Implications of Gamma-Ray Transparency Constraints in Blazars: Minimum Distances and Gamma-Ray Collimation

We develop a general expression for the gamma - gamma absorption coefficient, alpha(sub gamma(gamma)) for gamma-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, R(sub 0), and the inner radius, R(sub ms) which is the radius of marginal stability for a Schwarzschild black hole. We use our result for alpha(sub gamma(gamma)) to calculate the gamma - gamma optical depth, tau(sub gamma(gamma)) for gamma - rays created at height z and propagating at angle Phi relative to the disk axis, and we show that for Phi = 0 and z greater than or approx equal to R(sub 0), tau(sub gamma(gamma)) proportional to Epsilon(sup alpha)z(sup -2(alpha) - 3), where alpha is the X-ray spectral index and Epsilon is the gamma - 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 gamma-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 gamma - rays may have originated within less than or approx equal to 45 GM/sq c from a black hole of mass greater than or approx equal to 10(exp 9) solar mass, 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 gamma - 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 Phi. Our results indicate that the "focusing" of the gamma - rays along the disk axis due to pair production is strong enough to explain the observed degree of alignment in blazar sources. If the gamma - rays are produced isotropically in gamma - ray blazars, then these objects should appear as bright MeV sources when viewed along off-axis lines of sight.