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the variable star has oscillations due to the instability of the hydrostatic equilibrium and radiation pressure. How can opacity decrease with an increase in temperature

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    $\begingroup$ What variable star are you talking about? $\endgroup$ – Rob Jeffries Nov 18 '18 at 10:06
  • $\begingroup$ Opacity does not increase with an increase in temperature: the hot outer layers of a star are a plasma which is highly ionized. A photon travelling through a plasma can go only a very short distance before it gets scattered, which means that a hot plasma is essentially opaque to light. $\endgroup$ – niels nielsen Nov 18 '18 at 19:24
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How can opacity decrease with an increase in temperature?

A physical interpretation of this lies in the awareness of what opacity stands for in stellar physics.

Let's return to the definition of opacity. In stars, radiative heat transports takes place when photons are emitted by a slab of gas at a distance $r$ from the center and absorbed by a slab of gas at a distance $r + dr$. In fact, radiative heat transport can be described as a random walk process, in which photons have a mean free path defined by:

$l = \dfrac{1}{\kappa}$

Where $\kappa$ is the opacity. The mean free path $l$ of a photon represents the average distance that it may travel before it is absorbed by the medium and then re-emitted. This distance is very short in the inner core of a star: actually it is of the order of a few centimeters in the sun. This means that a photon emitted at the center of the sun would need approximately $10^{21}$ steps and a few thousand years to reach the corona.

But why/by what are photons absorbed?

Because the interior of a star is extremely dense and filled with a highly ionized plasma, which interacts strongly with photons. In fact, this is the very reason why stars are at thermodynamic equilibrium: matter and radiation field are heavily exchanging energy. This would not be possible if opacity was negligible.

There are several mechanisms in which atoms, ions and electrons may interact with photons in stars:

  • atomic electron transition
  • photoionization
  • thompson scattering

Now let's see how the efficiency of these mechanisms changes as the temperature increases:

  • At low temperature (at the surface of the star), matter is made of neutral atoms which interact weakly with photons: $\kappa(T)$ is low.
  • As you dive into the star the temperature grows to $T \approx 10^4~\mathrm{K}$, hydrogen is completely ionized and heavier atoms are partly ionized. At this stage, photons are massively absorbed and re-emitted by the electrons bound to ionized nuclei: $\kappa(T)$ is maximum in this region, where convective heat transport is superior to radiative heat transport.
  • If you go further in, as the temperature reaches $T > 10^5~\mathrm{K}$ it becomes harder for ions to absorb photons because they are more and more energetic (because the energy of thermal photons relates to $\sigma T^4$, in virtue of the Stefan-Boltzmann law). Indeed, remember that an electronic transition can only take place if the energy of the photon corresponds to the energy difference between the two quantum levels. Thus, $\kappa(T)$ decreases.
  • Once $ T > 10^6~\mathrm{K}$, there are only the free electrons on which very high energy photons can diffuse: $\kappa(T)$ is minimum in this region where radiative heat transport is dominating.

As you can see, for $T > 10^4~\mathrm{K}$ the opacity is decreasing as the temperature increases, because the interaction between ions and photons declines as the energy of photons grows too much.

As you implied in your question, this will play a role in variable stars. Indeed, as the oscillations of hydrostatic equilibrium relate to oscillations in temperature, the opacity will vary. On one hand, when temperature is low, opacity increases and the luminosity of the star decreases (the amount of photons reaching the outer shell - or photosphere - where light is radiated decreases because of absorption). On the other hand, when temperature is high, opacity decreases and the luminosity increases, as the photons can diffuse through the outer shell. This will result in a pulsating variable star.

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