<p>In several papers about Recombination, (for instance Peebles, Astrophysical Journal, vol. 153, eq. 39) it is claimed that the evolution of matter temperature follows the equation
\begin{equation}
\frac{dT}{dt} = -2\frac{\dot{a}}{a}T + \frac{2}{3 K_B n} \epsilon
\end{equation}
where the first term in the right hand side accounts for the expansion of the universe and $\epsilon$ is the energy injected into the gas per second per unit physical volume. How can I derive this equation?
I suppose that this equation can be derived applying the First Principle of Thermodynamics to a comoving volume in the expanding universe or maybe taking the second moment of a Boltzmann equation. Any help or reference in which this equation is discussed would be very helpful. </p>

CC BY-SA 4.0
2018-09-21T14:47:03.650Z
Are you talking about [this paper](http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1968ApJ...153....1P&amp;data_type=PDF_HIGH&amp;whole_paper=YES&amp;type=PRINTER&amp;filetype=.pdf)? If so, the equation you reference in your question is not (as far as I can tell) the same equation as in the paper. What's more, that paper pretty much derives the equation throughout.
zephyr
CC BY-SA 4.0
2018-09-21T19:28:38.887Z
The difference between the equation I wrote and the one in the paper is that in the latter what I called $\epsilon$ is explicitly calculated for Compton Scattering. Anyway I can't see where this equation is derived.
JohnNNNY
CC BY-SA 4.0
2018-09-28T22:17:09.793Z
I've voted for this question to be reopened - [see meta discussion](https://astronomy.meta.stackexchange.com/questions/450)
Chappo Hasn't Forgotten
CC BY-SA 4.0
2018-09-29T07:32:53.033Z
I found the answer anyway. The equation can be derived using the first principle of thermodynamics to a comoving volume and the equation of state for a perfect gas.
JohnNNNY
<thermodynamics>
Matter Temperature during Recombination
JohnNNNY
# Matter Temperature during Recombination

(Asked by: username_0 on 2018-09-19)

In several papers about Recombination, (for instance Peebles, Astrophysical Journal, vol. 153, eq. 39) it is claimed that the evolution of matter temperature follows the equation
\begin{equation}
\frac{dT}{dt} = -2\frac{\dot{a}}{a}T + \frac{2}{3 K\_B n} \epsilon
\end{equation}
where the first term in the right hand side accounts for the expansion of the universe and $\epsilon$ is the energy injected into the gas per second per unit physical volume. How can I derive this equation?
I suppose that this equation can be derived applying the First Principle of Thermodynamics to a comoving volume in the expanding universe or maybe taking the second moment of a Boltzmann equation. Any help or reference in which this equation is discussed would be very helpful.

**Tags:** thermodynamics
> **Comments:**
> **username_1**: Are you talking about [this paper](http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1968ApJ...153....1P&amp;data_type=PDF_HIGH&amp;whole_paper=YES&amp;type=PRINTER&amp;filetype=.pdf)? If so, the equation you reference in your question is not (as far as I can tell) the same equation as in the paper. What's more, that paper pretty much derives the equation throughout.  
> **username_0**: The difference between the equation I wrote and the one in the paper is that in the latter what I called $\epsilon$ is explicitly calculated for Compton Scattering. Anyway I can't see where this equation is derived.  
> **username_2**: I've voted for this question to be reopened - [see meta discussion](https://astronomy.meta.stackexchange.com/questions/450)  
> **username_0**: I found the answer anyway. The equation can be derived using the first principle of thermodynamics to a comoving volume and the equation of state for a perfect gas.  

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## 0 answers

