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 \tildeτ_ε (ω) denote the first exit time of the particle. For any interval I⊂ [0,2\pi), any r>0, and any δ >0, we estimate the moments of \tildeτ_ε on I and prove the asymptotic formula ∫_I \tildeτ^r_ε (ω) dω = c_r |I| ε^-r +O_δ (ε^-r+(1/8)-δ) as ε → 0⁺, where c_r is the constant (12/\pi²) ∫₀^1/2(x(x^r-1+(1-x)^r-1) +(1-(1-x)^r)/(rx(1-x)) - (1-(1-x)^r+1)/((r+1)x(1-x)))dx. 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.

All three authors were partially supported by ANSTI grant C6189/2000