THEORETICAL IDEAS

Axisymmetric Instabilities

	Beyond some limit, roughly defined by GM2/R2 - Mv2/R ~ 0 or, alternatively, as P2 ~ R3/(GM) ~ 1/density, axisymmetric instabilities set in (ring formation, mass shedding, ...).

Nonaxisymmetric Instabilities

Typically, nonaxisymmetric instabilities also set in at high rotation rates (but usually before the axisymmetric instabilities). Nonaxisymmetric instabilities break the symmetry which has important consequences (the first mode to set in drives an oblate object to a triaxial state), that is, it forms a bar.



Nonaxisymmetric modes ==> J can be re-distributed both internally and through coupling to external wave fields and if their growth is unchecked, they can the object to fission.

Kinds of Instability: Secular and Dynamic Instability

Secular Instability

The Jacobi and Dedekind ellipsoids have lower total energy than does the corresponding Maclaurin spheroid. So, if a dissipative mechanism exists ==> can drive a Maclaurin spheroid 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_R+if_imag]t), where |Q₁/Q₀| << 1,

if the frequency

f_imag < 0

==> perturbation grows exponentially on a dynamic time scale

Calculational Details and General Results.

In terms of T/|W|, where T is the total bulk kinetic energy of the model and W is total gravitational energy of the model, the essential stability results for the lowest order nonaxisymmetric mode, the bar mode (m=2 for a mode with harmonic azimuthal dependence), are:

T/|W| > 0.09-0.14 ==> secularly unstable

T/|W| > 0.27 ==> dynamically unstable

for all compressibilities and angular momentum distributions tested.


Q: Roughly, what is the order of the critical T/|W|'s?

Virial Theorem ==> 2(T_rot+T_thermal) + W = 0 ==> T/|W| = 1/2 for an object supported solely by rotation.

Quantitative comparisons between the eigenvalues found by linear and nonlinear numerical codes agree to within 1 % in the best cases and to 10 to 15 % in the worst cases (of those considered reliable). In addition, the linear, quasi-linear, and nonlinear techniques unambiguously determined the instability saturation mechanism which allows predictions of the nonlinear outcome of nonaxisymmetric instabilities to be made for a much larger range of parameter space than was previously possible.

Comments-- Amazingly, results based on incompressible fluids are relatively good predictors of the stability properties of even highly compressible objects.