BAT AGN Spectroscopic Survey -- XII. The relation between coronal
  properties of Active Galactic Nuclei and the Eddington ratio
  The bulk of the X-ray emission in Active Galactic Nuclei (AGN) is produced
very close to the accreting supermassive black hole (SMBH), in a corona of hot
electrons which up scatters optical and ultraviolet photons from the accretion
flow. The cutoff energy ($E_{\rm C}$) of the primary X-ray continuum emission
carries important information on the physical characteristics of the X-ray
emitting plasma, but little is currently known about its potential relation
with the properties of accreting SMBHs. Using the largest broad-band (0.3-150
keV) X-ray spectroscopic study available to date, we investigate how the corona
is related to the AGN luminosity, black hole mass and Eddington ratio
($\lambda_{\rm Edd}$). Assuming a slab corona the median values of the
temperature and optical depth of the Comptonizing plasma are $kT_{\rm e}=105
\pm 18$ keV and $\tau=0.25\pm0.06$, respectively. When we properly account for
the large number of $E_{\rm C}$ lower limits, we find a statistically
significant dependence of the cutoff energy on the Eddington ratio. In
particular, objects with $ \lambda_{\rm Edd}>0.1$ have a significantly lower
median cutoff energy ($E_{\rm C}=160\pm41$ keV) than those with $\lambda_{\rm
Edd}\leq 0.1$ ($E_{\rm C}=370\pm51$ keV). This is consistent with the idea that
radiatively compact coronae are also cooler, because they tend to avoid the
region in the temperature-compactness parameter space where runaway pair
production would dominate. We show that this behaviour could also
straightforwardly explain the suggested positive correlation between the photon
index ($\Gamma$) and the Eddington ratio, being able to reproduce the observed
slope of the $\Gamma-\lambda_{\rm Edd}$ trend.
