Envelopes and vertical amplitudes of disc-crossing orbits

Abstract

We recently found that regular orbits in axially symmetric galactic discs have their envelopes Z(R) accurately described by the relation Z(R)∝[Σ_I(R)]^-1/3, if their amplitudes are comparable to the disc thickness, where Σ_I is the surface density of the disc (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 disc. We compare existing Z(R) estimates: the razor-thin disc case, the adiabatic approximation (low-amplitude orbits in three-dimensional discs), and the integrated surface density estimate (high-amplitude orbits in three-dimensional discs) 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 disc and with an expression that 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 disc's vertically integrated surface density, without the need to obtain the gravitational potential for the system.