The context is the ionization zones in stellar atmospheres or interiors, the Sun, for instance.

The adiabatic exponent is the heat capacity ratio:

$$\gamma = \frac{c_P}{c_V} = \frac{C_P}{C_V}$$

And, for an ideal gas, it can be shown that $\gamma=(f+2)/f$ where $f$ is the degrees of freedom. Resulting in $\gamma\approx1.66$.

In most of the extent of the solar interior, $\gamma\sim1.66$ but it presents dips when ionization is present, like the H, HeI and HeII ionization zones. My question is why does this happen?

I thought it could be because when ionization, the heat is used to ionize atoms instead of increasing the temperature and, therefore, the heat capacity would be larger in those regions. But, because the $\gamma$ is the ratio between the heat capacities, the argument does not work.

Any suggestion?

  • $\begingroup$ Not sure I understand what you're asking. Yes, $\gamma$ is in general not a constant when ionization processes are involved. $\endgroup$ Commented Sep 25, 2018 at 20:25
  • $\begingroup$ @AtmosphericPrisonEscape - My question is why the ionization processes affect the value of $\gamma$, in particular, why do they suppress it? I clarified it in the question too. $\endgroup$
    – Stefano
    Commented Sep 26, 2018 at 16:57
  • $\begingroup$ I believe it is because of EM interaction. You might want to check that. $\endgroup$ Commented Sep 28, 2018 at 13:39

1 Answer 1


There are two things going on. (1) When you add heat to a gas that is on the threshold of ionisation or partially ionised, some of that heat goes into ionisation. This means that it takes a much larger amount of energy to produce a rise in temperature. (2) However, as the gas becomes ionised the number of particles per unit mass also increases and so the pressure increases at a given temperature. The net effect is that $C_V$ (heat capacity for a fixed volume) increases by more than $C_P$ (heat capacity for a fixed pressure) and so the adiabatic index decreases.

It is a tricky and detailed computation to work out how $C_V$ and $C_P$ change as ionisation proceeds. An example calculation for pure hydrogen gas is give on pp 123-126 of "Principles of stellar evolution and nucleosynthesis" (1983, D. Clayton). Both $C_V$ and $C_P$ increase drastically (by factor of 30-40) as ionisation increases, both peaking at around 50% ionisation before falling back to exactly twice their unionised values once the gas is completely ionised (when it just behaves like a monatomic perfect gas but with twice the number of particles per unit mass). However, the behaviour of $C_V$ and $C_P$ are slightly different, $C_P$ increases by less than $C_V$, and so the ratio of $C_P/C_V$ changes and goes through a minimum at around 50% ionisation before regaining the standard monatomic gas value once completely ionised.


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