We have mapped over 50 massive , dense clumps with four dense gas tracers : HCN J = 1 - 0 and 3 - 2 ; and CS J = 2 - 1 and 7 - 6 transitions .
Spectral lines of optically thin H ^ { 13 } CN 3-2 and C ^ { 34 } S 5-4 were also obtained towards the map centers .
These maps usually demonstrate single well-peaked distributions at our resolution , even with higher J transitions .
The size , virial mass , surface density , and mean volume density within a well-defined angular size ( FWHM ) were calculated from the contour maps for each transition .
We found that transitions with higher effective density usually trace the more compact , inner part of the clumps but have larger linewidths , leading to an inverse linewidth-size relation using different tracers .
The mean surface densities are 0.29 , 0.33 , 0.78 , 1.09 g cm ^ { -2 } within FWHM contours of CS 2-1 , HCN 1-0 , HCN 3-2 and CS 7-6 , respectively .
We find no correlation of L _ { IR } with surface density and a possible inverse correlation with mean volume density , contrary to some theoretical expectations .
Molecular line luminosities L ^ { \prime } _ { mol } were derived for each transition .
We see no evidence in the data for the relation between L ^ { \prime } _ { mol } and mean density posited by modelers .
The correlation between L ^ { \prime } _ { mol } and the virial mass is roughly linear for each dense gas tracer .
No obvious correlation was found between the line luminosity ratio and infrared luminosity , bolometric temperature , or the L _ { IR } / M _ { Vir } ratio .
A nearly linear correlation was found between the infrared luminosity and the line luminosity of all dense gas tracers for these massive , dense clumps , with a lower cutoff in luminosity at \mbox { $L _ { IR } $ } = \mbox { $ 10 ^ { 4.5 } $ } L _ { \odot } .
The L _ { IR } - L ^ { \prime } _ { HCN 1 - 0 } correlation agrees well with the one found in galaxies .
These correlations indicate a constant star formation rate per unit mass from the scale of dense clumps to that of distant galaxies when the mass is measured for dense gas .
These results support the suggestion that starburst galaxies may be understood as having a large fraction of gas in dense clumps .