What causes the dimensions of a star increase when its hydrogen fuel is exhausted?

What causes the dimensions of a star increase when its hydrogen fuel is exhausted? For example, the Sun is expected to increase its radius 250 times. What causes this if its temperature is expected to fall? How gas can expand if the temperature falls?

The Sun will never run out of Hydrogen. This is a common misconception.

At this moment the Sun is fusing Hydrogen into Helium. This fused Helium remains at the core until it will reach a critical mass. At this point the core will begin to collapse. This collapse increases the temperature and pressure around the core where Hydrogen is being fused causing the increase the Hydrogen fusion which also causes an increase of the radiation pressure making the outer layers to expand and cool.

Once the core reaches a temperature of about 100 million Kelvin Helium fusion starts dramatically (Helium Flash) with high radiation pressure in the core. Here is where the Sun will reach 250 times it's current radius.

When the Helium fusion stabilizes the radiation pressure at the cores decreases making the Sun's radius to shrink.

The temperature will only drop at the surface because as it's expands it's further from the core.

• According to The Disappearing Spoon, at least as I understand it, an increase in temperature results in a decrease in the rate of fusion--a fact which establishes the equilibrium temperature for a star of given composition. If helium is capable of supporting fusion at higher temperatures than hydrogen which is subjected to similar pressure, that would suggest a higher equilibrium temperature. Oct 9, 2014 at 18:13
• Temperature is not all what matters in fusion. There are high increases in pressures in the core. Oct 10, 2014 at 6:22
• Density is a function of pressure and temperature. As temperatures increase, density decreases. Further, by my understanding, in order for atoms to fuse, the difference in their velocity vectors must be in a certain range. If too small, the nuclei will be pushed apart by their electric charges. If too great, the kinetic energy remaining after the collision will carry them apart. Similar to the reason that nuclear reactors need something to moderate the speed of the neutrons flying around therein. Oct 10, 2014 at 15:15
• @supercat Not exactly. In degenerate matter pressure is almost completely independent of temperature. This is what critically happens in various stages of the evolution of various stars (such as a helium flash): the core becomes degenerate before it can begin fusing, so temperature can scale up (or down) drastically with little to no pressure change, resulting in an extremely rapid fusion of nearly the entire core once temperatures reach a certain point. Jun 5, 2015 at 2:48