PHASE LAGS IN CYGNUS X-1

Coronae of hot electrons are ubiquitous features in models of the inner regions of accreting black holes and nonmagnetic neutron stars. The scattering optical depth of these coronae inferred from observations is often tau similar to 3, so the energy spectrum of the disk in these regions is likely masked by the spectrum of upscattered photons in the corona and some of the disk properties are thus obscured. Observations of the dependence on Fourier frequency of phase or time lags between photons of different energies provide a window onto the disk not available with the energy spectrum. In our picture, the disk emission is modeled as a photon source injected into a Comptonizing corona; Comptonization may also occur in the disk, but for our purposes the disk emission is simply an input to the corona. We show that, contrary to some claims, the functional dependence of lag on Fourier frequency emerges intact from transit through the corona, module a multiplicative factor (which may in principle be negative, so that a phase lag can be changed to a phase lead), even if the properties of the corona vary with time. We also show that any frequency dependence of the lag due to variation in the corona itself is only second order in the amplitude of the variation, and cannot exceed the transit time similar to ms of the corona; thus, the lags of up to 0.1 s seen in several black hole candidates come from lags in the emission from the disk. Finally, we predict that plots of the time lag versus Fourier frequency in black hole candidates should have a ''shelf'' of constant lag equal to the coronal lag (similar to 1 ms), with the constant being proportional to ln (E(2)/E(1)) for the lag between energies E(2) and E(1). The lack of such a shelf in current observations of several galactic black hole candidates constrains the radii of the coronae to be R less than or equal to 10(8) tau cm.