= Solution
For <polytropic index> three, the <Lane-Emden mass formula> cancels the central <mass density>:
$$
M=4\pi\omega_3\left(\frac K{\pi G}\right)^{3/2},\qquad
\omega_3=-\xi_1^2\theta'(\xi_1)\simeq2.01824.
$$
Substituting the preceding constant $K$ gives
$$
\boxed{M=\frac{4\omega_3}{\sqrt\pi\,G^{3/2}}
\left(\frac3{a_r}\right)^{1/2}
\left(\frac{\mathcal R}{\mu}\right)^2
\frac{\sqrt{1-\beta}}{\beta^2}.}
$$
For fixed composition this is \b[$M\propto\sqrt{1-\beta}/\beta^2$], as required. Squaring and rearranging yields the <Eddington quartic relation>, $(1-\beta)/\beta^4\propto\mu^4M^2$. The larger masses in this model have smaller gas fractions and relatively more radiation support.
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