How do inert helium cores in sub giant stars create a pressure force? I'm reading about the sub giant branch (SGB) and the evolution to the red giant branch (RGB).
On the SGB stars have burned all hydrogen into helium, as a result, they have an inert helium core. The core begins to contract and the pressure increases in order to re-establish hydrostatic equilibrium. my question is, how? Nowhere have I found an explanation as to the reason behind the pressure rise in the core, so, I have tried to come up with my own; I just want to know if I am right or wrong.
Due to the inert core no energy is produced to counter gravity, so gravity begins to take over and the core contracts. For some reason or another the pressure of the core increases to counter gravity and regain hydrostatic equilibrium. This also causes the temperature of the core to rise and move hydrogen out into a thin shell surrounding the core, where it continues to burn. 
I understand that degenerecy sets in when the star reaches the RGB and not quite at the SGB. Its difficult for me to understand where the energy comes from to counter gravity in the core. 
 A: Don't forget that the inert core is still very hot, and doesn't necessarily have to start cooling down, because it's underneath the hydrogen-burning shell, which produces heat. It does start contracting later, but this is probably because the fluid can no longer configure itself to support the envelope without producing a bit more pressure, either by contracting or becoming degenerate. Also, countering gravity doesn't need you to produce energy: it needs you to establish a pressure gradient.
All this said, I want to point out that you're touching on what is sometimes referred to as the red giant problem. That is, it is not presently understood precisely why stars or stellar models expand so drastically after they exhaust hydrogen in their cores.
This often surprises people, so let me explain a bit further. The problem is not that the physics is missing, or we don't know enough about how stars work inside. Our models clearly do become red giants, at about the same time as their cores become inert, a large mean molecular weight gradient builds up at the core-envelope boundary, the hydrogen-burning shell becomes narrow and intense, and the envelope becomes convective.  But it's not clear which of these (or what combination) is to blame.
I usually try to explain this by turning to models of pure helium stars less massive than about 0.8 solar masses. In this range, the stars burn helium into carbon on a kind of "helium main sequence". Then, as in ordinary stars, the helium burning proceeds in a shell, and the carbon/oxygen core remains inert.  But the stars don't expand!  They get brighter, sure, but also hotter rather than cooler.
As far as I know, there's currently no answer. Many have been offered over the years (and I have my own ideas), but none has stood the test of time. Beware what you read: we know what happens, but I wouldn't trust anything that claims to know precisely why it happens.
