# Roemer and the speed of light

Two questions:

I saw a TV program recently which had an explanation for the first measurement of the speed of light, and it bothered me until I started looking it up (via Google...) I've now seen several instances agreeing with the TV guy saying that Roemer noticed that the observed revolution time of a moon of Jupiter varied according to whether Jupiter is near or far.

This is wrong, right? The revolution time depends on whether the distance is increasing or decreasing, and is the same when Jupiter is at the closest and farthest points. If this is true, then how does a bozo get on the science channel saying the former (Ok, ok. I don't really need an answer to this..)?

Secondly, the speed of light was calculated using the Earth's revolution speed as known at the time. Since that was 1676, I have no real idea whether the error (220,000 km/s, versus 300,000 km/s) was due to the inaccuracy of the observation times, or the value of the Earth's revolution time, or the absence of the revolution time of Jupiter in the calculation (which seemed a possible oversight in what I've read so far), or something else.

[As usual, please excuse my mediocre understanding of the subject matter.]

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#### jtbell

Mentor
Here's an example of the sort of thing Roemer looked at. I've made up the numbers out of thin air, and they have nothing to do with the actual numbers for Jupiter, but they illustrate the principle.

Suppose that when Jupiter is at its closest distance from Earth, we observe that one of its satellites completes an orbit every 24 hours, at midnight according to our clocks. Several months later, when Jupiter is at its farthest distance from Earth, we observe that the satellite still takes 24 hours for an orbit (as we should expect because planetary satellites don't change their orbits significantly over many years, let alone a few months), but each orbit starts at 1:00 a.m. instead of midnight. From this we can conclude that it takes an hour for light to travel the difference between the two Earth-Jupiter distances. A simple diagram shows that this difference is just the diameter of the Earth's orbit, so if we know that diameter, we can calculate the speed of light.

Or course, I assume that we're correcting for things like Daylight Savings Time. :-)

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#### cesiumfrog

jtbell, that's an interesting assumption: that Ole Romer measured the accumulated phase difference to simply compare with the distance difference. Like the OP, I had understood that he measured what is actually a doppler shift to compare with the relative velocity difference.

Do you have a reference in your support?

Just some quick calculations.. Our distance from Jupiter varies by as much as 2AU, so the light travel time will vary by about 15 minutes (over a year, hence the period must be known to a part in 40,000). Now, our relative velocity varies by roughly 50km/s (0.02% of c, 1.77day period), which gives only about a minute of difference (in one period, a part in 3000). Hence, I think the doppler effect is an order of magnitude more statistically significant than the phase delay.

Caveat: I only drag out my math skills every couple years...

cesiumfrog, you said that "our relative velocity varies by roughly 50km/s". Is it this

Earth's orbital speed = 30 km/s ,
If Jupiter's year is about 12 Earth years, then Jupiter's orbital speed is 30/12 = 2.5 km/s
( 30 - 2.5 ) * 2 = 55 km/s.

?

Regarding the significance of the possible measurements:

Since I have no idea how Roemer used occlusion, and how many degrees that represents, I'll estimate assuming you can accurately tell when Io is at the farthest points to right and left of Jupiter, or 180deg.
( Io orbital period = 42.3 hr ) / 2 = 21.15 hr per 180 deg = 76140 s for Io's half revolution

76,140 s * 27.5 km/s
= 2,284,200 km total distance accumulated over 180deg
/ ( c = 300,000km/s )
= time it takes for light to travel that distance
= 7 secs observed (sort of) difference in half Io's revolution, between observations when Earth is at 1/2 closest, and closest distance to Jupiter

I assert no claims to correctness or resemblance to Roemer's method :-)

Given that clocks at that time were accurate to about 10 secs/day, I'm not sure that the significance you calculated, of 1 part in 3000, is useful. He might have averaged readings together, but still, how many readings/years would that take?

This then begs the question of whether he would be able to more accurately know the closest and farthest points to measure the 15 min difference, as jtbell indicates.

I still haven't found any details on Roemer's method.

