4. THE STAR FORMATION

In spite of still many open questions, enormous progresses have been made in the last decade in simulating the process of star formation [17, 136, 25, 135, 16, 91]. However, the level of detail and the resolution reached by these works can not be matched by galactic simulations. Suitable parametrisations of the star formation need to be implemented. It is also worth mentioning that many papers dealing with simulations of galaxies do not self-consistently calculate the star formation, but use prescribed star formation rates (SFRs) or star formation histories (SFHs). These are either based on the reconstructed SFH of specific galaxies [228, 225], or are simple functions of time such as instantaneous bursts or exponentially declining SFRs [318, 159, 227, 79]. This is a viable possibility if the star formation process itself is not the focus of the numerical study.

A star formation law scaling with some power of the gas volume or surface density is often assumed. This relation is based on the observation of star formation indicators in local galaxies [113] and is often called the Kennicutt-Schmidt law. To be more precise, the Kennicutt-Schmidt law implies that:

(4)

where _SFR is the SFR surface density and _g is the gas surface density. The value of n reported by Kennicutt [113] is 1.4 ± 0.15. In many works, a dependence on the total volume density [298, 175, 64, 174] or on the molecular gas density [218, 137, 133, 100] is also assumed. A dependence on the molecular gas density appears to be particularly relevant because there is a tight correlation between the H₂ and the SFR surface densities [21]. Moreover, in spiral galaxies, often the Toomre criterium is used to identify regions prone to star formation [331], or _SFR is assumed to be _g, where is the circular frequency [345, 216]. Eventually, the spatial distribution of a molecular cloud seems to play a critical role in determining its star formation activity [140], but the dependence of the SFR on the structure of a molecular cloud appears to be very difficult to implement in numerical simulations.

In hydrodynamical simulations, many authors still follow the star formation recipes of Katz [110], namely (see also Katz et al. [111]):

• The gas density must be larger than a certain threshold
• The particle must reside in an overdense region
• The gas flow must be converging (∇ ⋅ v < 0)
• The gas particle must be Jeans unstable: h / c_s > 1 / √4 G, where h is the dimension of the gas particle (smoothing length for SPH simulations and the grid cell size for grid-based methods) and c_s is the local sound speed

With small variants, this recipe has been applied in most of galaxy simulations [334, 270, 288, 301, 89]. The Jeans criterium appears to be particularly relevant, otherwise artificial fragmentation and, hence, spurious star formation can arise [321, 38]. However, the implementation of this criterium some times leads to unrealistic SFRs [288].

Often, a star formation law of the type:

(5)

is assumed, where (t) is the SFR and c_* is the star formation efficiency [288, 248, 57]. Here t_dyn is a typical star formation timescale given by the free-fall timescale, the cooling timescale or a combination of both. Notice that the free-fall time scale is proportional to ^-1/2, thus a star formation very similar to the Kennicutt-Schmidt law can be obtained in this way (see also [71]). Notice also that observed laws (such as the Kennicutt-Schmidt law Eq. 4) involve surface densities, whereas theoretical models and simulations generally work with volume density laws such as Eq. 5 and not necessarily these two formulations are equivalent. Typically adopted values for c_* in Eq. 5 are quite low, ranging between 0.1 and 0.01 [288]. This is also the ratio between the gas consumption time scale and t_dyn. This assumption is in agreement with the conclusion, deduced from observations, that only a small fraction of gas in molecular clouds can be converted into stars [74, 196]. The star formation efficiencies are larger (of the order of 0.3) if one considers only the dense cores of molecular clouds [5]. Global star formation efficiencies tend to be even lower in DGs (see also Sect. 2 and below in this Section).

Since the cooling timescale depends on the gas temperature, a dependence of the star formation with the temperature is implicit in Eq. 5. It is of course very reasonable to assume that the SFR depends on the temperature, since star formation occurs in the very cold cores of molecular clouds. For this reason, some authors even assume a temperature threshold, above which star formation cannot occur [288, 230, 2]. However, one should be aware of the fact that simulations still do not have the capability to spatially resolve the cores of molecular clouds. The temperature of a star forming region is thus simply the average temperature of a region of gas, with size equal to a computational unit (gas particle in a SPH simulation or grid cell in grid-based codes), encompassing the star forming molecular cloud core. For this reason, typical temperature thresholds are of the order of 10³ - 10⁴ K, at least two orders of magnitude larger than typical molecular core temperatures.

Some authors adopt a more complex temperature dependence. For instance, Köppen et al. [124] derive:

(6)

where the transition temperature T_s = 1000 K implies that the star formation is very low in regions with T > T_s. Notice that, in this case, c_* does not have the same dimensions (and the same meaning) of the c_* introduced in Eq. 5. This star formation recipe, coupled with the feedback from stellar winds and dying stars (see Sect. 7), nicely leads to self-regulation of the star formation process. In fact, a large SFR increases the feedback, which in turn strongly reduces further star formation whereas, if the feedback is low, the temperature does not increase and star formation is more efficient.

Eventually, theoretical works [72] suggest that the star formation efficiency can depend on the external pressure, simply because gas collapse is favoured in environments with large pressures. This hypothesis is supported by the observational fact that the molecular fraction depends on the gas pressure [22, 149] and, as noticed above, the surface density of molecular gas strongly correlates with the SFR [21]. DGs are usually characterised by lower pressures compared to larger galaxies, thus the predicted star formation efficiency is lower. This finding is in agreement with other lines of evidence, showing that DGs are quite inefficient in forming stars (see Skillman et al. [267] for a review). The pressure dependence on the star formation efficiency has been used in Harfst et al. [94].