We derived in the class that, for a degenerate electron gas at $T=0$, the degeneracy pressure is given by
\begin{eqnarray}
P = K_{\rm NR}\,\rho^{5/3}~ \text{(non relativistic limit)}, ~~~
P = K_{\rm ER}\,\rho^{4/3} ~ \text{(extreme relativistic limit)}
\end{eqnarray}
\subsubsection{Problems}
\begin{enumerate}
\item Assume hydrostatic equilibrium and show that for the non-relativistic equation of state, white dwarfs follow the mass radius relation $R \propto M^{-1/3}$.
\item Show that for the relativistic case, the radius cancels and derive the Chandrasekhar mass scaling $M_{\rm Ch}\sim\left({hc}/{G}\right)^{3/2}{(\mu_e m_p)^{-2}}$.
\item In a real white dwarf, ions form a Coulomb plasma which modifies the pressure. Show that the Coulomb energy per ion scales as: $E_C \sim-{Z^2 e^2}/{a}$, where $a \sim n_i^{-1/3}$ is the inter-ion spacing. Show that the corresponding correction to pressure scales as:
$P_C \sim-\rho^{4/3}$. (Hints: Use $n_i \propto\rho$. Total Coulomb energy density $\epsilon_C \sim n_i E_C$. Pressure: $P = n^2\frac{d}{dn}(\epsilon/n)$). Comment on how Coulomb corrections modify the mass-radius relation.
\item Assume a polytropic equation of state: $P = K \rho^{\gamma}, ~~ \gamma={5}/{3}$. Write down the equations of hydrostatic equilibrium. Reduce these equations to the Lane--Emden equation for $n=3/2$. Solve numerically using the boundary conditions:
$\rho(0)=\rho_c, \quad M(0)=0$. (Hint: Use dimensionless variables: $\rho=\rho_c \theta^n, \quad r = a \xi$).
\item For a range of central densities $\rho_c =10^6$ to $10^{10} , \text{g cm}^{-3}$, compute the corresponding mass and radius.
\item Plot the mass--radius relation $R(M)$. Show that the numerical result agrees with the scaling that we estimated earlier: $R \propto M^{-1/3}$. (Hints: Stop integration when $\rho\to0$. Use dimensionless Lane-Emden solution and scale results).
