Astron. Astrophys. 363, 851-862 (2000)

2. The basics of our semi-analytic model

In the SAM ab initio approach, galaxies form from Gaussian random density fluctuations in the primordial matter distribution, dominated by CDM. Bound perturbations grow along with the expanding universe, until gravitation makes them turn around and (non-dissipatively) collapse. As a result, they end up as virialized halos. Then the collisionally-shocked baryonic gas cools down radiatively, and settles at the bottoms of the potential wells where it is rotationally-supported. Stars form from the cold gas and evolve. At the end of their lifetimes, they inject energy, gas and heavy elements back into the interstellar medium. The chemical evolution is computed, and the recipes developed in GHBM and in Paper I give the amount of optical luminosity that is absorbed by dust and thermally released at IR/submm wavelengths. Finally, overall SEDs from the UV to the submm are computed with STARDUST .

We refer the reader to GHBM for a detailed description of how to compute the mass distribution of collapsed dark matter halos from the peaks formalism introduced by Bardeen et al. (1986), and Lacey & Silk (1991) in an Einstein-de Sitter universe. We give in appendix A the quantities which enable us to extend this formalism to low matter-density universes with or without a cosmological constant. As we follow closely the prescriptions in GHBM, we only mention in the following subsections the quantities which differ from their work.

2.1. Gas

We assume that a universal "baryonic fraction" of the pristine gas gets locked up within each dark matter halo, where it is collisionally ionised by the shocks occurring during virialization. Because it can cool radiatively, gas then sinks into the potential wells of the halos. The cooling time depends on the gas metallicity. Here, we decide to adopt the cooling function given by Sutherland & Dopita (1993) for one third of solar metallicity. This choice is motivated by the fact that it is the average value that is observed today in clusters, and probably, as argued by Renzini (1999), the average value of the low-redshift universe as a whole. Under this assumption, the cooling time is underestimated in high-redshift halos where the gas is more metal-poor. However, these objects are also smaller and denser on an average, so that their cooling times are already very short.

We assume that the gas stops falling into the dark matter potential wells when it reaches rotational equilibrium, and forms rotating thin disks (see e.g. Dalcanton et al. 1997 and Mo et al. 1998). Following these authors, we adopt for the thin disk an exponential surface density profile with scale length , and truncation radius , such as:

where is defined as the minimum value between the virial radius , and . The free parameter defines the extent of the cold gas disk.

We then relate the exponential scale length of the cold gas disk to the initial radius , through conservation of specific angular momentum (Fall & Efstathiou 1980). As shown by Mo et al. (1998), stability criteria yield:

where is the well-know dimensionless spin parameter.

As this will be important later, we emphasise that the simple formalism used here does not allow us to form spheroids through mergers/interactions of galaxies. Therefore, in Sect. 5, we will define a "starburst mode" which phenomenologically accounts for this process.

2.2. Stars

The only time scale available in our gas disks is the dynamical time scale . Therefore, guided by observational data (Kennicutt 1998), we assume that the complicated physical processes ruling star formation lead, at least in a disk galaxy, to a global star formation rate (SFR) with the simple law:

where is the total mass of cold gas in the disk at time t. We introduce an efficiency factor as a second free parameter. The IMF is chosen to be Salpeter's, with slope between masses and .

The STARDUST spectrophotometric and chemical evolution model presented in Paper I is then used to compute metal enrichment of the gas as well as the UV to NIR spectra of the stellar populations produced with such star formation rates. Details on the stellar spectra, evolutionary tracks and yields can be retrieved from this paper and references therein.

2.3. Feedback

Along with producing metals, massive stars which, at the end of their lifetimes, explode in galaxies, eject hot gas and heavy elements into the interstellar and/or intergalactic medium. We focus here on the modelling of this "stellar feedback", which is inspired from Dekel & Silk (1986). The average binding energy of a mass of gas distributed within a truncated exponential disk at time t, which is gravitationally-dominated by its dark matter halo, is given by:

where is the gravitational potential of a singular isothermal sphere truncated at virial radius , and:

is its escape velocity at radius r.

As a result, the energy balance between the gravitational binding energy and the kinetic energy pumped by supernovae into the interstellar medium yields the fraction of stars that formed before the triggering of the galactic wind (at time ):

with:

where the energy available per supernova is , and the number of supernovae per mass unit of the stars that just formed is for a Salpeter IMF. The SN heating efficiency is a third free parameter. By taking the initial gas mass available for star formation to be the initial cold gas mass minus the gas mass lost in the galactic wind, one then approximate chemical evolution by the closed-box model described in Paper I. Note that we have neglected any dynamical effect due to mass loss in the previous analysis.

2.4. Dust

Part of the luminosity released by stars is absorbed by dust and re-emitted in the IR/submm range. We now briefly outline how we compute the luminosity budget of our objects within the SAM. We emphasise that this is a major improvement with respect to GHBM, as for the first time, stellar and dust emission are linked self-consistently. As in Paper I, we proceed to derive the IR/submm dust spectra with three steps: (i) computation of the optical depth of the disks, (ii) computation of the amount of bolometric energy absorbed by dust, and (iii) computation of the spectral energy distribution of dust emission.

The first step is easily completed because we know the sizes of our objects from Eq. 1 and the definition of the truncation radius , and we obtain the mass of gas and metallicity as a function of time through our model of chemical evolution. We then use the scaling of the extinction curve with gas column density and metallicity described in Guiderdoni & Rocca-Volmerange (1987) to compute the face-on optical depth of our objects at any wavelength:

where the mean H column density (accounting for the presence of helium) reads:

The second step is more delicate because it involves choosing a "realistic" geometry distribution for the relative distribution of stars and dust. We model galaxies as oblate ellipsoids where dust and stars are homogeneously mixed, and scattering is taken into account. As explained in Paper I, the model gives a decent fit of the sample of local spirals analysed by Andreani & Franceschini (1996).

Finally, the third step involves an explicit modelling of the dust grain properties and sizes. We use the three-component model described in Désert et al. (1990) for the Milky Way with polycyclic aromatic hydrocarbons, very small grains and big grains, and we allow a fraction of the big grain population to be in thermal equilibrium at a warmer temperature if our galaxies undergo a massive starburst. The weights of these four components are fixed in order to reproduce the relations of IR/submm colours with bolometric IR luminosity that are observed locally, as detailed in Paper I. Once the full (UV/submm) spectral energy distributions of individual objects are computed following such a method, we build populations of galaxies for which we derive galaxy counts and redshift distributions. We present these results in the following sections.

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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