Integrability of motion around galactic razor-thin disks

Abstract

We consider the three-dimensional bounded motion of a test particle around razor-thin disk configurations, by focusing on the adiabatic invariance of the vertical action associated with disk-crossing orbits. We find that it leads to an approximate third integral of motion predicting envelopes of the form $Z(R)\propto[\Sigma(R)]^{-1/3}$, where $R$ is the radial galactocentric coordinate, $Z$ is the z-amplitude (vertical amplitude) of the orbit and $\Sigma$ represents the surface mass density of the thin disk. This third integral, which was previously formulated for the case of flattened 3D configurations, is tested for a variety of trajectories in different thin-disk models.Comment: Version accepted for publication at Celestial Mechanics and Dynamical Astronomy. Replaces arxiv version arxiv:1206.6501. The final publication is available at Springer via http://dx.doi.org/10.1007/s10569-016-9705-

Vertical stability of circular orbits in relativistic razor-thin disks

“…As in the Newtonian case [35], where we have obtained for the Kuzmin disk unexpectedly large regions in which the approximate third integral is valid, we also note that for Vogt-Letelier disks with moderate density profiles (a/m ≫ 1/2) the same situation occurs. Varying the parameters a, E and L z , it is possible to investigate whether the aforementioned approximate third integral is valid along the whole parameter space.…”

“…It is worth pointing out that the vertical stability condition obtained here corresponds, in the Newtonian limit, to the expected result σ > 0 [35,36]. Since the strong energy condition can be considered as one of the possible relativistic extensions to the positivity of mass in Newtonian gravity, we may regard our criterion as a natural counterpart in GR of the Newtonian one.…”

“…Since the strong energy condition can be considered as one of the possible relativistic extensions to the positivity of mass in Newtonian gravity, we may regard our criterion as a natural counterpart in GR of the Newtonian one. On the other hand, the approximated third integral I 3 obtained in Section IV via the adiabatic approximation is not so directly related to the Newtonian result [35,36]. From Eqs.…”

“…an expression similar to the Newtonian case [35]. At this point, it is worth comparing the GR predictions for the shape of the envelopes (36) and for the approximate third integral (38) with the Newtonian equivalent ones.…”

“…Incidentally, all the usual energy conditions for the disk fluid are also verified in this case for the exterior region of (39) (f < 1). For the sake of comparison with the Newtonian limit [35], it is convenient to consider the quantity…”