The Local Tremaine-Weinberg Method for Galactic Pattern Speed: Theory and its Application to IllustrisTNG

Abstract

The Tremaine-Weinberg (TW) method and its variations provide the most direct means to measure the pattern speeds of galactic bars. We establish a unifying framework by deriving an integral form of the continuity equation over an arbitrary closed loop. This naturally defines a local pattern speed for any chosen region in a galactic disk (including bars and spirals). We demonstrate that this intuitive formalism recovers all standard variants of the TW method as special cases corresponding to specific choices of the integration loop. In this paper, we validate this framework and demonstrate its diagnostic power. By applying it to a diverse set of test cases from the TNG50 simulation, including face-on prototype barred galaxies and highly constrained Mock Milky Way standard configurations, we show that this formalism accurately recovers both constant global pattern speeds and radially varying profiles. Rather than relying on rigid geometric approximations, our method naturally differentiates coherent solid-body rotators (bars) from spirals. Our results validate that this unified integral framework provides a robust, geometrically flexible, and practically extensible tool for decoding complex dynamics of galactic structures.

Figures (14)

• Figure 1: Direct geometric interpretation of the local pattern speed. A coherent galactic pattern rotates around the origin with angular velocity $\Omega_{\rm p}$. We verify the mass balance within a small annular sector (bounded by red or black lines). During a time interval $\delta t$, the pattern rotates from the black sector ($A_1 \cup A_2$) to the red sector ($A_2 \cup A_3$). Provided that the tracer particles are neither created nor destroyed, the mass difference between the non-overlapping regions $A_1$ and $A_3$ must obtain a precise balance with the net mass flux crossing the sector boundaries.
• Figure 2: A unified view of the local pattern speed measurement based on the continuity equation, generalized from Figure \ref{['fig:local-geo']}. The local pattern speed $\Omega_{\rm p}$ is defined by line integrals around an arbitrary closed path $\partial S$ enclosing an area $S$. For the pattern to appear stationary in a co-rotating frame, the tracer mass must be conserved. This requires that the net physical mass flux out of the region, given by the numerator of Equation (\ref{['eq:general_loop_integral']}), $\oint \Sigma (\mathbf{v} \cdot \hat{\mathbf{n}}) dl$, must be perfectly balanced by the effective flux induced by the rigid rotation of the pattern, given by the denominator, $\Omega_{\rm p} \oint \Sigma ((\hat{\mathbf{z}} \times \mathbf{r}) \cdot \hat{\mathbf{n}}) dl$. Here, $\hat{\mathbf{n}}$ is the outward normal vector to the boundary element $d\mathbf{l}$. This formulation is universal, independent of the loop's shape, and directly connects the pattern speed to the observable kinematics ($\mathbf{v}$) and density ($\Sigma$) along any chosen boundary.
• Figure 3: Schematic illustrating the application of the local pattern speed framework to a region harboring multiple, radially varying pattern speeds. The annular sector, bounded by the solid red lines, is conceptually subdivided into a series of thinner annuli (separated by dotted red lines), with each sub-annulus $n$ assumed to host a pattern rotating with a distinct local speed $\Omega_{\rm p}(r_n)$. As derived in Section \ref{['sec:multiple_patterns']}, applying our integral formulation (Eq. \ref{['eq:general_loop_integral']}) across the entire outer boundary yields a single measured value, $\langle \Omega_{\rm p} \rangle$. This value is not a simple average, but a weighted mean of the individual speeds $\Omega_{\rm p}(r_n)$, where the weights are determined by the strength of the non-axisymmetric density signature within each sub-annulus. This highlights how the choice of integration path explicitly determines which dynamical components are being measured.
• Figure 4: Local pattern speed analysis for three distinct TNG50 barred galaxies. Each panel corresponds to a specific galaxy: (a) a barred galaxy with grand-design spiral arms, (b) a classic bar with no significant outer structures, and (c) a failed candidate for an ultrafast bar. The layout follows Figure \ref{['fig:radial_validation']}, showing the radial profile of $\Omega_p(R)$ (blue solid line) and circular frequency $\Omega_c(R)$ (orange dashed line).
• Figure 5: Local pattern speed analysis for three non-barred spiral galaxies in TNG50, illustrating three distinct dynamical regimes. The layout is identical to Figure \ref{['fig:radial_validation']}. The grey points represent the binned tangential angular velocity of stellar particles, $\Omega_\varphi(R) = \langle v_{\varphi}/R \rangle$. (a) A two-armed spiral where the pattern speed (blue) forms a distinct plateau significantly slower than the stellar material speed. (b) A multi-armed spiral where the pattern speed tracks the differential rotation of the stars. (c) A complex spiral where the pattern speed locally exceeds the material speed.