Sep 13, 2019 1:00 PM

Super Planetary-Motion Smackdown: Kepler v. Newton

In science, progress is all about building a better model—explaining more with less.

Science is always an unfinished project. That’s what makes it so much fun. The process—collecting data, building models to explain how the world works, and then dethroning them with new models—is full of spills and thrills. But perhaps the very best stories come from astronomy. So let’s look at part of that tale, the chapter where Isaac Newton got over on Johannes Kepler.

Of course, you first need the backstory. The ancient Greeks studied the earth and the sky, but their basic model had all the objects (sun, moon, and planets) moving in circles around us. Later, Nicolaus Copernicus said, "Hey, if you put the sun in the center, then you can explain this weird motion of Mars." After that, in the early 1600s, Kepler came up with his model for planetary motion. There was lots of fighting and crying in the middle of this, but I will leave that up to your imagination.

Kepler's model has three main ideas. (These are usually presented as "Kepler's three laws of planetary motion," but taking them together, it’s really just a model.)

Planets orbit the sun in elliptical (not circular) paths.
As a planet gets closer to the sun, it moves faster.
The orbital period (T ) is related to the orbital distance (a) by the expression T ² = a ³ (where T is measured in years and a is measured in units of the Earth-sun distance).

A couple of comments: First, this model is just based on the observational evidence available at the time—but it fit the data quite well. That was no easy task. Imagine just trying to plot the orbits of the planets. You’d do that by observing their location in the sky over the course of years. But then you had to account for the fact that the spot you were measuring from was also spinning through space.

There is another important thing to notice. The relationship between period and orbital distance gives a "1 = 1" equation for Earth. It takes Earth one year to orbit the sun, and it has an orbital distance of 1 AU (astronomical unit—distance from Earth to sun). It wasn't until much later that someone was able to actually determine the distance from Earth to the sun. This is crazy if you think about it.

Just so we’re all on the same page, here is a numerical model using Kepler's laws for some random planet orbiting the sun. It's just a gif below, but here is the code if you want to see it.

This is the best model of planetary motion we had before Newton. And, really, it's a fine model. You could even use it to find some new object orbiting the sun or to model the motion of a comet. But could it be more general? Is there a more fundamental model that could explain both the motion of a planet orbiting the sun and the motion of the moon orbiting Earth? Maybe even one that could also explain the motion of an apple falling from a tree?

OK, the legend of Newton’s apple incident may or may not be true, but that doesn't matter. Basically, he wondered if the same force that makes things like apples fall down instead of up could also be what caused the moon to orbit Earth. It might have seemed like a crazy question, since a falling apple has no obvious similarities to a moon. But Newton managed to create a model for gravity that works pretty much everywhere. That's why it's commonly called the Universal Law of Gravity. Here's how it works:

Suppose I have two masses (m₁ and m₂ ) that are some distance (r ) apart, like this:

Illustration: Rhett Allain

You can see that there is an attractive interaction between them. The force that m₁ exerts on m₂ (F₁₂) has the same magnitude (but opposite direction) as the force that m₂ exerts on m₁ (F₂₁). The magnitude of this interaction can be found with the following expression:

Illustration: Rhett Allain

The key here is the "inverse square" nature of the force. If you double the distance r between two objects, the magnitude of the force declines by a factor of 4 (because that's 2 squared). But what about that G? That is the universal gravitational constant. It has a value of about 6.67 x 10^-11 Nm²/kg². Although it's pretty important, Newton didn't actually know the value of this constant.

So how did Newton’s model work? How could it explain falling fruit and at the same time satisfy Kepler's planetary orbit model? Let's do this. I'm going to use the gravitational model to check Kepler's model. It's possible to do this on paper (an analytical solution), but that can get pretty messy. Instead, I’m going to use a method that wasn’t available to Newton: numerical calculation. This works by breaking the motion of a planet into short time intervals. During these short intervals, we can assume the gravitational force is constant (both in direction and magnitude) and use this constant force to update the velocity and position. Then we just repeat the same process for the next interval, and the next, and so on. With a computer, it's really not too hard. Of course, we need the relationship between force (F ) and acceleration (a ):

Illustration: Rhett Allain

I’m using the standard symbol a for acceleration; just to be clear, it’s not the same a as in Kepler’s laws, above. Those arrow symbols? They mean the variables are vectors, not single numbers. (If the word “vector” freaks you out, just pretend I didn't say it. You can still easily follow the math here.) Using that equation, I can find the acceleration of the planet. Then, with the acceleration, I can find the change in velocity, v. (The Greek letter Δ means "change in.")

