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23. An overview of the theoretical underpinnings: core helium ignition

The evolution of a post-main-sequence low-mass star up the red giant branch is one of the best-understood phases of stellar evolution (e.g., Iben & Renzini 1983). For the stars of interest to us in the context of the TRGB, a helium core forms at the center of the star, supported by degenerate electron pressure. Surrounding the core, and providing the entire luminosity of the star is a hydrogen-burning shell. The ``helium ash'' from the shell rains down on the core increasing its mass systematically with time. The temperature of the (basically isothermal) core and therefore the luminosity generation rate at its surface are simple functions of core mass and core radius. In analogy with the white dwarf equation of state and the consequent scaling relations that interrelate core mass (Mc) and core radius (Rc) for degenerate electron support, the core (= shell) temperature (Tc) and the resulting shell luminosity are simple functions of Mc and Rc:

Tc ~ Mc / Rc

Lc ~ Mc7 / Rc5

meaning that as the core mass increases (and the radius shrinks) the luminosity increases due to both effects. That is, the star secularly ascends the red giant branch to ever-increasing luminosities, and higher core temperatures. And it is the latter increase that stops the progression in luminosity: When Tc exceeds a physically well-defined temperature (set by nuclear physics), helium in the core will ignite. While this does provide a new and additionally powerful source of energy, helium core ignition does not make the star brighter, but rather it all but eliminates the shell source by explosively heating and thereby lifting the electron degeneracy within the core. This dramatic change in the equation of state is such that the entire internal structure of the star is rearranged so quickly that the core flash (which generates the equivalent instantaneous luminosity of the entire galaxy) is internally quenched in a matter of seconds, inflating the core and settling down to a helium core-burning main sequence, far removed in luminosity and effective temperature from the RGB. This phase change marks the TRGB. And nuclear physics fundamentally controls the stellar luminosity at which the RGB is truncated, essentially independent of the chemical composition and/or residual mass of the envelope sitting above the core. This is the underlying power of the TRGB: it is a physically based and theoretically well understood distance indicator.

Figure 21 Figure 21. A schematic representation of the time evolution of the interior of a low-mass star from core hydrogen burning (A-B-C) through shell-hydrogen burning (D), ending with abrupt core Helium ignition at 7.47 x 109 years (arrow). Adapted from Thomas (1967).

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