Gravitational Torques in Spiral Arms

• Gravitational torque from stellar spiral arms is the process where non-axisymmetric mass distributions drive angular momentum exchange in galactic disks, influencing evolution and radial migration.
• It involves swing amplification of density fluctuations and a self-regulating mechanism via the Toomre Q parameter that balances spiral arm growth and disk heating.
• These torques enable efficient radial migration and maintain long-lived spiral patterns, serving as a fundamental mechanism in the secular evolution of disk galaxies.

Gravitational torque from stellar spiral arms refers to the exchange and redistribution of angular momentum within galactic disks driven by the non-axisymmetric gravitational fields of spiral density perturbations. These torques play a fundamental role in secular disk evolution, stellar and gas radial migration, star formation triggering, and the overall dynamical coupling of baryonic and dark matter components. Extensive theoretical and numerical research demonstrates that spiral arm-induced torques are both a driver of disk heating and a key regulatory mechanism in maintaining spiral patterns over cosmic timescales.

1. Origin and Physical Basis of Gravitational Torque in Spiral Arms

Stellar spiral arms constitute large-scale, non-axisymmetric mass overdensities within galactic disks. The gravitational field generated by these overdensities is inherently asymmetric in azimuth, and thus exerts a tangential (azimuthal) force on other disk material—stars and gas alike. The torque, classically expressed as $T = -r\,\partial\Phi/\partial\phi$, where $\Phi$ is the gravitational potential, acts to transfer angular momentum between and within different radial zones of the disk.

Swing amplification of initial density fluctuations produces growing spiral arms. As they grow, these features scatter stars via gravitational interaction, altering their orbits and increasing the radial velocity dispersion, $\sigma_R$ (Fujii et al., 2010). This leads to an increase in the Toomre $Q$ parameter—limiting the arm amplitude and self-regulating further development.

Key analytic expressions quantifying the torque-related heating rate include: $$\frac{dQ}{dt} \simeq \frac{\pi \kappa\, G\, M_d\,\Omega\, \ln\Lambda}{11.6\,v^3}\sum_m \frac{|A_m|^2}{m^2}$$ where $Q$ is Toomre’s parameter, $v$ is the 3D velocity dispersion, $A_m$ are the Fourier amplitudes for mode $m$, $\kappa$ is the epicyclic frequency, and $M_d$ is disk mass (Fujii et al., 2010).

2. Self-Regulation and the Role of Toomre’s $Q$ Parameter

The maximum amplitude, and thus the gravitational torque produced by stellar spiral arms, is constrained by the disk’s dynamical state. The Toomre parameter,

$Q = \frac{\sigma_R\kappa}{3.36\,G\,\Sigma},$

with $\Sigma$ the surface density, acts as a feedback controller. As the random stellar motions (velocity dispersion) increase due to gravitational heating by arms, $Q$ rises, disk stability increases, and arm amplitude saturates or decays (Fujii et al., 2010, Baba et al., 2012). This self-regulation preserves spiral structure over many Gyr without recourse to external cooling—contradicting earlier views that cooling from an interstellar medium was essential for arm longevity.

The feedback loop is summarized as:

• Arm growth increases random stellar motion and $Q$
• Increased $Q$ suppresses further spiral amplification
• Arm amplitude and torque adjust dynamically, maintaining a marginally unstable disk over cosmological times

3. Gravitational Torque and Radial Migration

Spiral arms mediate angular momentum exchange, causing radial migration of stars and, analogously, gas. Stars trailing the arm (on the inside) are accelerated, migrating outward; those leading the arm (on the outside) are decelerated, shifting inward (Grand et al., 2012, Baba et al., 2012). Crucially, this migration occurs with minimal increase in random energy—stellar orbits remain dynamically cold.

Simulation results demonstrate that the pattern speed of the spiral arm closely matches the azimuthal velocity of stars across a broad range of radii, which allows prolonged, coherent torquing and efficient migration. The inclusion of a bar component can further amplify the pattern speed and enhance torque-induced migration (Grand et al., 2012). The exchange can be large—leading to orbital angular momentum changes up to $\sim$50% during individual arm events, as evidenced in L–$\Delta$L diagrams (Grand et al., 2012, Baba et al., 2012).

