In an effort to model the observed energy spectrum of Cygnus X-1 as well as its hard X-ray lag by Comptonization in inhomogeneous clouds of hot electrons with spherical geometry and various radial density profiles we discovered that: 1) Plasma clouds with different density profiles will lead to different Comptonization energy spectra even though they have the same optical depth and temperature. On the other hand, clouds with different optical depths can produce the same energy spectra as long as their radial density distributions are properly chosen. Thus by fitting the energy spectrum alone, it is not possible to uniquely determine the optical depth of the Comptonization cloud, let alone its density structure. 2) The phase or time difference as a function of Fourier frequency or period for the X-rays in two energy bands is sensitive to the radial density distribution of the scattering cloud. Comptonization in plasma clouds with non-uniform density profiles can account for the long standing puzzle of the frequency-dependent hard X-ray lags of Cygnus X-1 and other sources. Thus simultaneously fitting the observed spectral and temporal X-ray properties will allow us to probe the density structure of the Comptonizing atmosphere and thereby the dynamics of mass accretion onto the compact object.
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