1 Introduction

The formation of a galactic disk involves such complex physical processes that only numerical computations can give some insight in their relative importance. The essential force is gravity, which is nowadays easily controlled through N-body codes, based either on grid calculations (such as FFT), or on hierarchical grouping of particles (such as the tree-code). Both algorithms optimize the computing time, growing roughly as $N \ln(N)$. The main improvement in this domain is obtained by going towards higher and higher spatial resolution, with an ever-growing number of particles. The gravitational force is softened at small scales, to reduce unphysical two-body relaxation, resulting in a resolution equal or slightly smaller than the inter-particle distance (Romeo 1998; Dehnen 2001; Knebe et al. 2001).

It has been recognized for a long time that the dissipative component is also an essential feature in galaxy evolution, and the interstellar medium has been widely introduced in galaxy simulations, either as a continuous fluid (Eulerian grid codes, van Albada & Roberts 1981; Lagrangian SPH codes, Hernquist & Katz 1989), or through sticky particle algorithms (Combes & Gerin 1985). However, contrary to the pure gravity case, the result of simulations now depends not only on the spatial resolution and computer power, but on the physical assumptions made on the nature of the ISM and related processes.

Star formation and feedback, coupling the stars and gas by non-gravitational processes are essential (Katz 1992; Mihos & Hernquist 1994). Since the detailed processes involved in star formation are not yet well known, this introduces further uncertainties and liberties in the modeling. The most widely adopted recipe to control star formation is based on a local Schmidt law (i.e. the star formation rate is locally proportional to some power of the volume density), sometimes associated with a threshold for star formation (e.g. Friedli & Benz 1995). However, the local Schmidt law is not actually observed in galaxies, and the justification comes from an observed global empirical Schmidt law, averaged over the whole galaxy (e.g. Kennicutt 1998). Other star formation recipes are based on Jeans instability (e.g. Steinmetz & Müller 1994; Gerritsen & Icke 1997), or cloud-cloud collisions (Noguchi & Ishibashi 1986).

Due to the limited number of particles, each star "particle'' represents in fact a stellar cluster of the order of 10⁵ to  $10^6~M_\odot$. The conversion of a fraction of a gas particle into stars requires the creation of a large number of star-particles (e.g. Katz 1992), or the consideration of "hybrid'' particles, that are transiently containing both gas and stars, and have the same dynamics for a while (Mihos & Hernquist 1994). Alternatively, some "starlet'' particles are transiently created, and merged with the nearest neighbors (Jungwiert et al. 2001). Whatever the choice, some approximations are made, related to the limited resolution in time, mass and space, respectively of the order of 10⁶ yr, 10 $^6~M_\odot$ and 0.3 kpc for a typical giant spiral galaxy.

The same limitations occur when mass loss and feedback are considered. Either only stellar heating is considered (Gerritsen & Icke 1997), or the supernovae mechanical energy only (Mihos & Hernquist 1994), or more sophisticated models are used including several types of supernovae, planetary nebulae, stellar winds, evaporation and condensation (e.g. Theis et al. 1992; Samland et al. 1997; Berczik 1999).

Stellar nucleosynthesis can be followed to describe the detailed metal enrichment history of the galaxy (e.g. Lia et al. 2001). Models reveal that the frequently used instantaneous recycling is a too simple approximation (see also Jungwiert et al. 2001).

Of first importance is the thermal evolution of the gas, since thermal instabilities are responsible for cloud condensation, feeding the cold gas phase, and thus star formation (Hultman & Pharasyn 1999). Cooling and heating processes are taken into account depending on the temperature, density and metallicity of the gas (e.g. Dalgarno & McCray 1972; Sutherland & Dopita 1993). However, the time-scales involved in the thermal processes can be much smaller than the dynamical time-scales, and the spatial scales of the corresponding processes are much below the spatial resolution of the simulations. Phenomenological recipes are therefore used to convey at large-scales the resulting effects of the processes occurring below the resolution (e.g. Thacker & Couchman 2000): the heating energy is smoothed over the resolution scale, and the effective time-scales slowed down.

