Outline

14a. Oscillation

Modes [some from SISSA]
ε-modes: driven by nuclear reactions, usually involving electron degeneracy.
f-mode: surface wave, gravito-acoustic mode.
The frequency of this mode is between the lowest order g-mode (the highest frequency g-mode) and the lowest order p-mode (the lowest frequencu p-mode).
g-modes: bouyancy ('gravity') modes.
Low frequency branch of the stellar oscillation. The main restoring force is the buoyancy force. It means that this mode has non-zero frequency when the star has a stratified temperature or chemical composition. Eigenfunction of a mode with quantum number n has n radial nodes. As n increases, the frequency decreases. In the limit of short wave length the frequency of the mode goes to zero.
p-modes: acoustic ('pressure') modes.
High frequency branch of the stellar oscillation. The main restoring force is the pressure. Eigenfunction (for density and radial velocity perturbation) of a mode with quantum number n has n radial nodes. As n increases, the frequency increases. In the limit of short wave length this represents the acoustic waves traveling in the star. In this same limit the frequency of the mode goes to infinity.
r-modes: coriolis forces in rotating stars.
The main restoring force is the Coriolis force. In the slow rotation limit, these modes have zero frequency. The oscillation is velocity-dominant, i.e., the density perturbation appears only in higher orders than the (tangential) velocity in the slow-rotation expansion analysis. The so-called 'classical r-mode' has a simple eigenfunction structure and it is a generalization of the Rossby wave well-known in geophysical hydrodynamics.
w-modes: general-relativistic spacetime fluctuations with little movement of stellar matter.

Driving mechanisms
&kappa-mechanism: In most of a stellar interior, the opacity decreases as temperature increases (e.g. Kramer law ∝ρT −3.5). However, in a strong ionization region, the temperature dependence weakens and may even reverse. For example, in the hydrogen and first helium ionization zones, opacity increases as temperature increases.

Assume that the gas in the ionization zone is compressed by a perturbation. The density and temperature of the zone increases and the opacity also increases. Even if the power dependence on T is still negative, much of the work done to compress the gas goes into ionization, so the change in temperature is relatively less. Then, the luminosity from inside of the star is bottled by the increase in opacity. The region receives extra restoring force from this dammed heat energy.

When the gas in this region expands, the opacity decreases. The decrease allows the extra energy to escape and again results in an increased restoring force.

Note that above described mechanism works when the most of the luminosity is transferred by radiation. When convection dominates, pulsation will not occur. The onset of convective transport determines the red edge of the pulsation strip in the HR diagram.

If the effective temperature of a star is high enough, the density of the ionization region is too low and the destabilizing effect is too small. Shallow ionization determines the blue edge. Until the most recent opacity calculations, the theoretical blue edge was not in agreement with the observed edge. This discrepancy was the driving force behind the work of the Opacity and OPAL Projects.

&gamma-mechanism: The relative lack of temperature increase on compression in an ionization zone has another effect. The underlying fully ionized regions do increase in temperature on compression, which increases the temperature gradient and sends more luminosity into the ionizing region, again increasing the restoring force. This destabilizing effect is called the &gamma-mechanism. Both the &kappa- and the &gamma-mechanisms act in the same regions and cannot really be separated.

The Cepheid instability strip results from the He II → He III ionization zone. β-Cephei and other high-temperature pulsators result from the small bump in the opacity distribution near T = 2 MK due to higher Z elements.

Ryan Maderak's Crash Course in Stellar Pulsation.

Some future reference material:
All about oscillations.
Helioseismology review.
Solar introduction.