Recovery of transverse velocities of steadily rotating patterns in flat galaxies

The transverse velocities of steadily rotating, non–axisymmetric patterns in flat galaxies may be determined by a purely kinematical method, using two dimensional maps of a tracer surface brightness and radial current density. The data–maps could be viewed as the zeroth and first velocity moments of the line–of–sight velocity distribution, which is the natural output of integral– field spectrographs. Our method is closely related to the Tremaine–Weinberg method of estimating pattern speeds of steadily rotating patterns, when the tracer surface brightness satisfies a source–free continuity equation. We prove that, under identical assumptions about the pattern, two dimensional maps may be used to recover not just one number (the pattern speed), but the full vector field of tracer flow in the disc plane. We illustrate the recovery process by applying it to simulated data, and test its robustness by including the effects of noise.