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XKCD usually has solid (and often contemporary) science behind it. Lightning Difference, #2027 one says:

Q: What’s that trick for telling how many miles away lightning is?

A: Just count the seconds between the visible flash and the radio wave burst, then multiply by 5 billion.

Usually it's lightning versus thunder, and you divide the time by 5 (or thereabouts) to get the distance in miles.

Here though, light time for 1 mile (about 1600 meters) would be about 5.3E-06 seconds, and if the difference between the visible light flash and the radio burst were one five-billionths of a second (2E-10 seconds), that suggests a velocity difference of about 38 ppm.

What is the physics behind that 38 ppm difference?

enter image description here

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uhoh
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    Hint: index of refraction varies with frequency. – The Photon Aug 01 '18 at 14:41
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    The alt text of the image also hints in the same direction – Jasper Aug 01 '18 at 14:54
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    You seem to be asking "Why does the speed of EMR in a medium depend on frequency?" Possible duplicate of Why do prisms work (why is refraction frequency dependent)? – sammy gerbil Aug 01 '18 at 16:40
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    @uhoh always check the hover text on xkcds.. sometimes the entire joke is hidden in there :) – Bill K Aug 01 '18 at 17:15
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    @uhoh Looks like you have 2027 xkcds to reread. – Yakk Aug 01 '18 at 17:17
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    @Jasper to be pedantic, technically what (almost) everyone who reads xkcd calls "alt text" is "title text" as "alt text" is the text that is show when the image fails to render, not the text that is show on mouse-over. Though I doubt it would have been any more clear to those who are unfamiliar with it if you used the "correct" term. – reffu Aug 01 '18 at 19:17
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    @uhoh - Your question has alt-text for the linked xkcd cartoon. Unfortunately, it's the default enter image description here that's automatically written by the stackexchange software when one creates an image link. Alt-text provides an alternative to the image such as in the case the image can't be shown (but most browsers show a broken image image rather than showing the alt text), or it provides a detailed description for computer users who cannot see the imagery, or, in the case of xkcd, the alt-text (strictly speaking, title text) oftentimes contains the punch line of the joke. – David Hammen Aug 01 '18 at 21:11
  • @sammygerbil In this case the index of refraction is higher for the lower frequency, so your prism dupe is a faux dupe. If you are compelled to find a duplicate, at least make an effort to find one with the correct sign! – uhoh Aug 01 '18 at 22:42
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    @uhoh It does not mention radio waves either! The slope of the dispersion curve can change sign even within the visible range - there's plenty of range for changes of sign before reaching radio frequencies. The point is : why does the speed of EMR depend on frequency, which is what you are asking, isn't it? – sammy gerbil Aug 01 '18 at 23:36
  • @sammygerbil XKCD says the radio burst comes later than the light. – uhoh Aug 01 '18 at 23:39
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    @uhoh Just find a plugin to run the following script anytime you go to xkcd.com: $('<p>' + $('#comic img').attr('title') + '</p>').insertAfter($('#comic'));. It'll insert the title text below the comic. – maxathousand Aug 02 '18 at 14:26
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    You can always first go to Explain xkcd, in this case 2027: Lightning Distance. – Peter Mortensen Aug 02 '18 at 21:23

4 Answers4

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I think it's fair to say that explainxkcd.com is the authoritative source for questions regarding xkcd. In this case, a detailed discussion (including formulas) is taking place on the page for xkcd 2027.

Here's a quote from its current text:

According to Wikipedia and other sources, refractive index of air at 0°C is about 1.000277, which equates to a speed of light around 299709.4 km/s (186230.8 miles/s). According to this paper, refractive index for radio waves in similar conditions is 1.000315, which equates to a speed around 299698.1 km/s (186223.7 miles/s). This means that to get the distance, the time difference in seconds between visible flash and radio burst should be multiplied by about 4.9 billion for miles, or about 7.9 billion for kilometers. More details for the calculations are in the comments below.

As for why radio waves are slower in air than visible light - I don't know, and I didn't find any useful sources, but I guess it's because even in the troposphere some molecules are ionized, and the free electrons affect radio waves much more than waves of higher frequencies. What I read about the ionosphere and dispersion due to free electrons in the interstellar medium seems to support that idea. But it's just a guess - I may be completely wrong.

