katz_budnik_waxman_2010_eqn_41

\begin{matrix}
\rho = \rho_1 \delta^n = \rho_1 \left( \frac{R_* - r}{r} \right )^n
\\
M = \int_0^{R_*} 4 \pi r^2 \rho dr = 4 \pi \rho_1 \int_0^{R_*} r^2 \left( \frac{R_* - r}{r} \right )^n dr = 4 \pi \rho_1 R^3 \int_0^1 x^2 \left( \frac{1 - x}{x} \right )^n =
\\
 = \frac{1}{6} \left(n - 2 \right ) \left( n - 1 \right ) \frac{n \pi}{\sin n \pi} \cdot 4 \pi \rho_1 R_*^3 \Rightarrow \rho_1 = \frac{3}{2} \frac{\sin n \pi}{n \pi \left( n - 2\right) \left( n - 1\right )} \frac{M}{R_*^3}
\\

\end{matrix}