Equations of state inside stars

For species i, specify mass fraction Xi, nuclear mass number Ai, and ionic charge Zi in units of the elementary charge. The ion density is its share of the mass divided by the mass per nucleus. Full ionization supplies Zi electrons per nucleus. Summing both ions and electrons defines the dimensionless mean molecular weight mu, whereas summing only electrons defines mu electron. Neither symbol is a mass in kilograms. Charge neutrality determines electron number without creating additional baryonic mass.

For pure ionized hydrogen there are two pressure-carrying particles per baryon, giving mu equal to one half. For ionized helium there are three particles per four baryons, giving mu equal to four thirds. Replacing hydrogen by helium therefore reduces classical gas pressure at fixed density and temperature. For metals, the frequently used approximation of one electron per two baryons neglects a smaller ion contribution and depends on the chosen mixture; an exact species sum avoids silently assigning every metal the same atomic mass.

The classical ideal gas contributes particle number density times Boltzmann constant times temperature. Locally thermalized photons add one third of their energy density as radiation pressure. Their relative importance increases rapidly with temperature and decreases with material density. The sum is useful only when the material remains classical and weakly coupled and the radiation field is sufficiently close to local thermal equilibrium. It is not an equation of state for an arbitrary optically thin, anisotropic radiation field.

An equation of state table can include partial ionization, Coulomb corrections, pair creation, and other effects absent from the simple sum. Partial ionization stores heat in changing ionization rather than only random motion, altering heat capacities and adiabatic exponents. Radiation pressure likewise changes the relation between an adiabatic temperature change and compression. Using a monatomic ideal-gas adiabatic index everywhere merely because ideal-gas pressure was included would miss these mechanisms. Thermodynamic derivatives must be computed from the same pressure and energy model.

When electron states are densely occupied, the exclusion principle produces a filled momentum distribution up to the Fermi momentum in the cold limit. Electrons still have momentum and exert pressure even as their temperature approaches zero. At temperatures far below the Fermi temperature and momenta small compared with electron rest mass times light speed, the electron pressure scales as electron density to the five-thirds power. The ions may remain classical even while the electrons are degenerate.

As the Fermi momentum becomes much larger than electron rest mass times light speed, the limiting exponent changes to four thirds. Intermediate degeneracy and relativity require the appropriate Fermi integrals rather than switching formulas at an arbitrary density. Composition matters through electron number per unit mass. The cold electron law approximates white-dwarf support, but finite temperature, Coulomb effects, electron captures, and relativistic gravity eventually modify simple inferences about radii or limiting masses. A pressure law alone does not specify the complete evolutionary state.