Envelopes for orbits around axially symmetric sources with spheroidal shape

Abstract

We introduce a method to obtain the envelopes of eccentric orbits in vacuum axially symmetric potentials, Φ(R,z), endowed with z-symmetry of reflection, as it is usual in discoidal galaxies and other spheroidal-shaped astrophysical objects. By making the transformation z→a+a2+z2, with a>0, we compute the resulting mass density, referred here as the effective densityρef(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)∝[Σef(R;a≈0)]−1/3, where Σef(R;a) is the integrated surface density associated with ρef(R,z;a). As examples we consider the dynamics in two potentials: the monopole plus quadrupole and the Kalnajs disc.