= Inverse-square-opacity burning envelope
{title2=$y''=-\omega^2y^2,\quad F_s\propto\Sigma_b^{-6}$}
In a thin nondegenerate <hydrogen-burning white-dwarf envelope>, fix gravity $g$ and use $P=g\Sigma$, $\rho=B\Sigma/T$. With opacity $\kappa=\kappa_0\rho T^{-2}$ and specific heating $\epsilon=\epsilon_0\rho T^{15}$, set $y=T^7$, $x=\Sigma^2/2$. Then $y'=AF$ and $F'=-\epsilon_0By^2$, where $A=21\kappa_0B/(4ac)$ and $\omega^2=A\epsilon_0B$. Radiative-zero surface and zero core-flux base conditions give $(y')^2=(2\omega^2/3)(y_b^3-y^3)$. Thus $T_b\propto F_s^{2/21}$ and $F_s=12I^3/(A\omega^2\Sigma_b^6)$, with $I=\int_0^1(1-t^3)^{-1/2}dt$. At fixed column the specific heating varies as $T^{14}$ while radiative cooling varies as $T^7$, giving <thin-shell instability> in this approximation.
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