This is a hard test case in astrophysics problem, and it models a globular cluster of stars, which is highly non uniform. It is called the plummer distribution. To construct such distribution, first we construct a uniform points distribution on the unit sphere. Second, the radius is chosen according to the plummer distribution (double power law in astrophysics). We consider $u$ a random number between 0 and 1, then the associated radius is given by
\begin{equation*}
r = \sqrt{\frac{u^{2/3}}{u^{2/3}-1}}
r = 1.0/\sqrt{u^{-2/3}-1},
\end{equation*}
and the total mass is one. Then, $m_i =\frac{1}{Npt}$.
\begin{figure}[h]
\centering
\begin{minipage}{0.45\textwidth}%
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@@ -140,6 +140,8 @@ The corresponding potential is
\begin{equation}
\Phi_P(r) = - \frac{G M}{\sqrt{r^2+a^2}}
\end{equation}
In N-body units, $G = M =1$ and $a =3\pi/16\sim0.589$
\subsection{Diagonal Model}
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