Hey, guys. So now we know a little bit more about elliptical orbits, we're going to see how we use energy conservation to transition from a circular to an elliptical orbit. Let's check it out. Now the idea is that we're still changing from an orbit. In this case, a circular to an elliptical one, so some work has to be done. And by the way, if you're going from an elliptical to a circular, still some work to be done. So everything I say can be done backwards here as well. Now we're still talking about changing orbits, changing works, so we're going to use energy conservation. Now when we did this for circular orbits, we were able to combine these two kinetic and potential terms into a total amount of energy. So, this simplified our process. Now, the way this worked for circular orbits was that all we needed was this distance here of little r of our circular orbit, and we use this equation, gmm/2r. Now we already know that this equation doesn't work because what happens is that as this orbit or as this object travels in its orbit, it goes from the periapsis distance, which is the closest that it gets, to the apoapsis, which is the farthest. So what happens is as it travels throughout its orbit, this distance of little r is constantly changing. Alright? So we're going to need a new equation to work with this. Now, how have we done this before? We've always taken an elliptical orbit and broken it up its axis, the minor and the major axis. And we always said that the most important variable of an elliptical orbit is the semi-major axis, which is given by the letter a. Now, this is not the same thing as the apoapsis, which is the farthest distance over here. So don't get those two things confused. But all we did was that any time we had an equation that went from circular to elliptical, we just replaced the little r with a, and it's going to work the exact same way here. So we take this equation and we replace it with negative gmm/2a. So that means that these equations over here are going to be the equations that you plug into your energy conservation here. You just have to make sure that you understand that you're going from a circular to an elliptical, which one is your initial and which one's your final, and then plug in the appropriate expressions for those. And to see how that works, we're going to go ahead and do this quick example right over here.
So we've got a spacecraft and we're told what the mass of this thing is. We're told it's initially in a circular orbit, and we're going to be calculating the energy required to change it to an elliptical orbit and we're told some characteristics about that. So let's go ahead and just draw a quick little diagram right here. So I've got the Earth and I've got the initial circular orbit like that. Now, let's try it again. There we go. That's a circular orbit. And what happens is we're going to transition to an elliptical orbit over here. So we have the initial distance of this circular orbit, little r, and then we're told what the apogee distance is. Again, remember that is this farthest distance here, not the semi-major axis. Okay? So this is the apogee and we need to know how much work is done. So or how much, how much energy is required to do that. But it's also the same thing as the amount of work. So let's go ahead and set up our energy conservation equations. So we got initial energy plus the work is equal to the final energy. Now, we're going from a circular orbit all the way to an elliptical orbit, so we just have to make sure to plug in the correct equations for those. The total energy of a circular orbit is negative gmm/2r plus the work is equal to now negative gmm/2a. So we're we're looking for the amount of work that's involved. So all I have to do is just take this term and move it to the other side and just have to, plus it because I've got a minus sign right here. So I've got the work is equal to I've got gmm/2r and it's positive, minus gmm/2a. Now let's look through my variables, make sure I've got everything. So I've got all of these letters here are just constants. I can look them up in this table over here, and I have the mass of the spacecraft. Now I'm told what the r distance is. That's this 6.87×106. Now, the only other thing I'm told is the apogee distance, But I need to find out what little a is. That's the semi-major axis. Remember, a is not the same thing as the apoapsis distance. So I need another equation to figure out what a is. Let's go ahead and look at my equations over here. So I've got, this t2=4πa2 blah blah. I've got Kepler's 3rd law for elliptical orbits. Now I've got a, I've got g and m, but I don't have any information about the period of the satellite, so I can't use that. Both of these equations over here involve a, but they also related to the eccentricity of the orbit, which I also don't have. So I can't be using these equations either. So I'm going to use this equation over here, which is my apoapsis and periapsis distance divided by 2. So I've got ra+rp/2. Now the good thing is that I have the apogee distance. I'm just told what that is as a number over here. Now what about this periapsis distance? What is that value? Well, let's take a look at my diagram. Right? So I know that I've got the I've got the semi-major axis over here, which I'm trying to find. So that's going to be this piece over here. So that's a. And the apo sorry, the periapsis, which is going to be the closest distance, is going to be this piece over here. But notice how these two distance are actually the same. So this actually becomes a really really important point that you need to know. And that when you're going from a circular orbit like we did over here to an elliptical orbit, what happens is that your little r actually becomes r p or r a, and it depends on the size of your new orbit. So let me give you a couple of rules to follow. Now just like we did with this problem, if your new orbit is larger, then that means that your little r becomes r p, just like it did for this problem over here. Right? So these two things are the same exact variable. Now, if your new orbit were smaller, then it's the opposite. Your little r, which just becomes r a. So that means that we actually know what r p is. We can just go ahead and solve that. So r a is the apogee distance 2.0×107 plus, and then we've got 6.87×106, and we're going to divide that by 2. So that means that a is equal to 1.34×107, and that's meters. We're going to plug that in there for our equation. So now we can figure out what the work is. The work is just equal to Now what we can do is we can take out this gmm/2 as a common factor. So gmm/2. Now we've got one over and the r is going to be 6.87×106 minus 1 over I've got 1.34×107. So if you work all of this out in your calculator, you're going to get the work done is equal to 1.41×1011 in joules. So this is our final answer for the amount of work. And because it's positive, that makes sense because we ended up with a larger orbit. So all of those rules for orbit still make sense, in terms of work and positives and negatives. Now the last thing I want to mention is that we went in this example from a circular to an elliptical orbit. So you might be wondering what happens when you do it backwards. Well, when you're going from elliptical to circular, all the all the rules are backwards. So we've got this elliptical orbit over here and imagine we're going sort of in the in the clockwise direction. Now if we wanted a larger orbit like for a, we'd actually have to do a burn over here, so we'd have to increase our velocity in order to basically circularize that orbit. So if you take a look at the diagram here in our new in our old elliptical orbit, this distance over here is actually our apoapsis distance, and that becomes the radius or the distance of our new circular orbit. So if you're going from an elliptical to a circular, if your new orbit is larger, then that means that r a becomes your r. And if it's smaller, then r p will become the new r. So you'd actually have to do a burner over here. You'd have some delta v, and this is your periapsis distance that would become your new orbital distance in a circular orbit. Alright. So all the rules are just backwards. Alright, guys. That's it for this one. Let me know if you guys have any questions.