Direct determination of the epicycle frequency in the galactic disk, and the derived rotation velocity V0

Abstract

We present a method which allows a direct measurement of the epicycle frequency $κ$ in the galactic disk, using the large database on open clusters completed by our group. The observed velocity vector (amplitude and direction) of the clusters in the galactic plane is derived from the catalog data. In the epicycle approximation, this velocity is the sum of the circular velocity, described by the galactic rotation curve, and of a residual velocity, which has a direction that rotates with the frequency $κ$. If for some reason the clusters are formed with non-random initial perturbation velocity direction (measured for instance with respect to the direction of circular rotation), then a plot of the orientation angle of the residual velocity as a function of age reveals the epicycle frequency. The data analysis confirms that this is the case; due to the non-random initial velocities, it is possible to measure $κ$ for different galactic radii. Our analysis considers that the effect of the arms on the stellar orbits is small (the Galactic potential is mainly axis-symmetric); in this sense our results do not depend on any specific model of the spiral structure, like the existence of a given number of spiral arms, or on a particular choice of the radius of corotation. The values of $κ$ provide constraints on the rotation velocity of the disk and on its minimum beyond the solar radius; in particular, $V_0$ is found to be 226 $\pm$ 15 kms$^{-1}$ even if the short scale (R$_0$ = 7.5 kpc) of the galaxy is adopted. The mesured $κ$ at the solar radius is 42$\pm$4 kms$^{-1}$kpc$^{-1}$.