## Measuring the Speed of Light in 1676

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*Chance favors the prepared mind.* Louis Pasteur

In 1676, the Danish astronomer Ole Romer did something quite remarkable for his time – he measured the speed of light. Although his value was not very accurate, it was the first demonstration that light does not travel instantaneously, a belief that been held by almost everyone from Aristotle on down. In this post I will describe how Romer did it, and then describe my little experiment to reproduce his measurements and calculations.

Romer’s results were based on timings of eclipses of Jupiter’s innermost satellite Io. Consider Figure 2, which is from Romer’s paper.

Io orbits Jupiter at a constant rate, and once each orbit it passes into Jupiter’s shadow and disappears from view. A few hours later, it emerges into sunlight. Even with a modest telescope, it is easy to observe these eclipses and record the time that they occur. Now suppose the Earth is at L, and we observe Io leaving the shadow and note the time. A month or two later, Earth is at K, and we note the time when another eclipse end occurs (for simplicity, assume Jupiter hasn’t moved). In the interim, some number of eclipses, say 10, has occurred. The time between our two eclipses has to be simply \({10 \times T}\), where *T* is the (fixed) time for an Io orbit.

In tabulating eclipse times, Romer was surprised to find that the second eclipse he observed seemed to happen later than expected. The delay was only a few minutes, but it was too much to attribute to measurement error. He realized that the difference must be due to light having a finite speed. When the Earth was at K, Jupiter was further from Earth, so the light announcing the eclipse had arrived at Earth later than it would have if the distance had remained constant. If we divide the difference in distance by the time difference, the result should be the speed of light.

As confirmation of this theory, Romer found that as the Earth moved from F to G, the eclipse at G occurred *sooner* than expected, because light from this eclipse didn’t have as far to travel to reach Earth at G. In his 1676 paper, Romer computes a value for the speed of light that equates to 220,000 km/sec, which is 27% less than the correct value.

Romer’s work looks strange to the modern reader. Although he clearly understood the phenomenon just described, he does not list the timing values he used, nor does he show how he made his calculations. In our modern peer review system, his paper would have been rejected immediately.

** My Experiment, 434 years later **

I have a telescope, and I decided to see if I could reproduce (or even improve on) Romer’s results by timing some Io eclipses. The plan was to time two different eclipses when Earth was moving away from Jupiter and two others when Earth was moving toward Jupiter, as described above. I thought it would be best to treat both the speed of light and Io’s period as unknowns. This is because the two are confounded: measurements of Io’s period are affected by the (unknown) speed of light, and vice versa.

Io enters and exits Jupiter’s shadow over several minutes’ time, so extreme accuracy is neither possible or necessary – a watch that has been set accurately is sufficient. I averaged the time when Io began to get dim (entering the shadow) with the time when it disappeared from view (fully in the shadow), then rounded to the nearest minute. All these eclipse timings are actually available in tables published by Sky and Telescope and others. I used these tables to know about when I could look for an eclipse, but I recorded the times from my observations, not from the tables. In the end, my values differed from those in the table by only one or two minutes.

** The Setup **

These diagrams show the configuration at opposition, and then a few month afterward.

One recent opposition was 2009.07.09 at 8:07 GMT, and this was the base point for measuring time (in days), and position (in degrees). Distance is in Astronomical Units, so that the Earth is 1 unit from the Sun, and Jupiter is 5.2. Since an Earth year is 365.25 days, Earth moves 360/365.25 = 0.9856 degrees per day. Jupiter orbits the Sun in 11.6 Earth years, so it moves .9856/11.96=0.0831 degrees per day. Notice that we are assuming that the planets move in circles when, if fact, their orbits are slightly elliptical.

Knowing how many days have passed, we can compute the angles \(\theta_e\) and \(\theta_j\). Note the triangle formed by the darker lines. The side marked d is the Earth-Jupiter distance, which is needed for the calculations. Computing d is simple trigonometry – we can use the use the Law of Cosines:

\[d^2=1^2+5.2^2-2(1)(5.2)cos(\theta _e -\theta_j )\]

As mentioned, I timed two Io eclipses with the planets positioned as above – i.e., Earth moving away from Jupiter. Here are the results:

# | Date/Time | Relative Days | \(\theta_e\) | _{\(\theta_j\)} |
Earth-Jupiter Distance |

1 | 2009.09.05 0:56 GMT | 57.7007 | 56.87 | 4.87 | 4.65 |

2 | 2009.12.06 01:59 GMT | 149.7444 | 147.59 | 12.63 |
5.95 |

The two readings are 92.0424 days apart, and we know that Io’s orbit takes about 42.5 hours, or 1.771 days. Since 92.0424 / 1.771 is very close to 52, there must have been 52 orbits between the readings. If *T* is the actual time for an orbit, then the actual exact time for the 52 orbits was *52T* .

