We develop a general expression for the γ-γ absorption coefficient, αγγ, for γ-rays propagating in an arbitrary direction at an arbitrary point in space above an X-ray-emitting accretion disk. The X-ray intensity is assumed to vary as a power law in energy and radius between the outer disk radius, R0 , and the inner radius, Rm., which is the radius of marginal stability for a Schwarzschild black hole. We use our result for αγγ to calculate the γ-γ optical depth, Tγγ, for γ-rays created at height z and propagating at angle Φ relative to the disk axis, and we show that for Φ= 0 and z ≥ R0 , Tγγ oc E^a*z^-2α-3, where α is the X-ray spectral index and E is the γ-ray energy. As an application, we use our formalism to compute the minimum distance between the central black hole and the site of production of the γ-rays detected by EGRET during the 1991 June flare of 3C 279. In order to obtain an upper limit, we assume that all of the X-rays observed contemporaneously by Ginga were emitted by the disk. Our results suggest that the observed γ-rays may have originated within ≤45GM/c^2 from a black hole of mass ≥ 10^9 M0 , perhaps in active plasma located above the central funnel of the accretion disk. This raises the possibility of establishing a direct connection between the production of the observed y-rays and the accretion of material onto the black hole. We also consider the variation of the optical depth as a function of the angle of propagation Φ. Our results indicate that the "focusing" of the γ-rays along the disk axis due to pair production is strong enough to explain the observed degree of alignment in blazar sources. If the γ-rays are produced isotropically in γ-ray blazars, then these objects should appear as bright MeV sources when viewed along off-axis lines of sight.
Becker, P., Kafatos, M. (1995) Implications of Gamma-Ray Transparency Constraints in Blazars: Minimum Distances and Gamma-Ray Collimation, Astrophysical Journal, 453:pp. 83-94. doi: 10.1086/176372