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Michael Lee's Mathematics and Physics

Ole Romer and the Speed of Light

 

In 1676, shortly after Galileo's death, the Danish Astronomer Ole Romer (alternatively Roemer) calculated the speed of light as being 220,000 km/s; it's a pretty good estimate compared to 300,000 km/s by today's measurements. He accomplished this by timing the orbit of the moon Io around Jupiter and observed the Doppler effect 166 years before Christian Doppler figured it out; he must have had the equation below or he wouldn't have been successful and he must also have known the distance from the earth to the sun or one astronomical unit (AU).  Io and Jupiter act as a clock with a period of 42 hours.  He noticed that when the earth is moving towards Jupiter, owing to our orbit around the sun, the measured orbital period of Io was slightly shorter than its actual orbital period of 42 hours. 

 

Now, how is it possible to measure a difference of only about 13 seconds over a total period of 42 hours using a good clock when the earth is hurling directly towards Jupiter?  The rather crude clock Romer and Huygens' had was the best made during the seventeenth century, but how could they measure such a short interval of time of 13 seconds?  Even with a modern clock, measuring only 13 seconds is going to be difficult too.  Here's what we'll do, suppose you have a watch that runs one minute slow per hour.  You decide to verify this one day by synchronizing your watch to a good clock.  After only one hour, you would hardly notice the difference, but how about over a period of twenty four hours?  The difference of one minute per hour accumulates as the hours go by.  After twenty four hours, you'll very much notice that your watch is now a full twenty four minutes slow compared to the good clock.  So, to solve our problem, we measure, using a single good clock, not just one orbit of Io, but rather twenty or perhaps even more.

 

Here is the equation for the Doppler effect... 

 

 

The following diagram, not drawn to scale, illustrates the geometry.

 

 

 

Using the modern figure for the speed of light and a modern clock you would notice a maximum 13 second difference every 42 hours.