By Jove

7th January may be the date in 1610 that Galileo Galilei finally realised that the planet Jupiter was being orbitted by something(s) else and relegated the geocentric view of the universe to the dustbin of history.  If objects were orbitting planets, then planets were probably orbitting the other stars . . . and we were not alone in the universe. It's silly to cite a single date for the event because it required at least two separate observations to realise that the dots that were the moons of Jupiter moved relative to each other and relative to the planet.  Three observations (and having a brain as big as Gauss) are, of course, necessary to calculate the orbital frequencies.  It's pretty clear that Ganymede had been observed in China nearly 2000 years before the events of 1610.  But there is no evidence that the Chinese observers realised what they were seeing.  In the same way, Uranus is clearly if faintly visible to many naked eyes but it was never recognised as a planet by The Ancients (Chinese or otherwise) - that took William Herschel, his remarkable sister Caroline, and a state of the art telescope to discover the seventh planet out from the Sun while scanning the skies for double-stars.

When Galileo wrote up his observations in Sidereus Nuncius [The starry messenger] in March 1610 [rush job to establish priority!], he had identified four separate orbitting moons which he named Medicea Sidera [Medician stars] to honour his filthy-rich patron Cosimo de' Medici, and tribbed four members of the clan by naming the moons individually Cosimo, Francesco, Carlo, and Lorenzo. The printer of Sidereus Nuncius used ascii art to show how the moons moved on successive suitable-for-observation nights.  I've clipped out the data [R], to spare you having to translate the new-Latin in which science was communicated in those days. Because SN is so important in the development of human thought, you can get an English translation of the text including modern facsimiles of the art-work.

If you're thinking hmmm I've never heard of a body in the Solar System called Lorenzo, it's because Galileo's sycophantic names never really caught on beyond the Court of Florence. At the same time that Galileo was getting eye-strain in Tuscany, Simon Mayr [Marius] a German sky-watcher was doing the same thing 600+km further North in Ansbach, Bavaria.  It's probable that Marius nailed the problem a few days before Galileo but most of us have never even heard of him.  Marius consulted with Kepler and named the planets from inside track outwards: Io, Europa, Ganymede and Callisto but he didn't get his book Mundus Jovialis into print until 1614. They are cleverly named after various paramours/lovees/victims of a priapic Zeus/Jupiter and are an early example of BLT publicity because Ganymede was a boy.  As it happens Herschel also had trouble getting his preferred name Georgium Sidus [after George III] accepted for his new discovery: hmmmm perhaps the astronomical community would rather not acknowledge people whose only contribution to the project was inherited wealth and titles. Yes, I know you're dying to know, there is a Bob the Asteroid.  One really solid benefit from having Galileo and Marius looking at/for the same thing at the same time and recording the results, is that we have a replication of the experiment: the mutual agreement of the data transcends the priority spat.

Strangely enough the inner three Jovian Moons are in orbital resonance with relative period 1:2:4. Which is to say that Io goes round the planet precisely 2x for every turn made by Europa; and Europa 2x for every turn by Ganymede. You can see a moving graphic of the system. Inevitably you'll be looking to see when the three planets all line up together.  But you'll be waiting a long long time because of the Laplace resonance.

where  λ is the mean longitude of each moon . . . and nope! I don't understand it [Physics] either.