When $s=0$, part i gives the <polytropic index> $n=3$. <Stellar homology> gives
$$
\rho\sim\frac{M}{R^3},
\qquad
T\sim\beta\frac{\mu m_uGM}{k_BR},
$$
because gas pressure is the fraction $\beta$ of the pressure required by hydrostatic equilibrium. The <Kramers opacity law> therefore scales as
$$
\kappa\propto\rho T^{-7/2}
\propto\beta^{-7/2}M^{-5/2}R^{1/2}.
$$
With constant $\eta$, the opacity relation from part i gives
$$
L\propto(1-\beta)\frac{M}{\kappa}
\propto(1-\beta)\beta^{7/2}M^{7/2}R^{-1/2}.
$$

For an $n=3$ polytrope, $M\propto(K/G)^{3/2}$. The explicit $K(\beta)$ of part i obeys
$$
K^3\propto\frac{1-\beta}{\beta^4},
$$
so the <Eddington quartic relation> is
$$
\frac{1-\beta}{\beta^4}\propto M^2
$$
at fixed composition. Substitution gives
$$
\boxed{L\propto\beta^{15/2}M^{11/2}R^{-1/2}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 317 / 2 / iii / Solution

{title2=$n$}

The polytropic index $n$ is defined by $P=K\rho^{1+1/n}$. A monatomic convective ideal gas has $n=3/2$, while a radiation-supported Eddington model has $n=3$.


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 Incoming links: Polytropic index