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I'm asked to verify an expression for the ionization fraction of helium in a universe in which helium dominates baryonic matter.

I'm given that the ionization fraction $X = \frac{n_{He^+}}{n_{He^+}+n_{He}}$ with a given ionization energy $Q_{He}$, and number densities by the formula $n_i=g_i(\frac{m_ik_BT}{2\pi \hbar})^{3/2}\exp(-\frac{m_ic^2}{k_BT})$. Also the baryon-photon ratio in this case is $\eta=\frac{4(n_{He^+}+n_{He})}{n_\gamma}$.

I'm trying to show that by putting all this together I get something with the form:

$\ln(\frac{1-X}{X^2})=A+\ln(\eta)-B\,\ln(\frac{Q_{He}}{k_BT})+\frac{Q_{He}}{k_BT}$. where $A$ and $B$ are constants.

When I plug everything in, I got $\frac{1-X}{X^2}=\frac{n_\gamma}{8}\eta \, \exp(-\frac{Q_{He}}{K_BT})$, which gives:

$\ln(\frac{1-X}{X^2})=\ln(\frac{n_\gamma}{8})+\ln(\eta)+\frac{Q_{He}}{k_BT}$

Any ideas of where I might have gone wrong?

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    $\begingroup$ There are some folks here who could probably answer that - however, it seems to me more like a Q for the Physics stack. $\endgroup$ – Florin Andrei Mar 23 '17 at 22:48
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My answer is incomplete (but still possibly helpful, only posting because I don't have enough "points" to comment).

I did a bit of the math involved but did not substitute terms with n's and eta's. However, from the result at the bottom of that page, it looks to me like a Binomial Expansion (truncated Taylor Expansion). The link shows the highest order for your terms (which is okay since higher order terms can usually be regarded as negligible) with the same +/- pattern you expect in your answer.

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  • $\begingroup$ Thank you so much, I will work on this and let you know how it works out. I always forget about Taylor Expansion... $\endgroup$ – Spuds Mar 25 '17 at 15:28

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