How can I relate the temperature and pressure at the core of a star?

Question

There are many ways to estimate the temperature of the core of a star, one of which is by radiation pressure and the gravitational force of the star on itself. I am unable to relate both of them correctly to get the temperature. Note that by radiation pressure, I mean pressure due to the emission of photons and not any pressure of the form $P = cRT$.

Answer 1

First you need the equation of state for stellar matter. Deriving that is an extraordinarily difficult job: don't try this at home! Approximate EOS's have been written down, one of the experts here can point you to them.

Answer 2

In the interior of a star, the radiation pressure is given by $$ P_{\rm rad} = \frac{4\sigma}{3c}T^4\ , $$ where $T$ is the temperature and $\sigma$ is the Stefan-Boltzmann constant.

If radiation pressure is dominant, then it is related to the local density and gravity in a spherically symmetric star by $$\frac{dP_{\rm rad}}{dr}=- \rho g\ , $$ where $r$ is a radial coordinate.