Envelopes for orbits around axially symmetric sources with spheroidal shape

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