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Geosynchronous Orbits

Patrick Ford
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Alright, guys. So in this video, I want to talk about a special kind of satellite orbit that you might see in some problems called synchronous orbit. So let's go ahead and check it out. So imagine we had a satellite above the Earth and it was at some orbital distance, are right, and as it's in its orbit, it's traveling around with some orbital period t sat. We've got a bunch of equations here that'll give us tea Sat. What? What happens is this thing is orbiting around the Earth, but the earth is also spinning, and the rotation period of the earth is called T earth. That's the amount of time that it takes for it to spin. Once the idea of a synchronous orbit is that there's a special distance called our synchronous such that the satellite's orbital period T sat will synchronize with the Earth's rotation so that T SATs is equal to t earth. Now, By the way, this equation works for any planet, not just Earth on DSO. What happens is if you would actually go outside and look at a satellite at this specific distance. Then, as the Earth is rotating the satellite is constantly above the same place because the orbit period off the satellite matches the rotation period of the earth. So they're constantly synchronized. So this thing would basically appear above the same place at all times. And so sometimes, instead of synchronous orbit, you might see stationary orbit. We use these all the time in telecommunications because we want satellites to peer above the same place at all times. So this special distance here are synchronous has an equation. It's the cube root of G m t squared, divided by four pi squared. Now, I just want to remind you that this t, um is just the t of the planets and we can actually use it in any planet, not just the earth. We can actually see where this equation comes from. Pretty quickly from this T squared equation that we have, we have t squared equals four pi squared R cubed, divided by gm. So we just want to solve for this are right here. So move everything to the other side We get g m t squared divided by four pi squared equals r cubed and I have to do is just take the cube root So then you'll just get to this equation right here, right? Cool. So I want to point out, is that this is the Onley distance are where circular geosynchronous orbit. It's possible because what happens is if you were to increase, this are then your tea would increase right here, and it wouldn't be in sync with the Earth's rotation anymore. So there's only one specific distance here that's possible. And you could also make sense of that because all of these letters are capital. Letters are all big letters are all constants. So this is just a constant right here, all right, It's basically it. Let's go ahead and actually solve for what the height of earth geosynchronous orbit is. So if we want to figure out what the height is, we're trying to figure out what H is equal to all right, But we know forever solving for H first, we have to go ahead and solve for little our first, and then we can use big R plus h to solve for that. Now what am I using? I'm using the synchronous orbit equation, which is right here, So let's go ahead and write that out. I've got our synchronous equals the cube roots of G and I'm gonna use the mass of the earth. And then I've got t of the earth squared, divided by four pi squared. OK, well, what is that? T Earth? What is the rotation? Period of the earth? Well, we've got that. We've got all of these other letters we got G and m e. The rotation period of the earth is the amount of time that it takes to spin once once you should recognize that as Earth Day, that's 24 hours, right? Takes 24 hours for the Earth to spend once. But we want that in seconds. So we're just gonna have to multiply that by 3600 and we get that that's equal to 86,400. So now I just plug everything that we should give. Our synchronous is so our synchronous is equal to the Cube roots of a whole bunch of letters and numbers. 6.67 times 10 to the minus 11. And we got 5 97 times 10 to the 24 and then we've got to do 86,400. But we gotta square that and now we just divide it by four pi squared. Just make sure that you plug all of this stuff carefully into a calculator. Four pi squared should be in a little a parentheses. And what you should get is you should get 4.22 times 10 to the seventh. But we're not quite done yet because again, we've Onley starved for our synchronous We want the heights. So now to do that at last step, we just have to get a check is equal to r minus. Are we Get that from this equation right here. And you should get 4.22 times 10 to the seventh, minus the radius of the earth, which is 6.37 times 10 to the sixth. And if you do that, you should get 3.59 times, 10 to the seventh, and that's in meters. So if you actually put this into kilometers, you're gonna get 35,900 kilometers. You could actually google this the height of a stationary orbit. You're gonna find out it is 35,900 kilometers. That's how far it needs to be so that this satellite orbits every day, just like the Earth spins. Let me know if you guys have any questions with this.