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\sect{Fermions in gravitation field of a star}

Consider an artificial model of neutron star where the gas of ${N}$ neutrons 
is held together by a gravitational potential ${U=-GMm/r}$ generated by 
the solid core $M$ of the star, where ${G}$  is the gravitation constant,
and ${m}$ is mass of a neutron. The core has a radius ${r\sim0}$. 

\begin{itemize}

\item [(a)]
Assume the neutron gas as Fermi gas. 
Find the density ${n(r)}$ for $T=0$,   
and determine the Fermi energy ${\epsilon_{f} = -W(N)}$.

\item [(b)]
What is the radius $R$ that is occupied by neutrons?  

\item [(c)]
Find the flux of particles that escape the gravitational field 
assuming a low temperature $T$. Note the analogy to thermionic emission.

\item [(d)] 
Write a differential equation for the number ${N(t)}$ 
of remaining particles, assuming that the temperature 
is not changing. 

\item [(e)] 
The calculation of the flux in item (c) involves  
a Boltzmann approximation. Write the condition on $T$ 
for the validity of this assumption. 
Additionally, point out what is the systematic 
error that follows from the constant $T$ assumption.  

\end{itemize}

Note: In item (a) your answer depends on a numerical constant $C$ 
that you have to define in terms of an elementary definite integral.