Envelopes for orbits around axially symmetric sources with spheroidal shape

TL;DR

This paper presents a novel method to determine the envelopes of eccentric orbits in axially symmetric gravitational potentials by transforming the potential and analyzing the resulting effective density, with applications to specific astrophysical models.

Contribution

It introduces a new approach using a transformation to compute orbit envelopes in axially symmetric potentials, linking them to an effective surface density.

Findings

Envelope formula Z(R) ∝ [Σ_ef(R;a≈0)]^(-1/3)

Method applied to monopole plus quadrupole potential

Method applied to Kalnajs disc model

Abstract

We introduce a method to obtain the envelopes of eccentric orbits in axially symmetric potentials, $\Phi(R,z)$, endowed with $z$-symmetry of reflection. By making the transformation $z\rightarrow a+\sqrt{a^{2}+ z^{2}}$, with $a>0$, we compute the resulting mass density, referred here as the \emph{effective density} $\rho_{\rm ef}(R,z;a)$, in order to calculate the envelopes $Z(R)$ of orbits in the meridional plane $(R,z)$. We find that they obey the approximated formula $Z(R)\propto [\Sigma_{\rm ef}(R;a\approx 0)]^{-1/3}$, where $\Sigma_{\rm ef}(R;a)$ is the integrated surface density associated with $\rho_{\rm ef}(R,z;a)$. As examples we consider the dynamics in two potentials: the monopole plus quadrupole and the Kalnajs disc.