Aside from violent phenomena, regular forms of motions originate often in instabilities and the linear theory with terms ∞ exp (st) yields already significant information. The system, here a spherical star, will be the seat of an instability if R (s) > 0. In general, s will be complex as both conservative (adiabatic) and non-conservative (non-adiabatic) factors are present. However if the latter (small) are neglected, the eigen-values s2 often denoted -σ2 are real. If at least one σ2 < 0, then the star is dynamically unstable.

Radial perturbations. If an appropriate average value Γ1 > 4/3, then all σ2 are positive. If Γ < 4/3 (formation phase: ionization; late evolution: nuclear equilibrium; degeneracy in white dwarfs and neutron stars or radiation in very large masses plus general relativistic effects) the fundamental eigenvalue of only becomes negative.