Galileo’s Moons

In early 1610, Galileo Galilei was the first to observe the largest four moons (now often called the Galilean moons) of Jupiter. He recorded these observations with enough detail that we can reproduce them and evaluate his accuracy more than four centuries later.

Perhaps the best way to do this would be to trace the positions of the moons from a modern observation back to Galileo’s. This is possible with accurate modern ephemerides (Lainey, Duriez, and Vienne 2004). A second option is to use software such as the fantastic Stellarium which uses such ephemerides. However, we can also directly use Galileo’s observations to set up the initial conditions for the moons. Once we have these, we can use the modern orbital elements to advance their position over time. This is simpler, should be accurate enough for us, and allows us to dig more deeply into Galileo’s observations.

In January 1610, Jupiter was roughly $4.3\,{\rm AU} \sim 6.4 \times 10^8\,{\rm km}$ away. At that distance, 1 arcsecond subtends roughly 3100 km. Thus, the angular size of the orbits are

Given this information, we can identify Callisto in Galileo’s observation of January 17th.

30 minutes after sunset, the configuration was thus … one [star] was 11 minutes from Jupiter to the West.
(Galilei and Van Helden 2000)

Unless Galileo has the angular distance estimates very wrong this can only be Callisto. Note that, while Galileo realizes that these objects are satellites of Jupiter, he does not call them “moons”. He interchangably calls them “planets” or “stars”.

Similarly, we can identify Ganymede in his January 18th observation.

20 minutes after sunset, the appearance was thus. The eastern star was larger than the western one and 8 minutes distant from Jupiter, while the western one [already identified as Callisto from the day before] was 10 minutes from Jupiter.
(Galilei and Van Helden 2000)

Again, unless Galileo has significantly overestimated the angular distance, this must be Ganymede. Finally, using the observation of January 15th, we can identify Io and Europa.

The intervals between Jupiter and the next three stars were all equal and of 2 minutes [i.e., Io at 2 minutes and Europa at 4 minutes. Ganymede and Europa are already identified further out (6 and 10 minutes).]
(Galilei and Van Helden 2000)

We have a final constraint on the positions of the inner three moons. Io, Europa and Ganymede are in a Laplacian resonance (a ratio of their orbital periods of 4:2:1) with their phase locked such that, at all times,

$\lambda_{\rm Io} - 3 \lambda_{\rm Europa} + 2 \lambda_{\rm Ganymede} = 180^{\circ}$

where $\lambda$ is the mean longitude (the angle from a reference vector) in the orbit.

Having identified the moons in Galileo’s observations, and using this added constraint, we can set the initial conditions that best match Galileo’s observations. This optimization could be done numerically, but we just do it by eyeballing Galileo’s diagrams (which I find incredibly useful in understanding what he saw, see also Edward Tufte appreciating his diagrams of Saturn) and our predictions. There are some understandable discrepancies; in some observations a moon is missing (likely caused by imperfect atmospheric conditions and 17th century optics!) and in others the model and observations disagree on the exact positions of the moons (Galileo generally only gives the time of the observation to the nearest hour and our model is fairly simple). In general though, the accuracy of Galileo’s observations is astronishing.

Galileo’s Conclusions

Galileo did more than just mark the positions of these objects, he also made inferences about their physical properties. The first was that these were indeed satellites of Jupiter – the first moons discovered around another planet.

since they sometimes follow and at other times precede Jupiter by similar intervals, and are removed from him toward the east as well as the west by only very narrow limits, and accompany him equally in retrograde and direct motion, no one can doubt that they complete their revolutions about him while, in the meantime, all together they complete a 12-year period about the center of the world [the sun].
(Galilei and Van Helden 2000, pg 28R)

He also realized that their orbits had different radii.

Moreover, they whirl around in unequal circles, which is clearly deduced from the fact that at the greatest separations from Jupiter two planets could never be seen united while, on the other hand, near Jupiter two, three, and occasionally all four planets are found crowded together at the same time.
(Galilei and Van Helden 2000, pg 28R)

And that the moons with small orbits moved faster (this relationship between distance and period is now explained by Kepler’s third law).

It is further seen that the revolutions of the planets describing smaller circles around Jupiter are faster. For the stars closer to Jupiter are often seen to the east when the previous day they appeared to the west, and vice versa
(Galilei and Van Helden 2000, pg 28R)

He also makes a good estimate of the orbital period of Callisto.

the planet traversing the largest orb [the moon with the largest orbit, Callisto] appears to have a semimonthly period. [the modern value is 16.7 days]
(Galilei and Van Helden 2000, pg 28R)

Galileo used these new observations of a single system to make a general argument about the solar system. He claimed that this observation of another planet with its own satellites was an argument against Tycho Brahe’s geoheliocentric model in which the moon and sun orbit the earth, while the other planets orbit the sun.

