Kepler's First Law

Hey, guys. So for this video, I want to cover Kepler's first Law in a little bit more detail because you're gonna need to know some equations involving elliptical orbits. So let's check it out. So Kepler's first Law basically said that all orbits are elliptical and that includes circular ones. So a circular orbit is really just a special case like oven lips. Kind of like a square is always a rectangle, and the sun occupies one focus. So we've got a focus point right here. The sun occupies one. There's another one right here, but we don't need to worry about that because more of a mathematical point. There's nothing actually there. And we have the center of this lips that basically cuts the thing into two axes. Now the longer one, the one I've labeled in blue is called the major access, and that's always going to be the long one. And it has a distance because we've cut this thing into equal halves. The distance of these little pieces right here has led her little A, and both of these things are little a. So that means the length of the whole entire thing is to A. And there's two specific points that happened along this orbit because as this planet is traveling, it goes in towards the sun and then goes farther away. Well, at this closest distance right here, it's called the Perihelion or Perry axis. She's always, you know, that the perry dot, dot dot always corresponds to the closest one and the variable that we used to represent that is capital AARP. Then it gets much farther away and it start its farthest point here. And that's called the appeal Ian or Apple Kapsis. So that's gonna be app something. And the way I just remember that is that a corresponds to the longest distance, the major axis. And so anything that starts with a a pillion is gonna be the longest distance away from the planet from the sun. All right, so that distance that it's at right here is called are a capital R A. So you got the variables for those capital are a couple R p and these two special distances. If we look at it, actually make up the entirety of the major axis, right, it makes up the whole entire length. So that means that that is equal to two times a. And if we do that, if we basically divide by this to weaken, solve for the length of the semi major axis, which is the most important variable in elliptical orbits, that's just our A plus. AARP divided by two. And that's basically it. There's one other axis, which is that green one here that's called the minor axis on. The only thing you need to know about that is that it's always gonna be the short axis. And instead of having a length of little A, this has a length of little B. So that means the length of the entire thing is just to be. That's all you need to know. So Kepler was studying how weird these orbits these elliptical orbits are, how non perfectly circular are they are. And there's a word for that. It's called the Eccentricity Oven orbit. It's a number between zero and one and is represented by little e. Here, and it's basically a measure of how elliptical or how weird this orbit is. So I'll give you two examples right here. So let's say I have a perfect circle like this, and I have another highly elliptical orbit like this. So this circle right here, if the sun was at its center is not weird at all. It's perfectly circular orbit. And so it has a very low amount of eccentricity. So lower eccentricities near zero are going to be very circular. Whereas this orbit over here is extremely weird, has a high amount of elliptical nous and so e lift up and so highly eccentric orbits that air very near one are going to be highly elliptical. So this would probably have a eccentricity of basically zero and just throw your number here. This would have eccentricity of 0.95 very close to one very elliptical. And so the eccentricity basically relates these appeal Ian Perihelion and the semi major axis by these equations right here. Eso I've got a one plus e and then I've got a one minus e for the peri op sys. Now, the way I like to remember this because I always get confused is again. We see A is always the longer one. So this is gonna be one where you add the eccentricity because it's gonna be bigger. Whereas Pete, you're gonna subtract it and that's basically it. So let's go ahead and take a look at some equations of our own earth. So in our own solar system, So we've got that the closest distance to the earth to the sun from the earth is this distance right here and the farthest distance is this number right here. Now we know that farthest is always going to be the appeal, Ian, whereas the closest distance is gonna be the perihelion and in this first part were asked to calculate what the semi major axis is. So let's take a look at our equations. I've got three equations that involved a but I don't have what the eccentricity is. So it means I have to use the top equation first. So let's go and do that. A equals our A plus. RP divided by two. So I got the appeal, Ian Distance 1.5 to 1 times 10 to the 11th, plus the perihelion distance 1.471 times 31st divide by two. You get the semi major axis of the earth 1.496 times 10 to the 11th. You can actually look this up on Google. It the semi major axis of the earth and you're gonna get this number right here. So now, in part be here. I'm asked to find out what the eccentricity is, so let's take a look at my equations. I have both of these special distances, right? R and R. P. And now I have this semi major axis, so I could actually use any one of these equations to find it. So I'm just gonna use the top one as an example. So I've got are a equals one, Uh, a one plus e. Now I just got to solve for the right here, so I'm just gonna go ahead and divide by a and then I get one plus e. And now, finally, if I just minus the one to the other side, I get easy equal to R A over a minus one. Now, go ahead and substitute these values in. Remember, this are a corresponds to this number right here and now I have semi major axis, which is 1.496 blah blah and you subtract one. You should get the eccentricity of our orbit, which is 00167 So this actually a number that's very, very close to zero, which means that Earth's orbit around the sun is very nearly circular. This actually goes for most of the planets in our solar system. It's pretty cool. Let me know if you guys have any question.