[CII] $158\,\mu\mathrm{m}$ line emission from Orion A. I. A template for
  extragalactic studies?
  The [CII] $158\,\mu\mathrm{m}$ fine-structure line is one of the dominant
coolants of the neutral interstellar medium. It is hence one of the brightest
far-infrared emission lines and can be observed not only in star-forming
regions throughout the Galaxy, but also in the diffuse interstellar medium and
in distant galaxies. [CII] line emission has been suggested to be a powerful
tracer of star-formation. We aim to understand the origin of [CII] emission and
its relation to other tracers of interstellar gas and dust. This includes a
study of the heating efficiency of interstellar gas as traced by the [CII] line
to test models of gas heating. We make use of a one-square-degree map of
velocity-resolved [CII] line emission towards the Orion Nebula complex,
including M43 and NGC 1977. The [CII] intensity is tightly correlated with PAH
emission in the IRAC $8\,\mu\mathrm{m}$ band and far-infrared emission from
warm dust. The correlation between [CII] and CO(2-1) is affected by the
detailed geometry of the region. We find particularly low [CII]-over-FIR
intensity ratios towards large columns of (warm and cold) dust, which suggest
the interpretation of the "[CII] deficit" in terms of a "FIR excess". A slight
decrease in the FIR line-over-continuum intensity ratio can be attributed to a
decreased heating efficiency of the gas. We find that, at the mapped spatial
scales, predictions of the star-formation rate from [CII] emission
underestimate the star-formation rate calculated from YSO counts in the Orion
Nebula complex by an order of magnitude. [CII] emission from the Orion Nebula
complex arises dominantly in the cloud surfaces, many viewed in edge-on
geometry. [CII] emission from extended faint cloud surfaces may contribute
significantly to the total [CII] emission on galactic scales.
