The light of speed and 24 hours

Question

The notions of meter and second had been created much earlier than the speed of light was calculated. The speed of light is $299792458 ~ \text{m/s}$, i.e. it is really close to $300 ~000$ km/s. Also, the stellar day is ~ $23$ hours and $56$ minutes, i.e. it is pretty close to $24$ hours. These two numbers are pretty specific. Namely, the first one is "almost" a round number and the second one is almost an integer (in hours).

$\textbf{Question.}$ Is there a conceptual way of understanding this or is it just a coincidence? I've thought that the second was defined in such a way to make the speed of light "a good number" but the notion of the second was created more than 2000 years ago.

Answer 1

The phenomenon you're finding surprising is that the second and third digits of $c$ (expressed in appropriate units) are both 9's. The probability of that is 1%, or one in a hundred. One-in-a-hundred events happen to me all the time, so I don't find this too surprising. Indeed, Littlewood estimated that on average, you have about a million experiences per month that are noticeable enough to surprise you --- so you should expect a one-in-a-million level surprise once a month or so. If that's right, there should be about 10,000 occasions in the next month when you encounter something as surprising as the speed-of-light coincidence you're pointing to.

But you'd probably have been equally surprised if those digits had both been 0's. If so, this becomes a one-in-fifty event, which should be doubly common.

Compare this to the discovery that $e^{\pi\sqrt{163}}$ has 13 nine's after the decimal point, a one-in-$10^{13}$ event, which, on Littlewood's reckoning, should trigger the level of surprise you'd expect to experience once every 10,000,000 months, or (very roughly) once in a million years. Unlike the speed of light, this should trigger a strong sense that "there must be something going on here" and of course there very much is.

Answer 2

In the case of the speed of light, it just happens to be that number. We measured it as best as we could and that number popped out. I personally think it was a missed opportunity to not just set it to $3*10^8 m/s$ when we redefined the meter. It would have made calculations look more pretty.

As for the stellar day, it is just a feature of the difference between earths rotation around its axis and its orbit around the sun.

For a second lets just imagine the moon as the earth and the earth as the sun. Then the moons stellar day would be 1 month because that is how long it takes for it to rotate around its axis. While its regular day would be $∞$ because it is tidally locked to earth. If we were to increase the rotational speed of the moon, its day and stellar day would approach one another. One is the rotation on its own the other is rotation with respect to earth.

Back to earth. The stellar day is $1436$ min. Correct for the day lost on the year. $1436*( \frac{365}{364} ) ≈ 1440 min$ which matches the regular day.