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Astron. Astrophys. 335, 855-866 (1998)

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2. The chemical evolution model

2.1. Basic equations

The adopted model of galactic evolution is that outlined by Matteucci and Gibson (1995)(hereafter MG95), where extensive descriptions and references can be found. The evolution of the abundances of several chemical species (H, He, C, N, O, Ne, Mg, Si and Fe) in the gas is followed, taking into account detailed nucleosynthesis from stars of all masses and SNe of types Ia, Ib, and II. We assume that ellipticals can be considered initially as homogeneous spheres of gas with luminous mass in the range [FORMULA] [FORMULA]. A single zone interstellar medium (ISM) with instantaneous mixing of gas is assumed throughout. The adopted age for all galaxy models is [FORMULA] Gyr.

The fundamental equations can be written as:


where [FORMULA] is the volume gas density in the form of an element i normalized to the initial total volume gas density. The quantity [FORMULA] represents the abundance by mass of an element i and by definition the summation over all the elements present in the gas mixture is equal to unity.

The various integrals in Eq. (1) represent the rates at which SNe (I and II) as well as single low and intermediate mass stars and single massive stars restore their processed and unprocessed material into the ISM (for a detailed description of the integrals see MG95; Matteucci and Greggio 1986). We only remind here that the quantity [FORMULA] represents the fraction of a star of mass m which is restored into the ISM in the form of an element i and therefore contains the nucleosynthesis prescriptions that we assume to be the same as in MG95.

The star formation rate [FORMULA] is given by:


i.e. normalized to the initial total volume density. [FORMULA] is assumed to drop to zero at the onset of the galactic wind. The quantity [FORMULA] is expressed in units of Gyr-1 and represents the efficiency of star formation, namely the inverse of the time scale of star formation. The values adopted here for [FORMULA] are the same as in Matteucci (1992) and Arimoto and Yoshii (1987). In one case we have adopted the prescription for the inverse wind model of Matteucci (1994). The difference between the two cases is that in Matteucci (1994) the classic wind model assumes that the efficiency of star formation decreases with increasing total galactic mass, as in Arimoto and Yoshii (1987), whereas in the inverse wind model the efficiency of star formation increases as the total galactic mass increases (similar in spirit to the SFR efficiency parametrization of Tinsley and Larson 1979) leading to a situation in which more massive galaxies experience a galactic wind before the less massive ones.

The quantity [FORMULA] represents the lifetime of a star of mass m, and is taken from Padovani and Matteucci (1993).

2.2. Galactic winds

For gas to be expelled from a galaxy the following condition should be satisfied: the thermal energy of the gas heated by SN explosions should exceed the binding energy of the gas (Larson 1974). At this point the gas present in the galaxy is swept away and the subsequent evolution is determined only by the amount of matter and energy which is restored in to the ISM by the dying stellar generations. In particular, only low mass stars contribute to this evolutionary phase and, among the SNe, only SNe of type Ia (i.e. those SNe events whose progenitors have the longest lifetimes).

Therefore, in order to evaluate the time for the onset of a galactic wind we need to know the energy input from SNe and the binding energy of the gas as a function of time. The total thermal energy of the gas at time t, [FORMULA], and the binding energy of the gas in presence of a diffuse halo of dark matter are calculated as described in Matteucci (1992). In particular, [FORMULA] is calculated by assuming that [FORMULA] of the initial blast wave energy is transferred into the ISM as thermal energy by a SN remnant, if the time elapsed from the SN explosion is shorter than a SN remnant cooling time (Cox 1972). The percentage of transferred energy then decreases as a power law in time [FORMULA] for times larger than the cooling time.

This is the same formulation used by Arimoto and Yoshii (1987), Matteucci and Tornambè (1987) and MG95 with the exception that we consider a more realistic cooling time (expressed in years) which takes into account that the gas density is changing with time and that part of the interstellar gas is in the form of He:


Other formulations of the energy input from SNe can be found in Gibson (1997) and Gibson and Matteucci (1997).

The energetic input from stellar winds in massive stars is ignored, since for normal ellipticals it is negligible compared to the SN thermal energy contribution, as shown by Gibson (1994).

The galactic wind may last only for few [FORMULA] years or continue until the present time depending crucially on the assumptions about the thermal energy of the gas and the potential energy of the gas. Unfortunately, none of these quantities is well known. The time for the occurrence of the galactic wind, [FORMULA], either increases with the galactic mass as a consequence of the potential well incresing with galactic mass together with the efficiency of star formation decreasing with galactic mass (classic wind model), or decreases with the galactic mass if the efficiency of star formation is strongly increasing with galactic mass (inverse wind model), as can be seen in Table 3, and will be discussed in Sect. 4.

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© European Southern Observatory (ESO) 1998

Online publication: June 26, 1998