However, after a time equal to half of Jupiter's synodic period, the eclipses were occuring after a time nP + 16.6 minutes!

And after another half synodic period, the variation had disappeared, and the eclipses were again occuring after an integral number of periods.

This correlation with the relative position of Earth and Jupiter could only be due to the variation in their distance as they orbited the Sun.

Rømer realized that if light had a finite speed, the time to travel over an extra distance of two astronomical units would explain the variation.

The speed of light must then be:

c = D/t = 2 AU/16.6 min = 7.2 AU/h

The size of the astronomical unit wasn't determined until the 19th century, but once it was the speed of light could be expressed in terrestrial units:

c = 2.998 x 105 Km/s

Note the use of the special symbol c for the speed of light.

By the middle of the 19th century improved technology allowed the measurement of the speed of light in the laboratory, verifying this value.
 

Source: opencourse.info

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                                                How was the speed of light determined

Everything we know about the stars and other objects in the universe comes to us through their release of some form of light, both visible and invisible.
It is therefore important to understand light's basic properties to determine what that reveals about the stars.

Although light appears to travel instantaneously from its emission to its detection, it does in fact have a finite speed, which must be very, very fast.

The first person known to attempt a determination of the speed of light was Galileo.
Galileo and an assistant each took a shrouded lantern and stood within site of each other on two widely-separated hilltops.
Galileo opened his lantern, and when the assistant saw its light, he opened his, too.

Galileo measured the time it took from when he opened his lantern to when he saw his assistant's lamp, which should be the time for light to travel back and forth between the two hilltops.

But how would this information be used to determine light's speed?

Unfortunately the time Galileo measured could not be distinguished from the time it took to open the lanterns, again due to light's high speed.

 

These eclipses should occur periodically, so the nth eclipse should begin after a time nP.

For example, the closest moon Io has an orbital period of P = 42.5 h, so the second eclipse takes place after 2P = 85.0 h, the third after 3P = 127.5 h, etc.
The first determination of the speed of light was an astronomical one, since only in this circumstance will large distances result in a measurable travel time.

In 1676, the Danish astronomer Ole Rømer (1644-1710) was studying the orbits of Jupiter's Galilean satellites.

Over a period of many months, Rømer recorded the starting time of eclipses of these moons by Jupiter.