A Herschel [CII] Galactic plane survey III: [CII] as a tracer of star
  formation
  We study the relationship between the [CII] emission and the star formation
rate (SFR) in the Galactic plane and separate the relationship of different ISM
phases to the SFR. We compare these relationships to those in external galaxies
and local clouds, allowing examinations of these relationships over a wide
range of physical scales. We compare the distribution of the [CII] emission,
with its different contributing ISM phases, as a function of Galactocentric
distance with the SFR derived from radio continuum observations. We also
compare the SFR with the surface density distribution of atomic and molecular
gas, including the CO-dark H2 component. The [CII] and SFR are well correlated
at Galactic scales with a relationship that is in general agreement with that
found for external galaxies. By combining [CII] and SFR data points in the
Galactic plane with those in external galaxies and nearby star forming regions,
we find that a single scaling relationship between the [CII] luminosity and SFR
applies over six orders of magnitude. The [CII] emission from different ISM
phases are each correlated with the SFR, but only the combined emission shows a
slope that is consistent with extragalactic observations. These ISM components
have roughly comparable contributions to the Galactic [CII] luminosity: dense
PDRs (30%), cold HI (25%), CO-dark H2 (25%), and ionized gas (20%). The SFR-gas
surface density relationship shows a steeper slope compared to that observed in
galaxies, but one that it is consistent with those seen in nearby clouds. The
different slope is a result of the use of a constant CO-to-H2 conversion factor
in the extragalactic studies, which in turn is related to the assumption of
constant metallicity in galaxies. We find a linear correlation between the SFR
surface density and that of the dense molecular gas.
