It's probably not a big surprise to most of you to learn that the words for "moon" and "month" are related in English (and some other languages as well). Our Moon's orbital period of 27 days, 7 hours, and 41.1 minutes comes very close to the number of days you get when you divide the Earth's orbital period by twelve, and makes a nice natural division of time.
But have you ever thought about the moons of other planets? For example Mars' two moons, Phobos and Deimos, orbit their parent planet in just 7 hours 40 minutes and 30.3 hours respectively. Many of Jupiter and Saturn's close-in moons likewise orbit in less than an Earth day. In fact, there are dozens of moons with a shorter orbital period than our Moon.
On the flip side of the scale, there are also dozens of moons with longer orbital periods than our Moon. Jupiter and Saturn both also have lots of small, irregular moons that orbit far from their parent body, which can take months or even years to complete one orbit. Saturn's moon Phoebe, for instance, takes 550.3 days to make a complete circuit, nearly two Earth years. Prior to last week, I knew of a few Jovian moons with orbital periods measured in days in the 600's and 700's. Given Jupiter's humongous mass, you'd expect that it would be able to hold onto satellites further out than other planets, which would have correspondingly long orbital periods.
So you can imagine my surprise when I, on a whim, looked up the satellite with the longest orbital period and discovered it belonged to...Neptune?? And not just by a few days or even a few months – we're talking years here.
In fact, it turns out the four longest orbital-period moons all belong to Neptune. The two inner ones, Sao and Laomedia, have orbital periods of 7.97 and 8.68 years respectively. The two outer ones, Psamathe and Neso, take 24.84 and 26.67 years to orbit Neptune once, respectively.
I found this revelation absolutely mind-boggling. Neither of these moons has completed an orbit since I've been born. They have longer orbital periods than the first five inner planets. They orbit Neptune at a mean distance of around 48-49 billion kilometers (about 30 million miles), which is nearly a third of the distance from the Earth to Sun. At its furthest point, Neso can be further from Neptune than Mercury ever gets from the Sun!
If you wondered, like me, how Neptune and not Jupiter can have the furthest-out and longest-orbiting satellites, it has to do with something called the Hill sphere (named after 19th-century American astronomer and mathematician George William Hill). The Hill sphere is basically the region of space in which an object's gravitational pull dominates the attraction from other objects in the region. For a moon to remain in orbit about a planet, it must remain entirely inside the planet's Hill sphere, or it will eventually be pulled loose by the gravitational perturbations of other planets. This limits how long of an orbital period a moon (or other satellite) can have before it is no longer stably bound to its parent planet. For instance, the mathematics suggests that it is impossible for the Earth to have a satellite with an orbital period of longer than about seven months.
To get to the point, a planet's Hill sphere depends both on its mass, and its distance from the Sun (and other massive sources of gravitational perturbation). Jupiter, of course, is many times more massive than Neptune (and all the other planets combined), but Neptune is several times further from the Sun. Add in the inverse-square nature of gravity, and Neptune manages to eke out a victory in the "largest planetary Hill sphere" competition. (Interestingly, of the four outer planets, Jupiter has the smallest Hill sphere; it increases slightly but steadily in size from Jupiter through Saturn and Uranus on to Neptune. Turns out increased distance from the Sun is more important than decreasing mass.) Neso and Psamathe are orbiting nearly at the outer limit of Neptune's Hill sphere, so they are likely to remain the moons with the longest orbital periods for the foreseeable future.
Of course, they were only discovered in 2002 and 2003, respectively, so who knows what else could be out there! It's an exciting time for us lovers of planetary science and Solar System dynamics.
Anyway, I hope you found that as interesting as I did. If you're interested in other comparisons between the moons of the Solar System, this page on Wikipedia has a nice table that you can sort by various categories.