Why Do Stars Turn Red? II. Steady-State Envelope Solutions

Abstract

The physical origin of red giants (RGs) and red supergiants (RSGs) remains a fundamental question in stellar astrophysics. In Paper II of this series, we investigate the physical mechanisms governing envelope expansion toward the RG/RSG phase by systematically exploring the physically realizable configurations of stellar envelopes. We construct steady-state stellar envelope models by solving the time-independent stellar structure equations while neglecting the core. The inner boundary is defined by a fixed pressure condition motivated by MESA stellar evolution models presented in Paper I. Our models show three key features of envelope expansion toward the RG/RSG phase. (1) The refined mirror principle identified in Paper I is recovered: the post-main-sequence stellar radius varies inversely with the radius of the envelope's inner boundary, arising purely from hydrostatic equilibrium. (2) We identify an upper limit to envelope expansion corresponding to an effective temperature of $\sim 4000,{\rm K}$, characteristic of RG/RSG stars and consistent with the Hayashi limit. This temperature limit is regulated by H$^{-}$ opacity, whose sharp decline at low temperatures flattens the surface temperature gradient and drives a structural transition. (3) The yellow regime of intermediate radius corresponds to an instability zone, in which small displacements of the hydrogen-burning shell produce large variations in stellar radius, naturally accounting for the bifurcation of giants and supergiants into blue and red branches instead of remaining in the yellow regime.

Figures (9)

• Figure 1: Schematic diagram illustrating the problem setup in this study. We integrate the stellar structure equations inward from the surface down to the inner boundary of the envelope, which is defined by a termination pressure $P_0$.
• Figure 2: Radius ($R_{\rm in}$, top panel) and enclosed mass ($M_{\rm in}$, bottom panel) at the envelope's inner boundary for polytropic models of $K=5.0\times 10^{15}$ (in the appropriate cgs unit for the corresponding $\gamma$) but different $\gamma$.
• Figure 3: Pressure and density profiles for polytropic models with $\gamma = 1.3$ and varying stellar radius $R_*$. Profiles are shown as functions of both mass and radius coordinates.
• Figure 4: $R_{\rm in}$ versus $R_*$ (top panel), $M_{\rm in}$ versus $R_*$ (middle panel), and $R_{\rm in}$ versus $T_{\rm eff}$ (bottom panel) of $25\,M_{\odot}$ stars with different luminosities $L_0$.
• Figure 5: $R_{\rm in}$ versus $R_*$ (top panel), $M_{\rm in}$ versus $R_*$ (middle panel), and $R_{\rm in}$ versus $T_{\rm eff}$ (bottom panel) of $5\,M_{\odot}$ stars with different luminosities $L_0$.