When as a middle schooler I got my first real telescope (a beautifully-painted 4 ¼ inch reflector) I was fascinated by the four bright spinning moons of Jupiter – eclipsing, being eclipsed, disappearing, reappearing, seemingly moved by an invisible hand in a cosmic shell game. Watching Jupiter’s intriguing little moons was a favorite start to my summertime evening of observing. Now it’s commonplace knowledge, but in the first years of the 17th century nobody knew about any other moons but ours in our solar system. One can imagine Galileo’s excitement on the evening of January 7, 1610, when he turned his little ‘Venetian glass’ to Jupiter and made out three tiny stars floating near the planet, two on one side, one on the other. The next night they were all on the same side. Still later a fourth star appeared. He had discovered something completely new and important. Contrary to the teachings of Aristotle, the ‘fixed’ stars moved*. *The perfect unchanging cosmos was mutable – and of course any findings that opposed Aristotle and the Peripatetics was a *very* big deal. He named the little stars Sidera Medicea (“Medici’s stars”), in honor of the Grand Duke of Tuscany, Cosmo II de’ Medici. That spring he let the world know about it in a little book called *Nuncius Sidereus*, or *Starry Messenger*.

Four centuries and one year after Galileo’s great discovery, NASA launched its Juno mission. It arrived at the giant planet five years ago after five years in space. It has since been giving us dramatic looks of Jupiter and his brood. (See the latest at __https://www.nasa.gov/mission_pages/juno/images/index.html____.__) This month we see its largest satellite Ganymede close-up. There were earlier missions that took its picture, but Juno's thousand-kilometer buzz over Ganymede shows craters, pocks, wormy rills, and long scars raked across its face. Study the picture above and marvel at the detail.

Even though it’s advertised as ‘covered in water ice,’ Ganymede looks startlingly like our moon, replete with craters and maria, but it’s one and a half times fatter: Ganymede’s 5,262 km girth tops our moon’s 3,475 km. It’s the biggest of the four Galilean moons. And it’s bigger even than its closest competitor, Saturn’s big moon Titan (at 5,149 km in diameter), and larger than all the moons of Uranus (all of which are under the two-thousand-kilometer category), and Neptune’s moon Triton (at 2,707 km). Is it as big as a planet? Well, Mercury’s equatorial diameter is 4,879 km, so it’s bigger than Mercury. No asteroid is bigger. And it’s bigger than dwarf planets Eris and Pluto. It’s actually about four-fifths the size of Mars.

How about its mass relative to our moon? Let’s see, Ganymede’s *radius* is 1½ times larger than our moon, but our moon is about 5/3 more *dense*. So do we think Ganymede should be greater or less massive than the moon? Who wins out in the contest between greater radius (Ganymede > moon) and greater density (moon > Ganymede) when it comes to mass? The mass *m* of a spherical object is proportional to its density (Greek letter rho) times it’s volume, and the volume is proportional to the *cube* of the radius *r*:

The sideways loop means “is proportional to.” You can see that density has a linear effect on mass, but radius has a cubic effect on mass. So, the ratio of 1½ cubed to 5/3 is about …. two. Despite its lesser density, radius wins the contest by a long shot: Ganymede is twice as massive as the moon.

That gives us a sense of the size, mass, and uniqueness of Ganymede. Yet Ganymede is only part of the story that involves all the Galilean moons, Kepler, Newton and history.

Did you know that Ganymede and its sibling moons, Io, Europa, and Callisto had a big role to play in convincing the world that the force of gravitation was universal? A century after Galileo, they were in the news again. Now the question was a mathematical one relating to Isaac Newton’s new theory: whether the Jovian moons orbiting their planet – resembling through the telescope a miniature version of our solar system – obeyed the same laws as the planets did with respect to the sun. If the ‘centripetal force’ was universal, then Kepler’s law governing planetary periods around the sun should apply to Jupiter’s moons. Did they? Only observation would tell. The key was whether, in fact, they obeyed Kepler’s Third Law.

Here is a quick look at Kepler’s Third Law, first in words then in mathematical notation: *The squares of the periods (P) of the planets’ orbits are proportional to the cubes of their semi-major axes (a) of their orbits.* It’s perhaps easier to remember as an equation:

where *k* is a constant. This simple relationship between the period of a planet and the semi-major axis of its orbit is often called Kepler’s Harmonic Law. He found it by comparing the periods of the planets in the solar system with their distances from the sun. The semimajor axis is half the major (long) axis of an elliptical orbit. For near-circular orbits the mean radius is a good approximation. For the planets in our solar system, the unit of measure is typically years and astronomical units (AU), though as we’ll see, it doesn’t have to be. If the baseline unit is the earth, then the period P is one year and *a* is one AU, so *k* = 1.

