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I am trying to understand how the degenerated He core progressively increases its temperature as the star moves up through the Red Giant Branch.

A well-known property of the degenerate He core is that it is decoupled from the temperature and so, even if it shrinks, it should increase its pressure but not its temperature.

The derivation of the Virial theorem only relies on two equations: (1) the Hydrostatic Equilibrium equation and (2) the Mass Continuity equation. Two things that the degenerated He core can perfectly satisfy. Then, if the Virial Theorem applies, it should imply that as the core shrinks, half of the released gravitational energy will be radiated and the other half will be added to the internal energy of the degenerated He core. Does it mean that the degenerate He core will increase its temperature? Doesn't it contradict the statement that the temperature and pressure are decoupled?

Any thought is welcome.

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3 Answers 3

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Yes, the virial theorem applies to degenerate matter. As the He core shrinks, its temperature does increase. That is eventually why He fusion begins. The core is kept approximately isothermal by electron conduction. Most of the increase in internal energy goes into the kinetic energy of the non-degenerate ions, since they have a large heat capacity compared to the nearly degenerate electrons. The pressure due to both ions and electrons increases by about 2/3 of the increase in their respective internal energy densities.

The total energy density $$u = u_i + u_e.$$ The total pressure assuming a perfect ion gas and degenerate electron gas is $$P = n_i kT + P_e,$$ where $n_i$ is the ion density and the electron pressure $P_e = f(n_e)$ and is $\gg n_i kT$.

Now if we add $\Delta u$, this increases the gas temperature by $\Delta u/C_v$, where $C_v$ is the heat capacity per cubic metre. Most of this energy goes into raising the ion temperature. The electron temperature also rises and so does the pressure attributable to the ions, but in a degenerate electron gas the internal energy and pressure are almost independent of temperature. So long as $P_e \gg P_i$ there will only be a weak dependence of total pressure on temperature.

I should add, that the reason contraction can continue even when the helium core is supported by degeneracy pressure, is that it's mass is continually increased by the hydrogen burning shell above. This requires a more compact configuration to support the increased weight. However, changing the core mass also complicates the use of the virial theorem.

The bottom line is that although you can use the virial theorem to basically explain the behaviour, there is no substitute for a numerical model in the end.

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The Virial theorem applies to systems that are isolated in the sense that they conserve energy and angular momentum.

You can't treat the core of a heavy star in this way as it's temperature is set by the surrounding layer of fusing material.

In principle you can treat a white dwarf or neutron star (both systems that are largely degenerate) as a whole using the Virial theorem, but the work is going to be complicated because the potential is not simple.

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  • $\begingroup$ "You can't treat the core of a heavy star in this way as it's temperature is set by the surrounding layer of fusing material." Thanks for stating that. It is very useful. $\endgroup$
    – Stefano
    Commented Sep 25, 2017 at 17:50
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    $\begingroup$ I think the surrounding shell is just an additional term in the virial equation, but yes it makes things more complicated. $\endgroup$
    – ProfRob
    Commented Sep 25, 2017 at 20:54
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    $\begingroup$ Yeah, you can patch up the Virial theorem for various purposes when the violation of the assumptions is regular enough. That's also done for cases like dynamic cooling of clusters, too. But I might claim that you're using Virial++ or something like that. $\endgroup$ Commented Sep 26, 2017 at 2:18
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    $\begingroup$ Agreed. It's more complicated than just an external pressure, which is handled by the virial theorem. A further complication is that the core mass grows. $\endgroup$
    – ProfRob
    Commented Sep 26, 2017 at 6:48
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The Virial theorem is a relation between the potential energy and the kinetic energy of the system. The numerical factor by which they are connected will depend on the equation of state of the system and, since this equation is different for the degenerate matter, the Virial theorem for the degenerate matter will differ from the one for the non-degenerate matter.

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