jcsahnwaldt Reinstate Monica
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    I'll give this a read right away, thank you! I had no idea there was such a thing as an explainedxkcd! – uhoh Aug 01 '18 at 22:37
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    I love explainxkcd.com! And I love its motto: "Explain xkcd: It's 'cause you're dumb." ;-) Certainly applies to me. For at least a third of the comics, I need explainxkcd.com to understand what's going on, and in many other cases I need it to point out and explain all the nuances that I would otherwise miss. Great site! – jcsahnwaldt Reinstate Monica Aug 01 '18 at 22:42
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    So far this answer simply confirms what XKCD has already asserted, that radio is slower than visible light in air. Can you address the physics of why this is so? "What's the physics behind..." appears in the title and the body of the question. If not the magnitude, at least why radio is the slower of the two? – uhoh Aug 01 '18 at 23:30
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    I'm afraid I can't explain it any better than the answers to the question mentioned by @sammy gerbil: https://physics.stackexchange.com/questions/65812/why-do-prisms-work-why-is-refraction-frequency-dependent In a nutshell: refractive index depends on frequency, and on the details of the material the waves travel through, e.g. air humidity (or amount of rain) and (I guess) ionization rate of air / water molecules. It's complicated. – jcsahnwaldt Reinstate Monica Aug 01 '18 at 23:38
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    @uhoh I think this is a pretty good and accessible explanation: https://en.wikipedia.org/wiki/Refractive_index#Microscopic_explanation – jcsahnwaldt Reinstate Monica Aug 02 '18 at 00:04
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    There are plenty of atomic as well as band excitations and absorption edges at wavelengths somewhat shorter than visible light, thus most transparent materials made of atoms (glass, water, air...) have higher index for blue than for red light. But I'm surprised that for non-polar molecules like nitrogen and oxygen the index is even higher at radio frequency. I wonder if it's because there is some water assumed, or if even dry air has a higher index at radio frequency than for visible light? That answer doesn't go low enough in frequency to address this. Anyway, thanks for your answer! – uhoh Aug 02 '18 at 00:29
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    I kept searching the web, but I didn't find any useful discussion of the question why the refraction index in air of radio waves is higher than that of visible light. After I read a bit about the ionosphere, I guess it's because even in the troposphere some molecules are ionized, and the free electrons affect radio waves much more than higher frequencies. But that's just a guess. I may be completely wrong. – jcsahnwaldt Reinstate Monica Aug 02 '18 at 01:02
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    Maths doesn't check out. Although I am starting with spherical lightning in a vacuum. – Aron Aug 02 '18 at 07:40
  • @Aron I don't get it. What doesn't check out? – jcsahnwaldt Reinstate Monica Aug 02 '18 at 15:55
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    @JonaChristopherSahnwaldt It was a joke. :) Physics problems (in text books, etc) often include those kinds of simplifications. "If a piano falls from a 50-meter building, how long will it take to hit the ground? You may ignore air resistance and assume the piano is a 1-meter sphere." – yshavit Aug 02 '18 at 16:17
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    In this case spherical lightning is a rare and poorly understood physical phenomena. Whereas lightning in a vacuum is just nonsense. – Aron Aug 02 '18 at 18:42
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    @JonaChristopherSahnwaldt also lookup spherical cows. – muru Aug 03 '18 at 03:42
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    @jcsahnwaldt: Footnote 1 in https://what-if.xkcd.com/95/ makes the same joke. https://what-if.xkcd.com/120/ references spherical cows directly as a joke. Also, https://arstechnica.com/civis/viewtopic.php?f=23&t=43500 is basically an explain-xkcd for https://xkcd.com/669/, about working in a frictionless vacuum, and brings up the "assume a spherical" ... as a related kind of assumption. – Peter Cordes Aug 03 '18 at 20:28
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Well, without researching this at all, I’m going to go out on a limb and say it’s due to the difference in refractive indexes between visible light and radio waves in air. Air has dispersion like everything else, and so electromagnetic waves of different frequencies travel at different speeds through it. If you know the difference in refractive index, you can compute the time delay per mile.

Gilbert
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  • Okay well dispersion is just another word for different propagation speeds for different frequencies, but can you go a little deeper into the physics of why the speeds for light and radio have this particular difference? – uhoh Aug 01 '18 at 14:46
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    @uhoh Sure! Dispersion is also another word for different polarizabilities of the medium for different frequencies. Apparently, the air molecules are more easily polarizable for radio waves than light. Ultimately, this would be related to the density of air, the strength of optically active molecular vibration/rotational resonances, and the relative detuning of the different EM frequencies from those resonances. – Gilbert Aug 01 '18 at 14:56
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    @uhoh can you go a little deeper into the physics of why the speeds for light and radio have this particular difference? What is particular about this difference? It has to be some value. – sammy gerbil Aug 01 '18 at 16:44
  • @sammygerbil I guess what he is getting would be be "is the value of 5 billion reasonable?" – Tyberius Aug 01 '18 at 18:16
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    @Tyberius It seems to me uhoh is asking about the physics of the 38 ppm difference in velocity between visible and radio waves, rather than the mathematics of the calculation for the distance of the lightning. – sammy gerbil Aug 01 '18 at 18:23
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Note that the explanation from explainxkcd.com is not entirely correct. Not completely wrong but they make the common error to confuse the group index with the refractive index.

