Gravitational Potential in Spiral Galaxies

• Gravitational potential of spiral perturbations is the non-axisymmetric field produced by spiral density structures that modulate stellar and gas orbits in galactic disks.
• Advanced methods like Fourier decomposition and the WKB approximation reveal key features including phase lags, pitch angle variations, and efficient angular momentum transport.
• Numerical and analytic models highlight that edge effects and non-local torques critically influence the global dynamics and evolution of spiral galaxies.

The gravitational potential of spiral perturbations refers to the non-axisymmetric component of a galactic disk’s gravitational field generated by the presence of spiral density structures. This potential modulates the orbits of stars and gas in both stellar and gaseous disks, organizes large-scale morphological features, mediates angular momentum transport, and ultimately controls the formation and evolution of spiral structure and its connection to galactic dynamics and star formation.

1. Theoretical Foundations and Local Approximations

The canonical approach for modeling the gravitational potential of spiral arms is to consider the razor-thin disk limit and decompose the surface density perturbation into azimuthal Fourier components: $$\Sigma(\mathbf{R}) = \sum_{m} \Sigma_m(R) \, e^{i m \phi}$$ The resulting mid-plane potential in cylindrical coordinates is given by the two-dimensional Poisson integral: $$\Psi(\mathbf{R}) = -G \iint \frac{\Sigma(\mathbf{R}') \, d^2\mathbf{R}'}{|\mathbf{R} - \mathbf{R}'|}$$ For each Fourier mode with $m$ arms, this is recast using Laplace coefficients: $$\Psi_m(R) = -\pi G \int_0^\infty dR' \, \Sigma_m(R') \, b_{1/2}^m(R/R')$$ with

$$b_s^m(\alpha) = \frac{1}{\pi} \int_0^{2\pi} \frac{e^{i m \phi} \, d\phi}{(1 - 2\alpha\cos\phi + \alpha^2)^s}$$

This formulation is well-suited for accurate numerical calculation, especially when the surface density profile is arbitrary and the global structure or edge effects cannot be neglected (Dehnen, 22 Jul 2025).

The tight-winding (WKB) approximation is commonly employed for local analysis and is valid in the limit of small pitch angles $\alpha\lesssim 20^\circ$: $$\Psi(\mathbf{R}) \simeq -\frac{2\pi G}{k(R)} \Sigma(\mathbf{R})$$ where $k(R) = m/(R \sin\alpha)$ is the local horizontal wavenumber. This approach is efficient and physically transparent for tightly wound arms, though it erases any phase offset between potential and density.

2. Beyond First-Order: Potential-Density Pairs and Second-Order Approximations

The standard WKB relation only captures the leading-order ($\propto 1/k$) behavior, failing to account for the phase lag between potential and density or for global effects such as non-local torque. Kalnajs (1971) derived analytic potential-density pairs for scale-invariant spirals ($\Sigma\propto R^{-3/2}$), which are generalized in recent work to include arbitrary power-law amplitudes and logarithmic spiral phase dependence: $$\Sigma(R,\phi) = S_{m,0} (R/R_0)^{-\gamma} e^{i m(\phi - \lambda \ln(R/R_0))}$$ with corresponding potential

$$\Psi(\mathbf{R}) = -2\pi G \, K_m(\zeta) \, R \, \Sigma(\mathbf{R})$$

where $\zeta = m\lambda - i(\gamma-3/2)$ and $K_m(\zeta)\simeq (\zeta^2 + m^2 - 1/4)^{-1/2}$ (Dehnen, 22 Jul 2025).

Approximating a general spiral locally with such scale-invariant models yields a “second-order tight-winding approximation.” This includes finite phase offset $\delta\psi$ and more realistic angular momentum transport: $$\Psi(\mathbf{R}) \approx -2\pi G \frac{\sin\alpha}{m} \left[1 + i\sigma \frac{\sin2\alpha}{2m}(\gamma-3/2)\right] R\Sigma(\mathbf{R})$$ ($\sigma = \pm 1$ encodes trailing/leading arm sense.) Key predictions are:

• For tightly wound, trailing spirals ($\sigma=+1$), the potential lags density at small radii, leads at large radii,
• A phase offset $\delta\psi \sim -(\sigma/m^2)\,\alpha(\gamma-3/2)$ for small $\alpha$,
• Nonlocal gravitational torques $$\langle\tau_\phi\rangle \sim -(\sigma \pi G R S_m^2 \sin\alpha \sin2\alpha)/2m$$ enable outward angular momentum transport.

3. Edge Effects and Non-locality

When spiral arms are confined to a finite radial extent (i.e., not globally extended logarithmic spirals), the gravitational potential ceases to trace the tight winding of local density at the boundaries. Outside the spiral’s radial range, the potential “unwinds” (pitch angle approaches $90^\circ$), and its amplitude follows distinct scaling: $$\Psi(R)\propto \begin{cases} R^m, & R \rightarrow 0 \ R^{-1-m}, & R \rightarrow \infty \end{cases}$$ This transition cannot be captured by any purely local (WKB-type) approximation (Dehnen, 22 Jul 2025). Accurate treatment of spiral arm edges requires global numerical integration of the Poisson equation for the projected density—this is essential to describe outer-arm effects, non-winding behavior of the potential at the disk’s edge, and the correct global angular momentum budget.

4. Physical Implications: Phase Offset, Pitch, and Angular Momentum Transport

The phase lag and change in pitch angle between spiral potential and density have direct dynamical consequences. When a phase offset exists, the gravitational torque across an annulus is nonzero, resulting in radial angular momentum transfer. The analytic result for the second-order WKB approximation predicts that for trailing spirals:

• The potential gains a nonzero phase lag that grows with pitch angle and power-law index,
• The pitch angle of the potential increases with radius,
• The net gravitational torque leads to outward transport of angular momentum, regulating disk evolution.

These effects are absent in purely first-order (tight-winding) treatments and only emerge in numerical or improved analytic models that go beyond plane-wave approximations.

5. Numerical Implementation and Practical Considerations

The calculation of the spiral gravitational potential for arbitrary surface density profiles relies on:

• Decomposition of the density into Fourier azimuthal modes,
• Evaluation of Laplace coefficients $b_{1/2}^m(\alpha)$, often using upward/downward recursion or as tabulated via elliptic integrals,
• Addressing the logarithmic singularity at $R'=R$ by coordinate transformations $(R' - R)/(R' + R + a)$ and smoothing, e.g., substituting $x=u^3$,
• Efficient quadrature (e.g., Gauss–Legendre) for integration over radius.

The method is robust and recovers the standard WKB result in the appropriate limit, but quantitatively accurate for moderate-to-large pitch angles and necessary for assessing inner/outer zone behavior (Dehnen, 22 Jul 2025).

6. Physical and Observational Context

The development and refinement of the gravitational potential formalism for spiral arms, including accurate treatment of phase, edge, and torque properties, are critical for interpreting the secular evolution of spiral disks, morphologies of real galaxies, pattern speeds, and angular momentum redistribution. Observational diagnostics such as the pitch angle of potential, phase offset between observed arm tracers, and kinematic signatures may be used to test these theoretical expectations.

Notably, the limitations of WKB methods for $\alpha\gtrsim20^\circ$, the emergence of non-locality and secular torques, and the quantifiable phase shift all have direct observational and dynamical significance (Dehnen, 22 Jul 2025). These considerations anchor the theory of spiral perturbation potentials as a central tool for modern galactic dynamics research.