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# Treebeard's Stumper Answer18 September 98

Seeing the Light from Jupiter

The planet Jupiter was in opposition this week. The Earth is on a line between the Sun and Jupiter. You can easily see bright Jupiter rising in the east just as the sun sets. In the 1600s, the Danish astronomer Ole Rømer noticed that the moon Io's orbits of Jupiter lose time for about 6 months, and then gain time over the next 6 months, losing and gaining about 8 minutes each cycle between successive oppositions. From this, and the fact that the earth is about 93 million miles from the sun, Rømer made the first reasonable estimate of the speed of light. What did he realize?

Ole Rømer reported in 1675 that the orbits of Jupiter's moon Io lost and then gained about 8 minutes in just over a year between successive oppositions of Jupiter, when the Earth and Jupiter are closest. He realized that this was because the light from Jupiter had to travel an extra distance when the earth was on the far side of the sun. 93,000,000 miles divided by 8 minutes (480 seconds) is close to the official value of the speed of light of 186,000 miles per second. The fact can be measured. It takes genius to see the light and realize what it means!

Note: The time between successive oppositions with Jupiter is a bit over a year because Jupiter itself moves, and the earth has to catch up. Jupiter orbits the sun in almost 12 years, so it takes the earth almost an extra month to get to the same relative place. Graybear came up with a nice analogy for this: "In a way, Earth and Jupiter are like the minute and hour hands of an old-fashioned clock, except the relative lengths would change. One revolves 12 times faster than the other."

Graybear also sent this clear (and entertaining!) answer to the stumper:

Old Ole Romer was pretty ingenious to realize he could determine the speed of light from the orbit of Io! I guess he figured there must be a reason that the orbit appears to change, and set out to understand why.

Since Earth's orbit is about 93 million miles, we are about 186 million miles closer to Jupiter (and any of the outer planets) during opposition than when we are on the opposite side of the sun. Romer realized that the light from Io took sixteen minutes longer to travel that extra distance, so we can derive:

```    186,000,000 miles      1 minute
-----------------  X  ----------  = 193,750 miles/second
16 minutes         60 seconds
```

Not bad considering the approximations and that you can't see Jupiter when it is on the other side of the sun!

I figured the change in Io's orbit as plus and minus 8 minutes over the year, because those numbers provide a big hint. You can also think of Io as losing and then gaining back 16 minutes over the year, as Graybear does. It comes to the same thing. (I was actually measuring from the point of quadrature rather than opposition.)

This is actually a kind of slow Doppler Effect. Just as a car or train sounds higher in pitch as it approaches and lower in pitch as it recedes, so the time for Io's orbit runs a bit faster as we approach opposition and a bit slower as we leave it.

I've simplified history a bit in this stumper. Rømer didn't actually give a specific value for the speed of light based on his observations because the distance from the Earth to the Sun was not known with any certainty. He did explain the connection.

Rømer thought it took light about 22 minutes to cross from one side of Earth's orbit to the other. It is now accepted that the time is 16 2/3 minutes, or about 1000 seconds. 186,000,000 miles divided by 1000 seconds is 186,000 miles per second.

The historical details are available on the Web at http://www.rundetaarn.dk/engelsk/observatorium/light.htm. Michael Fowler gives an account of Rømer and other early attempts to measure the speed of light in his Physics 109 Lecture Notes.

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