Envelopes and vertical amplitudes of disc-crossing orbits

Abstract

We recently found that regular orbits in axially symmetric galactic disks have their envelopes Z(R) accurately described by the relation Z(R) ∝ [Σ I (R)] −1/3 , if their amplitudes are comparable to the disk thickness, where Σ 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.

Envelopes for Orbits Around Axially Symmetric Sources with Spheroidal Shape

“…on the vacuum solution Φ(R, z). The result presented above can be considered, at the same time, an application and an extension of the ideas introduced in the references Vieira & Ramos-Caro (2014, 2019, devoted to study the integrability of disc-crossing orbits, but here we focus on test-particle motion that takes place entirely in vacuum. A natural next step in this direction is to introduce this formalism in the realm of general relativity (as an extension of the envelopes in razor-thin disc spacetimes obtained in , in order to describe envelopes of geodesics associated with vacuum solutions of Einstein's equations with axial symmetry.…”

“…. Subsequently, such approach was extended to describe the envelopes of regular orbits with arbitrary vertical amplitudes inside the disc distribution (Vieira & Ramos-Caro 2019), improving the predictions of the adiabatic approximation.…”

Post-Newtonian Hamiltonian dynamics: applications to stationary spacetimes and statistical mechanics

“…Since the 1PN Hamiltonian H is essentially the function (10) written in terms of the canonical variables (x i , p j ), we have, keeping again only terms up to 1 c 2 -order,…”

“…Since the 1PN field equations ( 2)-( 4) are linear in the fields Φ, Ψ and ζ, we can linearly superpose their solutions in order to consider more general systems. In particular, we can superpose the razor-thin disk to axially symmetric structures such as a thick disk, a spheroidal bulge and a spheroidal (stellar + dark matter) halo, as in mass models for spiral galaxies [6,7,8,9,10]. Anyway, we can always analyze the thin-disk contribution separately from the contribution due to the surrounding matter.…”