Orbital motion | Physics | Khan Academy

If a satellite has just the right velocity, then we can make sure that the force of gravity will always stay perpendicular to that velocity vector. In that case, the satellite will go in a perfect circular orbit, because the gravitational force will act like a centripetal force. We've seen this before in our previous videos. Now, the goal of this video is to use insights from this and understand how planetary orbits work and talk about Kepler's laws.

So, let's begin. First of all, we can model our Earth and Sun to be very similar to that satellite. If the Earth has just the right velocity, then the force of gravity due to the Sun on the Earth will now act as a central force, making sure the Earth goes in a perfect circular path. But we can have a couple of questions over here. Our first question over here is: If the Sun puts a force on the Earth, does the Earth put a force back on the Sun? What do you think?

The answer is yes. Newton's third law states equal and opposite forces, right? So, shouldn't the Sun also accelerate? Well, it does, but because the Sun is so much more massive compared to the Earth, its acceleration is very tiny. But it's there, and because of that, the Sun would actually wobble. So, if I were to exaggerate that a little bit, it would look somewhat like this. Look at that; the Sun is wobbling a little bit as the Earth would go around it. But this is exaggerated because the Sun's mass is so much that its wobbling is so tiny that we can completely neglect it.

So, while modeling this, we can completely neglect the wobbling of the Sun, and we can assume that it's only the Earth that's going around in the orbit. Okay, but that now brings us to the next question. This only works if the Earth has that particular specific velocity. But now what happens if it doesn't? What happens if it has slightly bigger velocity or slightly smaller velocity?

Well, then you can kind of see if it has a bigger velocity, it'll go away from this circle. The path will look somewhat like this. If it has a smaller velocity, the path's going to look somewhat like that. In general, the orbit will not be a circle; instead, it will look like this. This is called an ellipse. You can think of an ellipse as a sort of squished-out circle.

But if you're wondering if there's a slightly better definition of an ellipse, I'm glad you asked. In fact, we can have some fun by actually drawing an ellipse. To do that, take a couple of thumbtacks and then put a thread around it. Put your pen or a pencil and draw that shape in such a way that the thread always stays tight. The shape that you get is an ellipse. If you keep the thumbtacks farther away, it becomes more elliptical, more squished out.

On the other hand, if you keep the thumbtacks closer to each other, it'll look more like a circle. Now, which means you can see that a circle is basically a special case of an ellipse, a case in which the two thumbtacks are pretty much at the same point. The more you squish it out, the more elliptical it gets. Now, there's a particular mathematical term that we introduce over here to talk about how elliptical, how squished out it is. We call that eccentricity. It's a number between 0 to 1. If it's a circular path, then we say it's not squished out at all, and we say its eccentricity is zero.

On the other hand, if you squish it out so much that it almost looks like a straight line, then its eccentricity is one. By the way, these two thumbtack points, well, they mathematically are called the foci; "foci" is plural, "focus" is singular. This is one focus; this is the other focus of the ellipse. When it comes to a circle, both the foci are at the same point, which becomes the center of the circle.

Okay, now the cool thing about our orbits is that it turns out that the Sun will always be at one of the foci. It will not be at the center of the ellipse, but it'll be at one of the foci of the ellipse. The first time I heard of this, I was like, "Wait a second! Is this the reason why we have seasons? When the Earth is closer to the Sun, we get summer, and when it goes farther away, we get winter?" Makes sense, right?

Well, actually, no, that's a big misconception. Because if that were true, then everybody on the planet would get summer and winter at the same time, but that doesn't happen, right? The reason it doesn't happen is because this is a highly exaggerated figure. It turns out that the orbits of most planets around the Sun actually have an eccentricity very close to zero, so they have almost circular orbits.

Okay, so the distance of the Earth from the Sun pretty much stays the same. The reason for the seasons has something to do with the tilt of the Earth's rotational axis. This we'll not talk much about that over here.

Anyways, putting it all together, we can now write down Kepler's first law, which says that all planets revolve around the Sun in an elliptical orbit with the Sun at one of the foci. It's one of the three laws that we're going to study in this video made by Johannes Kepler, who spent years recording the positions of the planets. Again, remember, most planets have very circular orbits, but there are other things, like for example, comets. They too orbit around the Sun, and they have highly elliptical, highly eccentric elliptical orbits.

Okay, moving on to the next part. Here's the next question for us: Can we now comment about what happens to the speed of the planet as it goes around the Sun? If it was a circular orbit, then we know the speed remains the same. But what about over here?

Well, one way is to draw the velocity and force vectors. So here are the velocity vectors. The velocity vectors are tangential, but I don't know the magnitude of the velocity, the speed. I don't know; that's what I'm trying to figure out. So don't concentrate on how big the vectors are. I've just drawn them randomly. But I also know the force vectors. The force will always be directed towards the Sun, and when you're closer to the Sun, you'll have a much larger force; and when you're farther away from the Sun, the force weakens.

Okay, okay. So just from this, how can we comment on the speed of the planet? Well, here's how I like to think about it. If I were to focus at this position, what I would do is drop a perpendicular to my velocity vector. The reason I do that is that I remember if a force is perpendicular to the velocity vector, it will not change the speed of the planet or the object; it will only make it curve. But here we see that the force is slightly tilted backwards. That means it's slightly pulling it back.

Therefore, because the force is slightly pulling it back, I know that this is going to slow down. So, right at this point, the force will slow down a little bit. We can do the same analysis everywhere, and we can comment on what happens to the speed. So it'll be a great idea to pause the video and see if you can do the analysis yourself.

