On the stability criteria for equatorial circular orbits in Galactic Dynamics: I.Newtonian Thin Disks

We make a revision of the stability criteria for equatorial circular orbits, obtained from the epicyclic approximation, which is widely used in Newtonian models for axisymmetric galaxies. We find that, for the case of thin disk models, the criteria of vertical stability must be reformulated, due to the discontinuity in the gravitational field. We show that, for a model characterized by a surface mass density $\Sigma$, the necessary and sufficient condition to have vertically stable orbits is that $\Sigma>0$. On the other hand, the criteria for radial stability is the same as in thick diks, i.e. that the epicyclic frequency is positive.