Carbon radio recombination lines from gigahertz to megahertz frequencies
  towards Orion A
  Context. The combined use of carbon radio recombination lines (CRRLs) and the
158 $\mu$m-[CII] line is a powerful tool for the study of the energetics and
physical conditions (e.g., temperature and density) of photodissociation
regions (PDRs). However, there are few observational studies that exploit this
synergy. Aims. Here we explore the relation between CRRLs and the 158
$\mu$m-[CII] line in light of new observations and models. Methods. We present
new and existing observations of CRRLs in the frequency range 0.15--230 GHz
with ALMA, VLA, the GBT, Effelsberg 100m, and LOFAR towards Orion~A (M42). We
complement these observations with SOFIA observations of the 158 $\mu$m-[CII]
line. We studied two PDRs: the foreground atomic gas, known as the Veil, and
the dense PDR between the HII region and the background molecular cloud.
Results. In the Veil we are able to determine the gas temperature and electron
density, which we use to measure the ionization parameter and the photoelectric
heating efficiency. In the dense PDR, we are able to identify a layered PDR
structure at the surface of the molecular cloud to the south of the Trapezium
cluster. There we find that the radio lines trace the colder portion of the
ionized carbon layer, the C$^{+}$/C/CO interface. By modeling the emission of
the $158$~$\mu$m-[CII] line and CRRLs as arising from a PDR we derive a thermal
pressure $>5\times10^{7}$ K cm$^{-3}$ and a radiation field
$G_{0}\approx10^{5}$ close to the Trapezium. Conclusions. This work provides
additional observational support for the use of CRRLs and the 158 $\mu$m-[CII]
line as complementary tools to study dense and diffuse PDRs, and highlights the
usefulness of CRRLs as probes of the C$^{+}$/C/CO interface.
