We observed a 10 \times 20 pc region of the molecular cloud M17 in the ^ { 12 } CO and ^ { 13 } CO J = 3 \rightarrow 2 and J = 2 \rightarrow 1 transitions at the James Clerk Maxwell Telescope to determine their global behavior and to assess the reliability of using ratios of CO line intensities integrated over an entire cloud to determine the physical conditions within the cloud .
Both the ^ { 12 } CO/ ^ { 13 } CO J = 2 \rightarrow 1 and J = 3 \rightarrow 2 line ratios correlate with the ^ { 13 } CO integrated intensity , with smaller line ratios observed at locations with large integrated intensities .
This correlation is likely due to variations in the column density from one position to another within M17 .
The ^ { 12 } CO and ^ { 13 } CO ( J = 3 \rightarrow 2 ) / ( J = 2 \rightarrow 1 ) line ratios show no significant variation from place to place within M17 , even on the peak of the photon-dominated region .
A Large Velocity Gradient analysis of globally averaged line ratios gives a kinetic temperature of 30 K , a density of 0.2 - 5 \times 10 ^ { 5 } cm ^ { -3 } , and a ^ { 12 } CO column density of 1 \times 10 ^ { 18 } cm ^ { -2 } .
This result is in reasonable agreement with the results obtained for individual lines-of-sight through the cloud , which suggests that the typical physical conditions in a molecular cloud can be determined using CO line ratios integrated over the entire cloud .

We estimate the global CO line ratios for M17 to be ^ { 12 } CO/ ^ { 13 } CO J = 2 \rightarrow 1 : 4.5 \pm 0.7 ; ^ { 12 } CO/ ^ { 13 } CO J = 3 \rightarrow 2 : 3.7 \pm 0.9 ; ^ { 12 } CO ( J = 3 \rightarrow 2 ) / ( J = 2 \rightarrow 1 ) : 0.76 \pm 0.19 ; ^ { 13 } CO ( J = 3 \rightarrow 2 ) / ( J = 2 \rightarrow 1 ) : 1.3 \pm 0.3 .
These line ratios generally agree quite well with previous measurements of individual Galactic molecular clouds .
There appears to be a clear trend of increasing ^ { 12 } CO/ ^ { 13 } CO J = 2 \rightarrow 1 and J = 3 \rightarrow 2 line ratios as one moves from Galactic molecular cloud cores to entire Galactic molecular clouds to normal galaxies , similar to the trend seen previously for the ^ { 12 } CO/ ^ { 13 } CO J = 1 \rightarrow 0 line .
These new observations of M17 show that the difference between the ^ { 12 } CO/ ^ { 13 } CO line ratios for Galactic molecular clouds and the disks of spiral galaxies occurs for all three of the lowest rotational transitions .
The most likely explanation of the high line ratios for normal galaxies is a significant contribution to the CO emission by low column density material , such as diffuse molecular clouds or the outer envelopes of giant molecular clouds .
Radial gradients in the relative contribution of low and high column density material in galaxies could be a significant source of uncertainty in derivations of the physical properties of molecular gas in external galaxies .