Why Stars Become Red Giants

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Abstract
The evolution of stars to red giants is revisited in order to promote a better understanding of the behavior of stars on the H-R diagram by separating the essential nonlinear characteristics of stellar structure from detailed effects of input physics such as chemical compositions, opacity, convection criterion, etc. It is shown that the red giant and dwarf structures are clearly discriminated in terms of the variation through the stellar interior of the ratio, W, between the mass interior to the relevant shell and the mass contained in the pressure scale height at the shell (≡ GM/4πr4P). For the simplest structures of dwarfs such as main-sequence stars, W increases monotonically from zero at the center to the greatest value at the surface. In later stages of evolution such as the red giants, W evolves to have a local extremum near the hydrogen-burning shell. Such a change in the distribution of W is a consequence of the increase in the ratio, Θ, of thermal and/or Fermi energy (P/ρ) between the core center and the envelope. In fact there exists a lower bound to Θ that is required by a red giant structure. During stellar evolution, the increase in Θ is brought about by the gravitational contraction of the core and by the fact that shell burning prevents the envelope from following the core contraction. Structures with an extremum of W correspond to an envelope of the condensed type, the properties of which are strongly regulated by the polytropic index N and the ratio, V [= (GMr/r)/(P/ρ)], between the local gravitational potential to the thermal energy in the lower envelope. A large polytropic index N 3 makes the envelope expanded, particularly when V N + 1 is realized in the lower envelope. On the basis of such an understanding, we are able to analyze the effects of input physics on the excursion of the star to a red giant. A key role is played by the gradients of the opacity and of the mean molecular weight. The stellar radius can be larger if the opacity is increasing outward to make N larger than 3. A moderate gradient in the mean molecular weight also allows the value of N to come into the range of 3-5, which is appropriate for V N + 1 to be realized in the bottom of the envelope, while a steep gradient yields too large a value of N. The so-called peculiar evolution of the SN 1987A progenitor, Sk -69°202, can be understood in such a context. In particular, a constraint is derived for the mechanism of the extra mixing that is responsible for the final red-to-blue excursion of the star.

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