All right, again, I'm going to drop a perpendicular to my velocity vector over here, and I see that my force is tilted slightly behind, slightly back, and therefore it's again going to slow down. What happens over here? Hey, here it's exactly perpendicular. So at this position, it neither slows down nor speeds it up.

What happens over here? Hey, here you can see it's the other way around. The force is tilted, you know, slightly forward. Can you see that? Therefore, it's now going to speed it up. What about here? Hey, it's again similar. The force is slightly tilted in the forward direction, speeding it up. And finally, over here, again, it's perfectly perpendicular.

So look at this from here. I know there's not going to be any change. So if I start from here, it slows down, slows down, slows down; it goes slower, slower, slower, and then it speeds up, speeds up, speeds up, and then again slows down.

Oh! That means that I should get the fastest over here because it speeds up, speeds up, fastest. Slows down, slows down, slows down, slows down; and the slowest over here. So if you could see it, it would look somewhat like this. It slows down, slows, slows, slows, slowest, and then increases speed like this and so on and so forth. So now we can draw the correct velocity vectors. You'll have the biggest vector over here, become smaller, smallest vector over here becomes bigger, and so on and so forth.

Okay, so here's a question to check our understanding. All right, so in this orbit, let's consider a point when the Earth is somewhere over here, and let's wait for some time—let's say about three months—for the Earth to come from here to here. Okay, then again we'll wait for some more time, and let's say we'll again look at when the Earth is over here, and we'll wait for three months.

Now, my question to you is: Now if we wait for another three months, the distance traveled by the Earth over here in the orbit, will it be the same as over here, more than over here, less than over here? Why don't you pause the video and think about that?

All right, we just saw that far away, the speeds are lower and closer to the Sun, the speeds are much higher. Therefore, over here, because the Earth is traveling much faster, we should expect it to cover more distance in that same amount of time, right?

Okay, now here's the cool thing. If we were to join these lines and find this area, we will find that this area is exactly the same as the area over here. We can get some intuition behind it because although this length is small over here, you can see this length is bigger over here. This length is small, but this length is bigger, so it all just works out in a beautiful way that the area stays the same.

And this now brings us to Kepler's second law, which says that a line joining the planet and the Sun, like this line over here, sweeps equal areas in equal time intervals. The three months was just an example; it doesn't matter what time interval you take, but as long as you take equal time intervals, you will find that regardless of where you take those timing intervals in the orbit, as long as it's equal, the area swept by this line will always stay the same.

But I'm sure you're wondering why does it work out that way? Without getting into all the proofs and stuff, there are some beautiful geometrical ways to think about it. But the main reason why this works out this way is because the force is directed towards the Sun. Turns out that whenever the force is directed at a specific point, you will always get this to be true. In fact, this is how Newton was able to use Kepler's second law to realize that the force on all the planets must be towards the Sun, and that actually helped him establish his law of universal gravitation.

Kepler's laws came before Newton's laws of gravitation.

Anyways, on to the final law, and for this, now we'll zoom out and start looking at all the planets. Now, since we've seen that most planets have circular orbits, we'll just stick to circular orbits now. Okay, this time we want to look at how long different planets will take to complete their orbit.

Again, we can try to think about this slightly intuitively. So if you consider a planet which is very close to the Sun, the force of gravity acting on it would be very large, and therefore the acceleration due to gravity would also be very large, right? So if you want to have a perfect circular orbit to make sure that it misses the Sun, then it just goes in a perfect circular orbit; we would need pretty high velocity.

So over here, when you're close to the Sun, you need very high velocity. But what happens if you go slightly farther away from the Sun? Well, then the force of gravity on this planet would be weaker. Remember the inverse square law: the force weakens as you go farther away, and therefore the acceleration due to gravity over here would also be much weaker than over here.

Now immediately you might say, "Well, wait a second, Mahesh, what about the mass of the object? Doesn't the mass also matter?" Well, no. Remember one of the cool things about gravity is that the acceleration that you get due to gravity is independent of the mass of the planets. Over here, it does depend on the mass of the Sun, of course, but not on the mass of the planets. We've seen that before, just like how when you drop a bowling ball and a feather, both of them have the same acceleration due to gravity on the Earth, right? Same is the case over here.

Okay, okay. Anyway, since the acceleration due to gravity is very weak over here, it requires a much smaller velocity to stay in a circular orbit because this force will not curve it that much; the acceleration is very little. Based on this piece of information, can you think about which of these two will take more time to complete their orbits?

All right, let's see. We know this is faster, and it's traveling a smaller distance. This is slower, and it's traveling a bigger distance. So clearly, this should take longer, isn't it? If you could see it, it would look somewhat like this. Clearly, this would take much longer. And the farther you go, the longer it will take to finish that orbit.

This finally brings us to Kepler's third law. I say third law because Kepler's third law is actually mathematical. We're not going to get into the math part of it, but the key part of that law is that the farther the planet is from the Sun, the longer its time period.

And there you have it—the Kepler's three laws that govern how the planets go around the Sun. Kepler's and Newton's laws are incredible at predicting any orbits, not just in the Sun and our solar system but any orbits in general. Unless things are very massive—if you're dealing with things like black holes and neutron stars and stuff—well, now these laws will not work anymore. We will need a much more accurate description of gravity that is given by the general theory of relativity.

But as long as you're not considering these extreme cases, Kepler's laws and Newton's laws are going to be super, super useful for us.