# Is the ratio between earth's distance from the sun and the speed of light just a coincidence?

I was doing some calculations to see how hard it would be to observe the speed of light and discovered an interesting correlation between the speed of light and the average distance from the earth to the sun. In my example, an object would be placed in orbit around the sun so that its shadow would pass over the earth at the speed of light. After doing some math, I found that an object roughly 1 km above the sun, orbiting at 2 m/s would achieve this. I though that these numbers were a bit odd, so I took a closer look at the speed of light (299,792,458 m/s) and AU (149,597,871 km.) I found that 299,792,458 / 149,597,871 = 2 within a .2% degree of inaccuracy. To me, this seems too close to be chance. Is there an established relationship here, or is this just luck?

• I remember some values with pi as the basis. The root of ten is pi. There are ten million times pi seconds a year. And so on. Within a % or so. Its just random mnemonics. But Johannis Kepler describes how he happened to recognize a certain number, 400 something, at two places in his endless manual calculations and how that made him curious and put him on track to realize that planets have elliptical orbits, which became the foundation of physics, so never say never. – LocalFluff Aug 7 '15 at 5:46
• The French who defined the meter 200+ years ago, knew both the speed of light and the distance to the Sun. Maybe it was considered? One in fourty million of the length of the equator is a bit odd choice for a definition. But the French do have a 20 kind of base in their language, as does English with the "teens", so maybe 40 is quite square to them? (Stupid spell checker suggests "forty" I refuse to use that ugly abomination). – LocalFluff Aug 7 '15 at 5:51
• Something 1km above the Sun does not "orbit at 2m/s". – ProfRob Aug 7 '15 at 7:22
• The square root of ten is not pi. They may be close but 10 is algebraic over the rationals where as pi is not. – TheBluegrassMathematician Aug 11 '15 at 15:49
• @Rob Jeffries, very true. It falls like Tech stock on Black Tuesday. – Hoytman Aug 13 '15 at 6:32

It is a co-incidence. The units are what you need to look at, in both the speed of light and the distance of the Sun the units of length are the same (m) (your ratio is $\approx 0.002 s^{-1}$ when you use metres as the unit for the distance to the Sun). The definition of the second is in a sense arbitrary and is a consequence of how we have chosen to sub-divide the day into smaller units of time (there is nothing cosmic in dividing the day into 24 units each subdivided into sixtieths which are again subdivided into sixtieths).

If we had chosen to take as our basic unit of time $1/100000$ of a day rather than $1/86400$ the nice round answer vanishes (if my calculations are right the $\approx 0.002$ becomes $\approx0.0023$)

Pure coincidence. Your result of 2 is better written as 2 meters/kilometer/second. If we had six fingers on each hand we would probably use base twelve rather than base ten. A base 12 metric system would have 1728 (in our base 10) meters in a kilometer. Your coincidence would vanish in a twelve-fingered world.

Our Earth takes 86400 seconds to make one rotation with respect to the Sun. Your coincidence would vanish if our day was longer or shorter (but we still divided the day into 86400 seconds).

Our division of the day into 86400 seconds is a historical oddity that results from

• The ancient Egyptians and their infatuation with base 12. They divided day and night into 12 parts each, hence the origin of our 24 hour day.
• The ancient Babylonians and their infatuation with base 60. They divided the circle into 360 parts (degrees), that into 60 parts each (minutes), and that into 60 parts each again (seconds), hence the origin of our division of an hour into 60 minutes, and a minute into 60 seconds.
• The not-so-ancient French revolutionaries and their infatuation with base 10 failing to catch on with regard to time. They tried to metricize time and angle, but unlike the length, area, volume, and mass, metric time and angle didn't take hold.

Your coincidence would vanish if the French metric time had caught on, or if those ancients hadn't been so numerologically inspired, or if they had been inspired by a different base.

• Here is another strange coincidence involving our system of time and distance: it takes about an hour for the earth's surface to rotate 1000 miles. – Hoytman Aug 8 '15 at 23:12