Lately, you might have noticed an abundance of pictures showing the rings of Saturn. These were recently captured by the Cassini space probe, whose mission will come to a dramatic end in September when it flings itself into the gas giant's atmosphere. One of the coolest things in these images, taken as the probe travels between Saturn's north pole and the edge of its main rings, are the gaps in those rings. But why do those gaps exist?

Why are there gaps?

A planetary ring is essentially millions of particles orbiting a planet in a flat plane. If their mass is small enough, the particles don't interact with each other. They simply orbit the planet. In the absence of a massive nearby object, the only force acting on those particles is gravitational force. You can determine the magnitude of the force like this:

Remember, this is simply the magnitude of the gravitational force—the direction matters too, but I left that off (for now). In this expression, G represents the universal gravitational constant with a value of 6.67 x 10-11 N*m2/kg2. Also, M p represents the mass of the planet, and m r is the mass of the ring particle.

If a ring-particle follows a circular orbit, this gravitational force must make the ring-particle accelerate toward the center of the planet. Given that this is the only force acting on the particle, the acceleration follows the same direction as the force. I can write this centripetal acceleration in terms of the angular velocity (ω) like this:

This says that the rings closer to the planet must orbit with a greater angular speed. When the particles are further away, the angular velocity decreases. And with this, you see that orbital mechanics dictates that a planetary ring cannot be solid.

OK, but what about the gaps between rings? Suppose a small moon also orbits the planet. In this case, both the moon and the planet exert gravitational forces on the ring-particle.

With the moon in this position, the net force is no longer of sufficient magnitude for circular motion at that orbital difference. Assuming the moon is relatively small (Earth's moon is fairly massive relative to the size of the planet), you'd see a tiny disturbance in the ring-particle's motion. But it shouldn't be a big deal. However, one set of ring-particles will exhibit a significant disturbance. If the orbital angular frequency of a ring-particle is an integer factor of the moon's frequency, then the moon will regularly be in a position to pull on the ring-particle in the same way. Let me offer an example. Let's say a moon orbits at a distance of r m so it has an orbital angular frequency of:

Now imagine a ring-particle with an orbital angular velocity twice that. It will have an orbital distance of:

With this double frequency, the ring-particle will have a consistent nudge that eventually pushes it out of its orbit. It's a bit like pushing a child on a swing at the right frequency. If you push every other cycle, the child climbs get higher and higher. Integer multiples of orbital frequencies are what causes the ring gaps.

Modeling Ring Gaps

Maybe I should make something clear. Although I understand the basics of gravity and orbits, I'm not an astrophysicist. I can create a model based on fundamental principles, but there is a chance that I might miss something important. This is what makes this so exciting. Heck—I'm not even sure of the term "ring gap," but I think you understand what I am saying. (Editor's note: Ring gap checks out, but planetary scientists call the largest gaps divisions. The biggest visible gap in Saturn's rings is the Cassini division.)

Here is the plan for the ring-model.

I am going to make four ring-particles. These four particles will start at different orbital distances, so it won't actually be a ring.

Each ring particle will interact with the moon and Earth, but not each other—I will assume they have masses small enough to ignore ring-ring interactions.

Of course this means I must model the three-body problem, which you can. You can learn about here. OK, technically this is more like a two-and-a-half body problem since the rings don't do anything to the moon or the Earth.

I won't use Earth's moon. Instead I'll use a fake moon with lower mass that's closer to Earth. I think this will make it easier to model a ring gap.

Finally, I am going to put these four ring-particles at a non-gap location, then a gap location.

Oh, I guess I should add that I started with a different plan. I thought I'd make a bunch of ring-particles and let them leave a trail. That way, over time I should see a gap form. Let's just say this plan didn't really work.

Let me start off with four particles centered around a position that is 0.8 times the calculated gap radius. Here's the code. Remember you can click "play" to run and "pencil" to edit—yes, you can edit the code if you want. It won't break anything (well, not permanently).

Really, you don't have to run that code—it's not terribly exciting. Here is the important part, a plot of the distance from the earth for the four ring-particles:

Note the variations in the orbits due to the interaction with the moon. That said, they essentially maintain the same orbit.

Next I will move the four ring-particles so that they are very near the ring gap distance. Here is a link to the code (which you can play with if it makes you happy), but really I simply want to show the plot for the orbital radius.

Clearly there is a difference with these four ring-particles. They do not have stable orbits like the ring-particles at a non ring gap position. Why the difference? Since the orbital frequency is around twice that of the moon, these ring-particles experience a regular nudge when they are close to the moon. The particles at the non-ring gap location get a nudge more infrequently.

Homework

Here are some problems for you to consider.