  We use the IRAM HERACLES survey to study CO emission from 33 nearby spiral
galaxies down to very low intensities. Using atomic hydrogen (HI) data, mostly
from THINGS, we predict the local mean CO velocity from the mean HI velocity.
By renormalizing the CO velocity axis so that zero corresponds to the local
mean HI velocity we are able to stack spectra coherently over large regions as
function of radius. This enables us to measure CO intensities with high
significance as low as Ico = 0.3 K km/s (H2_SD = 1 Msun/pc2), an improvement of
about one order of magnitude over previous studies. We detect CO out to radii
Rgal = R25 and find the CO radial profile to follow a uniform exponential
decline with scale length of 0.2 R25. Comparing our sensitive CO profiles to
matched profiles of HI, Halpha, FUV, and IR emission at 24um and 70um, we
observe a tight, roughly linear relation between CO and IR intensity that does
not show any notable break between regions that are dominated by molecular (H2)
gas (H2_SD > HI_SD) and those dominated by atomic gas (H2_SD < HI_SD). We use
combinations of FUV+24um and Halpha+24um to estimate the recent star formation
rate (SFR) surface density, SFR_SD, and find approximately linear relations
between SFR_SD and H2_SD. We interpret this as evidence for stars forming in
molecular gas with little dependence on the local total gas surface density.
While galaxies display small internal variations in the SFR-to-H2 ratio, we do
observe systematic galaxy-to-galaxy variations. These galaxy-to-galaxy
variations dominate the scatter in relations between CO and SFR tracers
measured at large scales. The variations have the sense that less massive
galaxies exhibit larger ratios of SFR-to-CO than massive galaxies. Unlike the
SFR-to-CO ratio, the balance between HI and H2 depends strongly on the total
gas surface density and radius. It must also depend on additional parameters.
