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

TL;DR

This paper introduces a unified local Tremaine-Weinberg method for measuring galactic pattern speeds, validated through simulations, capable of handling both bars and spirals with variable pattern speeds.

Contribution

It derives a general integral formalism that encompasses all TW variants, enabling local and radially varying pattern speed measurements in galactic disks.

Findings

Accurately recovers constant and radially varying pattern speeds in simulations.

Differentiates between coherent bars and spirals without rigid geometric assumptions.

Provides a robust, flexible tool for analyzing complex galactic dynamics.

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.