Stephanie White, of Kakanui School, asks :-
How do scientists know the temperature of the core of the sun?
Ian Craig, a mathematician who works on plasmas at Waikato University, responded.
I guess the first thing to mention is that scientists don't really know the temperature exactly. They can't measure it using a thermometer or similar device, not yet anyway. What they have to do is estimate it using physical arguments.
Here's an example for the sun. We know the sun's mass and radius from astronomical observations. This means we can estimate its average density (mass divided by volume).If we work this out we find that the average density is close to one gram per cc, very similar to the density of water! By using similar reasoning to this, but a bit more sophisticated, we can begin to answer your question.
In the first place the sun is neither growing or shrinking rapidly with time. Therefore the forces that hold it together must be in balance. What are these forces? Its obvious from the spherical shape of the sun that gravity must be trying to pull material into the core, and so there must be some force to stop the sun collapsing inward. This force is due to pressure gradients in the gas and pressure comes about through the random motions of molecules in the gas. The higher the temperature of the gas, the faster the molecules move and so the higher the pressure becomes. If the pressure became too large the sun would expand and maybe even explode. Too small and the star would collapse inward under gravity, maybe even forming a black hole. Black holes are a big deal these days!
To get much further we have to do a small calculation. I have enclosed details separately. From gas theory we can relate the pressure to temperature and hence the change in pressure between the core and the surface. This pressure gradient is equal to the force of gravity at the surface of the sun and we know this from Newton's law. When we put the numbers in we get about a hundred thousand degrees Celsius.
I agree that this is a pretty crude estimate but it should be in the right ballpark. Perhaps we shouldn't have used the surface gravity on the sun but a value for the gravity in the interior. That would make a bit more sense but its easy to check that the core temperature wouldn't be changed too much. I reckon this is good enough for a first bash.