 Radiative-envelope convection matching

For $P=C(T^b+K{)}^{1/p}$ with $p=n+1$ and $b=4+n+m$, the logarithmic radiative temperature gradient is $(p/b)(1+K/T^b)$. At a marginal boundary with an efficient ideal-gas convective envelope, the Schwarzschild criterion equates it to $(\gamma -1)/\gamma$. Hence $K=T_b^b[(b/p)(\gamma -1)/\gamma -1]$. The notation $K=T_0^b$ with real positive $T_0$ requires a nonnegative bracket. Vanishing $p$ or $b$ needs the corresponding logarithmic integration instead of this power form.