Model to predict temperature of the Sun?

Question

In a homework problem, we were to assume the Sun is a sphere with uniform density and use the hydrostatic equilibrium equation to compute the pressure at the center of the Sun? Upon doing this, I was able to derive this equation:

$ \frac{dP}{dr} = \frac{-GM(<r)}{r^2}\rho(r)$

$ \frac{dP}{dr} = -g(r)\rho(r)$

Now, a related part of the problem is asking that we assume the core of the Sun contains only ionized hydrogen. We are to use the equation derived in the first part of the problem for the central pressure and the ideal gas law to estimate the temperature at the core of the Sun, and compare it to its actual temperature of $ 1.6 \times 10^7 K $.

Can someone explain how to combine the equation for finding the central pressure with the ideal gas law? Also, why is it relevant that we assume the Sun is only composed of ionizable hydrogen?

Answer 1

I don't want to give the answer, so maybe this will help.

First, you've written one equation (twice) but probably also need the definition of $M(<r)$, which can be derived from $$\frac{dM}{dr}=4\pi r^2\rho$$ Usually, $\rho$ depends on $r$, but in this example it's assumed constant. That should give you enough to integrate inwards to get the central pressure. You don't need the ideal gas law at this point.

As mentioned in the comments, you'll then have the pressure $P$ and density $\rho$, from which you can derive the temperature $T$ from the ideal gas law $P=\rho kT/\mu$, where $k$ and $\mu$ are Boltzmann's constant and the mean molecular weight. The assumption that the Sun is only composed of ionized hydrogen allows you to work out the mean molecular weight $\mu$. I assume that's been discussed in your class. If not, you should be able to work it out from the definition: what's the mass per free particle of ionized hydrogen? (What's the mass? How many free particles are there? Divide them!)