Now at point 2, we were further from Jupiter by *5.95 – 4.65 = 1.30*. This means we would see the second eclipse later by an amount equal to 1.30 / c, where c is the speed of light. Thus, we can write the equation:

\[ 92.0424-52T=\frac{5.95-4.65}{c}\tag{1}\]

There are two unknowns in the equation – to solve for *T* and *c* we need a second equation. We get that from the second pair of observations made when Earth is approaching Jupiter, as shown in Figure 5.

Here is my second pair of timing data:

# | Date/Time | Relative Days | \(\theta_e\) | \(\theta_j\) | Earth-Jupiter Distance |

3 |
2010.05.28 10:36 GMT |
323.1935 |
318.46 |
27.24 |
4.93 |

4 |
2010.06.20 10:47 GMT |
346.2011 |
341.14 |
29.18 |
4.59 |

\[13T-23.0076=\frac{4.93-4.59}{c}\tag{2}\]

Solving (1) and (2), we get T=1.770 days, or 42.478 hours, and c = 199.9 AU’s per day. Converting c to kilometers per second, we have (drum roll please):

**c = 346,120 km/sec**

This is greater than the true value by 15%. It is more accurate than Romer’s result, but then again I wasn’t using 17th century technology.

#### Afterword

When the form of the two equations is examined carefully, it turns out that the computed value of c is extraordinarily sensitive to the value of T, Io’s period. Measurement error could cause us trouble, but there is also the fact that Io’s period actually varies a little, since it is gravitationally perturbed by the three outer moons, Europa, Callisto, and Ganymede. The upshot is that this is not a workable way to get an accurate value of c. I suspect that Romer’s result was off by 27% because he didn’t deal with Io’s period as well as he should have.

Finally, some might wonder how Romer could make measurements that were accurate to within a few minutes over a span of several months. The answer is simple; the apparent movement of the stars, caused by the Earth’s motion, is very constant, and easily measured.

January 5th, 2011 at 10:29 am

I am an 1st year undergraduate student. I am again doing the same experiment using a 4 inch reflecting telescope. can you please give me some suggestions regarding the experiment.

Thanks

January 6th, 2011 at 3:03 am

Here are some thoughts about the practical aspects of this project:

1) a four inch telescope should be adequate for the observations

2) it is extremely helpful to know when to look for an Io eclipse. Sky and Telescope tabulates these in a table:

http://media.skyandtelescope.com/documents/JphenTab10x.pdf

The events of interest in the table are marked “I.Ec.D” (Io disappears into shadow), and “I.Ec.R” (Io reappears from shadow). Depending on the relative position of Earth and Jupiter, just one of these types of event will be visible on any given date.

3) when you look through the telescope, it’s also helpful to know which moon is Io. That can be determined from Sky & Telescope’s monthly Jupiter moon positions at

media.skyandtelescope.com/documents/Wigglegram201012Jup.pdf

4) I created an Excel spreadsheet to solve the two equations – I just enter the date/time for each of the four events, and it computes c. Not that this is terribly hard to do manually.

5) The procedure and calculations that I used are my own. However, they must be basically correct, since the computed result isn’t too far off.

Good luck!

September 14th, 2011 at 6:02 am

great.. article…

is c = 346,120 km/sec is the real measurement? compare in these day technology.?

December 8th, 2011 at 6:19 pm

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October 30th, 2012 at 1:02 pm

T1 – 52T = d1/c

13T – T2 = d2/c

13T1 – 52T2 = 13d1 / c + 52d2 /c =>

c = [13d1 + 52d2] / [13T1 - 52T2]

d1 = 5,95-4,65 = 1.3; T1 = 92,0424;

d2 = 4,93-4,59 = 0.34; T2 = 23,0076

c = [13*1,3 + 52*0,34] / [13*92,0424 - 52*23,0076] = 34,58 / 0,156 = 221,67 = au/day = 385 000 km/s

Incorrect equations – observed periods T are not constant!

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