We can extend the theory of a free electron gas (Section 5.3.1) to the relativistic doma...

We can extend the theory of a free electron gas (Section 5.3.1) to the relativistic domain by replacing the classical kinetic energy, .

(a) Replace : in Equation 5.44 by the ultra-relativistic expression, lick, and calculate Eux in this regime.

(b) Repeat parts (a) and (b) of Problem 5.35 for the ultra-relativistic electron gas. Notice that in this case there is no stable minimum, regardless of R; if the total energy is positive, degeneracy forces exceed gravitational forces, and the star will expand, whereas if the total is negative, gravitational forces win out, and the star will collapse. Find the critical number of nucleons, No such that gravitational collapse occurs for N > Nc. This is called the Chandrasekhar limit. Answer: . What is the corresponding stellar mass (give your answer as a multiple of the sun's mass). Stars heavier than this will not form white dwarfs, but collapse further, becoming (if conditions are right) neutron stars.

(c) At extremely high density, inverse beta decay, ,converts virtually all of the protons and electrons into neutrons (liberating neutrinos, which carry off energy, in the process). Eventually neutron degeneracy pressure stabilizes the collapse, just as electron degeneracy does for the white dwarf (see Problem 5.35). Calculate the radius of a neutron star with the mass of the sun. Also Calculate the (neutron) Fermi energy, and compare it to the rest energy of a neutron. Is it reason-able to treat a neutron star nonrelativistically?