How can the orbit of Jupiter's moons be used to calculate the speed of light? It seems this was one of the first methods and goes back to 1656.
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From Roemer and Light Speed:
The orbital period of Io is now known to be 1.769 Earth days. The satellite is eclipsed by Jupiter once every orbit, as seen from the Earth. By timing these eclipses over many years, Roemer noticed something peculiar. The time interval between successive eclipses became steadily shorter as the Earth in its orbit moved toward Jupiter and became steadily longer as the Earth moved away from Jupiter. These differences accumulated. From his data, Roemer estimated that when the Earth was nearest to Jupiter (at E1), eclipses of Io would occur about eleven minutes earlier than predicted based on the average orbital period over many years. And 6.5 months later, when the Earth was farthest from Jupiter (at E2), the eclipses would occur about eleven minutes later than predicted.
Roemer knew that the true orbital period of Io could have nothing to do with the relative positions of the Earth and Jupiter. In a brilliant insight, he realized that the time difference must be due to the finite speed of light. That is, light from the Jupiter system has to travel farther to reach the Earth when the two planets are on opposite sides of the Sun than when they are closer together. Romer estimated that light required twenty-two minutes to cross the diameter of the Earth’s orbit. The speed of light could then be found by dividing the diameter of the Earth’s orbit by the time difference.
