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A. Arcones developed equilibrium bounds on the electron fraction in stellar matter with G. Martínez-Pinedo, L. F. Roberts and S. E. Woosley. Their 2010 study applied nuclear statistical equilibrium and beta equilibrium to obtain lower limits relevant to Type Ia supernovae and accretion-induced collapse.^[ref]

Beta equilibrium is a condition in which weak reactions interconverting neutrons and protons produce no net change of composition. For matter where neutrinos are thermally trapped, the reaction $n+{\nu }_e\rightleftharpoons p+e^{-}$ gives the chemical equilibrium conditionwhere each $\mu$ is the corresponding chemical potential. In cold neutrino-transparent neutron star matter, one usually takes ${\mu }_{{\nu }_e}=0$, giving ${\mu }_n={\mu }_p+{\mu }_e$. The escaping neutrinos do not form an equilibrated thermal bath in that approximation.^[ref]

Charge neutrality in matter containing only neutrons, protons and electrons additionally requires $n_p=n_e$. Given a total baryon density, these relations and an equation of state determine the composition. As an illustrative chemical-potential check, ${\mu }_n=1000$ MeV and ${\mu }_p=900$ MeV require ${\mu }_e=100$ MeV in the transparent approximation. At finite temperature, balancing emission and capture rates is the more general kinetic condition; nuclear statistical equilibrium under strong reactions alone does not impose beta equilibrium.

For electrically neutral, fully ionized stellar matter with electrons and Positrons as its charged leptons, the electron fraction in stellar matter is the net electron number per baryon:Here $n_i$ is the number density of nuclear species $i$, $Z_i$ its atomic number, $A_i$ its mass number, and $n_b$ the total baryon number density. Charge neutrality equates the net negative lepton charge to the total proton charge. The subtraction of Positrons matters in hot matter containing abundant thermally produced Positrons; ordinary electron number alone need not equal nuclear charge.^[ref]

Neglecting small nuclear mass defect corrections, isotope mass fractions $X_i$ give $Y_e={\sum }_i{X}_i{Z}_i/A_i$. Pure fully ionized helium-4 therefore has $Y_e=1/2$. Strong interactions and electromagnetic reactions preserve this proton fraction, whereas electron capture and beta-plus decay reduce it and beta-minus decay can increase it. The total proton-to-neutron number ratio is $Y_e/(1-Y_e)$; electron-positron pair production changes neither of those baryon counts. If other charged leptons such as muons are appreciably populated, their charge must also be included in the neutrality equation.

G. Martínez-Pinedo studied how nuclear composition constrains the electron fraction in stellar matter. With A. Arcones, L. F. Roberts and S. E. Woosley, he tested combined nuclear statistical equilibrium and beta equilibrium against presupernova models, rather than assuming every evolving plasma has reached equilibrium.^[ref]

L. F. Roberts helped calculate equilibrium constraints on the electron fraction in stellar matter. The 2010 study with A. Arcones, G. Martínez-Pinedo and S. E. Woosley sought limits less sensitive to uncertain weak-interaction rates, using nuclear statistical equilibrium together with beta equilibrium.^[ref]

S. E. Woosley investigated the electron fraction in stellar matter with A. Arcones, G. Martínez-Pinedo and L. F. Roberts. Their 2010 calculation combined nuclear statistical equilibrium with beta equilibrium conditions to constrain composition in presupernova and collapse environments.^[ref]