Many simulation models consider only one gas phase, treated as a continuous fluid at the virial temperature corresponding to a galaxy potential ($\sim$10⁴ K). A simple approximation is a strictly isothermal gas, since the cooling is very efficient above 10⁴ K (Barnes & Hernquist 1991; Mihos & Hernquist 1994). Some allow gas to cool down to low temperatures ($\sim$10 K), and to spread into several temperature phases (e.g. Gerritsen & Icke 1997; Yepes et al. 1997; Thacker & Couchman 2001). However, their dynamics is still that of a one-phase medium, given the insufficient density contrast that can be dealt with the numerical simulations on large-scales. Significant contrasts can be achieved only in simulations of small volumes, that approach more realistically the multiphase nature of the interstellar medium (e.g. Rosen & Bregman 1995; Wada & Norman 1999, 2001).

In the present paper, we implement most of the above processes to investigate the formation and evolution of disk galaxies (gravity, stellar and gas dynamics, star formation, feedback and massloss heating and cooling, metal enrichment...), and we focus on the large-scale multi-phase dynamics of the gas. The "warm gas'' component is modeled by a continuous fluid with an SPH code, and contains a large range of temperatures, up to hot gas re-injected by supernovae, though the bulk of the gas is found between 10⁴ and $2\times 10^4$ K. The "cold gas'' component corresponds to the cloudy and fragmented, essentially molecular medium between 10 K and 100 K. It is modeled as a separate phase, via individual cloud-particles, subject to inelastic collisions. Since at large galactic scales, the density contrast necessary to form such a fragmented structure (12 orders of magnitude) is far beyond the present computational power, the cloud-particle approximation appears well suited. For the cloudy component, the pressure forces or viscosity forces are negligible. The dissipation occurring during cloud collisions at AU scales, much below our spatial resolution, is modeled phenomenologically.

Figure 1: Schematic presentation of the model. The four phases are represented with the main physical processes responsible for mass and energy exchanges.

Two general classes of models have been proposed for the formation of galaxies: either big galaxies can form through the monolithic collapse of a very massive baryonic system (e.g. Eggen et al. 1962), or dwarf galaxies form first, and act as the building blocks of larger systems, that subsequently form through recursive mergers, in the hierarchical scenario (e.g. Kauffmann et al. 1999). The first scenario assumes that the gas experiences violent 3D star formation, so quickly that it has no time to settle into a disk. This has been proposed to form large elliptical galaxies or bulges. However, it is today recognized that many (if not all) elliptical galaxies can form through mergers of spiral galaxies, or smaller entities, as first suggested by Toomre & Toomre (1972), and developed further by Schweizer (1986), while bulges can form similarly by mergers, and also through secular evolution (e.g. Combes 2000). Although the most successful cosmological scheme today, based on inflation, promotes primordial density fluctuations that are scale-invariant, which together with the existence of cold dark matter, results in hierarchical formation of structures, the collapse of baryons inside dark matter halos is still an unsolved problem, with many uncertainties and free parameters. The rate at which stars are formed in a given system, and whether the stellar populations observed today can be attributed to a sudden event like the monolithic collapse, or a more progressive one like the gradual and recursive merging, is a matter of debate. It is likely that both point of views may correspond to some observed galaxy examples today, since the major merger of two gas rich giant spirals have some similarities with a monolithic event. More subtle diagnoses should be searched for, and large number statistics should be used to determine the dominant processes. In this work, we focus on the formation and evolution of a single galaxy. To achieved the required resolution we use the monolithic collapse scenario.

In Sect. 2 we detailed the assumptions of the multiphase model, and present the code and numerical methods in Sect. 3. In Sects. 4, 5 and 6 we present and discuss the results from our simulations. We give our conclusions in Sect. 7.