<p>I have found another source giving an actual number for the constant factor: The <a href="https://github.com/hongwanliu/DarkHistory/blob/0659cd06e3803091c7abce224a44659c25c26770/darkhistory/physics.py#L1262-L1266" rel="nofollow noreferrer">DarkHistory code</a> calculates the prefactor <span class="math-container">$4\pi e^4$</span> in these units (citing one of my sources above, <a href="https://ui.adsabs.harvard.edu/abs/2010MNRAS.404.1869F" rel="nofollow noreferrer">Furlanetto &amp; Stoever 2010</a>; note the “missing” factor of 10, since this is for a slightly different quantity). Plugging in the numbers from the code, I get
<span class="math-container">$$
4\pi e^4 = 2.61 \times 10^{-13} \,\mathrm{eV}^2 \mathrm{cm}^2
$$</span>
or in other words, exact agreement with my own calculation.</p>
<p>In light of this, and given that there are no other sources giving the same number as <a href="https://ui.adsabs.harvard.edu/abs/1985ApJ...298..268S" rel="nofollow noreferrer">Shull &amp; van Steenberg (1985)</a>, as well as the lack of anything that could explain an extra factor of 30 within the paper itself, I can only conclude that this is an error in <a href="https://ui.adsabs.harvard.edu/abs/1985ApJ...298..268S" rel="nofollow noreferrer">Shull &amp; van Steenberg (1985)</a>.</p>

CC BY-SA 4.0
2024-02-07T19:41:18.690Z
2024-02-07T19:41:18.690Z
Socob
<p>I’m a bit stuck trying to understand where a numerical constant in an old paper comes from. The number in question is in eq. (A1) in <a href="https://ui.adsabs.harvard.edu/abs/1985ApJ...298..268S" rel="nofollow noreferrer">Shull &amp; van Steenberg (1985)</a> (which gives the cross section for Coulomb collisions between electrons),
<span class="math-container">$$
\sigma_{ee} = (7.82 \times 10^{-11}) (0.05/f) (\ln \Lambda) E^{-2} \;\mathrm{cm}^2
\tag{A1}
$$</span>
specifically <span class="math-container">$7.82 \times 10^{-11}$</span>. There are two papers cited in the same sentence (<a href="https://ui.adsabs.harvard.edu/abs/1971ApJ...166..525H" rel="nofollow noreferrer">Habing &amp; Goldsmith 1971</a>, <a href="https://ui.adsabs.harvard.edu/abs/1979ApJ...234..761S" rel="nofollow noreferrer">Shull 1979</a>), and both give the equation
<span class="math-container">$$
\sigma_{ee} = 40\pi e^4 (0.05/f) (\ln \Lambda) E^{-2} \;\mathrm{cm}^2
$$</span>
(both in turn citing <a href="https://ui.adsabs.harvard.edu/abs/1969ApJ...158..161S" rel="nofollow noreferrer">Spitzer &amp; Scott 1969</a>, where they seem to have derived it from eq. (3)), which is also used in more recent papers (e. g. <a href="https://ui.adsabs.harvard.edu/abs/2010MNRAS.404.1869F" rel="nofollow noreferrer">Furlanetto &amp; Stoever 2010</a>, <a href="https://ui.adsabs.harvard.edu/abs/2012MNRAS.422..420E" rel="nofollow noreferrer">Evoli et al. 2012</a>).</p>
<p>This would seem to suggest that
<span class="math-container">$$
7.82 \times 10^{-11} \stackrel{?}{=} 40\pi e^4
.
$$</span>
An immediate issue are the units. For (A1), given the context it seems that <span class="math-container">$E$</span> should be given in <span class="math-container">$\mathrm{eV}$</span> (which is missing in the equation). Apart from that, it looks like the equations are in the cgs unit system, although I think the “<span class="math-container">$\mathrm{cm}^2$</span>” at the end of the equation with “<span class="math-container">$40\pi e^4$</span>” is actually wrong, since <span class="math-container">$e^4/E^2$</span> in cgs should already have units of <span class="math-container">$\mathrm{cm}^2$</span>. It’s a mess!</p>
<hr />
<p>Anyway, given that, I’ve found that
<span class="math-container">$$
40\pi e^4
= 2.61 \times 10^{-12} \,\mathrm{eV}^2 \mathrm{cm}^2
= (7.82 \times 10^{-11} \,\mathrm{eV}^2 \mathrm{cm}^2) \operatorname{/} 30
$$</span>
So I can almost get it consistent, except for an extra factor of 30! Where could this be coming from? Maybe there’s a mistake somewhere in the mess with the units?</p>

