A new approach to the problem of modes in the Mestel disk
We examine the modes admitted by the Mestel disk, a disk with a globally flat rotation curve. In contrast to previous analyses of this problem by Zang (\cite{1976PhDT........26Z}) and Evans & Read (\cite{1998MNRAS.300...83E}, \cite{1998MNRAS.300..106E}), we approximate the orbits to obtain almost closed expressions for the kernel of the integral equation governing the behaviour of the modes. Otherwise we, like them, follow Kalnajs' programme to simultaneously solve the Boltzmann and Poisson equations. We investigate the modes admitted by both the self-consistent and a cut-out Mestel disk, the difference being that in the latter, a part of the matter in the disk is immobilised. This breaks the self-similarity and produces a pronouncedly different picture, both technically and in terms of the disk properties. The self-consistent disk is governed by a Cauchy integral equation, the cut-out disk by an integral equation that can be treated as a Fredholm equation of the second kind. In general, our approximation reproduces the results of the previous works remarkably well, yielding quantities mostly within 5% of the values reported by Zang and Evans & Read and thus also the basic result that in a ``standard'' cut-out disk, only one-armed modes are unstable at the limit of axisymmetric stability. In the self-consistent disk, relatively compact expressions for the kernel allow an intuitive understanding of most of the properties of neutral (non-rotating, non-growing) modes there. We finally show that self-consistent Mestel disks do not admit growing or rotating modes in this sort of stellar-dynamical analysis.