On the nature of the radial orbit instability in spherically symmetric
  collisionless stellar systems
  We consider a two-parametric family of radially anisotropic models with
non-singular density distribution in the centre. If highly eccentric orbits are
locked near the centre, the characteristic growth rate of the instability is
much less than the Jeans and dynamic frequencies of the stars (slow modes). The
instability occurs only for even spherical harmonics and the perturbations are
purely growing (aperiodic). On the contrary, if all orbits nearly reach the
outer radius of the sphere, both even and odd harmonics are unstable. Unstable
odd modes oscillate having characteristic frequencies of the order of the
dynamical frequencies (fast modes). Unstable even harmonics contain a single
aperiodic mode and several oscillatory modes, the aperiodic mode being the most
unstable.
  The question of the nature of the radial orbit instability (ROI) is
revisited. Two main interpretations of ROI were suggested in the literature.
The first one refers to the classical Jeans instability associated with the
lack of velocity dispersion of stars in the transverse direction. The second
one refers to Lynden-Bell's orbital approach to bar formation in disc galaxies,
which implies slowness and bi-symmetry of the perturbation. Oscillatory modes,
odd spherical harmonics modes, and non-slow modes found in one of the models
show that the orbital interpretation is not the only possible.
