Estimating The Sun's Central Pressure and Temperature

Let's use the equation of hydrostatic equilibrium to (very crudely!) estimate the central pressure in the sun.

Remember we have

Homer estimates dP/dr to be

(how did he do this?)

So that he derives

We know these numbers, so we can calculate the pressure:
Being Homer, he was only a factor of 100 too low. To do it right we need to actually integrate the equation of hydrostatic equilibrium:
But we don't know M(r) and rho(r) explicitly, so this is hard. But it can be done through models, and we get the better value of

What about the central temperature?

Use the equation of state (the ideal gas law) to estimate the central temperature:

Make assumptions:

and we get

which, actually isn't too bad. The actual "best" value is more like 15 million K.