\item High temperature - for the P-P Chain, .
This is an excellent question. We know the temperature at the Surface of the Sun is far too cool for fusion. What about as we go deeper towards the center? This is the subject of a topic called Stellar Structure.
It's all based on the concept of .
The pressure exerted by a gas increases as the temperature of the gas increases. If you think about a container filled with gas atom or molecules, the thermal pressure exerted on the walls of the container is due to all the atoms and molecules bouncing off the walls. Remember, Temp is a measure of the mean kinetic energy of the gas particles.
At each level moving down towards the center of the Sun, the total column of mass above every point increases so the Gravitational pressure increases as you move in from the surface of the Sun. To maintain Hydrostatic Equilibrium, the Thermal Pressure must also increase with depth which requires that the Temperature increase toward the center of the Sun.
With computers it is possible to build (we think) very accurate models of the Sun and other stars based on the the requirement of hydrostatic equilibrium and the laws of chemistry and physics. We can predict the temperature and density of the Sun at each radius from the center to the surface.