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

TL;DR

This paper introduces a numerical method to calculate the gravitational potential of thin spiral structures in galaxies, evaluates the tight-winding approximation's accuracy, and derives analytic models for spiral potentials with various amplitudes.

Contribution

It provides an efficient numerical approach, assesses the approximation's validity, and derives analytic potential models for razor-thin spiral perturbations.

Findings

Tight-winding approximation is accurate for pitch angles up to 20°.

Derived analytic potential for logarithmic spirals with arbitrary amplitude.

Potential behavior beyond spiral edges is non-winding and decays with radius.

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.