Evolution of Stars During the Main Sequence and the Transition to the Red Giant Phase

Abstract

We derive a simple analytical description for the structure and evolution of $3$--$10 M_\odot$ stars throughout main-sequence hydrogen burning. We obtain an analytical relation for the convective core mass, $\frac{M_\star}{M_c}=1+2.1\left(\frac{μ_c}{μ_e}\right)^2$, where $μ$ is the mean molecular weight of the core and envelope. Using this relation, we analytically derive the hydrogen abundance profile outside the convective core. We find that $μ(m)\propto m^{-0.7}$, and show that this profile is important for an analytical description of these stars. Within this region of variable $μ$, the temperature, density, and pressure are well approximated by power laws of radius. We derive analytical expressions for the core and stellar radii, stellar luminosity, and effective temperature as functions of $μ_c$. We provide a simple physical explanation for the main-sequence hook, defined by the minimum in effective temperature. We show that the hook occurs when the hydrogen mass fraction in the core is $x_c\simeq0.045$, and stress that the same convective-core burning physics governs the subsequent evolution. In that sense, at the hook hydrogen is not yet fully exhausted. During late main-sequence evolution, we find that the ratio of nuclear luminosity between the core and the surrounding hydrogen-rich shell is $\simeq4000x_c$. Hence, the main sequence terminates only once $x_c\simeq2.5\times10^{-4}$, when the surrounding layers become as luminous as the core itself and $M_c\simeq0.11 M_\star$. Although this terminal core mass is numerically similar to the Sch"onberg--Chandrasekhar limit, we show that the two are physically unrelated, since the core remains far from isothermal even at this stage. We validate all analytical results using MESA simulations.