We modelled \mathrm { { H } _ { 2 } } and CO formation incorporating the fractionation and selective photodissociation affecting CO when A _ { V } \lesssim 2 mag .
UV absorption measurements typically have N ( \mathrm { { } ^ { 12 } CO } ) /N ( \mathrm { { } ^ { 13 } CO } ) \approx 65 that are reproduced with the standard UV radiation and little density dependence at n ( H ) \approx 32 - 1024 ~ { } { cm } ^ { -3 } : Densities n ( H ) \lesssim 256 ~ { } { cm } ^ { -3 } avoid overproducing CO. Sightlines observed in mm-wave absorption and a few in UV show enhanced \mathrm { { } ^ { 13 } CO } by factors of 2-4 and are explained by higher n ( H ) \gtrsim 256 ~ { } { cm } ^ { -3 } and/or weaker radiation .
The most difficult observations to understand are UV absorptions having N ( \mathrm { { } ^ { 12 } CO } ) /N ( \mathrm { { } ^ { 13 } CO } ) > 100 and N ( CO ) \gtrsim 10 ^ { 15 } ~ { } { cm } ^ { -2 } .
Plots of { W } _ { CO } vs. N ( CO ) show that { W } _ { CO } remains linearly proportional to N ( CO ) even at high opacity owing to sub-thermal excitation . \mathrm { { } ^ { 12 } CO } and \mathrm { { } ^ { 13 } CO } have nearly the same curve of growth so their ratios of column density/integrated intensity are comparable even when different from the isotopic abundance ratio .
For n ( H ) \gtrsim 128 ~ { } { cm } ^ { -3 } , plots of { W } _ { CO } vs N ( CH ) are insensitive to n ( H ) , and { W } _ { CO } /N ( CO ) \approx 1 \mathrm { K km s ^ { -1 } } / ( 10 ^ { 15 } ~ { } { CO } ~ { } { cm } ^ { -2 } ) : This compensates for small CO/ \mathrm { { H } _ { 2 } } to make { W } _ { CO } more readily detectable .
Rapid increases of N ( CO ) with n ( H ) , N ( H ) and N ( \mathrm { { H } _ { 2 } } ) often render the CO bright , ie a small CO- \mathrm { { H } _ { 2 } } conversion factor .
For n ( H ) \lesssim 64 ~ { } { cm } ^ { -3 } CO enters the regime of truly weak excitation where { W } _ { CO } \propto n ( H ) N ( CO ) . { W } _ { CO } is a strong function of the average \mathrm { { H } _ { 2 } } fraction and models with { W } _ { CO } =1 \mathrm { K km s ^ { -1 } } fall in the narrow range < { f } _ { \mathrm { { H } _ { 2 } } } > 0̄.65-0.8 , or < { f } _ { \mathrm { { H } _ { 2 } } } > 0̄.4-0.5 at { W } _ { CO } 0̄.1 \mathrm { K km s ^ { -1 } } .
The insensitivity of easily-detected CO emission to gas with small < { f } _ { \mathrm { { H } _ { 2 } } } > implies that even deep CO surveys using broad beams may not discover substantially more emission .