We analyze the relationship between maximum cluster mass , and surface densities of total gas ( \Sigma _ { gas } ) , molecular gas ( \Sigma _ { H _ { 2 } } ) , neutral gas ( \Sigma _ { HI } ) and star formation rate ( \Sigma _ { SFR } ) in the grand design galaxy M 51 , using published gas data and a catalog of masses , ages , and reddenings of more than 1800 star clusters in its disk , of which 223 are above the cluster mass distribution function completeness limit .
By comparing the 2-D distribution of cluster masses and gas surface densities , we find for clusters older than 25 Myr that M _ { 3 rd } \propto \Sigma _ { HI } ^ { 0.4 \pm 0.2 } , where M _ { 3 rd } is the median of the 5 most massive clusters .
There is no correlation with \Sigma _ { gas } , \Sigma _ { H 2 } , or \Sigma _ { SFR } .
For clusters younger than 10 Myr , M _ { 3 rd } \propto \Sigma _ { HI } ^ { 0.6 \pm 0.1 } , M _ { 3 rd } \propto \Sigma _ { gas } ^ { 0.5 \pm 0.2 } ; there is no correlation with either \Sigma _ { H _ { 2 } } or \Sigma _ { SFR } .
The results could hardly be more different than those found for clusters younger than 25 Myr in M 33 .
For the flocculent galaxy M 33 , there is no correlation between maximum cluster mass and neutral gas , but we have determined M _ { 3 rd } \propto \Sigma _ { gas } ^ { 3.8 \pm 0.3 } ; M _ { 3 rd } \propto \Sigma _ { H _ { 2 } } ^ { 1.2 \pm 0.1 } ; M _ { 3 rd } \propto \Sigma _ { SFR } ^ { 0.9 \pm 0.1 } .
For the older sample in M 51 , the lack of tight correlations is probably due to the combination of the strong azimuthal variations in the surface densities of gas and star formation rate , and the cluster ages .
These two facts mean that neither the azimuthal average of the surface densities at a given radius , nor the surface densities at the present-day location of a stellar cluster represent the true surface densities at the place and time of cluster formation .
In the case of the younger sample , even if the clusters have not yet traveled too far from their birthsites , the poor resolution of the radio data compared to the physical sizes of the clusters results in measured \Sigma s that are likely quite diluted compared to the actual densities relevant for the formation of the clusters .