Why Is Our Solar System Flat?

Some of my favorite things to look for in the night sky (besides the International Space Station) are the planets. If you need help finding them, I recommend Heavens-Above. It’s a website where you can enter your location and it will show you what’s currently visible in your area.

But one trick you can use on your own is to scan along the ecliptic. This is a line in the sky that arcs from one spot on the horizon to another, basically following the path that the sun travels from where it rises to where it sets. The cool thing is that the planets (and the moon!) also appear along this same line.

Why are they in a line? It’s because the solar system is flat. I know, last week I outlined two experiments you can do to prove that the Earth is not flat. But the solar system really is flat. All the planets and most of the asteroids orbit the sun in roughly the same plane, so that when you view them from Earth they line up neatly.

Just for fun, here’s a Python model showing part of the solar system (up to Jupiter). If you want to see how to make this, I have a video for you.

But why is the solar system flat? I’m glad you asked! Now we can talk about some physics.

The Gravity of the Situation

There is an attractive gravitational force between any two objects that have mass. You have mass and the Earth has mass, and it is that interaction that pulls you toward the Earth, so your feet stay on the ground. The magnitude of this gravitational force (F) depends on the values of the two masses (let’s call them M and m) and the distance between them (r).

But the solar system contains more than two masses. In fact, it started as a big cloud of dust without any planets and without the sun, and every speck of dust had an attractive interaction with every other speck. That’s a lot of complicated stuff going on, but there’s a trick we can use to simplify it. If the dust is evenly distributed, then a particle on the outside of the cloud would experience a gravitational force as though all of the other dust was concentrated at a single point in the middle of the cloud.

So what would this giant cloud of dust do? Well, each piece would experience a force pulling it toward the center of the cloud. It would essentially collapse in on itself. Just to get a feel for what this would look like, I built a computational model using 100 masses to represent all the dust. Here’s what it would look like:

Of course, that does not look like our solar system. The reason is that the cloud of dust that formed our solar system started off with a slight rotation. Why does that matter? In order to answer that, we have to think about what happens when an object moves in a circle.

Going in Circles

Imagine you have a ball attached to a string, and you swing it around in a circle. As the ball moves, its velocity changes direction. Since we define acceleration as the rate of change of velocity, this ball must have an acceleration. Even if it’s moving at a constant speed, it will be accelerating because of its circular motion. We call this centripetal acceleration—which literally means “center pointing,” since the direction of the acceleration vector is toward the center of the circle. See, words make sense sometimes.

We can also find the magnitude of this centripetal acceleration. It depends on both how fast the object is moving (v) as it speeds around the circle and the radius of the circle (r). However, sometimes it’s more useful to describe circular motion with angular velocity (ω).

The linear velocity (v) measures how far an object travels in a unit of time (e.g., meters per second). Angular velocity measures how much of the circle it traverses in a unit of time. How can we measure that? If you drew a line from the center of the circle to a starting point and another line to the ball’s position after one second, those two lines would define an angle. So angular velocity measures the angle that the ball covers (in radians per second). It basically tells you how fast an object rotates around a center point. With that, we get the following two definitions for centripetal acceleration (a_c).

This is important for dust particles in the early solar system. If they’re moving in a circle, then they are accelerating.

But now let’s do something a little dangerous—let’s think about fake forces. First, imagine you’re standing in a stationary elevator. There are two forces acting on you: the downward gravitational force and the upward-pushing “normal” force from the floor. Newton’s second law says the net force on an object (like you) is equal to the mass times the acceleration: F_net = ma. If the elevator is stationary, the acceleration is zero, and that means the floor has to push up with a force equal to the gravitational force so that the net force is zero.

Now suppose you push a button and the elevator starts to zoom upward. Since the acceleration is no longer zero, the floor has to push up more than gravity pulls down, in order to have a net force that is not zero. That’s all fine. Nothing fake about those forces.

But there is another way to look at this. It’s possible to describe how stuff moves with respect to the “reference frame” of the elevator. However, the elevator is accelerating, and Newton’s second law doesn’t work unless everything is measured with respect to a non-accelerating reference frame (we call this an inertial frame).

We can make things work in the non-inertial frame by adding in a fake force. The fake force would be in the opposite direction and would equal your mass multiplied by the acceleration of the elevator.

