Critical Metallicity of Cool Supergiant Formation. II. Physical Origin

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.