Consider a white dwarf of mass M and radius R. Let the electrons be degenerate but non- relativistic; the protons are non-degenerate.

(a) Show that the order of magnitude of the gravitational self-energy is – GM²/R, where G is the gravitational constant. (If the mass density is constant within the sphere of radius R, the exact potential energy is – 3GM²/5R).

(b) Show that the order of magnitude of the kinetic energy of the electrons in the ground state is

Where m is the mass of an electron and M_H is the mass of a proton.

(c) Show that if the gravitational and kinetic energies are of the same order of magnitude (as required by the virial theorem of mechanics), M^1/3 R ≈ 10²⁰ g^1/3 cm.

(d) If the mass is equal to that of the Sun (2 x 10³³ g), what is the density of the white dwarf?

(e) It is believed that pulsars are stars composed of a cold degenerate gas of neutrons. Show that for a neutron star M ^1/3 R ≈ 10¹⁷ g^1/3 cm. What is the value of the radius for a neutron star with a mass equal to that of the Sun? Express the result in km.

h?N5/3 h M5/3 mMn5/3R21 mR?