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Try to imagine what must have gone through Galileo Galilei's mind one January evening in 1610 when he first realized that the four points of light he saw through his new telescope were, in fact, worlds circling Jupiter. The thrill of discovery would have been magnified by the simultaneous realization that an unshakable truth that all worlds revolved around the Earth had just collapsed. Although viewing these same moons might not shake up your own worldview, you can at least relive some of Galileo’s excitement by discovering them for yourself with nothing more than binoculars or a small telescope.
Surprisingly, ordinary binoculars are optically superior to anything Galileo viewed through. Although binoculars typically magnify far less than Galileo’s telescopes, their sharp views will more than compensate for the lower power. Most binoculars will allow you to see the four largest Jovian moons Io, Europa, Ganymede, and Callisto (in order of increasing average distance from the planet). Binoculars that magnify 10x will have an advantage over those that magnify 7x. Either way, to get more than a fleeting glimpse you'll need to mount them on a tripod or some other support. Some binoculars have provisions for doing this; they're equipped with a standard ¼-20 threaded mounting hole located on the front hinge to facilitate the use of a tripod adapter. If yours lack the mounting hole, you can either securely tie the binoculars directly to the tripod head or simply steady yourself by leaning against a post or rail. Of course, if you’re using a telescope with a decent tripod, you’re already set.
Half the difficulty in seeing the Galilean moons for the first time is knowing what to look for. Although all four are major solar-system bodies (Europa being the only one that's smaller than our Moon), they are distant enough that at low magnification they look just like stars.
Knowing in advance where the moons will be makes finding and identifying them much easier, but keep in mind that each one becomes temporarily invisible when passing in front of or behind Jupiter, or when passing through its shadow. Don’t be surprised if you see only two or three moons most of the time. The easiest to catch are Ganymede and Callisto. They orbit farthest from their host and spend days at a time beyond Jupiter’s glare. By contrast, Io and Europa never venture far from the planet's disk.