A Method For Determining Two-Dimensional Orbits In Galactic Disks
  We propose a method for determining the two-dimensional orbits of interstellar matter (ISM) in galactic disks using observed line-of-sight velocity fields. Although four components, consisting of two of position and two of velocity, are necessary to determine two-dimensional orbits, the velocity perpendicular to the line of sight can hardly be determined from observations. We show that the ISM orbits can be approximately determined by assuming that (1) the angular momentum is conserved along each orbit, (2) all orbits are closed in a single rotating frame, and (3) the system is symmetric through the origin. Since the angular momentum can be estimated on the major axis of a disk, a velocity perpendicular to the line of sight can be determined according to the conservation of angular momentum, and thus two components of velocity are obtained. The ISM orbits can be calculated by solving differential equations given by an obtained two-dimensional velocity field when a certain pattern speed is given. The typical deviation from the correct orbit is less than the observational resolution according to our test calculations for modeled velocity fields. We applied this method to the observed data of NGC 4569 and obtained possible orbits for this galaxy. Furthermore, we calculated the mass distribution using our result. 
Hiroyuki Nakanishi