The Paradox of the Scale-Free Disks

Scale-free disks have no preferred length or time scale. The question has been raised whether such disks have a continuum of unstable linear modes or perhaps no unstable modes at all. We resolve this paradox by analysing the particular case of a gaseous, isentropic disk with a completely flat rotation curve (the Mestel disk) exactly. The heart of the matter is this: what are the correct boundary conditions to impose at the origin or central cusp? We argue that the linear stability problem is ill-posed. From any finite radius, waves reach the origin after finite time but with logarithmically divergent phase. Instabilities exist, but their pattern speeds depend upon an undetermined phase with which waves are reflected from the origin. For any definite choice of this phase, there is an infinite but discrete set of growing modes. Similar ambiguities may afflict general disk models with power-law central cusps. The ratio of growth rate to pattern speed, however, is independent of the central phase. This ratio is derived in closed form for non self-gravitating normal modes. The ratio for self-gravitating normal modes is found numerically by solving recurrence relations in Mellin-transform space.