Question

Jupiter and its four large Galilean moons are sometimes termed as a ‘mini-solar system’ because Jupiter is composed of mainly hydrogen and helium like stars and the Galilean moons seem like planets revolving around it. The total angular momentum of a system is contributed by the sum of orbital and rotational angular momenta of the central body and the bodies orbiting around it.

a. Derive an equation for rotational angular momentum and calculate the same for the Solar System and the Jupiter system. (Ignore the contribution of all satellites, asteroids and dwarf planets for the Solar System and the smaller Jovian satellites for the Jupiter system)

b. Derive an equation for orbital angular momentum and calculate the same for the Solar System and the Jupiter system. (Ignore the contribution of all satellites, asteroids and dwarf planets for the Solar System and the smaller Jovian satellites for the Jupiter system)

c. Explain how this analogy of the ‘mini-solar system’ breaks down.

Answer 1

Now for this analogy -

Jupiter is the fifth planet outward from the Sun and the largest planet in our Solar System. In fact, Jupiter has something like a whole planetary system of its own, with spectacular rings and the most moons of any planet.

Jupiter is the most massive planet in our Solar System and it has dozens of moons. In fact, with its numerous moons and rings, the Jupiter system resembles a miniature Solar System.

The planet itself is like a small star. If Jupiter had been only fifty or a hundred times more massive, it would have become a star rather than a planet.

Jupiter's rings surprised astronomers when they were discovered by the Voyager-1 probe in 1979 and confirmed by Galileo in 1998. The planet's system of rings apparently relates to its many moons. Scientists wonder if the rings may have been formed by dust kicked up as interplanetary meteoroids smashed into the giant gas planet's four small inner moons.
The sun is a star and Jupiter is a planet. Specifically, Jupiter is the largest planet that revolves around the sun, and it has several characteristics that make it similar to the sun, including composition and its own mini-system. However, despite these similarities, there are important differences that make the sun a star and Jupiter a planet, specifically in considering what occurs in their cores.

Star vs. Planet

The defining characteristic of a star is that it is hot enough and dense enough to have nuclear fusion occur in its core. Nuclear fusion occurs when the protons from hydrogen atoms combine to create helium atoms; photons and energy are released as a byproduct of nuclear fusion. Jupiter, despite being an extremely large planet (all the other planets in the solar system could fit inside of it), is not nearly as large as the sun, and it does not have nuclear fusion occurs in its core.

Composition

Jupiter and the sun are both very similar in their overall composition, as they are both made up almost entirely of hydrogen and helium. The sun has a core that is so hot that it causes hydrogen to separate into individual electrons and protons; Jupiter's core is made of liquid metallic hydrogen. Both the sun and Jupiter are similar in composition to what the solar system originally was like, which was almost entirely hydrogen and helium. The primary difference here is that the sun is much larger than Jupiter.

Solar System

The size difference between Jupiter and the sun is so large that the sun has the ability to hold distant objects in its gravitational field -- as shown in Newton's Universal Law of Gravitation, the more massive an object is, the farther out smaller objects are drawn to it. In addition to holding eight planets in its orbit, the sun has several smaller, more remote objects (such as comets) that revolve around it. The sun is so large that despite all the objects in its revolution, it still makes up over 99 percent of the mass in the solar system.

Jupiter's Mini-System

Despite being much smaller than the sun, Jupiter is still large enough to exert its own gravitational field, and as a result, it has several moons that orbit it. The four largest moons (Io, Europa, Ganymede and Callisto) were discovered by Galileo in 1610; a dozen smaller moons have been discovered since then. In addition to its satellites, Jupiter also has a thin ring system that was first seen by the Voyager I spacecraft.

• The solar system has a set of four inner planets (Mercury, Venus, Earth, and Mars); a main belt of asteroids, composed of rocky bodies; a set of four larger outer planets (Jupiter, Saturn, Uranus, and Neptune), and then another belt of small bodies (including Pluto and other dwarf planets), surrounded by a number of icy objects orbiting further out.
• Jupiter has a set of four inner moons, (Metis, Adrastea, Amalthea, and Thebe), th main ring composed of small rocky objects, a set of four larger outer moons (Io, Europa, Ganymede, and Callisto), and then a number of smaller icy objects orbiting further out.

Jupiter has a total of seventy-nine known moons as of the end of 2018. Notice that I said “known” moons. A year ago, we only knew about sixty-nine of them, and a year before that, we only knew about sixty-seven. Before the year 2000, we only knew about seventeen. That means that in the past few decades, we’ve been discovering an average of a little more than three new Jovian moons every year. At that rate, by the end of 2020, we’ll have discovered around eighty-six, and by 2025 we’ll hit the one-hundred mark. My point is simply that Jupiter has a lot of moons that we already know about, and almost certainly a lot more that we just haven’t found yet. Four of Jupiter’s moons are planet-sized. These are the ones that are so big, they were discovered by Galileo Galilei over four hundred years ago with one of the very first telescopes ever invented. They’re so big that you can easily see them in a thirty-dollar telescope you buy from a toy store (or even with reasonably good binoculars).

In addition to its dozens of moons, Jupiter also has a ring system composed mostly of rocky, dusty objects similar to the composition of the inner asteroid belt. Experts estimate the average size of these objects to be between 0.1 and 10 μm. Scaling up by a factor of ten again would give us relative sizes between 1 and 100 meters across for these objects. Interestingly enough, although there are a small number of fairly large asteroids, the best estimates put the largest number of asteroids in the asteroid belt—over 100 million of them—below the 100-meter threshold. Once again, when adjusting for scale, the objects that make up the “rings” orbiting Jupiter are eerily similar in relative size to the objects that make up the “belts” orbiting the sun.

But Jupiter’s moons and rings aren’t the only objects that are completely dominated by Jupiter’s gravitational supremacy. There are literally hundreds of thousands of objects—known as “Trojan asteroids”—that orbit within Jupiter’s gravitational Lagrangian points