We study the relationship between the gas column density ( \Sigma _ { gas } ) and the star formation rate surface density ( \Sigma _ { SFR } ) for a sample of 23 extremely faint dwarf irregular galaxies drawn from the Faint Irregular Galaxy GMRT Survey ( FIGGS ) .
Our sample galaxies have a median HI mass of 2.8 \times 10 ^ { 7 } { M } _ { \odot } and a median blue magnitude M _ { B } \sim - 13.2 . \Sigma _ { SFR } is derived from GALEX data , while \Sigma _ { gas } is derived from the GMRT based FIGGS HI 21 cm survey data .
We find that \Sigma _ { gas } averaged over the star forming region of the disk 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 ( 23 ) .
This deviation is indicative of an environmental dependence of star formation efficiencies , since the ( 23 ) relation was derived from data on the central regions of large spiral and star burst galaxies .
The star formation efficiency in small galaxies may be more relevant to modeling of star formation in gas rich , low metallicity systems in the early universe .
We also use our data to look for small scale ( 400 pc 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 \sim 1.4 derived for large galaxies by ( 23 ) .
Further , as for the globally averaged quantities , the \Sigma _ { SFR } at 400 pc resolution is in general lower than that predicted by the ( 23 ) 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 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 ( 23 ) relation .
Conversely , truncating the IMF at the high mass end would decrease the deviation from the ( 23 ) relation .
For the 5 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 UV bright regions and the HI high column density maps .
Four of these five galaxies have HI masses near the lower end of our sample distribution , while the remaining galaxy has a large central HI hole .
We have 200 pc resolution images for 10 of our galaxies .
At this resolution , the offsets between the peaks in the HI and UV images are more pronounced , and a power law parametrization is possible for only 5 of the 10 galaxies .