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2024-01-31T06:53:34.503Z
Couldn’t the article (in the formulas, to be more precise) miss some physical constant similar to $\frac{1}{30!}$? Just out of curiosity, I divided the number $\frac{1}{30!}$ at $2\pi$ and I got something very close to Planck’s constant: $6.0001217e^{-34} \approx 6.62607015e^{−34}$. Coincidence?
ayr
CC BY-SA 4.0
2024-01-31T17:41:59.900Z
@dtn Ah sorry, that was supposed to be an actual exclamation mark, not a factorial! The actual factor is 30, as written in the title. (I had put in a space in an attempt to make it clearer, but I’ve just removed it now since it obviously caused confusion.)
Socob
CC BY-SA 4.0
2024-01-31T17:51:47.557Z
Conceivably the factor of 30 could actually be a factor of $\text{"3"}\times 10 = 2.99792458 \times 10$.  You don't have enough sigfigs to tell the difference between these numbers, and factors of "3" crop up all the time in conversions between SI and CGS.
Michael Seifert
CC BY-SA 4.0
2024-01-31T18:41:56.323Z
@MichaelSeifert Hm yes, that seems like a good hint. I usually don’t have to deal with the CGS system, so I don’t have a feeling for what factors commonly show up. I’ll see if I can find something that makes sense!
Socob
CC BY-SA 4.0
2024-01-31T19:19:41.650Z
This seems more like a physics question than an astronomy question.
Darth Pseudonym
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2024-01-31T20:45:10.203Z
@DarthPseudonym It’s really more about literature and numbers than physical content… I considered whether to post this on physics or here, but since all the papers are in astrophysical journals, I thought this would be the more appropriate place in case there is some relevant convention (e. g. about units) in astrophysics that could explain it.
Socob
CC BY-SA 4.0
2024-02-03T01:53:28.337Z
@Socob **It's perfectly fine right here.** Someone's personal "more like a" view doesn't matter. The question author decides where they want to post and the community decides whether it is on-topic where it is posted. It's not the community's job to think about "better asked on" things.
uhoh
CC BY-SA 4.0
2024-02-05T19:58:22.700Z
The cross section is in my opinion incorrect anyway, as the 'Coulomb Logarithm' $ln(\Lambda)$ is the result of an erroneous derivation (by mistakenly identifying the scattering angle with the polar angle of a coordinate system fixed in the target center). See my page https://www.plasmaphysics.org.uk/coulomb.htm for more . With the 'Coulomb Logarithm' included, the cross section would be become so high that for instance any details of the electron spectrum in the Earth's ionosphere would be washed out, in contradiction to observations (see https://www.plasmaphysics.org.uk/research/elspec.htm )
Thomas
<astrophysics>
Mysterious factor of 30 in Coulomb cross section for electron–electron collisions
Socob
# Mysterious factor of 30 in Coulomb cross section for electron–electron collisions

(Asked by: username_0 on 2024-01-31)

I’m a bit stuck trying to understand where a numerical constant in an old paper comes from. The number in question is in eq. (A1) in [Shull & van Steenberg (1985)](https://ui.adsabs.harvard.edu/abs/1985ApJ...298..268S) (which gives the cross section for Coulomb collisions between electrons),
$$
\sigma\_{ee} = (7.82 \times 10^{-11}) (0.05/f) (\ln \Lambda) E^{-2} \;\mathrm{cm}^2
\tag{A1}
$$
specifically $7.82 \times 10^{-11}$. There are two papers cited in the same sentence ([Habing & Goldsmith 1971](https://ui.adsabs.harvard.edu/abs/1971ApJ...166..525H), [Shull 1979](https://ui.adsabs.harvard.edu/abs/1979ApJ...234..761S)), and both give the equation
$$
\sigma\_{ee} = 40\pi e^4 (0.05/f) (\ln \Lambda) E^{-2} \;\mathrm{cm}^2
$$
(both in turn citing [Spitzer & Scott 1969](https://ui.adsabs.harvard.edu/abs/1969ApJ...158..161S), where they seem to have derived it from eq. (3)), which is also used in more recent papers (e. g. [Furlanetto & Stoever 2010](https://ui.adsabs.harvard.edu/abs/2010MNRAS.404.1869F), [Evoli et al. 2012](https://ui.adsabs.harvard.edu/abs/2012MNRAS.422..420E)).

