The gravitational potential of spiral perturbations I. The 2D (razor-thin) case

Abstract

I developed an efficient numerical method for obtaining the gravitational potential of razor-thin spiral perturbations and used it to assess the standard tight-winding approximation, which is found to be reasonably accurate for pitch angles $\alpha\lesssim20^\circ$. I derived the analytic potential of razor-thin logarithmic spirals with an arbitrary power-law amplitude. Approximating a spiral locally by one of these models provides a second-order tight-winding approximation that predicts the phase offset between the spiral potential and density, the resulting radially increasing pitch of the potential, and the nonlocal outward angular-momentum transport by gravitational torques. Beyond the inner and outer edge of a spiral with $m$ arms, its potential is not winding ($\alpha=90^\circ$), decays like $R^m$ and $R^{-1-m}$, respectively, and cannot be predicted by a local approximation.