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:
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. 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).