Sin Aviatur la Concesión en Amacayacu sería otro cuento

As pointed out by Eddington (1926), to maintain pulsations, stars must operate pretty much as thermodynamic engines with heat being added to matter at a high temperature only to be withdrawn at a low temperature. The question is now, how does this exactly occur in stars? Accounting to (Catelan and Smith 2015), the equation for the stability coefficient κ in the m-th pulsation mode can be written as

κm =− R M (δT /t)_m,adδeffdm 2ω2 mJm + R M (δT /t)_m,rad ∂δL_∂M
m dm 2ω2 mJm (2.63) with Jm is the corresponding oscillatory moment of inertia.

The first term in Equ. (2.63) is associated with energy generation, and the second with energy transfer. In the first case, when pulsations are excited, one refers to the  mechanism, whereas in the latter case, the κ and γ mechanisms are at play.

The  Mechanism

Within the region of the star where the thermonuclear reactions take place, the temperature increases during compression. This leads to an increase in the rate of energy generation, and vice versa during the expansion. Thus, energy is gained by these layers during the compression, and released during the expansion. As stated by Catelan and Smith (2015), this mechanism works exactly as required to establish pulsational instability according to Equ. (2.56) and (2.63). Thermonuclear reactions show a strong dependence on temperature. If the amplitude of the temperature fluctuations in the energy-generation regions is sufficiently high in the course of pulsations, such a supply of energy will indeed fluctuate over time, thus being naturally able to maintain the pulsations. This is the so-called  mechanism of stellar pulsation, where the  is the nuclear energy generation rate. In classical pulsators, such as RR Lyrae and Cepheids, the 

mechanism does not play an important role, whereas in other types of pulsating stars, it has been claimed to be of considerable interest (Catelan and Smith 2015).

The κ and γ Mechanism

For most stars, the energy transfer, rather than the energy generation, is the main cause for pul- sation. Pulsation will be excited when the stability coefficient κ in Equ. (2.63) becomes negative. The second term in Equ. (2.63) then requires that during maximum compression (i.e., δT /T > 0), ∂δL/δm being negative, implying an increase in δL with increasing Mr (i.e., towards the surface

of the star). Thus, at least some layers of the star must gain energy during compression, and release energy during expansion to maintain pulsation. Such layers are called driving layers. They are typically associated with H and He partial ionization zones.

Assume now, that the Rosseland mean opacity Equ. (2.12) in a given layer of the star can be approximated for simplicity by

κR∝ ρnT−s. (2.64)

In the case of free-free absorption in a non-degenerate, fully ionized gas, the so-called Kramers opacity law can be applied, by setting n = 1, s = 7/2:

κR∝ ρT−7/2 (2.65)

that was derived by Eddington (1926) based on Kramer’s opacity law. According to this expres- sion, there is a tendency that during compression, opacity decreases in the layers of a star, caused by the rise in temperature.

However, there are a few “bumps” in the opacity, caused by the ionization of H and partial ioniza- tions of He. In these regions, there is a tendency for the opacity to actually increase with increasing temperature, so the s value in Equ. (2.64) becomes negative. The consequence of an increasing opacity during compression is that the corresponding region of the star will “concentrate” energy during compression, and more easily release it during the expansion, leading to pulsation. This increase in the opacity is known as κ mechanism (Baker and Kippenhahn 1962). The effect was first studied by S. A. Zhevakin and J. P. Cox (Cox and Whitney 1958; Cox 1960; Zhevakin 1963).

The increased ability of the same layers participating in the κ mechanism to gain heat during compression is called γ mechanism (Cox 1963).

The classical κ and γ mechanisms explain the excitation of pulsation instabilities in stars within the instability strip, such as RR Lyrae, Cepheids and δ Scuti stars.

Non-Radial Pulsations

The mechanisms presented so far describe radial pulsations. However, not all pulsating stars pulsate radially.

The discovery of the Sun’s 5-minute-oscillations (Leighton 1960; Leighton et al. 1962) hinted that stars might pulsate in non-radial modes. Furthermore, it has led to the assumption that similar pulsations might be detected in other stars when observational techniques improve.

Nowadays, the study of oscillation in stars – asteroseismology – requires the detection of a huge range of pulsation modes, many of which are non-radial. Nowadays, asteroseismological studies of stars have grown enormously in the course of the past several years. The results of the CoRoT (Auvergne et al. 2008) and Kepler (Borucki et al. 2010) missions, among others, have enabled us to gain insight into the physical processes of star interiors.

The κ and γ mechanisms, as described before, are successful in describing the pulsation of stars located within the instability strip, but fail for many different types of other pulsating stars, most of them non-radial pulsators. In hot pulsating stars, the metal opacity bump (due to an increase in the heavy element contribution), rather than the opacity bump discussed before being associated with the H and He partial ionization zones, is responsible for driving the oscillations (metal bump mechanism, Simon (1982); Cox et al. (1992).