In this Appendix we describe a class of EOSs for uniform matter in neutron star cores composed of nucleons, electrons and muons. These EOSs are based on analytic expressions for the energy per nucleon (excluding the rest-mass energy) quadratic in neutron excess,
where u = nb/n0 is the dimensionless baryon number density, xp = np/nb is the proton fraction; W (u) and S(u) are, respectively, the energy per nucleon in symmetric nuclear matter and the symmetry energy (assumed to be given by analytic functions). The total energy per nucleon is then E = EN + EN0 + Ee + Em, where EN0 is the nucleon rest-mass contribution, while Ee and Em are the electron and muon contributions (also given analytically because electrons and muons constitute almost free Fermi gases). In this case, the total energy E is presented in an analytic form which allows one to avoid ambiguities of interpolation (of otherwise tabulated values of En) and to strictly satisfy thermodynamic relations and conservation laws.
The beta equilibrium conditions are given by relations between the chemical potential of nucleons and leptons,
The local electric neutrality requires xp = xe + xM, where xe = ne/nb and xM = nM/nb. The electron and muon chemical potentials are equal to the appropriate Fermi energies,
where pFj = h (3n2nbxj)1//3 with j = e or H- At a fixed nb under the natural simplified assumption that mp = mn the beta equilibrium conditions reduce to a set of two equations xm + xe - 1+ A xl/3 = 0 , (D.4a)
x2/3 - x2J3 -B = 0 , (D.4b) where A and B are dimensionless functions of nb,
A = he (3n2nb)1/3/(8S(nb)) , B = (m^c/h)2/(3n2nb)2/3 . (D.5)
Beta equilibrium depends on S(u) but not on W(u). For a given nb, one can easily solve Eqs. (D.4a) and (D.4b) and determine all particle fractions. After that one can use standard thermodynamic relations, derive the analytic expressions for the energy density (pee2) and the pressure, and calculate p and P at given nb and the particle fractions. In this way one constructs a semi-analytical EOS; the only simple numerical procedure consists in solving Eqs. (D.4a) and (D.4b). The numerical accuracy of this EOS for an employed nuclear interaction model (D.1) can be formally very high.
Table D.1. Three sets of parameters for W (u) models of Prakash et al. (1988)
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