3. Theoretical prescriptions
To calculate the chemical evolution of the star forming region of DIG and BCG, we used an improved version of the model presented in Marconi et al. (1994). The main features of our chemical evolution model are the following:
where is the gas mass in the form of an
element i normalized to a total luminous mass
and is the present
The star formation rate we assume during a burst, , is defined as:
where is the star formation efficiency (expressed in units of ), and represents the inverse of the timescale of star formation, namely the timescale necessary to consume all the gas in the star forming region.
The rate of gas loss via galactic winds for each element is assumed to be simply proportional to the amount of gas present at the time t:
where is the abundance of the element i in the wind and in the interstellar medium (ISM); is a free parameter describing the efficiency of the galactic wind and expressed in . We have considered normal and differential winds: in the first case, all the elements are lost at the same efficiency (). In the second case, the value of has been assumed to be different for different elements: in particular, the assumption has been made that only the elements produced by type II SNe (mostly -elements and one third of the iron) can escape the star forming region. We have made this choice following the conclusions of Marconi et al. (1994), who showed that models with differential winds can better explain the observational constraints of blue compact galaxies in general. The conditions for the onset of the galactic wind are studied in detail and described in paragraph 3.2.
The chemical evolution equations include also an accretion term:
where is the abundance of the element i in the infalling gas, assumed to be primordial, is the time scale of mass accretion is the galactic lifetime, and C is a constant obtained by imposing to reproduce at the present time . The parameter has been assumed to be the same for all dwarf irregulars and short enough to avoid unlikely high infall rates at the present time ( years). The presence of infalling gas in these systems can be justified by the existence of large HI halos.
Finally, the initial mass function (IMF) by mass, , is expressed as a power law:
We considered two cases for the IMF:
A is the normalisation constant, obtained with the following condition:
3.1. Nucleosynthesis prescriptions
The term of the Eq. (1) represents the stellar contribution to the enrichment of the ISM, i.e. the rate at which the element i is restored into the ISM from a stellar generation (see Marconi et al. 1994). This term contains all the nucleosynthesis prescriptions:
As already said, nitrogen is a key element to understand the evolution of galaxies with few star forming events since it needs relatively long timescales as well as a relatively high underlying metallicity to be produced. The reason is that N is believed to be mostly a secondary element (secondary elements are those synthesized from metals originally present in the star and not produced in situ , while primary elements are those synthesized directly from H and He). Up to now N has been considered secondary in massive stars and mostly secondary and probably partly primary in low and intermediate mass stars (Renzini and Voli 1981). However, some doubts exist at the moment on the amount of primary nitrogen which can be produced in intermediate mass stars due to the uncertanties related to the occurrence of the third dredge-up in asymptotic giant branch stars (AGB). In fact, if Blöcker and Schoenberner (1991) calculations are correct, the third dredge-up in massive AGB stars should not occur and therefore the amount of primary N produced in AGB stars should be strongly reduced (Renzini, private communication).
As a consequence, the only way left to produce a reasonable quantity of N during a short burst (no longer than 20 Myr) is to require that massive stars produce a substantial amount of primary nitrogen. This claim was already made by Matteucci (1986) in order to explain the [N/O] abundances in the solar neighbourhood. Recently, Woosley and Weaver (1995) have indicated that massive stars can indeed produce primary nitrogen, and Marconi et al. (1994) and Kunth et al. (1995) have taken into account this possibility. In this paper we also have considered this possibility and, since quantitative predictions are not yet available, we have used the same parametrization adopted by Marconi et al. (1994) and Kunth et al. (1995).
We define as "galactic wind" any gas flow carrying material outside the galaxy. Such flows, which are not necessarily hot, have been recently detected in several blue compact galaxies (Kunth et al. 1998).
Our model presents a new formulation of galactic winds relative to previous published models of this type (Matteucci and Tosi 1985 and Marconi et al. 1994). In particular, we adopted the prescriptions developed for elliptical galaxies (Matteucci and Tornambé 1987; Gibson 1994, 1997): galaxies develop galactic winds when the gas thermal energy , exceeds its binding energy , i.e. when:
A detailed treatment of the energetics of the ISM is considered, in order to compute the gas thermal energy:
we calculated the energy fraction deposited in the gas by stellar winds from massive stars , and by supernova explosions . Here the supernova contribution is given by:
and are the rates of supernova (II and Ia) explosion, is the IMF and is the star formation rate. The type Ia and II SN rates are calculated according to Matteucci and Greggio (1986). The energy injected and effectively thermalized into the ISM from supernova explosions () and stellar winds from massive stars (), are given by the following formulas:
where is the total energy released by a supernova explosion, is the energy injected into the ISM by a typical massive star through stellar winds during all its lifetime, and and are the efficiencies of energy transfer from supernova and stellar winds into the ISM, respectively. We adopted a formulation for and which is described in detail in appendix.
Our formulation refers to an ideal case characterized by an uniform ISM, and no interaction with other supernova explosions or interstellar clouds. When a supernova explodes the stellar material is violently ejected into the ISM and its expansion is gradually slowed down by the ISM. The energy released by the supernova explosion is assumed to be , while the energy effectively transferred and thermalized into the ISM is given by Eq. (13) and depends on the assumed value for . The main parameters used to derive , as described in the appendix, are: the initial blast wave energy , the interstellar gas density , and the isothermal sound speed in the ISM .
For typical values of these parameters , and we obtained the following range of possible values for :
In particular, we have adopted an intermediate value of which we assume to be the typical SN energy transfer efficiency.
The energy injected into the ISM by a typical massive star through stellar winds during all its lifetime is estimated to be:
where is the time the star spends in Main Sequence and is the stellar wind luminosity. By using the arguments developed in the appendix, we obtained:
if we assume a 20 star as a typical massive star contributing to the stellar winds then , with and , we obtain:
Therefore, in our formulation we adopted:
It is worth noting that the assumed value for is in agreement with that calculated by Gibson (1994) for a star of initial mass .
The introduction of dark matter halos with variable amounts and concentrations of dark matter is considered when we compute the binding energy of interstellar gas, :
The two terms on the right of the equation take in account the gravitational interaction between the gas mass , and the total luminous mass of the galaxy , and between the gas mass and the dark matter :
G is the gravitational constant, and
is the ratio between the effective radius of luminous matter and the
effective radius of dark matter. These equations are taken from Bertin
et al. (1991) and are valid for S defined in the range
© European Southern Observatory (ESO) 1998
Online publication: August 17, 1998