The pattern speed of the nuclear disk of M31 using a variant of the Tremaine-Weinberg method

The twin peaks in the nucleus of M31 have been interpreted by Tremaine as a thick, eccentric, disk of stars orbiting a massive dark object; the required alignment of the apoapsides of the stellar orbits could be maintained by self-gravity, and the whole structure might be a discrete, nonlinear eigenmode. The pattern speed of this mode could, in principle, be determined by the Tremaine-Weinberg (TW) method, which requires measurements of the surface brightness, and radial velocity along a strip parallel to the line of nodes. However, spectroscopic observations along the line of nodes are not available. We propose a variant of the TW method, which exploits a basic feature of the eccentric disk model, to extract estimates of the pattern speed from {\it Hubble Space Telescope} spectroscopic data, taken along the line joining the two peaks. Within limitations imposed by the data, we estimate that the pattern rotates in a prograde manner and, for an assumed disk inclination of ${77}^{\circ }$, the pattern speed $\mathrm{\mid }{\mathrm{\Omega }}_p\mathrm{\mid }<30\mathrm{k}\mathrm{m}{\mathrm{s}}^{-1}\mathrm{p}{\mathrm{c}}^{-1}$, or period more than $200,000$ years.