Solve chemical and charge constraints together

Consider cold matter in which neutrinos escape, so their chemical potential vanishes. Weak equilibrium among neutrons, protons and electrons requires $\mu_n=\mu_p+\mu_e$. The chemical potentials include the appropriate rest energies. Charge neutrality relates proton density to the sum of negative-lepton densities. If muons are present and equilibrated, their chemical potential equals the electron value. The baryon density is neutron plus proton density in this restricted nucleonic model.

A trapped-neutrino newborn star has a different chemical relation and a lepton-number constraint. Hyperons, condensates, or quark matter introduce additional species and reaction conditions. Even the nucleonic interaction energy is uncertain at high density. It is therefore misleading to quote one proton fraction as an equation-of-state-independent property of every neutron star. Temperature, neutrino trapping, particle content, and the assumed interaction model must accompany the calculation.

An illustrative parabolic nuclear-energy model writes energy per baryon as a symmetric-matter part plus $S(n)(1-2x_p)^2$. Differentiation gives an approximate neutron–proton chemical difference $4S(n)(1-2x_p)$, neglecting the small vacuum mass difference in this estimate. Ultrarelativistic electrons supply a chemical potential set by their Fermi momentum. These two relations and charge neutrality must be solved at the same density; choosing their numerical values independently can fail that consistency check.

The direct-Urca decay and inverse reaction also need momentum conservation near the degenerate Fermi surfaces. At low temperature the neutrino momentum is small compared with the fermion Fermi momenta. For electron-only charge-neutral npe matter, proton and electron Fermi momenta agree. The triangle inequality then yields the familiar proton-fraction threshold one ninth. This is a kinematic result for the specified composition, not a universal density threshold. Muons redistribute the charge and modify the condition.

Interactions determine the symmetry energy and effective masses, thereby altering chemical potentials, reaction phase space, and neutrino emissivities. Opening a direct-Urca channel can accelerate cooling, but superfluid pairing may suppress it and an insulating envelope changes the observed surface temperature. A cool surface alone cannot uniquely measure the proton fraction. Conversely, a measured gravitational mass does not specify whether the star has crossed a particular microscopic reaction threshold.

Muons appear once their equilibrium chemical potential can exceed their rest energy, with threshold near 106 MeV in the free-lepton approximation. Their finite mass means their density is not generally equal to the electron density at the same chemical potential. Any calculation that adds a muon species but retains the electron-only relation $n_e=n_p$ violates charge neutrality. The useful procedure is to solve all chemical, charge, and baryon constraints and then test each reaction triangle separately.