How do you estimate the core temperature of a star? Given a star's mass, radius and average composition (e.g. 90% H, 10% He), is there a formula to estimate the core temperature of that star?
I only found one for a lower bound but that wasn't very accurate.
 A: There are approximate formulas for this, but to do this right you really need an equation of state for matter in the stellar core. Writing that down is really, really hard, which is why predicting the temperature and density profiles for stellar cores is really, really hard.
A: Yes, the formula you quoted in comments is an application of the viral theorem, that says for a star in equilibrium that twice its internal kinetic energy plus it's potential energy is zero. This can be written
$$\Omega = - 3 \int P\ dV = -3\int P\ \frac{dm}{\rho}$$
where $\Omega$ is gravitational potential, $P$ is pressure, $\rho$ is density and $dm$ is a mass element.
A rough solution is to assume the star is a uniform sphere with an average pressure, density and temperature. In which case
$$-\frac{3GM^2}{5R} = -3\frac{PM}{\rho}$$
where $M$ is the stellar mass and $R$ the radius. Writing $\rho = 3M/4\pi R^3$ and assuming an ideal, perfect gas with a mass per particle of $\mu$ and temperature $T$, then
$$\frac{GM}{5R} = \frac{\rho k_B T}{\mu \rho}$$
$$T = \frac{G\mu}{5k_B}\frac{M}{R}$$
This gives the right proportionality but the numerical coefficient of 0.2 is not accurate because the star is not uniform; the gravitational potential is not that of a sphere and the density, temperature and pressure vary with radius. A 1 solar mass, 1 solar radius star, with the composition you mention ($\mu = 0.58\times 1.67\times 10^{-27}$ kg) we get $T = 0.2 \times 1.3\times 10^7$ K. This is not so bad for an average temperature, but not close to the core temperature.
A more accurate approximation comes from assuming a polytropic equation of state with $P \propto \rho^{\alpha}$. For a star like the Sun (or of higher mass), where the energy is largely transported radiatively, it turns out $\alpha \simeq 4/3$. Solving the Lane-Emden equation and assuming an ideal gas then gives a new numerical coefficient of 1.17 (for a largely convective star with $\alpha =5/3$, the dtar is more centrally condensed and the coefficient would be 1.86).
Thus your answer for a sun-like star is
$$ T_c \simeq 1.17 \frac{GM\mu}{R}\ .$$
and for a lower mass main sequence star, where convection is dominant
$$T_c \simeq 1.86 \frac{GM\mu}{R}\ .$$
If you want something more accurate than this then a precise stellar evolution model is required to solve for energy transport at each radius, to account for the non-perfect nature of the gas, radiation pressure and accurately describe the pressure and density profile.
