A simple formula for the third integral of motion of disk-crossing stars
  in the Galaxy
  We present an analytical simple formula for an approximated third integral of
motion associated with nearly equatorial orbits in the Galaxy:
$I_{3}=Z\Sigma_{I}^{1/3}$, where $Z(R)$ is the vertical amplitude of the orbit
at galactocentric distance $R$ and $\Sigma_{I}(R)$ is the integrated dynamical
surface mass density of the disk, a quantity which has recently become
measurable. We also suggest that this relation is valid for disk-crossing
orbits in a wide variety of axially symmetric galactic models, which range from
razor-thin disks to disks with non-negligible thickness, whether or not the
system includes bulges and halos. We apply our formalism to a Miyamoto-Nagai
model and to a realistic model for the Milky Way. In both cases, the results
provide fits for the shape of nearly equatorial orbits which are better than
the corresponding curves obtained by the usual adiabatic approximation when the
orbits have vertical amplitudes comparable to the disk's scale height. We also
discuss the role of this approximate third integral of motion in modified
theories of gravity.
