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I just (February 28th, 2021) heard the news on progress how to measure the temperature of super giants:

Red supergiants are a class of star that end their lives in supernova explosions.

Their lifecycles are not fully understood, partly due to difficulties in measuring their temperatures. For the first time, astronomers develop an accurate method to determine the surface temperatures of red supergiants.

The original article by Daisuke Taniguchi and team is called Effective temperatures of red supergiants estimated from line-depth ratios of iron lines in the YJ bands, 0.97-1.32 μm but is not available (yet).

Maybe somebody can help me understanding the following quote intuitively:

However, by looking at the ratio of two different but related lines - those of iron - we found the ratio itself related to temperature. And it did so in a consistent and predictable way.

How could the ratio between the intensity of spectral lines correlate with temperature? Furthermore: How could we actually show that correlation if we do not know the temperature in first place?

References

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    $\begingroup$ The accepted paper is available on astro-ph. It should be very close to the final MNRAS version, minus any minor editorial corrections found in the typesetting. $\endgroup$ – astrosnapper Mar 2 at 18:30
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Maybe somebody can help me understanding the following quote intuitively:

However, by looking at the ratio of two different but related lines - those of iron - we found the ratio itself related to temperature. And it did so in a consistent and predictable way.

A particular atom can only be at integer quantum states (Hydrogen is depicted here for simplicity).

enter image description here

When it decreases its quantum state, it emits a photon at an exact known frequency. The higher the 'n', the more energy the photon has with a corresponding higher frequency. Here is an example set of frequencies:

enter image description here

When we move to the more complicated case of molecules, the photons are again emitted at particular frequencies, but the quantum mechanics gets much more complicated depending on the spin states and configuration of atoms in the molecule. Here is an example with ammonia:

enter image description here

Note that the higher frequency emissions correspond to higher temperature gas. The same is true for iron. The hotter the temperature of the Red Giant, the higher the spectral line frequencies will be for iron photon emissions. As the temperature rises, the spectral lines for lower transition states will get lower and lower. We don't need an a priori estimate for temperature to do this analysis. The results can be predicted for temperature based on spectral frequencies.

The emissions from iron are extremely complex and varied. Here is a paper with tables of thousands of lines of different frequencies for iron from stars

Acknowledgements: The figures for this post were taken from this website: https://www.cv.nrao.edu/~sransom/web/Ch7.html.

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Stars behave like blackbodys. Not perfect idealized blackbodies, however, the spectrum of a star is close enough to the standard blackbody spectrum. Reason why you can use the Wien's Law to calculate an estimate of its surface temperature:

$\lambda_{\rm max} = (0.29 {\rm\, cm\, K}) / T$

Where $\lambda_{\rm max}$ is the frequency of maximum measured emission of light. Problem is, measuring a star’s spectrum is not always easy. The most common approach is to observe the star with two different filters, a blue one and a visual yellow one. Measuring the difference of the Blue bigthness and the Yellow one gives you the actual color of the star. In some cases more color filters are used (UV, Near Infrared, V, ...).

Now, as of why the used the iron lines: When a Star ages, its core gets more and more iron rich due to nuclear fussion (iron is the last element that produces more energy when getting fussioned than it takes for the reaction to happen). When the balance between outwards radiation and gravity gets unstable, the star collapses. In the case of Red giants a supernova most likley follows. The Red giant that was beign messured in the paper had probably a lot of iron, so the spectal lines of iron where the most "stable" ones that did not get altered by the stars upper atmosphere. They probably used two frequencys of different iron configurations to propperly estimate the $\lambda_{\rm max}$ of the star.

Hope this clarifys things a bit.

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