This would seem to suggest that
$$
7.82 \times 10^{-11} \stackrel{?}{=} 40\pi e^4
.
$$
An immediate issue are the units. For (A1), given the context it seems that $E$ should be given in $\mathrm{eV}$ (which is missing in the equation). Apart from that, it looks like the equations are in the cgs unit system, although I think the “$\mathrm{cm}^2$” at the end of the equation with “$40\pi e^4$” is actually wrong, since $e^4/E^2$ in cgs should already have units of $\mathrm{cm}^2$. It’s a mess!

---

Anyway, given that, I’ve found that
$$
40\pi e^4
= 2.61 \times 10^{-12} \,\mathrm{eV}^2 \mathrm{cm}^2
= (7.82 \times 10^{-11} \,\mathrm{eV}^2 \mathrm{cm}^2) \operatorname{/} 30
$$
So I can almost get it consistent, except for an extra factor of 30! Where could this be coming from? Maybe there’s a mistake somewhere in the mess with the units?

**Tags:** astrophysics
> **Comments:**
> **username_1**: Couldn’t the article (in the formulas, to be more precise) miss some physical constant similar to $\frac{1}{30!}$? Just out of curiosity, I divided the number $\frac{1}{30!}$ at $2\pi$ and I got something very close to Planck’s constant: $6.0001217e^{-34} \approx 6.62607015e^{−34}$. Coincidence?  
> **username_0**: @dtn Ah sorry, that was supposed to be an actual exclamation mark, not a factorial! The actual factor is 30, as written in the title. (I had put in a space in an attempt to make it clearer, but I’ve just removed it now since it obviously caused confusion.)  
> **username_2**: Conceivably the factor of 30 could actually be a factor of $\text{"3"}\times 10 = 2.99792458 \times 10$.  You don't have enough sigfigs to tell the difference between these numbers, and factors of "3" crop up all the time in conversions between SI and CGS.  
> **username_0**: @MichaelSeifert Hm yes, that seems like a good hint. I usually don’t have to deal with the CGS system, so I don’t have a feeling for what factors commonly show up. I’ll see if I can find something that makes sense!  
> **username_3**: This seems more like a physics question than an astronomy question.  
> **username_0**: @DarthPseudonym It’s really more about literature and numbers than physical content… I considered whether to post this on physics or here, but since all the papers are in astrophysical journals, I thought this would be the more appropriate place in case there is some relevant convention (e. g. about units) in astrophysics that could explain it.  
> **username_4**: @Socob **It's perfectly fine right here.** Someone's personal "more like a" view doesn't matter. The question author decides where they want to post and the community decides whether it is on-topic where it is posted. It's not the community's job to think about "better asked on" things.  
> **username_5**: The cross section is in my opinion incorrect anyway, as the 'Coulomb Logarithm' $ln(\Lambda)$ is the result of an erroneous derivation (by mistakenly identifying the scattering angle with the polar angle of a coordinate system fixed in the target center). See my page https://www.plasmaphysics.org.uk/coulomb.htm for more . With the 'Coulomb Logarithm' included, the cross section would be become so high that for instance any details of the electron spectrum in the Earth's ionosphere would be washed out, in contradiction to observations (see https://www.plasmaphysics.org.uk/research/elspec.htm )  

---

## 1 answers

### Answer by username_0 on 2024-02-07. Score: 3 | Accepted answer

I have found another source giving an actual number for the constant factor: The [DarkHistory code](https://github.com/hongwanliu/DarkHistory/blob/0659cd06e3803091c7abce224a44659c25c26770/darkhistory/physics.py#L1262-L1266) calculates the prefactor $4\pi e^4$ in these units (citing one of my sources above, [Furlanetto & Stoever 2010](https://ui.adsabs.harvard.edu/abs/2010MNRAS.404.1869F); note the “missing” factor of 10, since this is for a slightly different quantity). Plugging in the numbers from the code, I get
$$
4\pi e^4 = 2.61 \times 10^{-13} \,\mathrm{eV}^2 \mathrm{cm}^2
$$
or in other words, exact agreement with my own calculation.

In light of this, and given that there are no other sources giving the same number as [Shull & van Steenberg (1985)](https://ui.adsabs.harvard.edu/abs/1985ApJ...298..268S), as well as the lack of anything that could explain an extra factor of 30 within the paper itself, I can only conclude that this is an error in [Shull & van Steenberg (1985)](https://ui.adsabs.harvard.edu/abs/1985ApJ...298..268S).


---

