  We present spectra of the extended 12CO and 13CO J=1-0 emission along the
major axes of 17 nearby galaxies. Spatial variations in the ratio of CO and
13CO integrated intensities, R, are found in nearly every galaxy observed.
There is an overall variation in R of 20-40% from the inner 2 kpc to the disk.
Roughly a third of the survey galaxies have such gradients in R detected above
the 2-sigma confidence level. Though some galaxies show a lower central value
of R, on average R inside 2 kpc is 10-30% higher than R outside of 2 kpc.
  An increase in the CO/13CO intensity ratio from disk to nucleus may imply
that the conversion factor between CO intensity and H2 column density, X, is
lower in galactic nuclei. Also variations in physical conditions, most notably
the gas kinetic temperature, affect both R and X. Abundance variations probably
do not cause the gradient in 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 H2 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.