Kepler’s 1619 discovery of the Harmonic Law was a tremendous advance for planetary astronomy, yoking the planets together under one law. And this small equation, as economical as a calling card, hinted at an even larger truth: Newton found that he could derive an inverse square law of attraction from Kepler’s Harmonic Law, and deduce the latter from the former. (An inverse square law in this context indicates the way in which the strength of a force, such as gravity or magnetism, varies with distance from the center of attraction – specifically, it *decreases* with the *square* of *increased* distance from the source, and conversely.) Newton proved that any system of rotating bodies that obeyed Kepler’s Harmonic Law must also obey an inverse square law. Yet Newton, like any good scientist, was cautious about making speculative leaps. Kepler showed that his law applied to the sun and its planets, but was that a unique situation? Was gravitation truly universal? All Newton needed to clinch the argument was a test case outside of the usual heliocentric solar system. And that’s where Jupiter’s Galilean satellites return to the stage.

Newton’s first job would be to determine the distance (the semimajor axis) of each satellite from Jupiter, a task helped by the fact that their orbits are almost perfectly circular – Ganymede’s eccentricity is .001; then, to measure their periods as they zip around it. With that information, he could see if there is a common proportionality constant, *k*, applicable to the four satellites.

Newton was not an astronomer; he was a mathematician. Newton relied for his data on reports of other observers and particularly on John Flamsteed (1646–1719), the first Astronomer Royal of Greenwich Observatory in England. Flamsteed had measured the motions of Jupiter’s satellites and determined their distances from the center of the planet. In Book III of the *Principia*, Newton used data from Flamsteed and other astronomers to confirm the expected result that the moons of Jupiter (and Saturn) obeyed Kepler’s Harmonic Law. Newton concluded that the moons acted under the influence of a centripetal (toward-the-center) force with the planet as the center, which force fell off with the inverse square of the distance from that center.

The data on the table below is taken as presented in Newton's *Principia*, Book III, Phenomenon I. The order of the Galilean satellites follows their distances from the planet: Io, Europa, Ganymede, and Callisto (remember the mnemonic, “I Eat Gram Crackers”?); They are also known in their Roman numeral order, I, II, III and IV. The bottom row of the table contains the distances (in Jovian radii) of the satellites’ orbits that Newton *calculated *from the periodic times by the application of Kepler’s Third Law; they match closely the distances derived from observations, shown in the rows above it.

From these findings, Newton could (and did) make his case in Book III of his *Principia* that the inverse square law (of gravity) applies to the satellite systems of Jupiter and Saturn just as it does to the planets of the solar system.

In Newton’s data we see the ratios of each periodic time squared to distance cubed are .017, which is the proportionality constant *k* referred to above. I thought it would be interesting to use modern data and compare the results. From https://nssdc.gsfc.nasa.gov/planetary/factsheet/joviansatfact.html we see that the four Galilean satellites have these distances (in Jovian radii) and periods (in days):

Notice how respectable Newton’s distance data is (along his bottom row) when contrasted to modern data (first column above) developed centuries later. Using this modern data, I divide the period squared by the distance cubed for each satellite and get .015 for the proportionality constant *k*. Thus, we get the same ratio (.015) for each satellite, just as Newton, using slightly different figures, arrived at the same ratio (.017) for each satellite, each tightly consistent with Kepler's Harmonic Law, each thereby exhibiting 'Keplerian motion.'

Actually, Jupiter has a slew of other moons that were invisible until the era of spacecraft. What about the four tiny moons inward of Io? Will Kepler’s Harmonic law apply snugly to them? If so, what do you think the proportionality constant should be for them?

If you do the drill and divide the period squared by the distance cubed for each satellite using the above data, surprise! you again get .015 for the proportionality constant *k*. That should really *not *be a surprise by now because they are in the same gravitational system as the Galilean moons.

Kepler’s Harmonic Law applies, albeit with different proportionality constants for each gravitational system (e.g., solar, Jovian, Saturnian, etc.) no matter what the units are and no matter where the system is. Indeed, finding the Keplerian motion of bodies elsewhere, even in exoplanet systems, is a highly useful tool. The Harmonic Law allowed led Edmond Halley to deduce the initial parameters of the Great Comet of 1680, an adventure also described in Part III of Newton’s *Principia*. Does the Harmonic Law apply to our galaxy? Well, yes to a point, but the *failure* of Keplerian motion in the rotation of stars in the outer parts of our galaxy is what led American astronomer Vera Rubin to her revolutionary hypothesis of the existence of dark matter.

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