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

TL;DR

This paper develops an efficient numerical method to compute the gravitational potential from razor-thin spiral perturbations in the disc plane and uses it to rigorously test the traditional tight-winding (WKB) approximation. By extending scale-invariant spiral models to general power-law amplitudes and deriving a second-order tight-winding form, the author demonstrates improved predictions for the potential phase, pitch, and the non-local angular-momentum transport via gravitational torques. The analysis reveals that, for typical spirals with exponentially declining amplitudes, the conventional first-order WKB approach is accurate in amplitude for small pitch angles (\alpha \lesssim 20^{\circ}) but fails to capture phase offsets and radial variation, especially near spiral edges; the second-order model provides substantial improvements. The work further shows that the net torque must vanish globally, yet trailing spirals still drive outward angular-momentum transport locally, and that many existing spiral-potential models in the literature can be physically inconsistent or unreliable if derived from inaccurate potentials. Overall, the study provides a robust framework for computing spiral potentials and torques, clarifies the limits of common approximations, and emphasizes the need for careful 2D/3D modeling of spiral perturbations in galactic discs.

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 $α\lesssim20^\circ$. 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 ($α=90^\circ$), decays like $R^m$ and $R^{-1-m}$, respectively, and cannot be predicted by a local approximation.

Figures (10)

• Figure 1: Assessing the accuracy of the approximation \ref{['eq:K:asymp']} for $m=2$ (for $m>2$ the accuracy is better). The real (imaginary) parts are even (odd) functions of $\gamma-3/2$ (hence the relations for $\gamma<3/2$ are hidden in the top panel). I also show (dashed) the relation given by Kalnajs1971.
• Figure 2: Assessing the tight-winding (WKB) approximation for scale-invariant spirals. The relative error of the density amplitude and the error in the phase offset $\delta\psi\equiv\psi_{\Psi}-\psi_{\Sigma}$ are plotted vs. pitch angle $\alpha$ for $m=2$ and various values of the exponent $\gamma$ (for these scale-invariant models, the errors are the same at each radius). Since the first-order approximations give $\delta\psi_{\mathrm{approx}}=0$, their error reflects the actual phase offset of the models.
• Figure 3: Assessing the ability of the first-order (left) and our novel (right) tight-winding approximations to provide a gravitational potential for the target density \ref{['eq:Sigma:exp:target']} of a logarithmic spiral with pitch angle $\alpha$ and exponentially declining amplitude. From the density that is actually generated by the approximate potential, I plot the radial profiles of the relative amplitude error (top) and the errors in phase (middle) and pitch angle (bottom), which for the first-order methods are also the respective offsets between the potential and density because to first order, $\psi_{\Psi}=\psi_{\mathrm{target}}$.
• Figure 4: Properties of a spiral with density \ref{['eq:Sigma:exp:target']} and its (numerically computed) potential. Radial profiles of the amplitude ratio to the first-order tight-winding approximation (Eqs. \ref{['eq:WKB']} and \ref{['eq:WKB:k:sin']}), the offsets of phase and pitch angle between the potential and density, and the torque per annulus ($<0$ if angular momentum is lost for stars in a galaxy with a trailing spiral).
• Figure 5: Properties of the (numerically computed) density that generates the potential \ref{['eq:Psi:Kuzmin']} with the phase $\psi=\psi_{\mathrm{R79}}(R)$ (Eq. \ref{['eq:psi:Roberts']}) for $R_{\mathrm{sp}}=1.8a$ (grey vertical line) and $N=5$ (as used by Roberts1979) and various values for $\alpha_\infty$ (Roberts1979 used $\alpha_\infty=20^\circ$). For $\alpha_\infty=90^\circ$, $\psi=0$, and the density is given by Eq. \ref{['eq:Sigma:Kuzmin']}.