## Info

Computed

Modern theory

Io

1d1S±h

1 d 18; 28, 26 h

1 d 18; 28, 34 h

Europa

3 d 13 3 h

3 d 13; 20, 51 h

3d 13; 17, 42h

Ganymede

7d 4 h

7d 3; 55, 14h

7d 3; 58, 48 h

Callisto

16 d 18 h

16d 17; 56, 14h

16 d 18; 0, 0 h

Fig. 7.5. Eclipses of the moons of Jupiter.

apogee (perigee) would be observed at A1 (Pi) or A2 (P2) and the angular difference 0 can be as large as 23°.

After incorporating the corrections implied by this realization, Galileo published (in his Discourse on Floating Bodies in 1612) his computed periods for the four moons he had observed (shown in Table 7.1). The first column gives the values Galileo published, and these were rounded from those he computed (shown in the second column). For comparison, the third column lists values computed from modern theory for the period 1610-14. Galileo's calculations clearly represent a significant achievement.

Galileo also noticed, when timing the invisibility of a satellite during an occultation, that sometimes the moon would remain invisible for rather longer than his calculations suggested it should, and he soon realized the cause. The satellite was in the shadow of Jupiter, just as our Moon goes into the shadow of the Earth during a lunar eclipse. The situation is illustrated schematically in Figure 7.5. One might expect the satellite to be invisible from the Earth as it

36 Galileo's values are taken from Swerdlow (1998b); whereas the numbers in the final column are from Johnson (1931).

revolves around Jupiter on the orbit ABCD, during the times between A and C when it is hidden from view behind the planet. But between B and D it is in the shadow of the Sun and so, in total the satellite will remain invisible from A to D.

An eclipse, unlike an occultation or a transit, is a phenomenon that does not depend on the position of the Earth in its orbit. Galileo realized because of this that eclipses of the Jovian satellites could be used to help solve one of the great problems of navigation, i.e. the determination of longitude. The local times of eclipses could be tabulated for some reference point (e.g. Florence) and then the differences in the local time at which these were observed would correspond to a difference in longitude from Florence. In 1612, Galileo sent a proposal to the king of Spain, who had been offering a reward for anyone who could 'discover the longitude', but nothing came of it. He proposed later the same thing to the Dutch in 1632, again with no success. The use of the satellites of Jupiter to determine longitude exercised the minds of astronomers and navigators for the next 200 years, however, and was very successful as a method for determining accurately the longitude of points on land.

Investigations into the motions of the moons of Jupiter led to another major astronomical discovery later in the seventeenth century. A number of astronomers noticed that the predicted times of the eclipses of Io did not correspond to the actual times, with the errors being greatest when Jupiter was at conjunction, and least at opposition. The Danish astronomer Ole Christenson Romer concluded that this must be due to the as yet undetected finite speed of light, with light taking about 22 min to cross the orbit of the Earth (the modern

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