Molecular Gas Density Measured with H$_2$CO and CS toward a Spiral Arm
  of M51
  Observations of various molecular lines toward a disk region of a nearby
galaxy are now feasible, and they are being employed as diagnostic tools to
study star-formation activities there. However, the spatial resolution
attainable for a nearby galaxy with currently available radio telescopes is
$10-1000$ pc, which is much larger than the scales of individual star-forming
regions and molecular cloud cores. Hence, it is of fundamental importance to
elucidate which part of an interstellar medium such spatially-unresolved
observations are tracing. Here we present sensitive measurements of the H$_2$CO
($1_{01}-0_{00}$) line at 72 GHz toward giant molecular clouds (GMCs) in the
spiral arm of M51 using the NRO 45 m and IRAM 30 m telescopes. In conjunction
with the previously observed H$_2$CO ($2_{02}-1_{01}$) and CS ($2-1$ and $3-2$)
lines, we derive the H$_2$ density of the emitting regions to be
$(0.6-2.6)\times10^4$ cm$^{-3}$ and $(2.9-12)\times10^4$ cm$^{-3}$ for H$_2$CO
and CS, respectively, by the non-LTE analyses, where we assume the source size
of $0.8-1$ kpc and the gas kinetic temperature of $10-20$ K. The derived H$_2$
density indicates that the emission of H$_2$CO and CS is not localized to
star-forming cores, but is likely distributed over an entire region of GMCs.
Such widespread distributions of H$_2$CO and CS are also supported by models
assuming lognormal density distributions over the 1 kpc region. Thus,
contributions from the widespread less-dense components should be taken into
account for interpretation of these molecular emission observed with a
GMC-scale resolution. The different H$_2$ densities derived for H$_2$CO and CS
imply their different distributions. We discuss this differences in terms of
the formation processes of H$_2$CO and CS.
