β = P_{g} / P

is spatially constant, with

0 < β < 1

. This is a sufficient condition for a stellar polytrope. Since

P_{g} = \frac{ρ k _{B} T ^{1 + s}}{μ _{1} m _{u}}, P_{rad} = \frac{a _{rad} T ^{4}}{3},

P_{rad} / P_{g} = (1 - β) / β

gives

ρ = C_{β} T^{3 - s}, C_{β} = \frac{a _{rad} μ _{1} m _{u}}{3 k _{B}} \frac{β}{1 - β} .

Therefore

P = \frac{a _{rad}}{3 ( 1 - β )} T^{4} = K ρ^{1 + 1/ n},

with

n = \frac{3 - s}{1 + s}, K (β) = \frac{a _{rad}}{3 ( 1 - β )} C_{β}^{- 4/ (3 - s)} .

For an ordinary positive polytropic index one also assumes

0 \leq s < 3

\frac{d P _{rad}}{d r} = - \frac{κ ρ L _{r}}{4 π c r ^{2}} .

Because

P_{rad} = (1 - β) P

and

β

\frac{d P _{rad}}{d r} = - (1 - β) \frac{G m _{r} ρ}{r ^{2}} .

Equating them and using

η = (L_{r} / L) / (m_{r} / M)

yields

κ η = 4 π c G (1 - β) \frac{M}{L} .

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