  Although gravitational waves tend to erase eccentricity of an inspiraling
binary system, ellipticity can be generated in the presence of surrounding
matter. We present a semi-analytical method for understanding the eccentricity
distribution of binary black holes in the presence of a supermassive black hole
in a galactic center. Given a matter distribution, we show how to determine the
resultant eccentricity analytically in the presence of both tidal forces and
evaporation up to one cutoff and one matter-distribution-independent function,
paving the way for understanding the environment of detected inspiraling black
holes. We furthermore generalize Kozai-Lidov dynamics to situations where
perturbation theory breaks down for short time intervals, allowing more general
angular momentum exchange, such that eccentricity is generated even when all
bodies orbit in the same plane.
