# Measuring the relativistic limit: c

A few months ago, I got a friend in facebook from Pakistan, who is probably doing her graduation in physics. She told me that she had got an assignment where she had to  write an essay on the applications of the Maxwell’s equations, and she asked me to suggest some applications. If I am not losing my memory, the first thing that came in mind is that Maxwell’s equations correctly predict the speed of light.

On another day I was discussing with one of my colleagues about the elegance of Maxwell’s equations and I told her the same, i.e. the best part of the equations is the accurate theoretical prediction of the speed of light. Then through our discussion, it turned out that the speed of light is known to people long before Maxwell proposed, during 1861-62, his equations and I wondered that how people did measurement of such a large speed (3 x 108 m/s)!

I just realised that I didn’t have known the answer to such a simple question.

My dear reader, do you know how speed of light was measured first (at least up to a realistic order)?

If you feel yourself as poor as I was at that time, then I’m here to ease your pain (You don’t need to google since I have already done it.).

Here it goes.

The measurement was first done by Ole Christensen Rømer in 1676 (Newton’s gravitational law came a few years later).

Rømer spotted the full shadow of the moon of Jupiter: Io (see the Hubble shot above). The full shadow is easy to figure out on a bright astronomical object. This happens when Io comes in between Jupiter and the Sun, causing the solar eclipse of Jupiter. Rømer found Io follows a periodic orbital motion and hence he expected the eclipse to repeat same time after it happens once. However, marking the first observation, about 6.5 months later, he found a delay of 22 minutes from the predicted time! So once we figure out the extra path

light might have travelled while the second eclipse was observed, we can determine its speed by dividing the path distance by the measured time delay.

From the schematic picture below, we can see that we don’t need to measure the Jupiter’s at distances (J1 and J2) from earth at all. Only we need to measure the distance between earth’s position during two consecutive observations (E1 and E2). Since the second observation happens about half of the earth’s period of revolution, E2 position is almost opposite to E1 position with respect to Sun (S). So the extra path will be just twice the earth-to-Sun distance, i.e. approximately 2 astronomical units (a standard unit of length in astrophysics and cosmology).

Let’s mathematically do the calculation. Simple algebra, never mind!

Time delay:
$\Delta t=22\, \text{min} =22 \times 60\,\text{s}$

Path difference: $\Delta x=c \Delta t=2\, \text{A.U.}\sim 2\times150\times10^6\,\text{km}$

So the speed of light:
$c=\Delta x/\Delta t$
$=300 \times 10^9/(22\times 60)\,\text{m/s}=2.27 \times 10 ^8\,\text{m/s}$

Well, this is quite below the value we know today (~3 x 108 m/s). This happened because Rømer measured the value of the delay incorrectly (it should be 16.7 min, not 22).

Later more sophisticated measurements (of course after Maxwell’s discovery) have been done, e.g. the famous rotating mirror experiment by Albert Michelson and Edward Morley in 1887. I found a site dedicated on c: http://www.speed-light.info/. Specifically one can look about its mention in the holy Quran (12000 lunar orbits/earth day).

Anyway, did you notice one important thing? How did people in Rømer’s time come to know about the astronomical unit, or in other words, how did they measure the distance between earth and the Sun?

To get an answer from me, wait for my next post.