# Saha equation (experiment verification?)

1. Nov 29, 2014

### ChrisVer

Hi,
The Saha equation for a procedure $a^+ + e \leftrightarrow a + \gamma$ is:
$\frac{1-X}{X^2} = \eta \frac{g_a}{g_{a+}g_e} \frac{4\sqrt{2} \zeta (3)}{\sqrt{\pi}} \Big( \frac{T}{m_e} \Big)^{3/2} \exp \Big[ \frac{E_{binding}(a)}{T} \Big]$

where $X = \frac{n_{a+}}{n_B}$ and $\eta = \frac{n_B}{n_\gamma}$

I worked with this equation for $\eta = 5.5 \times 10^{-10}$ and for $a=H$ and $a=^4He$. The results I got for $\frac{1-X}{X^2}$ are given the figure picture.

The above Helium line was taken by the assumption that baryonic matter consists only out of Helium at recombination time...

From the log(2) we can see the recombination temperature for both these cases.

I have one question though. Today the temperature is $T \approx 2.725~K$. So the above line today is going to give me the fractional ionization $X$ for today, right? Is it possible to calculate this fraction though experimentally?
If not, what are the experimental informations you can take from such a diagram?
Also without knowing the value today, how can people determine either the recombination temperature or the fraction of $\eta = n_B/n_\gamma$?

2. Nov 29, 2014

### Staff: Mentor

Only if the universe is in thermal equilibrium. Do you expect this?
The ionization fraction in the early universe.
You have the densities now and can extrapolate that backwards to the early universe.

3. Nov 29, 2014

### ChrisVer

So suppose that there are two hypothesis and you want to check them:
1. the baryon number during recombination consisted mainly of He
2. The baryon number during recombination consisted mainly of H
By the plot above, and measurements you can do today, can you verify either the Hyp#1 or Hyp#2?

4. Nov 29, 2014

### Staff: Mentor

The ratio H/He today is nearly the same as during recombination, so it is easy to get that ratio.
To see which type was type did recombination later (=what produced the CMB), you can look at the plot in post 1.

5. Nov 29, 2014

### ChrisVer

So in fact, if baryonic number consisted mainly of Helium, the universe would have been transparent to the photons at higher temperatures (=earlier) and thus the radiation temperature from CMB would have been larger today? :)

What I am trying to see is in this:
http://www.helsinki.fi/~hkurkisu/cosmology/Cosmo6.pdf
page 71 in his numbering (or 11 in scrolling)
lowest paragraph
where he states that 90% should have been protons (hydrogen) and the rest Helium..

6. Nov 29, 2014

### Staff: Mentor

The temperature at the (last) recombination would have been higher, but it would have happened earlier, so the universe would still have had time to cool down afterwards. I guess it would depend on the details like the actual helium density (do you convert all hydrogen to helium, or do you just remove hydrogen?).

Yes, that's still what we see. 90% hydrogen atoms and 10% helium atoms correspond to roughly 75% hydrogen by mass.

7. Nov 29, 2014

### ChrisVer

The one plot [for Hydrogen] was taken by considering what they state in the link I sent: there was no Helium (Helium neglected)...
The second plot [for Helium] was taken by considering again that there was only Helium (nothing else consisted the baryonic number).
So I guess I what I did was to remove the Hydrogen in the calculations.
I may also be wrong by trying to figure anything out of that graph...

But if the universe became transparent at the recombination time, and that is the time we see in CMB, if it had became earlier we would see its "earlier" stage and not the later one (which occured at Hydrogen). And so it would appear hotter.

8. Nov 29, 2014

### Staff: Mentor

Hmm, those assumptions will influence η. You also get some inequality between number of helium atoms and number of electrons.
Right.
Why? It would have cooled down more. There is no reason to assume it would appear hotter.

9. Nov 29, 2014

### ChrisVer

Oh yes, you are right... so CMB would appear cooler.

10. Nov 29, 2014

### Staff: Mentor

Why cooler?

I'm not sure if it would change anything at all. The thermal energy was dominated by photons anyway, the recombination just changed that number by a tiny amount.

11. Nov 30, 2014

### ChrisVer

hmm I guess I haven't really understood the mechanisms of temperature evolution then. But I think I solved my problem with it and you are right...the photon temperature almost always (with some small changes) scaled as 1/R even if CMB was produced by He or by H...

I have one further question, if you are able to help...
If the photons were too many combared to baryons and matter, as the $\eta$ parameter tells us, then why were they trapped into the primodial plasma? I know that plasma works as a trap or perfect reflector (by Compton or Ionization) for photons, but is that the case even if the photon number is over-dominating all the rest constituents?
The $\eta \sim 10^{-9}$ means that to 1 proton you have 1 billion photons... and the number of electrons=number of protons (neutrality) so to 1 proton and 1 electron you have 500 million photons... would all the 500million photons interact with these 2 particles in such a rate that they would get trapped? [or an expression for the interaction rate]

Last edited: Nov 30, 2014
12. Nov 30, 2014

### Staff: Mentor

What do you mean with trapped? There was plasma everywhere, how could you escape something that is everywhere?