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

<p>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 $\eta=\frac{4(n_{He^+}+n_{He})}{n_\gamma}$.</p>

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

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

<p>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:</p>

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

<p>Any ideas of where I might have gone wrong?</p>

CC BY-SA 3.0
2017-03-23T22:48:45.020Z
There are some folks here who could probably answer that - however, it seems to me more like a Q for the Physics stack.
Florin Andrei
CC BY-SA 4.0
2023-05-02T16:13:43.077Z
How did you put $Q$ in. You need to give the details of what you have done and why you expect to get the solution you think you should.
ProfRob
<cosmology><early-universe><nucleosynthesis><helium>
Ionization Fraction of Helium in alternate, He dominated early universe
Spuds
# Ionization Fraction of Helium in alternate, He dominated early universe

(Asked by: username_0 on 2017-03-23)

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?

**Tags:** cosmology, early-universe, nucleosynthesis, helium

> **Comments:**
> **username_1**: There are some folks here who could probably answer that - however, it seems to me more like a Q for the Physics stack.  
> **username_2**: How did you put $Q$ in. You need to give the details of what you have done and why you expect to get the solution you think you should.  

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## 0 answers

