Star formation in extremely faint dwarf galaxies

We study the relationship between the gas column density (Sigma(gas)) and the star formation rate (SFR) surface density (Sigma(SFR)) for a sample of 23 extremely faint dwarf irregular galaxies drawn from the Faint Irregular Galaxy GMRT (Giant Metrewave Radio Telescope) Survey (FIGGS). Our sample galaxies have a median H i mass of 2.8 x 10(7) M-circle dot and a median blue magnitude M-B similar to -13.2. Sigma(SFR) is derived from Galaxy Evolution Explorer (GALEX) data, while Sigma(gas) is derived from the GMRT-based FIGGS H i 21-cm survey data. We find that Sigma(gas) averaged over the star-forming region of the disc lies below most estimates of the 'threshold density' for star formation, and that the average Sigma(SFR) is also lower than would be expected from the 'Kennicutt-Schmidt' law. This deviation is indicative of an environmental dependence of star formation efficiencies, since the Kennicutt relation was derived from data on the central regions of large spiral and starburst galaxies. The star formation efficiency in small galaxies may be more relevant to modelling of star formation in gas-rich, low-metallicity systems in the early universe. We also use our data to look for small-scale (400 and 200 pc) correlations between Sigma(SFR) and Sigma(gas). For 18 of our 23 galaxies, we find that Sigma(SFR) can be parametrized as having a power-law dependence on Sigma(gas). The power-law relation holds until one reaches the sensitivity limit of the GALEX data, i.e. we find no evidence for a 'threshold density' below which star formation is completely cut off. The power-law slopes and coefficients however vary substantially from galaxy to galaxy, and are in general steeper than the value of similar to 1.4 derived for large galaxies by Kennicutt. Further, as for the globally averaged quantities, the Sigma(SFR) at 400 pc resolution is in general lower than that predicted by the Kennicutt relation, with the deviation decreasing with increasing Sigma(gas). Our computation of Sigma(SFR) uses a calibration that assumes solar metallicity and a standard Salpeter initial mass function (IMF), similarly, the Sigma(gas) we use is not corrected for the molecular gas density. Incorporating corrections for molecular gas and low metallicity will increase the deviation from the Kennicutt relation. Conversely, truncating the IMF at the high-mass end would decrease the deviation from the Kennicutt relation. For the five galaxies for which a power law does not provide a good parametrization of the (Sigma(SFR), Sigma(gas)) relation, there are substantial offsets between the ultraviolet (UV) bright regions and the H i high column density maps. Four of these five galaxies have H i masses near the lower end of our sample distribution, while the remaining galaxy has a large central H i hole. We have 200 pc resolution images for 10 of our galaxies. At this resolution, the offsets between the peaks in the H i and UV images are more pronounced, and a power-law parametrization is possible for only five of the 10 galaxies.