We have moreover an excellent and splendid argument for taking away the scruples of those who, while tolerating with equanimity the revolution of the planets around the Sun in the Copernican system, are so disturbed by the attendance of one Moon around the Earth while the two together complete the annual orb around the Sun that they conclude that this constitution of the universe must be overthrown as impossible. For here we have only one planet revolving around another while both run through a great circle around the Sun.
(Galilei and Van Helden 2000, pg 28R)

The discovery that Jupiter has moons removes Brahe’s reason – unhappiness with the moon orbiting the earth while the earth orbits the sun – for making a compromise to the Copernican system (Robison 1974).

Marius’ Claim

While Galileo is almost universally recognized as the discoverer of Jupiter’s moons, a German astronomer, Simon Marius, claims that he was the first to observe them.

However, as Jupiter was then retrograding, and still I saw these stars accompanying him throughout December, I was at first much astonished; but by degrees arrived at the following view, namely, that these stars moved round Jupiter, just as the five solar planets, Mercury, Venus, Mars, Jupiter, and Saturn revolve round the Sun. I therefore began to record my observations. The first was taken on December 29 [1609], when three stars of this description were visible in a straight line from Jupiter towards the west. At this time, as I frankly confess, I thought that there were only three such stars accompanying Jupiter, since I several times saw three of them near Jupiter. … Accordingly, from this time until January 12, I gave my diligent attention to these Jovian stars, and somehow ascertained that there were four such bodies, which themselves revolved about Jupiter.
(Marius 1614) translation from (Gaab and Leich 2018, pg 5)

While these observations are claimed to be done before Galileo’s, these results were published almost four years after and well after Marius was aware of Galileo’s findings.

Verifying Marius’ claims is impossible as, unlike Galileo, he did not leave an accurate record of his observations. However he does describe one observation in enough detail for us to verify.

The first [recorded observation] was taken on December 29 [1609 (this is in the Julian calendar, equivalent to January 8 1610 in the Gregorian calendar used by Galileo)], when three stars of this description were visible in a straight line from Jupiter towards the west.
(Marius 1614) translation from (Gaab and Leich 2018, pg 5)

On that date, as shown by Galileo above, three moons were indeed visible to the west.

Others have investigated this claim in far more detail than I can here. (Bosscha 1907) finds Marius’ claims compelling, but (Gaab and Leich 2018, chap. 16) gives reasons why this might be politically motivated; Bosscha was a patriotic Dutchman who felt that Galileo had been given credit for his study of motion over Simon Stevin, a Flemish physicist. Given the lack of detail from his early observations it is unlikely that the question of priority will ever be fully settled.

Marius does at least have the distinction of having named the Galilean moons – we are fortunately not stuck with “the Medicean stars” which Galileo propsed in honor of his patron.

Jupiter is much blamed by the poets on account of his irregular loves. Three maidens are specially mentioned as having been clandestinely courted by Jupiter with success. Io, daughter of the River Inachus, Callisto of Lycaon, Europa of Agenor. Then there was Ganymede, the handsome son of King Tros, who Jupiter, having taken the form of an eagle, transported to heaven on his back. … [These names] were suggested to me by Kepler, Imperial Astronomer, when we met at Ratisbon fair in October 1613.
(Marius 1614) translation from (Gaab and Leich 2018, pg 16)

Model Details

Here are some technical details of the model we use to predict the satellites position. Understanding this is not necessary to appreciate anything that we have done so far!

The largest source of error in the model will be caused by the changing positions of the Earth and Jupiter over the two months of observations which we do not account for. Jupiter was at opposition around December 7th 1609 and in quadrature around March 3rd 1610. The changing viewing angle will affect the observed position in the orbit of the satellites by roughly $10^{\circ}$ . The increasing distance (roughly 4.3 to 5.1 AU during Galileo’s observations) will also decrease the observed angular size of the orbits by roughly 20%. Another effect of the increasing distance (that is negligible but pretty cool) is the increasing light travel time from Jupiter to Earth. The satellites would appear lag in their orbits by roughly 6 minutes over the course of the two months (for Io this is roughly $0.5^{\circ}$ ).

Two further assuptions that will result in far smaller errors are: We assume all orbits are circular (fairly valid as no moon has $e > 0.01$ ). We assume all orbits are in the same plane (no inclination). This would have little effect on the general results (as no moon has $i > 0.5 ^{\circ}$ ), but would be interesting to include as Galileo occasionally mentions that a certain “star” is displaced North or South from the orbital plane (see e.g., his observation of January 13th).

We do model (roughly) the effect of the change in sunset time. This is roughly an hour over the course of Galileo’s observations.

References