A neutron star consists mainly of neutrons, but it also contains protons and electrons in thermal equilibrium as a result of the reactions

n \to p + e^ -   + \bar \nu and p + e^ -   \to n + \nu .

(The antineutrino or neutrino on the right side of these reactions escapes the star, hence do not reach equilibrium with the other particles, and their chemical potentials can be set to zero.) In the model of a neutron star to be treated here, the neutrons, protons, and electrons are all taken to be ultrarelativistic [E\left( {\vec k} \right) = \hbar c\left| {\vec k} \right|]. (Real life is more complicated.)

(a)Obtain an expression for the Fermi energy E_F as a function of the number density N/V for an ultrarelativistic gas of identical fermions.

(b)Find the thermal equilibrium relation between E_{F^n }, E_{F^p}, and E_{F^e} for a neutron star, where the superscripts on E_F describe particle types.

(c)Using the results of the first parts of the problem, find in our model the relative amounts of neutrons, protons, and electrons.


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