Envelopes for orbits around axially symmetric sources with spheroidal shape

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.