Large scale radial stability density of Hill's equation Henk Broer, Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Nijenborgh 9, 9747 AC Groningen, The Netherlands, Mark Levi, Department of Mathematics, Pennsylvania State University, University Park, PA, USA, Carles Simo, Departament de Matem\`atica Aplicada i An\`alisi, Gran Via 585, 08007 Barcelona, Spain Abstract This paper deals with large scale aspects of Hill's equation $\ddot{x}+(a+bp(t))x=0$, where $p$ is periodic with a fixed period. In particular the interest is the asymptotic radial density of the stability domain in the $(a,b)$-plane. It turns out that this density changes discontinuous at a certain direction and exhibits and interesting asympotic fine structure. Most of the paper deals with the case where $p$ is a Morse function with one maximum and one minimum, but also the discontinuous case of square Hill's equation is studied, where the density behaves differently.