In a unit circle, we draw the diameter $A C$, and let $B$ be the point on the arc $A C$ such that $\angle B O A = 30^{\circ}$ (where $O$ is the center of the circle). The tangent at $A$ intersects the line $C B$ at $D$. What is the difference between the length of the arc $A B$ and the segment $A D$?