Critical Metallicity of Cool Supergiant Formation. II. Physical Origin

Abstract

This study investigates the physical origin of the critical metallicity required for the formation of cool supergiants, as revealed by stellar evolution models. Using grids of stellar models, we show that the terminal-age main-sequence (TAMS) radius, $R_{\rm TAMS}$, defines a threshold that determines whether a star of a given mass can evolve into the red supergiant (RSG) phase. Metallicity influences the supergiant outcome because it modifies $R_{\rm TAMS}$ through its effects on opacity and nuclear energy generation, as demonstrated by our stellar models and dimensional analysis based on homology relations. The value of $R_{\rm TAMS}$ sets the initial radius for post-main-sequence expansion and therefore controls the envelope radius reached at subsequent core-evolution stages. Higher-metallicity stars develop larger $R_{\rm TAMS}$ and rapidly expand into the stable RSG regime during core helium burning. In contrast, lower-metallicity stars have smaller $R_{\rm TAMS}$ and advance to more evolved core helium or carbon-burning stages while retaining compact envelopes, thereby preventing expansion into the RSG regime during core helium burning. Our results explain the origin of the critical metallicity and offer insight into the evolution of metal-poor massive stars in the early universe.

Figures (18)

• Figure 1: Maximum stellar radius during the core He-burning stage ($R_{\rm max}$) for Grid (b), which includes models with an initial mass of $25\,M_{\odot}$, varying in metallicity ($Z$) and opacity parameter ($\zeta_{\kappa}$). The boundary between red and blue supergiants depends on $\zeta_{\kappa}$ in the range $\zeta_{\kappa} \gtrsim 10^{-3}$, but becomes nearly insensitive to $\zeta_{\kappa}$ when $\zeta_{\kappa} \lesssim 10^{-3}$.
• Figure 2: Maximum stellar radius during the core He-burning stage ($R_{\rm max}$) for Grid (c): models with an initial mass of $25\,M_{\odot}$ and metallicity $Z=0.001$, computed for varying values of the CNO-cycle reaction rate (scaled by $\eta_{\rm CNO}$) and opacity parameter ($\zeta_{\kappa}$). Both $\eta_{\rm CNO}$ and $\zeta_{\kappa}$ influence the evolutionary outcome, determining whether the star becomes a red or blue supergiant.
• Figure 3: Maximum stellar radius during the core He-burning stage ($R_{\rm max}$) for Grid (d): models with an initial mass of $25\,M_{\odot}$ and metallicity $Z=0.001$, computed for varying values of the CNO-cycle reaction rate during the shell H-burning stage (scaled by $\eta_{\rm shellCNO}$) and opacity parameter ($\zeta_{\kappa}$). For comparison, the parameter $\eta_{\rm CNO}$ used in Figure \ref{['fig:CNO-zbase']} is applied throughout all evolutionary stages, whereas $\eta_{\rm shellCNO}$ in this figure is applied only during the shell H-burning stage. While $R_{\rm max}$ exhibits some scatter, the overall trend indicates that $\eta_{\rm shellCNO}$ has little impact on the supergiant outcome.
• Figure 4: Maximum stellar radius during the core He-burning stage ($R_{\rm max}$) for Grid (e): models with an initial mass of $25\,M_{\odot}$ and metallicity $Z=0.001$, computed for varying values of the triple-alpha reaction rate (scaled by $\eta_{3\alpha}$) and opacity parameter ($\zeta_{\kappa}$). Increasing $\eta_{3\alpha}$ does not promote evolution toward the RSG phase and may even suppress it in the low-opacity regime ($\zeta_{\kappa} \lesssim 10^{-3}$).
• Figure 5: Effective temperature ($T_{\rm eff}$), central temperature ($T_{\rm c}$), surface pressure ($P_{\rm surf}$), central pressure ($P_{\rm c}$), and stellar radius ($R_*$) at the ZAMS and TAMS for $25\,M_{\odot}$ models from Grid (c). In each panel, the ZAMS value is shown on the X-axis and the TAMS value on the Y-axis. Points are colored by the maximum stellar radius ($R_{\rm max}$) reached during core He burning, with red indicating RSG-scale radii and blue indicating BSG-scale radii. The bottom-right panel shows a smoothed contour representation of the same $R_*$ data displayed in the bottom-left panel. Notably, models with TAMS radii $\gtrsim 12\, R_{\odot}$ predominantly evolve into RSGs, whereas those below this threshold remain BSGs.