It is the group index that is responsible for the delay of a burst, not the refractive index! *

While in air the group index differs only slightly from the refractive index, in the RF domain, where are many resonant absorption lines due to water, the group index can differ significantly from the refractive index. Also due to absorption lines, the group index is itself strongly frequency dependent in the RF domain.

The actual delay that is observed for a RF wave hence depends on H2O concentration and also on the actual frequency distribution of the wave packet. There is also the influence of the ionosphere.

I found this thesis which does an experimental study of the delay between the lighting and the receive of a frequency burst. Although it is not strictly RF domain, but at lower frequencies, they find different group velocities depending on the environmental conditions (e.g. day vs. night). At least I understand it that way.

*) In fact the refractive index can be below one or even negative without any violation of relativity ("no information can travel faster than the speed of light in vacuum"). The phase propagation (the refractive index refers to phase velocity) is not able to carry any information. Information propagation requires a modulated wave and there the group velocity (and group index) comes into play.

EDIT: Strictly speaking the group velocity is also not always the speed a wave packet travels. This is only true in weakly absorbing media. Since air qualifies as weakly absorbing the group index is in my opinion indeed the right quantity for the problem here, but for completeness I will explain the full story:

Group velocity is the velocity of the envelope of a wave packet. If the absorption is so strong that the shape of the wave packet envelope itself changes during propagation, then group velocity is no longer appropriate to describe the propagation speed. One the other hand it is very difficult to asses the speed of something that changes its shape during propagation. That is why there are other definitions of velocity. Depending on the criteria one uses there is e.g. Front velocity or velocity of energy transport. The velocity that can never exceed the speed of light in vaccuum is the Front velocity. However this is also a bit difficult to work with, both experimentally and theoretically.

As references to the topic one has the book by Brillouin and Sommerfeld "Wave Propagation and Group Velocity" (1960) and the article "The Velocities of Light" by R. Smith (1970) (Thanks to David for pointing this out).

