We study the global star formation law - the relation between gas and star formation ( SF ) rates in a sample of 181 local galaxies with infrared ( IR ) luminosities spanning almost five orders of magnitude ( { 10 ^ { 7.8 } -10 ^ { 12.3 } ~ { } L _ { \odot } } ) , which includes 115 normal spiral galaxies and 66 ( ultra ) luminous IR galaxies [ ( U ) LIRGs , L _ { IR } ~ { } \geq~ { } 10 ^ { 11 } ~ { } L _ { \odot } ] .
We derive their atomic , molecular gas and dense molecular gas masses using newly available HI , CO and HCN data from the literature , and SF rates are determined both from total IR ( { 8 - 1000 ~ { } \mu m } ) and 1.4 GHz radio continuum ( RC ) luminosities .
In order to derive the disk-averaged surface densities of gas and SF rates , we have taken a novel approach and used high-resolution RC observations to measure the radio sizes for all 181 galaxies .
In our sample , we find that the surface density of dense molecular gas ( as traced by HCN ) has the tightest correlation with that of SF rates ( \Sigma _ { SFR } ) , and is linear in \log - \log space ( power-law slope of N = 1.01 \pm 0.02 ) across the full galaxy sample .
The correlation between surface densities of molecular gas ( \Sigma _ { H _ { 2 } } , traced by CO ) and \Sigma _ { SFR } is sensitive to the adopted value of the CO-to- { H _ { 2 } } conversion factor ( \alpha _ { CO } ) used to infer molecular gas masses from CO luminosities .
For a fixed Galactic value of \alpha _ { CO } , a power law index of 1.14 \pm 0.02 is found .
If instead we adopt values for \alpha _ { CO } of 4.6 and 0.8 for disk galaxies and ( U ) LIRGs , respectively , we find the two galaxy populations separate into two distinct \Sigma _ { SFR } versus \Sigma _ { H _ { 2 } } relations .
Finally , applying a continuously varying \alpha _ { CO } to our sample , we recover a single { \Sigma _ { SFR } } - \Sigma _ { H _ { 2 } } relation with slope of 1.60 \pm 0.03 .
The { \Sigma _ { SFR } } is a steeper function of total gas \Sigma _ { gas } ( molecular gas with atomic gas ) than that of molecular gas \Sigma _ { H _ { 2 } } , and are tighter among low-luminosity galaxies .
We find no correlation between global surface densities of SFRs and atomic gas ( H I ) .