Mass-Radius Relation of Degenerate Configurations

3The maximum baryon number for equilibrium is determinedby setting E = 0:N max ∼(¯hcGm 2 b) 3/2∼ 2 × 10 57M max ∼ N max m B ∼ 1.5 M ⊙ .Note that the mass is independent of the fermion’s mass.The radius at equilibrium is set by the condition E F ≥ mc 2 ,R ≤ ¯hcmc 2 (¯hcGm 2 B) 1/2∼{5 × 103 km, m = me3 km, m = m n .At sufficiently high density, neutronization and pyconuclearreactions can occur. Thus, both A and N −Z will increase withdensity. The neutronization threshold for 56 Fe is about 10 9 gcm −3 . At this density, the Fermi energy of an electron is aboutm c c 2 +3.695 MeV, the threshold for the inverse beta-decay 56 Fe+ e − → 56 Mn + ν e . The Mn immediately electron captures:56 Mn + e − → 56 Cr + ν e . The Cr is stable until densities above10 10 g cm −3 are reached.Lighter nuclei have other thresholds: 4 He is at 20.6 MeV,12 C is at 13.4 MeV, 16 O is at 10.4 MeV and 20 Ne is at 7.0MeV. The loss of electrons softens the EOS: the Chandrasekharmass decreases. A white dwarf at these densities will begin togravitationally collapse. Thus the maximum density < ∼ 10 10 gcm −3 , with a minimum radius > ∼ 1500 km.