3. The free parameters
In addition to the cosmological parameters h, , , and , and to the choice of the IMF, which is assumed to be constant throughout a Hubble time, we basically have three astrophysical free parameters in the current version of our simple semi-analytic model: the star formation efficiency , the SN heating efficiency , and the disk truncation parameter which is used to compute the gas column density and the face-on optical depth. As a matter of fact, there is not much freedom in the choice of these parameters.
First, the value of the star formation efficiency deduced from Kennicutt's data (1998) is about for our definition, and is valid for galaxies ranging from quiescent objects to very active starbursts. We refer to GHBM for a discussion of how this prescription actually compares to the data, especially to the so-called "Roberts times" in nearby disks, and just mention that the difference between the value in this paper and the one used in GHBM () stems from the different prescriptions used to compute disk sizes, which result in our disks being about twice as large as theirs. As mentioned by Kennicutt, there is a lot of scatter in the data ( 30-50 %), which, along with plausible systematics in the calibration of the different star formation estimators, should make the value of uncertain by at least 20 %. Increasing decreases the normalisation and slope of the optical and IR counts, because star formation is lower, and takes place at lower redshifts.
Second, recent numerical simulations (Thornton et al. 1998) suggest that the SN heating efficiency is . However, there is much uncertainty on the actual efficiency of SN explosions in a disk galaxy, because SN bubbles can blow their energy out of the disk without altering the cold gas (see e.g. De Young & Heckman 1994, and Lobo & Guiderdoni 1999 for an examination of the issue within a SAM). Consequently, could be very low. We adopt in the following. Increasing decreases the normalisation and slope of the optical and IR counts, since star formation is quenched in galaxies with still higher masses, that form at still lower redshifts.
Third, the average value of the disk truncation parameter , which measures the gaseous disk extension, is around 6, from the sample of spiral galaxies with various morphological types observed by Bosma (1981), and used by GHMB. However, this number is probably uncertain by about a factor 2, and it can be adjusted within this range in order to match the UV/optical/near-IR counts as well as IRAS counts, as far as this parameter fixes the amount of dust absorption in a disk galaxy. Increasing increases the normalisation of the optical counts and decreases that of the IR counts, since extinction decreases.
As explained in Paper I, the set of STARDUST spectra depends on (i) the mass of baryons in the galaxy, (ii) the star formation timescale , (iii) the age t of the stellar population, and (iv) the parameter called that links the gas mass fraction to the gas surface density, and is used in the computation of the face-on optical depth. In the SAMs, these quantities are computed directly from the cold gas mass , the dynamical time , the collapse redshift and observed redshift z, and the disk exponential length , provided the values of the cosmological parameters are chosen, and the astrophysical parameters , and are fixed. Thus the implementation of the STARDUST spectral energy distributions within our SAM is very straightforward and does not bring new free parameters.
The above-mentioned values of the free parameters define the so-called "quiescent mode" of star formation, similar to what is observed in local disks. As a reference point, we list, for the CDM cosmology, the properties of a disk galaxy hosted by a halo of about , which collapses at a redshift (meaning that the age of the "Milky Way"-class spiral galaxy that sits in this halo is about 11.4 Gyr) with a spin parameter . At redshift 0, such a galaxy has turned about 88 % of its total 7.5 of cold gas into stars. Its disk exponential scale length is about 3.5 kpc, yielding a gaseous disk extending to 21 kpc with an average hydrogen column of and a metallicity of 0.02. This, in turn implies a face on optical depth in the B band of 0.7 resulting in values of and for the face-on absolute B magnitude and the bolometric IR luminosity (between 3 and 1000 µm) respectively.
Although this discussion is only valid, strictly speaking, for a given cosmology, it is unlikely that different cosmological parameters will significantly affect physical parameters like star formation or feedback efficiency. We therefore consider our astrophysical parameters as independent of the cosmological model. In the next section, we keep the same values of the astrophysical parameters, and we study the predictions of the SAM for the sets of cosmological parameters that are displayed in Table 1.
Table 1. Parameters of the different cosmologies.
© European Southern Observatory (ESO) 2000
Online publication: December 5, 2000