Find the limiting mass of an ideal relativistic electron star

Keep SI constants explicit and assume a cold, free electron gas with fixed $Y_e$. In the ultrarelativistic regime the occupied electron energy is approximately pc, and pressure is one third of its kinetic energy density. The resulting pressure exponent is four thirds rather than five thirds. Ionic rest mass still supplies most of the Newtonian gravitating density in this ideal white-dwarf model; treating that density as the energy density of an ultrarelativistic electron gas would describe another object.

The ultrarelativistic law applies to the dense interior of a limiting sequence, while the full finite-mass electron dispersion connects smoothly to the nonrelativistic outer material. One cannot obtain the correct limiting coefficient from a uniform-density energy estimate alone. Such an estimate explains the competition of inverse-radius energies, but the radial profile changes their numerical factors. The polytropic solution provides that missing structural calculation.

Write $P=K\rho_0^{1+1/n}$, so the ultrarelativistic law has polytropic index three. In the Lane–Emden variables its radial scale is proportional to $K^{1/2}\rho_c^{-1/3}$, where the central rest density is rho c. Stellar mass contains the combination of the cube of that scale and the central density. Their density powers cancel, leaving a mass proportional to K to the three halves. The radius continues to change along the formal sequence.

The numerical index-three solution gives an ideal limiting mass approximately $5.83Y_e^2$ solar masses when the atomic mass unit defines the baryon density. For carbon or oxygen, Ye is close to one half and this expression is about 1.46 solar masses. The familiar rounded value near 1.4 reflects the physical context and corrections, not a license to equate every measured compact-object mass with this ideal limit. Specify whether a quoted mass is baryonic, gravitational, or this Newtonian approximation.

Exact marginality of the four-thirds law makes neglected effects important. Finite electron mass supplies the approach to the ideal limit; general relativity, Coulomb interactions, and composition changes can destabilize a real star before an arbitrarily compressed ideal configuration is realized. Accreting carbon-rich stars can ignite thermonuclear burning, while other compositions may favor electron captures. A mass limit is therefore distinct from a prediction that every white dwarf collapses quietly into a neutron star.

Rotation can add support but changes the equilibrium family and its stability question; differential rotation is not equivalent to uniform rotation. Strong magnetic stresses can introduce anisotropy and cannot be represented by simply increasing an isotropic scalar pressure. Claims of a single revised universal Chandrasekhar mass need their assumptions exposed. Retain the ideal result as a benchmark for an equation of state and a structural scaling, then calculate corrections within a specified stellar model.