The dynamical ambiguity

Each line of sight along which an observation is made crosses a circle of gas at a given distance from the center of galaxy in two points. Thus an ambiguity arises, because the Doppler velocities with respect to the Sun corresponding to these two points are the same. To resolve this ambiguity is needed to resort to a probabilistic approach. If the line of sight has a galactic latitude different from 0, then the two points with the same velocity lie at different heights on the galactic plane (z). The matter density in the disk decreases exponentially with z, with different scale height for the different components. Then for a given line of sight, we can estimate that the probability of finding matter at distance r is proportional to

$$\zeta = e^{- \frac{1}{2} \frac{r^2 cos^2 b}{z_h^2}} %% controllare$$ (2.6)

where $z_h$ is the scale height. The values we have used in our model are $z_h=100$ pc and $z_h=60$ pc respectively for HI region and for molecular clouds, according to recent estimates of the disk thickness of the two components. The density derived from the radio signal, at a velocity range that gives ambiguity, will be split between the two ambiguity points according to the weighting functions:

$$w_{near} = \frac{ \zeta_{near} }{\zeta_{near} + \zeta_{far} }$$ (2.7)

$$w_{far} = \frac{ \zeta_{far} }{\zeta_{near} + \zeta_{far} }$$ (2.8)

where $\zeta_{near}$ and $\zeta_{far}$ are the functions 2.6 calculated for the near and the far distance giving the same Doppler shift.