It’s a fake force since it’s not actually due to an interaction between two things—but humans like these types of forces. It’s a simple way to explain why you feel heavier in an accelerating elevator.

What if instead of a human in an elevator, you were a ball attached to a string moving around in a circle? Using the reference frame of the ball, there are two equal forces. The string pulls toward the center of the circle, and there is an outward-pushing fake force. In this case, we call the fake force the “centrifugal force,” which means “center fleeing.” It’s the fake force you experience that pushes you outward when you take a fast turn in a car. It’s just like the centripetal force but in the opposite direction. If you increase the angular velocity, the centrifugal force will increase. If you make the circle that you’re implicitly traveling around smaller (like a hairpin turn versus a long gradual curve), the centrifugal force would also increase.

A Rotating Cloud of Dust

What if we want to create a solar system from a giant cloud of dust that is slightly rotating? Let’s look at just two of these particles. One is on the equator of the rotating cloud and another is near the top, close to the axis of rotation. Both experience the same magnitude of the gravitational force, but in different directions, and they have the same angular velocity (since they are in the same rotating cloud). In the reference frame of the rotating cloud, the two particles of dust will also have a fake centrifugal force acting on them. This diagram should help:

Look at the particle on the equator. The gravitational force (F_G) is pulling it to the left, but the centrifugal force (F_C) is pushing it away to the right. If these two opposing forces balance out, the piece of dust will move around the solar system in a nice circular orbit.

But now look at the piece near the axis of rotation (the yellow line). This one is also pulled toward the center of the cloud by gravity, and it also has an outward pushing centrifugal force due to the rotation. But this time the two forces are not in opposite directions. Here the centrifugal force points away from the axis of rotation, not away from the center of the cloud. Also, this dust speck is closer to the axis of rotation (smaller value of r), so that (with the same angular velocity) the centrifugal force has a smaller magnitude than at the equator. It’s not possible for these two forces to be balanced and keep that object in the same location in the cloud. Instead, this piece will accelerate down toward the equatorial plane. Voilà!

Flattening Out

Now let’s build this into our model. I’m going to start with 100 dust particles randomly distributed in a spherical shape. They will all have the same mass and start with a velocity consistent with a single angular velocity for the whole cloud. I can calculate the gravitational force on each particle due to each other particle (yes, that’s a lot of calculations) to get a net force. With the net force, I can determine how the velocity and position will change over a short time interval. After that, I just recalculate the force and repeat the process for as long as I want. This is what we call a numerical calculation. It’s a legitimate method for solving complicated problems.

If you want ALL the details going into building this computational model, check out this video. Here’s what it looks like:

You can see in this model that the cloud does indeed flatten out, as we predicted. The masses closer to the axis of rotation, at the top and the bottom, accelerate toward the equatorial plane, forming a giant disk. But wait—when they get to the middle, they don’t stop. Those masses just move through the disk until the net gravitational force slows them down to a stop. At that point, they again start accelerating toward the disk. What we have here is a flat but oscillating solar system. That kind of motion would make life on Earth very confusing.

So, how do we fix this? The answer is that the masses need to have collisions. Suppose you have two masses at opposite poles of the rotating dust cloud. One mass is moving down and the other is moving up. It’s possible that these two masses could collide and stick together. After the resulting collision, you would have a larger mass, but the vertical momentum of the two objects would essentially cancel. So, it is these collisions that prevent that up-down oscillation for our collection of spinning balls.

If we include these type of collisions in our numerical model, we can get something that looks like this.

You can see the clumping of masses that stick together after they collide. This makes it more like our actual solar system. It’s not perfect, but it’s much flatter.

OK, let’s review. The solar system started from a slightly rotating cloud of dust. Particles near the equator found a stable circular orbit as the inward-pulling gravitational force was balanced by the outward-pushing centrifugal force. Particles near the poles had less centrifugal force, so they got pulled down toward the middle, forming a big flat disk. Those at the pole had no centrifugal force at all, so they got pulled into the center of the solar system, where the sun would eventually form.

Like I said, there’s really no such thing in physics as a “centrifugal” force; it’s just a feeling you get when you’re moving around in a circle, like on some crazy spinning carnival ride. But you know what? Sometimes we can tell a true story using fake forces.