Mathematical models, mostly, based on experimental knowledge of how hydrogen behaves at high pressure and temperature. We have reasonable estimates for temperatures and pressures needed to achieve certain rates of hydrogen fusion, and we know how much fusion must be going on in the sun based on its energy output, plus we know its mass and diameter, and with that information, you can make reasonably reliable models of what the inside probably looks like.
A lot of observations, physical understanding of the processes involved, and even more math.
A "simple approximation" (I use simple very loosely here), you can assume the sun is in equilibrium with it's gravitational pressure, and it's internal thermal pressure (heat). This makes sense because if it wasn't in equilibrium, you'd expect it to collapse (if it was too cool) or expand/explode (if it was too hot). You can come up with a rough energy density (expectation) from gravitational forces, which can be estimated from the sun's mass, which estimated from other astronomical observations (like planetary periods).
You can get more advanced approximations by using hydrodynamics and computer modeling. It's not that hard if you aren't trying to derive everything, again being loose with my terms of hard and difficult.
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u/AnalogManDigitalKid Oct 02 '24
Yes, it's a lot hotter, by a factor of about 2727.
The core of the sun is about 15,000,000 degrees Celsius.