ionization: stars and the universe

The textbook CO Chap. 8 contains much relevant information. For people struggling with CGS-SIconversion,consult
this nice text from Jason Harlow from the physics department.For physicalparameters of the Sirius A/B sytem, see this
wiki page.

1. In the lecture notes, Fig. 2.4 (and 2.5) shows the separating line between ionized and neutral (atomic) hydrogen
   (i.e., 50% of hydrogen is ionized).It hugs closely to T = 104 K and only rises slightly with density.Now with the
   tool of Saha equation (eq.4.7), can you derive the position and slope of this line? Recall that the ionization energy
   of ground-state hydrogen is 13.6 ev.Write your answer in the following illuminating form, with (α, β) constants to
   be obtained from your exercise,
   T
   104 K
   = α ×
   ρ
   10−8 g/ cm3
   β
   . (.1)
   You full solution may not be of this form, but you could numerically fit your solution close to the above density and
   temperature to obtain the values of α and β.
2. Sirius, the brightest star at our night sky, is an A-type star with an effective tempreature of 9940K. We would like
   to ascertain the ionization state of its photosphere.But to do that, we need the photospheric density.This is how
   we willl get that. Sirius has a mass of 2M and a radius of 1.7R . We also know that photosphere pressure can
   be expressed as iP = 2g/3κ (eq. 5.9 in Notes), and that the atmosphere is idealgas. Moreover,we can write
   the photospheric opacity κ = f × κ 0, where f is an unknown factor and κ0 is the electron opacity. The value of
   f ranges from unity to as large as 102
   , depending on the levelof ionization – if the photosphere is very ionized,
   one expects f ∼ 1 (dominated by electron scattering); on the other hand, if it is very neutral, f can be as large as
   102
   (contributions from bound-free and other opacities, Fig.4.3). Now combine all above information to determine
   the rough value of f , and therefore,figure out whether the atmosphere of Sirius should be mostly ionized, mostly
   neutral, or about half/half.
3. Interestingly,in main-sequence stars,the Balmer absorption series (electrons at n = 2 levelabsorbing photons to
   jump to higher n states) are also the most prominent around spectral type A0. Determine the fraction of (neutral)
   hydrogen atoms that are in the n = 2 state for Sirius’ atmosphere.You should get a smallvalue. The question is
   then, why shouldn’t stars hotter than Sirius (say, B-type stars) show prominent Balmer absorption lines?
4. Now Sirius B, the white dwarf companion to Sirius A, has aeff T= 25, 000 K (there is an interesting mystery related
   to Sirius B, see here).Now, consider the photosphere to be ideal gas (due to its relatively low density), repeat the
   same exercise as question 2 to deduce the ionization state of its atmosphere (you can use the approximate formula).
5. Bonus question – and also the easiest question of all Look up the spectra for Siriua A and B. Explain the main
   characters of these spectra, based on your above physical understandings.
6. Now let us boldly go to the universe. The epoch of recombination,during which hydrogen in the universe turns
   from being mostly ionized to mostly neutral, occurs when the universe has a temperature of∼ 4000 K. Use the
   originalSaha equation (not the above approximate form) to obtain the density of normal matter (not dark matter)
   at that time, express it in the unit of number of hydrogen nucleiper cubic metre. Reflect on the dramatic change
   in landscape (from stellar atmosphere to the universe), but the (relatively) small change in ionization temperature