**Francis Berthomieu**

**"EAAE Summerschools" Working Group**

**CLEA, France**

# Abstract

Trying to find a good timekeeper allowing an accurate determination of longitude, Galileo and later Cassini thought that the periodicity in the revolution of planet Jupiter's sattelites was a valid solution.

However, observations done by Cassini, Picard and Roemer showed some strange irregularities in the orbital movements of these sattelites.

Roemer then suggested that these irregularities could be interpreted by a finite value of the velocity of light.

Examining the Ephemerides of satellite Io and the original Roemer's statements, and althought the complete explanation is something delicate, we will try to give simply a correct justification of the phenomenon.

# Jupiter's satellites and the measurement of longitudes in the 17th century

In 1610, Galileo discovered the four biggest satellites of Jupiter: Io, Europe, Ganymede and Callisto.

The orbits of these satellites have very small eccentricities (they are nearly circular) and are close to the equatorial plane of Jupiter.

Jupiter's equator and its orbit are not much inclined over the plane of the Ecliptic.

At the end of the 16th century, the determination of longitudes could not be very accurately done for lack of stable and accurate timekeepers. Galileo then had the idea of using Jupiter's satellites as time indicators: their motions are practically circular and regular, their periods are short enough and the instants of mutual eclipses do not depend on the location of the observer.

This idea was taken up again by Cassini in 1668. It eventually met with success, due to the perfectioning of observational instruments and the invention of the clock (Huygens, 1657). During the winter of 1671-1672, Picard and Roemer (from Uraniborg on the island of Hven, the site of Tycho Brahe's observatory, now in Sweden) and Cassini (from the Paris Observatory), observed simultaneously the moments of eclipses of Io by Jupiter. From these measurements, they measured the difference of geographical longitude between Uraniborg and Paris.

From 1672 on, Roemer worked at the Paris Observatory and continued the observation of the eclipses of Jupiter's satellites. It was from these measurements that Roemer was able to prove for the first time that the velocity of the light is finite.

# Roemer proves that the velocity of the light is finite

This proof was published in 1676 in the "Journal des Sçavans" (document 1 and document 2).

In what follows, we will assume that all motions occur in the same plane (that of the Ecliptic) and that Jupiter does not move (in first approximation) during the observations.

The discussion refers to the figure 1 which shows the orbit of the Earth around the Sun (in the Ecliptic Plane).

S: Sun J: Jupiter.

When the Earth is located in *O'*, conjunction.

When the Earth is in *O*, opposition.

Moreover, *AJ = f BJ = f' CJ = d DJ = d'*

*T* is the period of revolution of Io around Jupiter.

*T* = 1,769 day = 1d 18h 28min.

**1 - The Earth is between an opposition and a conjunction**

Let us assume that the end of an eclipse occurs at an epoch (time) *t* and the end of the next one at an epoch *t + T*. Meanwhile, the Earth has moved away from Jupiter.

If the light would reach the Earth instantaneously, we would observe from Earth a period equal to *T*.

However, if the velocity of light is finite, let *c* be its velocity. We observe the end of an eclipse at an epoch *t + f / c* (point A). Then we observe the end of the next eclipse at the epoch *t + T + f ' / c* (point B). The difference in time between the ends of two consecutive eclipses is equal to *T + ( f ' - f ) / c*.

Now, *f' > f*, so *T + (f' - f) / c > T*

**2 - The Earth is between a conjunction and an opposition**

The beginning of an eclipse occurs at epoch *t'* and the beginning of the next one at the date *t' + T*. In between, the Earth has moved towards Jupiter.

If light would reach Earth instantaneously, we would observe on Earth a period equal to *T*. However, if the velocity of light is finite, we observe the beginning of an eclipse at an epoch *t' + d / c*, (point C) and the beginning of the next one at the epoch *t' + T + d' / c* (point D). The time interval between the beginnings of two consecutive eclipses is equal to *T + (d' - d) / c*.

Now, *d' < d*, so *T + (d' - d) / c < T*

# Exercises

Here follow some exercises which are related to Io's motion around Jupiter and which put you in position similar to that of Roemer.

**The Io Phenomena**

Examine the Figure 2. The Earth is between a conjunction and an opposition. The radius of the orbit of Io is equal to six times the radius of Jupiter. The angle SJE = a is always smaller than 11 deg.

On this figure, indicate the points where the beginning and the end of the following four phenomena take place:

- Eclipse = the satellite enters the shadow of Jupiter.
- Occultation = the satellite, as seen from Earth, goes behind Jupiter.
- Shadow = the shadow of the satellite is seen on the planet.
- Passage = the satellite, as seen from Earth, passes in front of the planet.

~ Now explain why only the beginnings of the eclipses can be observed from Earth.

~ What happens when the Earth is close to a conjunction?

~ Draw another figure that shows what happens when the Earth is between an opposition and a conjunction.

~ Explain why then only the ends of the eclipses can be observed from Earth.

**Some calculations with data from the Ephemerides**

You will find, on document 3, document 4, document 5 and document 6, tables with actual timings of Io eclipses. Use these values for the calculations explained below.

~ Find the date of a Jupiter opposition. (Use for instance the opposition on July 4, 1996) ~ Note the dates and the hours of:

- The beginning of eclipses (before opposition) number X (the last one before opposition), as well as numbers X - 20; X - 40; X - 60.
- The end of eclipses (after opposition): numbers 0; 20; 40; 60.

~ Find the total duration (in days, hours and minutes) of 40 observed periods:

- Before opposition:

from X - 40 to X ; let D be this duration.

from X - 60 to X - 20 ; let D' be this duration. - After opposition:

from 0 to 40 ; let F be this duration.

from 20 to 60 ; let F' be this duration.

~ Explain why F > D and F' > D'.

~ Explain why F - D > F' - D'.

**Data**

The mean distance between the Earth and Jupiter and the time of an opposition is 4.2 Astronomical Units (AU).

1 AU = 149,600,000 km (radius of the Earth's orbit).