#### cesiumfrog

Caveat: I only drag out my math skills every couple years...
You ought polish them more often

Earth's orbital speed = 30 km/s ,
If Jupiter's year is about 12 Earth years, then Jupiter's orbital speed is 30/12 = 2.5 km/s
You've incorrectly assumed that Jupiter traverses the same circumference as Earth. You should compare with http://en.wikipedia.org/wiki/Jupiter" [Broken] (which lists the orbital speed as 13km/s, and includes other data relevent here).

To find the relative speed, yes you would first look at the relative angular velocity. Basically, because Jupiter is so far from the sun and takes so very long to orbit, we can approximate it is as stationary. More precisely, relative to Jupiter, Earth completes it's 1AU radius motion every 400 days (synodic period). From that corresponding speed, Earth's velocity relative to Jupiter varies from +27km/s to -27km/s.

In response to your first question, yes, obviously Io's observed period is equal at the points where the Earth is closest and furtherest from Jupiter, shorter as we approach, longest at the point of fastest recession. Like the soundwaves of a siren.

In response to your second question, it isn't an "oversight" to neglect Jupiter's motion (the difference between the 399 day synodic period and our 365 day year is not significant). You'd have to research some of the old letters to determine whether how much of his error can be attributed to uncertainty in other astronomical/orbital "constants" versus the simple experimental error in timing Io to such precision (I think proper error analysis is a relatively modern invention).

But this all misses the point. The result differs from all ancient "cannon and mirror" experiments and predates Bradley's "driving into the rain" by more than 50 years. It was still a couple of centuries until Fizeau, Foucault and Maxwell. The point wasn't really to measure the speed of light; Ole Romer is remembered because his was the first experimental evidence that light has a speed at all (contrary to the previous assumption that sight is instantaneous).

Since I have no idea how Roemer used occlusion, and how many degrees that represents, I'll estimate assuming you can accurately tell when Io is at the farthest points to right and left of Jupiter, or 180deg.
No, it's a good bet that astronomers marked times of the (edge-)points of occlusion. Likewise, if you time the motion of any pendulum, it's very inaccurate to try to pick the extremes of motion (velocity minima). Moreover, it is probable that Romer used averaged data from many nights (and decades) to obtain the best fit.

7 secs observed (sort of) difference in half Io's revolution, between observations when Earth is at 1/2 closest, and closest distance to Jupiter
Which is a difference of 26 seconds, between the length of the period when Earth is gaining most on Jupiter and the length of the period when Jupiter is receding fastest behind Earth. This is the same order of magnitude as I used ("about a minute").

Given that clocks at that time were accurate to about 10 secs/day, I'm not sure that the significance you calculated, of 1 part in 3000, is useful. He might have averaged readings together, but still, how many readings/years would that take?

This then begs the question of whether he would be able to more accurately know the closest and farthest points to measure the 15 min difference, as jtbell indicates.
Astronomers have always been the people with the best clocks (having the heavens to synchronise with) and most data (perhaps only weather compares, at least prior to the age of supermarkets). How long is a piece of string? One way or another, he clearly succeeded.

Nonetheless, it's easier to measure a 1.77 day period to seconds accuracy (and repeat the same a couple seasons later) than it is to measure a 399 day time-period to an accuracy of minutes (if timing an orbit of Io and then extrapolating a year, you'd need to measure the occulation times at an accuracy faster than the human reaction to visual stimuli).

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You ought polish them more often
Yeh, I suppose it's more fun for me than any bystanders
....

But this all misses the point. The result differs from all ancient "cannon and mirror" experiments and predates Bradley's "driving into the rain" by more than 50 years. It was still a couple of centuries until Fizeau, Foucault and Maxwell. The point wasn't really to measure the speed of light; Ole Romer is remembered because his was the first experimental evidence that light has a speed at all (contrary to the previous assumption that sight is instantaneous).
It was amazement that drew me to focus on his achievement. Not only did he prove that light had a speed, but he was damn close, considering he was the first, and what he had to work with.

My curiosity is about what piece of the puzzle he was missing.

