PP Chain and CNO cycle relationship At what temperature would the energy generation rates of the PP-Chain and CNO cycles be roughly equivalent? The dependences are so vastly different that I am wondering how and by what equations they could be related.
 A: The amount of energy liberated per gram of material per second in the fusion reactions depends on the density, the mass fraction (hydrogen, $X$, helium, $Y$, and all others $Z$) and temperature:
$$
\epsilon = \epsilon(\rho,X, Y, Z, T)
$$
Typically we express the energy generation rate as a power law, 
$$
\epsilon\propto\rho^\alpha T^\delta.
$$
though the "true" forms are not too far off (usually containing an exponential term).
For the pp-chain, the energy generation rate takes the form,
$$
\epsilon_{pp}\sim1.05\times10^{-5}\rho X^2T_6^4 \,\rm erg/g/s\tag{1}
$$
where $X$ is the mass fraction of hydrogen and $T_6=T/10^6\,\rm K$. For the CNO cycle, the energy generation rate is
$$
\epsilon_{CNO}\sim8.24\times10^{-24}\rho XX_{CNO} T_6^{19.9}\,\rm erg/g/s\tag{2}
$$
where $X_{CNO}$ is the mass fraction of carbon, nitrogen & oxygen. If we assume that $\rho X^2\approx \rho XX_{CNO}$, then we can iteratively compute the equilibrium temperature to be around 
$$
T_{eq}\sim16,500,000\,\rm K
$$
In the figure below, which plots the energy generation rates as a function of temperature (in $T_6$ units), the matched temperature looks to be around $T_6\sim17$, which is slightly above the equilibrium temperature I found above.

(source)
Note that the figure uses $\epsilon_{CNO}\propto T^{17}$ while (2) contains $\epsilon_{CNO}\propto T^{19.9}$. This is because the book I used, Carroll & Ostlie (Amazon link), gives the latter form that I've used and I trust that book.
