Two scenarios of the radial orbit instability in spherically symmetric collisionless stellar systems

The stability of a two-parameter family of radially anisotropic models with a nonsingular central density distribution is considered. Instability takes place at a sufficiently strong radial anisotropy (the so-called radial orbit instability, ROI). We show that the character of instability depends not only on the anisotropy but also on the energy distribution of stars. If this distribution is such that the highly eccentric orbits responsible for the instability are “trapped” in the radial direction near the center, then the instability develops with a characteristic growth time that exceeds considerably the Jeans and dynamical times of the trapped particles. In this case, the instability takes place only for even spherical harmonics and is aperiodic. If, however, almost all of the elongated orbits reach the outer radius of the sphere, then both even and odd harmonics turn out to be unstable. The unstable modes corresponding to odd harmonics are oscillatory in nature with characteristic frequencies of the order of the dynamical ones. The unstable perturbations corresponding to even harmonics contain only one aperiodic mode and several oscillatory modes, with the aperiodic mode being always the most unstable one. Two main interpretations of the ROI available in the literature have been analyzed: the “classical” Jeans instability related to an insufficient stellar velocity dispersion in the transversal direction and the “orbital” approach relying heavily on the analogous Lynden-Bell bar formation mechanism in disk galaxies. The assumptions that the perturbations are slow (compared to the orbital frequency of stars) and that the shape of the perturbed potential is symmetric are inherent integral conditions for the applicability of the latter. Our solutions show that the orbital approach cannot be considered as a universal one.