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I am an early graduate student in astronomy and have hard time understanding why do post-MS stars move up the RGB.

Here is what I understand about post main sequence evolution of stars. As their hydrogen core is exhausted, the core shrinks under its own gravity. This lets a region of previously too cold hydrogen enter hotter regions, thus starting a hydrogen shell burning process. While this process can keep the luminosity constant, it expands the envelope, hence lowering the temperature and moving the star to the right in the HR diagram.

Then suddenly, the star starts moving up on the RGB: slight or no temperature change, but sudden increase in luminosity. What makes the star move up the HR diagram all of the sudden? This is before the helium flash when the core starts burning helium.

Thank you!

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If your question is why the star starts moving up the red giant branch, it's in essence because of the behaviour of the surface opacity and the development of a substantial convective envelope in order to meet the surface boundary condition. It's basically the Hayashi track in reverse.

We can say this because if you create a model of a star and artificially suppress convection in the equations, then the model just keeps getting colder and colder at roughly constant luminosity, instead of ascending the giant branch. See e.g. Fig. 2 of Stancliffe et al. (2009). Note that the star still becomes large in terms of radius, just not bright.

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As the hydrogen supply in the core is exhausted, the pressure supporting against gravitational collapse lessens, and the core begins to collapse, which causes the inner temperature to rise. As a result, hydrogen in the less-processed regions outside the core starts to burn in a shell. Stellar models predict that, at this stage, there is a huge expansion of the outer layers of the star (the understanding of which is difficult to grasp intuitively, but it's robustly predicted by the equations of stellar structure). The increase in luminosity, due to the gravitational contraction and the hydrogen shell burning, moves the star up on the H-R diagram, while the increase in radius lowers the effective temperature, moving the star to the right on the diagram. This is the so-called "red giant phase".

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