= Solution
Using the <enclosed mass> integral, $R=a\xi_1$, and the Lane-Emden equation,
$$
\begin{aligned}
M
&=4\pi\int_0^R\rho(r)r^2\,dr\\
&=4\pi a^3\rho_c\int_0^{\xi_1}\theta^n\xi^2\,d\xi\\
&=4\pi a^3\rho_c
\left[-\xi^2\theta'(\xi)\right]_{\xi=\xi_1}
=4\pi a^3\rho_cY_n.
\end{aligned}
$$
Substituting the value of $a$ from part (b) gives the <Lane-Emden mass formula>
$$
\boxed{
M=\frac1{2\sqrt\pi}
\left(\frac{(n+1)K}{G}\right)^{3/2}
\rho_c^{(3-n)/(2n)}Y_n,
\qquad
Y_n=-\xi_1^2\theta'(\xi_1).}
$$
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