The X-ray spectra of accreting black hole systems generally contain components ( sometimes dominating the total emission ) which are well-fit by thermal Comptonization models with temperatures \sim 100 keV .
We demonstrate why , over many orders of magnitude in heating rate and seed photon supply , hot plasmas radiate primarily by inverse Compton scattering , and find equilibrium temperatures within a factor of a few of 100 keV .
We also determine quantitatively the ( wide ) bounds on heating rate and seed photon supply for which this statement is true .

Plasmas in thermal balance in this regime obey two simple scaling laws : \Theta \tau _ { T } \simeq 0.1 ( l _ { h } / l _ { s } ) ^ { 1 / 4 } ; and \alpha \simeq 1.6 ( l _ { s } / l _ { h } ) ^ { 1 / 4 } .
Here the hot plasma heating rate compactness is l _ { h } , the seed photon compactness is l _ { s } , the temperature in electron rest mass units is \Theta , and the Thomson optical depth is \tau _ { T } .
The coefficient in the first expression is weakly-dependent on plasma geometry ; the second expression is independent of geometry .
Only when l _ { s } / l _ { h } is a few tenths or greater is there a weak secondary dependence in both relations on \tau _ { T } .

Because \alpha is almost independent of everything but l _ { s } / l _ { h } , the observed power law index may be used to estimate l _ { s } / l _ { h } .
In both AGN and stellar black holes , the mean value estimated this way is l _ { s } / l _ { h } \sim 0.1 , although there is much greater sample dispersion among stellar black holes than among AGN .
This inference favors models in which the intrinsic ( as opposed to reprocessed ) luminosity in soft photons entering the hot plasma is small , or in which the hard X-ray production is comparatively distant from the source of soft photons .
In addition , it predicts that \Theta \tau _ { T } \simeq 0.1 – 0.2 , depending primarily on plasma geometry .
It is possible to construct coronal models ( i.e . models in which l _ { s } / l _ { h } \simeq 0.5 ) which fit the observed spectra , but they are tightly constrained : \tau _ { T } must be \simeq 0.08 and \Theta \simeq 0.8 .