Illustration: Rhett Allain

Finally, using the velocity I can find the new position of the planet:

Illustration: Rhett Allain

It might seem strange, but it's fairly common to use the distance symbol, r, for position. However, there’s a problem with this last expression. It uses the velocity of the object, which I just updated. So I'm technically using the velocity at the end of the time interval—and this is wrong. But it's only "sort of wrong." If the time interval is small enough, the error doesn't cause a problem. Oh, and by "small time interval" I mean something like an hour; we’re not talking about microseconds here. That won’t work for earthbound modeling, but we’re talking about huge distances in astrophysics. Planets don’t move so much (relatively speaking) in an hour that the force changes.

So that’s the basic idea of numerical calculation. Now you can see how I implement it to plot the trajectory of an orbiting planet. Click the Play button to run the simulation. This is actual code. You can click the pencil icon to see it, and I've put some comments in there to suggest things you could change for fun. Go crazy, see how you change the universe. You can't break anything (at least not permanently).

Try changing the starting position of the planet (line 12) and the starting velocity (line 21). What happens? I've dramatically enlarged the size of both the planet and the Sun so you can see them.

What about Kepler? Right away, it should be at least plausible that the trajectory of the planet is an ellipse. Yes, you can get a circular orbit, but you would need to either change the starting velocity or the starting position. (I put a hint in the code.) That's good enough for Kepler's first law.

The second law isn't too bad. Again, you should be able to see that the planet increases in speed as it gets near the sun. Here is a plot of the magnitude of the planet's velocity as a function of orbital distance. You can see that for lower orbital distances, it is indeed faster.

Now, if you’ve studied Kepler’s laws, you might raise an objection here: "What about the equal areas in equal times?" Yes, the most common way of stating Kepler's second law is that a planet will "sweep out" the same area in a given amount of time no matter where it is in its orbit. When it's closer to the sun, it has a small orbital radius but moves faster. The "wedge" it sweeps out will be wide and short. But this wedge will have the same area as when the planet is far away—where it will have a long skinny wedge. If you want to calculate areas, go ahead. I like my plot of speed vs. orbital distance.

The last part of Kepler's model is the relationship between orbital period and orbital distance. OK, again you caught me cheating a little bit. How do you find the orbital distance for a planet that isn't moving in a circle? There are several methods, but I'm going with the easiest. I'm going to plot a trajectory of the planet's path and then just measure the distance from the center to the "skinny" side of the ellipse. This is called the semi-major orbital axis. (In general, if you measure the diameter of the ellipse in the long direction—along the “major axis”—the semi-major axis is half of that.)

I can also get the orbital period by just looking at the simulation time at the point where the planet gets back to where it started. That means I can create a few different planets with different orbits to get this plot:

Here you can see a plot of the orbital period squared (in units of years) vs. the semi-major axis cubed (in units of AU). The data isn't perfect, because I just roughly measured the semi-major axis, but you can see that this is a linear function. More important, the slope of the linear fit is 1. That means that using the Newtonian gravitational model, I do indeed get Kepler's third law.

Wait! There's one more thing to check. Does Newton's gravitational model work with falling apples? If an apple falls from a tree, it will speed up as it moves downward. The acceleration of this falling apple will be –9.8 m/s² if it's near the surface of the Earth. Let's do this with a numerical calculation. I'm going to use the universal gravitational model with the apple starting 2 meters above the ground. Here is the code, and here is what I get:

Illustration: Rhett Allain

So there you have it. Kepler started with a very basic model to map out the motions of the planets. Newton took the next step and built a much more general model of gravity. Although Newton's model of gravity is awesome, it still had to agree with the existing data for planetary motion and falling apples. So, is Newton correct? Who knows? Science is about building models. If you have another model of the gravitational interaction—that's cool, but it can't contradict the old stuff.

Old Isaac wasn't known for his humility—and why should he be? He's probably the greatest scientist and mathematician of all time. But even he had this to say, in a letter to Robert Hooke in 1675: "If I have seen further, it is by standing upon the shoulders of giants."