# Roemer and the speed of light

(notes by R. Bigoni - home italian english)

## 1. Jupiter's satellites and the measurement of the longitude in the 17th century.

In 1610, Galileo discovered the four largest Jupiter satellites: Io, Europa, Ganymede and Callisto.

The orbits of these satellites have a vary small eccentricity (for example, the eccentricity of Io is 0.00041); so their motion can be considered with good approximation circular and uniform. Furthermore the orbits are very close to the equatorial plane of Jupiter. The equator of Jupiter and its own orbit have a very small inclination with respect to the eclyptic plane so, when seen from the Earth, the four satellites, at every revolution,

• are obscured (or eclipsed) by the shadow of the planet, like the Moon when there is a lunar eclipse;
• are hidden by the disk of the planet: this is an occultation.

Eclipses and occultations are different phenomena: an eclipse can begin before a successive occultation and finish during the occultation, so its end cannot be seen from the Earth; in an analogous way an eclipse can begin during an occultation and finish after the end of the occultation.

You may see a nice Java applet that shows the motion of Jupiter's satellites at The Castle Point Astronomy Club.

The motions of these satellites are so regular that Galileo thought that the accurate registration of them could have not only scientific but also practical interest with respect to a most important problem not still resolved around the beginning of the 17th century, that is the measure of the longitude. But Galileo could not carry out his idea for the lack of reliable clocks.

The aim of Galileo was successfully realized in 1668 by Cassini who could make use of better astronomical instruments and of the clock invented by Huygens(1657).

In the winter of 1671-1672, Picard and Roemer (at Uraniborg, the observatory of Tycho Brahe on the island of Hven, (then in Denmark, now in Sweden) and Cassini (at the observatory of Paris), simultaneously measured the beginning of an eclipse of Io. From this measurement they could calculate the difference of the longitude between Uraniborg and Paris.

From 1672 Roemer was at the observatory of Paris and, continuing the observation of the eclipses of Io, found that the intervals of time between two successive beginnings (or endings) of the eclipses resulted different around the year: when the Earth was moving away from Jupiter they were greater than when the Earth was moving nearer to Jupiter.

## The speed of light is finite.

Thanks to this discovery Roemer could demonstrate that the speed of light is finite and put an end to a controversy that divided the european scientists in the 17th century: on one side there were those, like Kepler, Descartes, Cassini, who, following the opinion of Aristotle, believed that the speed of light is infinite; on the other side there were those, like Galileo, who believed the contrary but couldn't experimentally validate their persuasion.

Roemer published his results in 1676.

Here we apply the method of Roemer to a set of data collected in 1995.

For simplicity's sake, we assume that the Earth's orbit around the Sun is circular.

When the Earth is in O, Jupiter (J) is in opposition to the Sun.
When the Earth is in O', Jupiter is in conjunction to the Sun.

Furthermore we assume that if the displacements of the Earth are small, Jupiter can be considered as immobile.

Time intervals between the beginnings (or the endings) of two consecutive eclipses of Io in the shadow of Jupiter result constant if the measurements are made when the Earth is near O or O': in these cases one obtains

But when the Earth is in an intermediate place between O and O', for example in the point A (between an opposition and a conjunction) or in the point C (between a conjunction and an opposition) time intervals measures are greater (in A) or lesser (in C) than those ones made near O or O'.

It must be noted that when the Earth is between an opposition and a conjunction (A) we can see only the endings of the eclipses; vice versa, when the Earth is between a conjunction and an opposition, we can see only the beginnings.

Roemer understood that the different results were ascribable to the relative motion of the Earth with respect to Jupiter and to the finiteness of the speed of light.

The Earth's radial motion relative to Jupiter is unimportant when the Earth is near O or O', but it is noteworthy in the intermediate positions. When the Earth is in A, it is going away from Jupiter; when it is in C it is approaching the big planet.

### a) The Earth is between an opposition and a conjunction.

If t is the instant of the end of an eclipse of Io and in this moment the Earth is in A, at the end of the following eclipse the Earth, rotating counterclockwise, reaches the point B.

Since the speed of light c is finite, from the Earth the end of the eclipse in A isn't seen at the time t but at the time

where is the time needed by light to cover the distance dA.

