The average length of a trajectory in a certain billiard in a flat two-torus

We remove a small disc of radius ε > 0 from the flat torus T² and consider a point-like particle that starts moving from the center of the disk with linear trajectory under angle ω. Let ~τ(_ε)(ω) denote the first exit time of the particle. For any interval I . [0, 2π), any r > 0, and any δ > 0, we estimate the moments of ~τ on I and prove the asymptotic formula (equation presented), where c_r is the constant (equation presented). A similar estimate is obtained for the moments of the number of reflections in the side cushions when T² is identified with [0, 1)².