= Luminosity integral of an index-five polytrope
{title2=$L=12\pi\sqrt3\epsilon_0\alpha^3\rho_c^2T_c^q\int_0^1u^2(1-u^2)^{(q+5)/2}du$}
For a gas-pressure-supported <polytrope of index five> with uniform <mean molecular weight>, $T/T_c=\theta$. A specific <stellar energy-generation rate> $\epsilon=\epsilon_0\rho T^q$ therefore integrates to $4\pi\epsilon_0\alpha^3\rho_c^2T_c^q\int_0^\infty\xi^2\theta^{10+q}d\xi$. Substituting $u=\xi/\sqrt{3+\xi^2}$ gives the displayed formula, convergent for $q>-7$. At $q=11$ its dimensionless coefficient is approximately $1.02942$. The scale factor is a volume, $\alpha^3$, rather than an area.
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