Star Opacity and Balmer Break

[desperate_student]

Homework Statement

Write an expression for κ(3647+)/κ(3647−), the ratio of the opacities just above and below the Balmer break for (i) low-temperature atmospheres where the opacity above the Balmer break is due to the H− ion, and (ii) high-temperature atmospheres where the opacity above the Balmer break is due to the n = 3 level of neutral hydrogen. Your results should show that at low temperatures, the Balmer discontinuity depends on both temperature and electron pressure, while at high temperature, it depends only on temperature.

Relevant Equations

Saha Equation, Boltzmann Equation, Kramers Opacity for free-free and free-bound and any other relevant equations.

I am not sure where to start for this. Considering it needs to be demonstrate Balmer Break, I am assuming it needs to be wavelength based. As a result I am assuming I cannot use mean (Kramers) Opacity but rather express in terms of Opacity k= n*sigma/rho.
My thoughts are to use Boltzmann Equation to obtain n_h,n=2,3 and Saha Equation to find n_h- and sub into n*sigma/rho and find k ratio based on that

 

[mark726]

and then plot a graph of k vs wavelength.

Hello there,

Thank you for sharing your thoughts on how to approach this task. I would like to offer some additional suggestions that may help you in your research.

Firstly, you are correct in assuming that the Balmer Break is a wavelength-based phenomenon. The Balmer Break refers to the discontinuity in the spectrum of a star caused by the absorption of hydrogen atoms in the star's atmosphere. This absorption occurs at specific wavelengths, known as Balmer lines, which are determined by the energy levels of the hydrogen atoms.

In order to demonstrate the Balmer Break, you will need to obtain a spectrum of the star in question. This can be done using a spectrometer, which measures the intensity of light at different wavelengths. You can then use this spectrum to identify the Balmer lines and their corresponding wavelengths.

Next, as you mentioned, you can use the Boltzmann Equation to calculate the number of hydrogen atoms in the star's atmosphere at energy levels n=2 and n=3. This will give you the population of hydrogen atoms at these energy levels.

To further understand the Balmer Break, you can also use the Saha Equation to determine the ratio of the number of hydrogen atoms at energy level n=2 to the number of hydrogen atoms at energy level n=3. This will give you an idea of the relative population of these two energy levels.

Finally, you can plot a graph of the opacity (k) vs wavelength using the values obtained from the Boltzmann and Saha Equations. This will allow you to see how the opacity changes at the Balmer lines, demonstrating the Balmer Break.

I hope this helps in your research. Good luck!