A Theoretical Study of the Stellar Parameters and Surface Nucleosynthesis of the Extremely Metal-Poor RGB Giant BD$-18^{\circ}$5550 - Part I: Effects of $\alpha$-enhancement and Internal Mixing
We present a detailed stellar evolution study of BD$-18^{\circ}$5550, an extremely metal-poor red giant star in the Galactic halo. Such stars are among the oldest known objects and preserve chemical signatures of the earliest stages of Galactic history. BD$-18^{\circ}$5550 exhibits $\alpha$-enhancement, $\rm[\alpha/Fe]=+0.4$, typical of Population II halo stars, with enhanced abundances of oxygen, magnesium, and other $\alpha$-elements relative to iron. Using the \texttt{MESA} stellar evolution code, we perform the first direct determination of its mass, age, radius, and luminosity by simultaneously fitting eight observed stellar properties and multiple chemical abundance ratios. Our analysis yields a current mass of $\rm M=0.76,M_{\odot}$, radius of $R=38.4,R_{\odot}$, luminosity of $\log L=2.8,L_{\odot}$, and an age of $t=11.87$ Gyr, with a 7-fold improvement in age precision. The model reproduces all observed constraints, supporting our treatment of stellar physics, nuclear reactions, and internal mixing. We further investigate the predicted surface chemical composition up to iron and compare it with observations. The surface abundances of elements up to iron remain essentially unchanged throughout the evolution, while nitrogen shows gradual enrichment, indicating the onset of mixing processes at intermediate RGB stages. The preservation of the other surface abundances provides a valuable window into the chemical conditions of the early Galactic environment. Our results establish BD$-18^{\circ}$5550 as a benchmark star for constraining stellar evolution and early Galactic chemical evolution. Its derived age places it among the oldest stars in the Milky Way, providing important constraints on the epoch of early star formation and the initial assembly and evolution of our Galaxy.