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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) (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, Shull 1979), 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, where they seem to have derived it from eq. (3)), which is also used in more recent papers (e. g. Furlanetto & Stoever 2010, Evoli et al. 2012).

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?

Socob
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  • 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? – dtn Jan 31 '24 at 06:53
  • @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 Jan 31 '24 at 17:41
  • 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 Jan 31 '24 at 17:51
  • @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 Jan 31 '24 at 18:41
  • This seems more like a physics question than an astronomy question. – Darth Pseudonym Jan 31 '24 at 19:19
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    @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 Jan 31 '24 at 20:45
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    @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 Feb 03 '24 at 01:53
  • 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 Feb 05 '24 at 19:58

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I have found another source giving an actual number for the constant factor: The DarkHistory code calculates the prefactor $4\pi e^4$ in these units (citing one of my sources above, Furlanetto & Stoever 2010; 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), 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).

Socob
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