= Solution
Both gas and radiation pressures are nonnegative, so $0\le\beta\le1$. Rearranging the constant-fraction result gives
$$
\boxed{L=(1-\beta)\frac{4\pi cGM}{\alpha}\le L_E,
\qquad L_E=\frac{4\pi cGM}{\alpha}.}
$$
This is the <Eddington luminosity> for the effective <opacity> constant of this model. Positive gas support gives a strict inequality, with equality only in the radiation-only limiting case. \b[Exceeding the limit would require a negative gas-pressure fraction], which is incompatible with the assumed <equation of state>.
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