Fermi gases in oil astrophysics. 

(a) Given M = 2 x 10³³ g for the mass of the Sun, estimate the number of electrons in the Sun. In a white dwarf star this number of electrons may be ionized and contained in a sphere of radius 2 x 10⁹ cm; find the Fermi energy of the electrons in electron volts. 

(b) The energy of an electron in the relativistic limit ε >> mc² is related to the wave vector as ε ≡ pc = hkc. Show that the Fermi energy in this limit is ε_F ≈ hc (N/V)^1/3, roughly 

(c) If the above number of electrons were contained within a pulsar of radius 10 km, show that the Fermi energy would be ≈ 10⁸eV. This value explains why pulsars are believed to he composed largely of neutrons rather than of protons and electrons, for the energy release in the reaction n → p + e^– is only 0.8 x 10⁶eV, which is not large enough to enable many electrons to form a Fermi sea. The neutron decay proceeds only until the electron concentration builds up enough to create a Fermi level of 0.8 x l0⁶eV, at which point the neutron, proton, and electron concentrations are in equilibrium.