Support a white dwarf with occupied electron states

Use SI units here. Let $\rho_0$ denote baryonic rest-mass density, $m_u$ the atomic mass unit, and $Y_e$ the number of electrons per baryon in fully ionized neutral matter. Then the electron density is fixed by composition rather than by temperature. Spin degeneracy two fills a sphere in momentum space up to the Fermi momentum. Coulomb interactions, crystallization corrections, and finite-temperature tails are omitted in this first electron equation of state.

The nonrelativistic limit requires $p_F\ll m_ec$, while strong degeneracy requires $k_BT\ll E_F$, with the kinetic Fermi energy used in the latter test. These are independent inequalities. A low temperature does not guarantee nonrelativistic electrons: sufficiently high density raises their Fermi momentum. Conversely, massive nuclei can remain nonrelativistic while their supporting electrons have become relativistic. Always identify the species and energy scale to which a regime statement refers.

For nonrelativistic electrons, integrating the occupied kinetic states gives energy density proportional to $n_e^{5/3}$. Differentiating energy with respect to volume at fixed electron number yields two thirds of that kinetic energy density as pressure. Thus the pressure is a polytrope with exponent five thirds at fixed composition. It is momentum-space occupation, rather than a mechanical repulsion between stationary electrons, that generates this leading degeneracy pressure.

In a cool low-mass white dwarf, combine this pressure with Newtonian spherical hydrostatic balance. The enclosed mass is dominated by ions, so their rest density enters gravity even though their thermal pressure may be small. Dimensional balance gives a radius that decreases approximately as mass to the negative one third for a fixed nonrelativistic polytrope. This scaling is not an exact stellar solution; the numerical coefficient requires the radial density profile and a surface boundary.

A real white dwarf has an atmosphere and a temperature-dependent outer envelope, even when its interior electrons are strongly degenerate. Ionic heat capacity can control cooling while contributing little to mechanical support. Coulomb corrections alter the pressure and crystallization releases energy during cooling. These statements concern different derivatives of the thermodynamic energy: dominance of one species in pressure does not imply dominance in heat capacity or luminosity.

Increasing mass compresses the star and eventually invalidates the nonrelativistic electron approximation. Electron capture can also change $Y_e$, and nuclear burning can intervene in an accreting star. Rotation, strong fields, and general relativity require additional checks. The next lesson follows the electron dispersion crossover, but a limiting mass derived from an ideal polytrope should never be interpreted as a universal ignition mass or a complete prediction of the endpoint of stellar evolution.