
\sect{Fermions in gravitation field of a star}

Consider a neutron star as non-relativistic gas of non-interacting neutrons 
of mass ${\mass}$ in a spherical symmetric equilibrium configuration. 
The neutrons are held together by a gravitational potential ${- G M \mass /r }$ 
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)]
Give an expression for $n(r)$ at $T > 0$ using Li functions.

\item [(b)]
Consider the neutrons as fermions at $T = 0$ and find $n(r)$, for a given $n(r_0)$. 

\item [(c)]
Calculate it explicitly in the Boltzmann approximation.

\item [(d)]
Repeat items (b) and (c) for a general potential ${-A/r^{\alpha}}$.

\item [(e)]
For ${T=0}$, what is the upper bound on $n(r_0)$ and on the total number $N$ of neutrons 
if the chemical potential is increased towards zero. Distinguish ${\alpha>2}$ from ${\alpha<2}$.

\end{itemize}


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