We present spectra of the extended ^ { 12 } CO and ^ { 13 } CO J = 1 \rightarrow 0 emission along the major axes of 17 nearby galaxies .
Spatial variations in the ratio of CO and ^ { 13 } CO integrated intensities , { \cal R } , are found in nearly every galaxy observed .
There is an overall variation in { \cal R } of 20–40 % from the inner 2 kpc to the disk .
Roughly a third of the survey galaxies have such gradients in { \cal R } detected above the 2 \sigma confidence level .
Though some galaxies show a lower central value of { \cal R } , on average { \cal R } inside 2 kpc is 10–30 % higher than { \cal R } outside of 2 kpc .
The average CO/ ^ { 13 } CO intensity ratio within the central 2 kpc of the survey sources is 11.6 \pm 0.4 ( based on the noise ) \pm 1.5 ( based on systematic uncertainties estimated from daily variations in CO and ^ { 13 } CO intensities ) .
The 1 \sigma dispersion in { \cal R } between galactic nuclei of 4.2 is also quite large .
The average value of { \cal R } outside 2 kpc is 9.8 \pm 0.6 \pm 1.2 with a standard deviation of 4.5 .

An increase in the CO/ ^ { 13 } CO intensity ratio from disk to nucleus may imply that the conversion factor between CO intensity and H _ { 2 } column density , X , is lower in galactic nuclei .
Also variations in physical conditions , most notably the gas kinetic temperature , affect both { \cal R } and X .
Abundance variations probably do not cause the gradient in { \cal R } , though we do not rule out a decrease in effective cloud column densities in galactic nuclei possibly caused by destructive starburst superwinds .
A modest rise in temperature ( less than a factor of 2 or 3 ) from outside a 2 kpc radius towards the nucleus can easily account for the observed gradient .
These results support previous work implying that X is lower in the center of the Milky Way and probably most galactic nuclei .
Therefore calculating H _ { 2 } masses using the standard Galactic X-factor , especially within the central few kpc of galaxies , overestimates the true mass by factors of a few .
The standard X-factor still appears to be appropriate for galactic disks .