PHASE LAGS IN CYGNUS X-1

Coronae of hot electrons are ubiquitous features in models of the inne r regions of accreting black holes and nonmagnetic neutron stars. The scattering optical depth of these coronae inferred from observations i s 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. Observation s of the dependence on Fourier frequency of phase or time lags between photons of different energies provide a window onto the disk not avai lable with the energy spectrum. In our picture, the disk emission is m odeled as a photon source injected into a Comptonizing corona; Compton ization may also occur in the disk, but for our purposes the disk emis sion 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 chang ed to a phase lead), even if the properties of the corona vary with ti me. We also show that any frequency dependence of the lag due to varia tion 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 co rona; thus, the lags of up to 0.1 s seen in several black hole candida tes come from lags in the emission from the disk. Finally, we predict that plots of the time lag versus Fourier frequency in black hole cand idates should have a ''shelf'' of constant lag equal to the coronal la g (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 s helf 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.