The end of the following eclipse isn't detected at the time t+T but at the time

therefore the time interval TAB between two successive eclipses, measured by a terrestrial observator, results

### b) The Earth is between a conjunction and an opposition.

If t is the instant of the beginning of an eclipse of Io and at this time the Earth is in C, the following eclipse will begin when the Earth will reach the point D.

In this case the beginning of the eclipse is seen from the Earth at the time

and the beginning of the following eclipse is detected at the time

therefore the time interval TCD between the two eclipses results

## 3. Astronomical measurement of the speed of light.

We have seen at the point a) of the previous section that, when the Earth is between an opposition and a conjunction, the time interval between two successive eclipses in A and B, as measured by a terrestrial observator, is

Let C, D, E,... be other successive points of the Earth's orbit between the same opposition and the same conjunction. In an analogous way we shall have

The sum of the left sides of the equalities gives

therefore

If we apply this procedure to the data of the following table, reporting the times of the ends of 99 eclipses registered between an opposition of Jupiter to the Sun and the following conjunction, we obtain with good approximation,

where the expression between parentheses represents the time Δ elapsed between the first and the last observation (times are in the form dd/mm/yy,hh.mm), T is the orbital period of Io and au is the astronomical unit, that is the semi-major axis of Earth's orbit.

Δ amounts to 14986680 seconds, 98 T amounts to 14985670 seconds. The difference Δ-98 T amounts to 1010 seconds. The astronomical unit is about 1.5·1011m.

Therefore we obtain

Termination times of the eclipses of Io
between the opposition Jupiter-Sun on 01/06/1995, (11.00h)
and the conjunction Jupiter-Sun on 18/12/1995 (22.00h)
ord.datetimeord.datetimeord.datetime
0002/06/9508h06 3401/08/9512h20 6830/09/9516h40
0104/06/9502h34 3503/08/9506h48 6902/10/9511h09
0205/06/9521h03 3605/08/9501h17 7004/10/9505h37
0307/06/9515h31 3706/08/9509h56 7106/10/9500h06
0409/06/9510h00 3808/08/9514h15 7207/10/9518h35
0511/06/9504h28 3910/08/9508h44 7309/10/9513h04
0612/06/9522h57 4012/08/9503h12 7411/10/9507h33
0714/06/9517h25 4113/08/9521h41 7513/10/9502h01
0816/06/9511h54 4215/08/9516h10 7614/10/9520h30
0918/06/9506h22 4317/08/9510h39 7716/10/9514h59
1020/06/9500h51 4419/08/9505h08 7818/10/9509h28
1121/06/9519h20 4520/08/9523h26 7920/10/9503h57
1223/06/9513h48 4622/08/9518h05 8021/10/9522h26
1325/06/9508h17 4724/08/9512h34 8123/10/9516h54
1427/06/9502h45 4826/08/9507h03 8225/10/9511h23
1528/06/9521h14 4928/08/9501h328327/10/9505h52
1630/06/9515h43 5029/08/9520h00 8429/10/9500h21
1702/07/9510h115131/08/9514h29 8530/10/9518h50
1804/07/9504h40 5202/09/9508h58 8601/11/9513h18
1905/07/9523h09 5304/09/9503h27 8703/11/9507h47
2007/07/9517h37 5405/09/9521h56 8805/11/9502h16
2109/07/9512h06 5507/09/9516h25 8906/11/9520h45
2211/07/9506h35 5609/09/9510h54 9008/11/9515h13
2313/07/9501h03 5711/09/9505h22 9110/11/9509h42
2414/07/9519h32 5812/09/9523h51 9212/11/9504h11
2516/07/9514h01 5914/09/9518h20 9313/11/9522h40
2618/07/9508h30 6016/09/9512h49 9415/11/9517h09
2720/07/9502h58 6118/09/9507h18 9517/11/9511h37
2821/07/9521h27 6220/09/9501h47 9619/11/9506h06
2923/07/9515h56 6321/09/9520h15 9721/11/9500h35
3025/07/9510h24 6423/09/9514h44 9822/11/9519h04
3127/07/9504h53 6525/09/9509h13
3228/07/9523h22 6627/09/9503h42
3330/07/9517h51 6728/09/9522h11

last revision: 23/10/2015