Hey, everyone. So in this video, we're going to cover Kepler's first law in a little bit more detail because there are some important equations you need to know about elliptical orbits. So let's go ahead and check this out and we'll do a quick example. Alright? So, Kepler's first law basically just says that all orbits are elliptical, even the circular ones. It's kind of just a special case of an ellipse, kind of like a square is always a rectangle. Right? And the sun occupies one of the foci points. So there are 2 foci. The sun is one of them. And then there's nothing physical at the other focus point. It's just sort of a mathematical point right there. So, the center of this ellipse is basically divided into 2 axes. Now, the major axis is always going to be the long one here, and this long axis is cut into sort of 2 halves. And each of these halves here has a number, has a length, which is going to be a, which means that the whole major axis has a length of 2a. Now, there are 2 important points that happen. As the Earth gets closer in its orbit, it approaches the closest distance here. This is called the periapsis or perihelion here. So this peri means the closest axis or closest distance here. And one way I like to remember that is the word perimeter means something that's relatively close by, so that's perihelion. Alright? Now, this length here is going to be RP. That's the variable that we use for that. Then, what happens is as it goes farther out, the opposite happens, and it gets to its farthest distance, which is going to be over here, and this is called the aphelion or the apoapsis. So, anything that starts with "ap" means it's the farthest away. This distance over here is going to be given by the letter RA. And what you'll notice here is that these two lengths combine to represent the entire length of the major axis. So, in other words, RA+RP=2a. Right? So these are both the same exact thing. Now if you rearrange this equation, you can actually come up with an expression for a, which is called the semi-major axis, and it's the most important variable that you need to know for elliptical orbits. It's basically just that the RA+RP÷2. Alright? Now the only other thing you need to know is that the minor axis is always going to be the shorter axis, so it's always going to be the shorter one. And, basically, it has a length of 2b, so it cuts it up into 2 equal sides of length b. That's really all you need to know. Not a whole lot of calculations there.

Alright. So, what Kepler was studying is he was studying how elliptical or circular orbits in our solar system were, and he came up with a word or a way to measure it called the eccentricity. It's given by the letter little e. Remember, it's just a number. It's usually a decimal between 0 and 1, and it's a measurement of how elliptical or how weird the orbits are, so how weird orbits. So, in other words, lower numbers that are very near 0 are going to be nearly circular. So an example of this is like the moon orbiting the Earth or most of the planets in our solar system. Basically, they're very circular orbits. So this would correspond to an eccentricity very close to 0. Alright? On the other hand, a very high number, very near 1, is going to be a very elliptical or very weird orbit. A good example of this is comets in our solar system. Comets have very weird strange orbits where they get very close to the sun and they get very far away, and it takes them a long time to get out there. Alright? So this I'm just going to make up a number. It would probably be an eccentricity that's like 0.95 or something like that. Alright? So, in this sense, this is a very elliptical weird orbit.

Now, this eccentricity here basically relates the aphelion and perihelion, basically, the closest and farthest distances with the semimajor axis from, that's the variable little a. I'm just going to go ahead and give you the equations. This is going to be a(1+e), and this is going to be a(1-e). One way I like to remember this is that Rp, p means the longest axis, longest distance, is going to be, is going to happen when you add the eccentricity. The periapsis is going to happen when you subtract the eccentricity. You're going to get a smaller number. Okay? Basically, what this means here is that if you have these two distances, right, your RP and your RA, you'll notice that in circular orbits, these two things are very nearly the same. So, in other words, if you have your major axis and both these RA, what you'll notice is that if you have a 0 eccentricity orbit, then basically this number will be 0, and this just means that both of your RA and RP will just be a. Right? So that's kind of one special quirk of those equations.

Alright, folks. So that's it for this one. Let's go ahead and just go ahead and get started with some real numbers in our solar system. So we have Earth's closest distance to the sun is this number. Remember, closest distance means this is the perihelion. Alright? And then the farthest distance is going to be this number over here. Remember, this means the aphelion. Alright? So we want to calculate the semi-major axis in the first part. Remember, that's going to be that variable little a. So how do we do that? Well, a appears in all of these three equations here, but you will notice that the bottom 2 involve the eccentricity and we actually don't have what that number is. We're going to be calculating that in part B. So instead, what I'm going to use in this first part is I'm going to use this equation over here. So a is just equal to the farthest and closest distances divided by 2. Alright? So this is going to be 1.521×1011+1.471×1011÷2. Now, one of the things you notice about these two numbers is they both have the same exponent. They're both times 1011, and the numbers themselves are actually very close to each other. Right? So the closest and farthest distance aren't a whole lot different. So if you go ahead and work this out, what you're going to get is 1.496×1011. That's going to be the semimajor axis, and you can actually look this up in a table, in your textbook, or something like that, and you should get something that's pretty close to this number.

Alright? So now let's move on to the eccentricity. We're going to calculate this e here. And now that we actually have what this a value is, we can actually use either one of these equations. It doesn't matter. You're going to get the right answer either way. I'm going to go ahead and just use, ra. So this is going to be ra=a(1+e). I've got what this number is. This is going to be the 1.521×1011. If you divide over the semimajor axis, what you'll get is that you'll get divided by 1.496×1011, and this is going to equal 1+e. Now, one shortcut to use, by the way, is if you notice that these things have the same exponent, you can actually skip plugging them into your calculator. You don't have to do it because they kind of just sort of cancel each other out. Right? So that's kind of one way to make it a little easier on yourself. And what you get out of this is you're going to get 1.017=1+e. So when you work this out, what you're going to get is that the e is equal to 0.017. Notice how this number here is very small. It's very near 0. So what that means is that Earth's orbit around the sun is very nearly circular, and this is actually true. This is true of most planets in our solar system. So, basically, this means that as the Earth goes around the sun, the difference between the farthest and closest distance is very, very tiny. That's why these numbers here are very close to each other. Alright. So that's it for this one, guys. Let me know if you have any questions.