- #1
ChrisVer
Gold Member
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Hi,
The Saha equation for a procedure [itex] a^+ + e \leftrightarrow a + \gamma [/itex] is:
[itex] \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] [/itex]
where [itex]X = \frac{n_{a+}}{n_B}[/itex] and [itex]\eta = \frac{n_B}{n_\gamma} [/itex]
I worked with this equation for [itex]\eta = 5.5 \times 10^{-10}[/itex] and for [itex]a=H[/itex] and [itex]a=^4He[/itex]. The results I got for [itex]\frac{1-X}{X^2}[/itex] 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 [itex]T \approx 2.725~K[/itex]. So the above line today is going to give me the fractional ionization [itex] X [/itex] 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 [itex]\eta = n_B/n_\gamma[/itex]?
The Saha equation for a procedure [itex] a^+ + e \leftrightarrow a + \gamma [/itex] is:
[itex] \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] [/itex]
where [itex]X = \frac{n_{a+}}{n_B}[/itex] and [itex]\eta = \frac{n_B}{n_\gamma} [/itex]
I worked with this equation for [itex]\eta = 5.5 \times 10^{-10}[/itex] and for [itex]a=H[/itex] and [itex]a=^4He[/itex]. The results I got for [itex]\frac{1-X}{X^2}[/itex] 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 [itex]T \approx 2.725~K[/itex]. So the above line today is going to give me the fractional ionization [itex] X [/itex] 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 [itex]\eta = n_B/n_\gamma[/itex]?