No, it's a good bet that astronomers marked times of the (edge-)points of occlusion.
....
I know. For my estimation, I couldn't figure out where the edge points were.

Which is a difference of 26 seconds, between the length of the period when Earth is gaining most on Jupiter and the length of the period when Jupiter is receding fastest behind Earth. This is the same order of magnitude as I used ("about a minute").
Yes, but in terms of accuracy, if you combine the two measurements, you also have to combine the errors (though you get a bump from the averaging, so I dunno).

Astronomers have always been the people with the best clocks (having the heavens to synchronise with) ....
..but navigators had more money to buy the best, but I suppose you can count them as astronomers too.

Nonetheless, it's easier to measure a 1.77 day period to seconds accuracy (and repeat the same a couple seasons later) than it is to measure a 399 day time-period to an accuracy of minutes (if timing an orbit of Io and then extrapolating a year, you'd need to measure the occulation times at an accuracy faster than the human reaction to visual stimuli).
The thing I was wondering about is that if you knew which day of the year to do the measurement, then you wouldn't have to time the whole year. If it were a known day, like a solstace or something, then the answer would be clear, since celestial calendars were good enough for that. Then you would be dealing with 15 minutes out of a day, or about 1 part in 100. However, the relative position to Jupiter throws in a wrench. It seems like if he could calculate the right day, then he could make a much more accurate measurement of the occultations, given the clocks availaible to him.

Here looks to be some good info:

http://www.rundetaarn.dk/engelsk/observatorium/light.htm" [Broken]

It describes how the Doppler effect can be used, but also states:

COMPUTATION OF THE SPEED OF LIGHT
Rømer never did compute the speed of light, maybe because he felt that the distance between the Earth and the Sun was not known too well. However, there is a value in "Adversaria" (his notebook, Royal library, Copenhagen, fol. αb, he writes: 1091 earthdiameter per minute).
and that an article was published, but not by him, which had some "misunderstandings":

http://www.texts.dnlb.dk/TouchantDeLaLumiere/1.html"

This is the article with the most oftenly used diagram. Sadly, I don't read Dutch or whatever that is.

It was Huygens and Newton who later use Roemer's findings to calculate light speeds.

It also says that
If we integrate ΔT from one conjunction between the Earth and Jupiter to the following opposition, we get an accumulated Doppler shift in the period of 16.7 minutes, near the value found by Rømer in 1676 (published in Journal des Sçavans 1676)
Which suggests a measurement like jtbell said, but these are the words of the web page author, and don't seem conclusive about which type of measurement Roemer made.

The following page indicates that the measurement wasn't at opposition, but at an intermediate point:

http://www.pbs.org/wgbh/nova/einstein/ance-c.html" [Broken]
By the late summer of 1676, Roemer had an exact figure for how many extra minutes light took to fly that extra distance when Earth was far from Jupiter. At the public forum of a journal all serious astronomers read, he proclaimed a challenge: Io would appear from behind Jupiter the following November 9 not at 5:27 p.m., as Cassini calculated, but ten minutes later, at 5:37.

On November 9, observatories in France and across Europe had their telescopes ready. 5:27 p.m. arrived. No Io. 5:30 arrrived. Still no Io. 5:35 p.m. And then it appeared, at 5:37 and 49 seconds exactly. And yet Cassini declared he had not been proven wrong! It was so far away, so hard to see exactly, that perhaps those clouds from Jupiter's upper atmosphere were producing a distorting haze.

Roemer had performed an impeccable experiment, with a clear prediction, yet Europe's astronomers still did not accept that light traveled at a finite speed. Cassini's supporters won: the official line remained that the speed of light was just a mystical, unmeasureable figure.

Roemer gave up and went back to Denmark. Only 50 years later did further experiments convince astronomers that he had been right. The value Roemer had estimated for light's speed was close to the actual speed of light, which is about 670,000,000 mph.
It also shows how yet another dude (Cassini) with a big reputation and ego to match, set back science, although only 50 years, in this case, which is relatively short.

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