# Solar luminosity proportional to $G^7$?

A book I recently purchased describes Dirac's proposed variation of Newton's G, and why that specific proposal wouldn't work because the earth would have been too hot for the known history of life. A footnote explains that the luminosity of the sun is proportional to the seventh power of G, which seems incredible to me. I know that this has to be a simplification, since stellar dynamics are annoyingly complex, but if we put that aside, is this essentially correct, or is it a typo? What would be the theoretical justification for this in broad terms? (eg, "Gravitational pressure is proportional to this exponent of G, but since that also shrinks the size of the star, which increases the pressure, it's actually this exponent of G, and the rate of fusion is proportional to the gravitational pressure squared, so you end up with 7").

• Can you clarify what you mean by "G"? – Dean Aug 15 '16 at 14:42
• It sounds like whatever you're talking about is a simple scaling law. Something akin to the stellar mass-luminosity relation, $L\propto M^3$ or the stellar expected lifetime-mass relation, $T_{star} \propto M^{-2.5}$. Despite the complexity of a given physics scenario, there are often basic scaling laws one can boil the system down to with enough approximations. – zephyr Aug 15 '16 at 15:50
• Can you provide a small excerpt from the book of relevant material to this question? – zephyr Aug 15 '16 at 15:55
• @Dean: Newton's universal gravitational constant. – Keith Thompson Aug 16 '16 at 14:59

I think the argument is the following. The central temperature can be estimated from a form of the virial theorem. At least dimensionally speaking, total thermal energy $MkT/\mu$ (where $\mu$ is the average mass per particle) is proportional to gravitational potential energy $GM^2/R$. So $T \propto GM/R$.
But for a given mass, standard polytropic theory for a star, says that $$M \propto \rho_c^{(3-n)/2n} G^{-3/2}$$ $$R \propto \rho_c^{(1-n)/2n} G^{-1/2}$$ where $n=3$ is an approximate polytropic index for the Sun and $\rho_c$ is the central density. See for example Does Eddington's variable polytropic index better fit data from the Standard Solar Model?
Putting this together we have $$T \propto GMR^{-1} \propto G^2 \rho_c^{-1/3}$$
The power generation in the solar core from the pp cycle goes approximately as $L \propto \rho_c^2 T^4$ and using the proportionality for $T$ derived above, we have $L \propto G^8 \rho_c^{2/3}$.
Finally, we note that for a $n=3$ polytrope, that $\rho _c$ and $M$ are independent, but that $\rho_c^{2/3} \propto G^{-1} R^{-2}$. So for a fixed solar radius we have $L \propto G^7$.