Andreas H.
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    I had a hunch there was some real physics here, thank you! Yes, with a dispersive medium and a broadband transient, what timing the speed of the "burst" means deserves some thought! I'll give that thesis a looks this week, thank you! – uhoh Aug 03 '18 at 09:06
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    @uhoh: Yes speed of wave bursts is a difficult topic with subtle nuances. BTW the thoughts of famous physicists on speed of propagation even fill a full book "Wave Propagation and Group Velocity" by Léon Brillouin. – Andreas H. Aug 03 '18 at 09:55
  • @Andreas H. Since you mention Brillouin (& Sommerfeld)'s work, why don't you detail the last part of your answer to say that even the group velocity is not the speed at which information/ signal is propagated (in all generality,https://en.wikipedia.org/wiki/Front_velocity)? It would be a good addition to show that it is indeed a complex problem. Another nice reference: https://aapt.scitation.org/doi/10.1119/1.1976551 – EigenDavid Aug 03 '18 at 12:14
  • The cited thesis is not relevant to this question - it measures the influence of the ionosphere (thus the day-night difference), but we're strictly interested in light and radio waves travelling in a straight line from lightning to observer. – jcsahnwaldt Reinstate Monica Aug 03 '18 at 12:36
  • @JonaChristopherSahnwaldt: Well, I would say that this depends on the receiver or antenna one uses. But yes it is a good point. The multi-path propagation will also influence the dispersion (in fibers this is known as modal dispersion). So the velocity should also be dependent on the antenna pattern, and how well one points the antenna to the source. So for practical reasons one would perhaps use an omidirectional (or something close to that) antenna and accepts the contributions from ionosphere. – Andreas H. Aug 03 '18 at 12:51
  • The point is that we want to measure the time difference between the arrival of the visible light and the radio waves from lightning and thus calculate the distance. For distances of a few kilometers, this time difference will be on the order of a nanosecond, but since the distance between the surface and the ionosphere is about 60 to 150 km, the time difference due the ionosphere will be at least an order of magnitude larger - harmful rather than helpful. (The time differences given in the thesis are on the order of milliseconds, so they are even less useful for this question.) – jcsahnwaldt Reinstate Monica Aug 03 '18 at 13:24
  • I mean, the whole idea is silly (it's xkcd after all), since it won't work for actual lightning (a flash lasts for about 60 microseconds, so we won't be able to measure signal differences of a nanosecond). But data from ionosphere reflection will only add more noise, no useful signal. – jcsahnwaldt Reinstate Monica Aug 03 '18 at 13:43
  • I think the difference between refractive index and group index is irrelevant in this case, because they will have the same numerical value. Their relation includes the term dn/dlambda - the derivative of n with respect to lambda. All sources I could find about refractive index values for radio waves give one value for a very large frequency range (e.g. from 1 Hz to about 1 GHz), so that derivative should be zero, and thus refractive index and group index should have the same value. – jcsahnwaldt Reinstate Monica Aug 03 '18 at 13:56
  • @AndreasH. re spherics: this and this. That link might be helpful in one or both places, as part of a comment at least, if not part of an answer. – uhoh Aug 03 '18 at 15:05
  • "many resonant absorption lines due to water" - AFAIK, that's only correct for wavelengths < ~10 cm (SHF, EHF and higher frequencies). Most of the radio spectrum (up to and including UHF) is not affected by water vapor. Sources: https://en.wikipedia.org/wiki/Electromagnetic_absorption_by_water - http://wgbis.ces.iisc.ernet.in/envis/Remote/introfile6.htm - http://wgbis.ces.iisc.ernet.in/envis/Remote/introfile6_files/AtmosWindows.jpeg - https://www.nap.edu/read/21729/chapter/3 - https://www.nap.edu/openbook/21729/xhtml/images/p15.jpg and many others... – jcsahnwaldt Reinstate Monica Aug 03 '18 at 18:48
  • The radio spectrum of lightning flashes diminishes quickly at frequencies of a few dozen MHz or higher and is thus not affected by absorption lines due to water vapor. Sources: https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19870001225.pdf - https://link.springer.com/article/10.1007/BF01042441 and many others – jcsahnwaldt Reinstate Monica Aug 03 '18 at 19:32
  • None of the points made in this answer are relevant to the question. 1. group index / refractive index: irrelevant because they have the same value in the relevant range (refractive index curve is flat). 2. ionosphere: irrelevant because reflected radio waves add only noise, no useful signal for comparison with direct, non-reflected light. 3. absorption lines: irrelevant because RF emission of lightning has frequencies up to ~50MHz, but there are no absorption lines < ~1GHz. – jcsahnwaldt Reinstate Monica Aug 04 '18 at 09:40
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The question of "why" radio waves have a lower speed in air than light might be due to the interactions of radio waves with diatomic molecules ( O1 and N1 ). The radio photons energy are going to be closer to the "available" transition energy for rotations. The Hyperphysics page has a discussion. Think of it as the cumulative effect of many photons causing transitions, but the molecules are then re-emitting the photons with a slight delay.

Visible light has a much higher energy per photon and the probability of an interaction with the rotational modes of diatomic molecules is much lower, so the gas is less "prismatic" to visible light. (I changed the adjective describing refractivity from "transparent" to "prismatic" since transparency could also describe the absorptive index.

This page give the refractive index of various gases for visible and radio frequency photons.

I was happy to find that Feynmann's blackboard was photographed and transcribed in a discussion of the semi-classical underpinnings of refraction, but with a footnote that references the QM basis.

DWin
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  • Thank you for the physics answer to my physics question!! While vibrational transitions are often observed in the infrared w/ FTIR, it's important to remember how low the rotational states can be. However, the first sentence there says "...provided they have an electric dipole moment." Do diatomic, symmetrical N2 and O2 have a dipole moment though? This is why I keep wondering about the humidity being a factor. I suppose if we had an n or \epsilon for pure N2 or O2, that would finally clear this up. – uhoh Aug 03 '18 at 01:25
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    Vibrational is not the same as rotational. To be honest, I also noted the dependence of the RF "spectrum" on H[2]O and CO[2]. There is also a body of experimental evidence re: the dielectric constant of gases. (I realize CO[2] is "non-polar" but it does have a quadrupole moment.) – DWin Aug 03 '18 at 02:32
  • All the data I found about absorption lines of RF waves in (dry or humid) air looks like there is no interaction between molecules and radio waves below ~1GHz. The radio waves emitted by lightning flashes are almost all below ~50 MHz. I guess the interaction of radio waves with molecules is not relevant in this case. What do you think? – jcsahnwaldt Reinstate Monica Aug 04 '18 at 22:28
  • If you don’t think the waves are interacting with the molecules of air, then what do you think is happening? The interactions do not necessarily be based on the absorption maxima. Quite the opposite in fact. I was suggesting a mechanism that would not have caused much net absorption. I was instead suggesting delay caused by the time between a transition to a higher state and re-radiation back to a lower state. – DWin Aug 05 '18 at 18:46