, G-dwarf metallicity distribution (TGD) has the same expression as in simple models, in connection with the main body of the TAMR. An application is made for the solar neighbourhood, and the input parameters are constrained to both the empirical age-metallicity relation (EAMR) and the empirical, differential, G-dwarf metallicity distribution (EGD). A different, power-law, initial mass function (IMF) is allowed during and after disk formation. A power-law steeper than the Salpeter IMF, which fits the Scalo IMF, is necessary to ensure a lower stellar mass limit above the stellar Jeans mass. An acceptable fit to the EAMR is provided by models where the ratio of oxygen yield, and lower stellar mass limit, during and after disk formation, range from about one half to about three halves. On the other hand, a fit to the EGD allows models where the IMF changes only slightly, or remains unchanged, during and after disk formation. Then a solution to the G-dwarf problem demands, in the light of the current model, nothing but an initial, substructured disk made of pre-enriched material, with same metallicity as the lower limit detected in G dwarfs belonging to the sample under consideration. Several runs are made in different situations, related to either constant number of regions or constant mass of a region, separately during and after disk formation. Significant statistical fluctuations are found when the number of regions is low enough, i.e. of the order of ten. A sharp peak exhibited by the EGD cannot be interpreted in terms of statistical fluctuations, in the context of the current model, unless about 3% of G dwarfs in the solar neighbourhood were born during a burst of star formation within quiescent regions. To avoid contradiction with both the TAMR and TGD, the above mentioned burst would have been occurred 2-4 Gyr ago in connection with a metallicity, Z/Z_sun ~ 0.5-0.6, consistent with both a chemical and kinematical transition observed in the disk.