How was the core temperature of the Sun estimated?
It was estimated that the heat inside the core of the Sun inside around 15 000 000 °C - this value is extremely enormous. How did scientists estimate this value?
I just want to point out this very enlightening article about the difficulty in find a "simple" means of computing the Sun's structure (and thus central temperature), Solar structure without computers. This is probably why you've yet to get an answer with a simple algebraic expression for the central temperature.
The composition can be determined by taking spectra. Additionally, the mass can be determined through dynamics. If you combine these two, under the assumption that the star is in a state of hydrostatic equilibrium (which means that the outward thermal pressure of the star due to fusion of hydrogen into helium is in balance with the inward tug of gravity), you can make statements about what the temperature and density must be in the core. You need high densities and high temperatures in order to fuse hydrogen into helium.
Remember what's happening: Temperatures are hot enough for hydrogen in the core to be completely ionized, meaning that in order to fuse these protons into helium nuclei, you need to overcome the electromagnetic repulsion as two protons come close (like charges repel). Below is a diagram of the process of one particular type of fusion (Proton-proton chain reaction).
The other fusion reaction which occurs at the cores of stars is called the carbon-nitrogen-oxygen (CNO) cycle, and is the dominant source of energy for stars more massive than about 1.3 solar masses. Below shows this process.
Somebody pointed out that this doesn't actually answer the question at hand - which is true. Forgetting how to do some of the basic back of the envelope calculations myself (I admit, stellar astrophysics is definitely not my specialty), I've stumbled across a very crude and simple estimation of how to calculate the central pressure and temperature of the sun from. The calculation does however point out the correct values and what one would need to know in order to get the details correct.
This answer doesn't really answer the question as to how the temperature value of ~10^7 K is determined.
@ Guillochon Yea, you're right. I was a bit too general. I'll try to update with a more specific answer.
@Guillochon I've added a link. Feel free to modify/edit my answer if you have better information at hand.