Gravitational instability and star formation in disc galaxies

We present a general star formation law in which the star formation rate depends upon the ef®ciency a and the time-scale t of star formation, and the gas component jg of the surface mass density. The current nominal Schmidt exponent nS for our model is 2 < nS < 3. Based on a gravitational instability parameter QA and another dimensionless parameter fP ˆ …P=Gj 2 c† , where P ˆ pressure and jc ˆ column density of molecular clouds, we suggest a general equation for the star formation rate which depends upon the relative contributions of the two parameters for various physical circumstances. We ®nd that QA turns out to be a better parameter for the star formation scenario than does the Toomre Q-parameter. The star formation rate in the solar neighbourhood is found to be in good agreement with values inferred from previous studies. In the closed box approximation model, we obtain a relationship between the metallicity of the gas and the ef®ciency of star formation. Our model calculations of metallicity in the solar neighbourhood agree with earlier estimates. We conclude that the metallicity dispersion for stars of the same age may result from a change in the ef®ciency with which different stars in the sample were processed. For no signi®cant change of metallicity with age, we suggest that all stars in the sample were born with similar ef®ciencies. Key words: instabilities ± stars: formation ± galaxies: evolution ± galaxies: general ± galaxies: ISM. 1 I N T R O D U C T I O N It was realized by Kennicutt (1989) that there is a non-linear increase in the star formation rate near the threshold surface density corresponding to the Q-parameter. The star formation rate (R) is high in spiral arms, mostly because there is a lot of gas present. The ef®ciency varies by a lesser amount, such as a factor of 2. For example, in M51 and NGC 6946 (Rydbeck, Hjalmarson & Rydbeck 1985; Lord 1987; Tacconi-Garman 1988), the gas densities in the spiral arms are larger than elsewhere by a factor of 2, indicating deviations in the usual power-law exponent (n q 2) of Schmidt (1959, 1963). In fact, star formation in many spiral galaxies under extreme conditions of low gas density and low disc self-gravity presents a challenge to all current theories for disc star formation (Ferguson et al. 1996). New star formation laws have therefore been proposed (see e.g. Talbot & Arnett 1975; Dopita 1985; Wyse 1986; Silk 1987, KoÈppen & FroÈhlich 1997). However, the concept of a global star formation law has been put in doubt (Hunter & Gallagher 1986). For a general star formation scenario, one may refer to Zinnecker & Tscharnuter (1984) and Zinnecker (1989, and reference therein). Many interesting variants on the simple star formation laws include, for example, selfpropagating star formation (stochastic) (Gerola & Seiden 1978; Seiden & Gerola 1982; Dopita 1985; Coziol 1996), self-propagating star formattion (Arimoto 1989; Hensler & Burkert 1990a,b), and star formation bursts (stochastic) (Matteucci & Tosi 1985). KruÈgel & Tutukov (1993) and Tutukov & KruÈgel (1995) have used a onezone dynamical code without radial dependence of the variables to study the conditions for bursts of star formation. In the latter paper, using a one-zone code, they studied types of bursts of star formation in a galactic nucleus that were different from periodic bursts. Furthermore, the surface gas density threshold for star formation has been discussed by Kennicutt (1989). Wyse & Silk (1989) have discussed an extended Schmidt model with R-dependence on the surface gas density jg and the local angular frequency Q…r† for atomic and molecular gases respectively, with n ˆ 1 and 2. Wang & Silk (1994) have recently presented a self-consistent model (considering the total gas surface density) for global star formation based on the gravitational instability parameter Q < 1 due to Toomre (1964). In the solar neighbourhood, the model agrees with (i) the observed star formation rate, (ii) the metallicity distribution among G dwarfs, and (iii) the age± metallicity relation for F dwarfs. The model results may be compared to the star formation rate in galactic discs with a Schmidt law with an exponent of about 2. The star formation rate also depends on the epicyclic frequency. A natural cut-off for Q ˆ 1 in the star formation rate results. However, Wang & Silk's analysis is heavily based on the Q < 1 criterion, which has been questioned in Mon. Not. R. Astron. Soc. 306, 181±190 (1999)