Orthosystoles and orthokissing numbers

For hyperbolic surfaces with geodesic boundary, we study the orthosystole, i.e. the length of a shortest essential arc from the boundary to the boundary. We recover and extend work by Bavard completely characterizing the surfaces maximizing the orthosystole in the case of a single boundary component. For multiple boundary components, we construct surfaces with large orthosystole and show that their orthosystole grows, as the genus goes to infinity, at the same rate as Bavard's upper bound.