\sect{Baruch's  B23.}

Consider a neutron star as non-relativistic gas of
non-interacting neutrons of mass $m$ in a spherical symmetric
equilibrium configuration. The neutrons are held together by a
gravitational potential  ${-mMG/r}$ of a heavy object of mass $M$
and radius ${r_{0}}$ at the center of the star (${G}$ is the
gravity constant and ${r}$ is the distance from the center).

\begin{itemize}
\item [(a)]
Assume that the neutrons are classical particles at temperature
${T}$ and find their density ${n\left(r\right)}$ at ${r>r_{0}}$
for a given $n(r_0)$. Is the potential confining, i.e. is there a
solution with ${n\left(r\right)\rightarrow 0}$ at ${r\rightarrow
\infty}$ ?
\item [(b)]
Consider the neutrons as fermions at ${T=0}$ and find
${n\left(r\right)}$. Is the potential confining? [Hint: classify
solutions according to their chemical potential $\mu$.]

\item [(c)]
Is the potential confining for fermions at ${T\neq 0}$? when is the result (a) valid?\\
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%