The gravitational potential of spiral perturbations

I. The 2D (razor-thin) case

Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Mönchhofstraße 12-14, 69120 Heidelberg, Germany

^⋆ Corresponding author: walter.dehnen@uni-heidelberg.de

Received: 16 June 2025
Accepted: 1 August 2025

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 α ≲ 20°. 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 (α = 90°), decays like R^m and R^{−1 − m}, respectively, and cannot be predicted by a local approximation.