4. Nonlinear Coupling, Pattern Transience, and Longevity

Stellar spiral arms in pure disks are fundamentally non-steady and recurrent. Swing amplification seeds growth, but as the arm saturates and differential rotation winds it up, nonlinear effects (including the interplay of Coriolis and gravitational forces) cause arm dissolution and reformation elsewhere in the disk (Baba et al., 2012). The Coriolis force can exceed the restoring gravitational force, ejecting stars from the arm as the velocity field reverses from inflow to outflow across the crest.

This inherently nonlinear, non-stationary behavior is responsible for observed features:

• Persistent, recurrent spiral structure over 10+ Gyr with high numerical resolution, even absent a gaseous component (Fujii et al., 2010, Baba et al., 2012)
• Disk heating rate and arm amplitude self-regulate through the $Q$ limit, preserving the morphology while avoiding excessive randomization of stellar orbits

5. Gravitational Torques in Realistic Disk Potentials

Analytical and numerical models of perturbed disk potentials further clarify the nature of spiral arm torques. In models employing spiral-arm-shaped Gaussian groove potentials, stable orbits crowd along the imposed perturbation, aligning density maxima with potential minima between key resonances—especially the ILR and 4:1 resonance (Junqueira et al., 2012). The torque per unit mass can reach a significant fraction (3–6%) of the axisymmetric force for perturbation amplitudes in the range 400–800 km$^2$ s$^{-2}$ kpc$^{-1}$, matching observed arm density contrasts. Resonant phenomena, such as bifurcations at the 4:1 resonance and bar-like structures induced by the central bulge + spiral interaction, arise naturally in this framework (Junqueira et al., 2012).

In the context of the self-gravitating filament equation, gravitational forces with a curvature dependence act to “pull” material toward the galactic center and backward with respect to rotation. The resulting torques drive the coherent winding and maintenance of the spiral pattern in concert with differential disk shear (Buck, 2013).

6. Implications for Galaxy Evolution and Observability

Gravitational torque from stellar spiral arms is pivotal for secular disk evolution. By enabling long-lived spiral features even in collisionless, purely stellar disks, these torques facilitate angular momentum redistribution, disk heating, and radial migration, all while sustaining both disk dynamical coldness and spiral structure over cosmological times (Fujii et al., 2010). This scenario obviates the need for strong external or ISM-based cooling mechanisms.

Observationally, quantitative connections are found between spiral amplitude, disk heating, pattern speed, and resonance locations. Disk galaxies displaying recurrent, non-stationary arms and secular radial mixing in stellar populations, as evidenced by simulations and integral field observations, are interpreted as being governed by the principles outlined above. The predicted amplitude–pitch angle relation of spiral modes, as well as the full pattern of angular momentum redistribution and longevity, aligns with the observed properties of local spiral disk galaxies (Baba et al., 2012).

7. Summary Table: Key Dynamical Relationships

Quantity	Formula / Relation	Physical Role
Toomre $Q$ Parameter	$Q = \frac{\sigma_R\kappa}{3.36G\Sigma}$	Disk stability, controls max spiral amplitude
Disk heating by arm torques	$\frac{dQ}{dt} \propto \sum_m |A_m|^2 / m^2$	Rate set by squared spiral amplitudes
Arm amplitude vs. $Q$	$A_m \sim 3.5 C - 1.0 - 0.75 Q^2$	Max density contrast limited by $Q$
Radial migration mechanism	Stars on trailing gain, leading lose $L_z$ (50% shifts)	Migration with weak disk heating
Torque per mass (local arm)	$T = -r\,\partial\Phi/\partial\phi}$	Drives $L_z$ exchange and disk heating

The gravitational torque from stellar spiral arms emerges as a self-limited, recurrent, and robust agent of angular momentum transport and secular evolution in disk galaxies, governed by a cycle of arm-induced disk heating and subsequent stabilization, which preserves spiral morphology and regulates galactic structural and kinematic evolution over cosmological timescales.