The rates of type Ia supernovae - I. Analytical formulations

The aim of this paper is to provide a handy tool to compute the impact of type Ia SN (SNIa) events on the evolution of stellar systems. An effective formalism to couple the rate of SNIa explosions from a single burst of star formation and the star formation history is presented, which rests upon the definition of the realization probability of the SNIa event (A(Ia)) and the distribution function of the delay times (f(Ia)(tau)). It is shown that the current SNIa rate in late type galaxies constrains A(Ia) to be on the order of 10(-3) (i.e. 1 SNIa every 1000 M(circle dot) of gas turned into stars), while the comparison of the current rates in early and late type galaxies implies that fIa ought to be more populated at short delays. The paper presents analytical formulations for the description of the fIa function for the most popular models of SNIa progenitors, namely Single Degenerates ( Chandrasekhar and Sub- Chandrasekhar exploders), and Double Degenerates. These formulations follow entirely from general considerations on the evolutionary behavior of stars in binary systems, modulo a schematization of the outcome of the phases of mass exchange, and compare well with the results of population synthesis codes, for the same choice of parameters. The derivation presented here offers an immediate astrophysical interpretation of the shape of the fIa functions, and have a built in parametrization of the key properties of the alternative candidates. The important parameters appear to be the minimum and maximum masses of the components of the binary systems giving rise to a SNIa explosions, the distribution of the primary mass and of the mass ratios in these systems, the distribution of the separations of the DD systems at their birth. The various models for the progenitors correspond to markedly different impact on the large scales; correspondingly, the model for the progenitor can be constrained by examining the relevant observations. Among these, the paper concentrates on the trend of the current SNIa rate with parent galaxy type. The recent data by Mannucci et al. (2005, A& A, 433, 807) favor the DD channel over the SD one, which tends to predict a too steep distribution function of the delay times. The SD scenario can be reconciled with the observations only if the distribution of the mass ratios in the primordial binaries is flat and the accretion efficiency onto the WD is close to 100%. The various models are characterized by different timescales for the Fe release from a single burst stellar population. In particular the delay time within which half of the SNIa events from such a population have occurred, ranges between 0.3 and 3 Gyr, for a wide variety of hypothesis on the progenitors.