Title Page

Axisymmetric Instabilities

	Beyond some limit, defined roughly by GM2/R2 - Mv2/R ~ 0 or, alternatively, as P2 ~ R3/(GM), axisymmetric instabilities set in.

Axisymmetric instabilities ===> ring formation, mass shedding, ... .

Nonaxisymmetric Instabilities

Typically, nonaxisymmetric instabilities also set in at high rotation rates (but usually before the axisymmetric instabilities).

Nonaxisymmetric modes break axial symmetry ==> J can be re-distributed both internally and through coupling to external wave fields. If their growth is unchecked, they may lead to fission.

Secular vs. Dynamic Instability

Secular Instability

The Jacobi and Dedekind ellipsoids have lower total energy than does the corresponding Maclaurin spheroid. If a dissipative mechanism exists ==> a Maclaurin spheroid will be driven to the lower energy state on a dissipative time scale

t_sec >> t_dyn

Dynamic Instability

Some modes become dynamically unstable in that for small perturbations,
Q = Q₀ + Q₁ exp(i[f_osc+if_g]t), where |Q₁/Q₀| << 1,

if the frequency

f_g < 0

==> a perturbation grows exponentially on t_dyn

Comment -- General Bar-like Mode Results

The instability limits for the bar-like mode (the m=2 mode) are:

T/|W| ~ 0.038-0.14 ==> secular instability

T/|W| ~ 0.04-0.27 ==> dynamical instability