The Pattern Speed of the OH/IR Stars in the Milky Way

We show how the continuity equation can be used to determine pattern speeds in the Milky Way Galaxy (MWG). This method, first discussed by Tremaine & Weinberg in the context of external galaxies, requires projected positions, $(l,b)$, and line-of-sight velocities for a spatially complete sample of relaxed tracers. If the local standard of rest (LSR) has a zero velocity in the radial direction ($u_{\rm LSR}$), then the quantity that is measured is $\Delta V \equiv \Omega_p R_0 - V_{\rm LSR}$, where $\Omega_p$ is the pattern speed of the non-axisymmetric feature, $R_0$ is the distance of the Sun from the Galactic centre and $V_{\rm LSR}$ is the tangential motion of the LSR, including the circular velocity. We use simple models to assess the reliability of the method for measuring a single, constant pattern speed of either a bar or spiral in the inner MWG. We then apply the method to the OH/IR stars in the ATCA/VLA OH 1612 MHz survey of Sevenster et al, finding $\Delta V = 252 \pm 41$ km/s, if $u_{\rm LSR} = 0$. Assuming further that $R_0 = 8$ kpc and $V_{\rm LSR} = 220$ \kms, this gives $\Omega_p = 59\pm 5$ km/s/kpc with a possible systematic error of perhaps 10 km/s/kpc. The non-axisymmetric feature for which we measure this pattern speed must be in the disc of the MWG.