For an ideal gas of fermions of mass $m$ in volume $V$, and at temperature $T$ and chemical potential $\mu$, the number density $n$ and kinetic energy $E$ are given by

$$n=\frac{4 \pi g_{s}}{h^{3}} \int_{0}^{\infty} \bar{n}(p) p^{2} d p, \quad E=\frac{4 \pi g_{s}}{h^{3}} V \int_{0}^{\infty} \bar{n}(p) \epsilon(p) p^{2} d p$$

where $g_{s}$ is the spin-degeneracy factor, $h$ is Planck's constant, $\epsilon(p)=c \sqrt{p^{2}+m^{2} c^{2}}$ is the single-particle energy as a function of the momentum $p$, and

$$\bar{n}(p)=\left[\exp \left(\frac{\epsilon(p)-\mu}{k T}\right)+1\right]^{-1}$$

where $k$ is Boltzmann's constant.

(i) Sketch the function $\bar{n}(p)$ at zero temperature, explaining why $\bar{n}(p)=0$ for $p>p_{F}$ (the Fermi momentum). Find an expression for $n$ at zero temperature as a function of $p F$.

Assuming that a typical fermion is ultra-relativistic $\left(p c \gg m c^{2}\right)$ even at zero temperature, obtain an estimate of the energy density $E / V$ as a function of $p_{F}$, and hence show that

$$E \sim h c n^{4 / 3} V$$

in the ultra-relativistic limit at zero temperature.

(ii) A white dwarf star of radius $R$ has total mass $M=\frac{4 \pi}{3} m_{p} n_{p} R^{3}$, where $m_{p}$ is the proton mass and $n_{p}$ the average proton number density. On the assumption that the star's degenerate electrons are ultra-relativistic, so that $(*)$ applies with $n$ replaced by the average electron number density $n_{e}$, deduce the following estimate for the star's internal kinetic energy:

$$E_{\mathrm{kin}} \sim h c\left(\frac{M}{m_{p}}\right)^{4 / 3} \frac{1}{R} .$$

By comparing this with the total gravitational potential energy, briefly discuss the consequences for white dwarf stability.