[Physics] What’s the physics behind XKCD #2027 (time between lightning flash and radio wave burst)

electromagnetic-radiationradioradio frequencyrefractionvisible-light

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

Best Answer

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.

Related Question