# Envelopes and vertical amplitudes of disk-crossing orbits

## TL;DR

This paper develops a unified formula for the envelopes of disk-crossing orbits in galactic disks, valid across all vertical amplitudes, linking previous models and relying only on observable surface density.

## Contribution

It extends existing models to describe regular orbits with arbitrary vertical amplitudes using a continuous, observable-based formula.

## Key findings

- Unified formula valid for all vertical amplitudes
- Connects previous thin-disk and high-amplitude models
- Depends only on observable surface density

## Abstract

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.