Envelopes and vertical amplitudes of disk-crossing orbits
  We recently found that regular orbits in axially symmetric galactic disks
have their envelopes $Z(R)$ accurately described by the relation
$Z(R)\propto[\Sigma_I(R)]^{-1/3}$, if their amplitudes are comparable to the
disk thickness, where $\Sigma_I$ is the surface density of the disk (integrated
over its whole vertical range). Moreover, the usual adiabatic approximation
gives a good description of the orbits' envelopes for low vertical amplitudes.
However, these two approaches are apparently disconnected, since their
expressions differ qualitatively. Our purpose in this paper is to fill this gap
by extending these previous formulae to regular orbits with arbitrary vertical
amplitudes inside the disk. We compare existing $Z(R)$ estimates: the
razor-thin disk case, the adiabatic approximation (low-amplitude orbits in
three-dimensional disks), and the integrated surface-density estimate
(high-amplitude orbits in three-dimensional disks) in order to establish a
connection between them. The formula presented here links the aforementioned
results in an elegant and continuous way, being valid for vertical amplitudes
throughout the whole vertical extension of the disk and with an expression
which has the same form for all regimes. The advantage of the present formalism
is the dependence of $Z(R)$ only on observable quantities, namely the disk's
vertically integrated surface density, without the need to obtain the
gravitational potential for the system.
