In March 2006, two small satellites were discovered orbiting Pluto, one at a distance of 48,000 km and the other at 64,000 km. Pluto already was known to have a large satellite Charon, orbiting at 19,600 km with an orbital period of 6.39 days. Assuming that the satellites do not affect each other, find the orbital periods of the two small satellites without using the mass of Pluto

Video transcript

Welcome back everybody. We are looking at kepler 36. Now around kepler 36. There are two highly close planets called kepler 36 B. And kepler 36 C. Let me go ahead and label these just real quick just to make sure that we do don't get lost here because we are going to be comparing kind of the distances and the orbits of all the different planets here. Right. So we are told a couple of different things. Looking at kepler 36 B. Here we are told that its orbit around is about 13.9 days. That is one period for it and the distance between 36 B. And 36 is 0.1153 astronomical units. Now let's go to 36 C. Here we are told that the period is 16.2 days. And we are tasked with finding What it what the distance is from 36 c. to 36. Wonderful. Okay, so we are looking at elliptical circular orbit here and we know that The orbit of any two given planets or or an orbiting body around another body is going to be equal to two pi Are to the 3/2ves all over the square root of Newton's gravitational constant times the mass of the planet orbits. Now we are tasked with one more of course with finding this R. C. But we have to do it without using this big end here. So we have to figure out how to get rid of that big m which is the mass of our K. 36 planet that the other two bodies are orbiting around with this formula right here, let's create other two formulas for the other two planets. Right? So we are going to have that the orbit of 36 B. Its period is going to be equal to two pi times the distance of B. From 36 to the three halves, all over the square root of G times the mass of big M, dividing both sides by this distance here we get that period divided by the distance between the two to the three halves is equal to two pi over the square root of G times M. Well gee we know is the gravitational constant. M. Is just the mass of the big planet. So that's not going to change. So this is just equal to some constant. C. You'll see how that comes into effect in just a second here, let's go and set up our second formula. So we have that T. C. Is equal to. Well, we're gonna use this same logic here except just replace all the bees with CS. Right? So we have T. C. Over the distance between C. To the three halves is equal to two pi over the square root of G. M. Which is once again equal to that constant. Since both of these equations are equal to that constant, let's set these equations equal to one another. So we have the T C over r C. Two the three halves is equal to E. B. Over R. B. To the three halves. Now we are looking for our C. And we're not doing it in terms of the mass. So we've already achieved the first half of our goal here. I'm going to divide both sides by TB and then take uh the power I guess raise each side to the power of two thirds to get rid of this three halves right here, right? This is going to give us that T. C over T. B. Raised to the two thirds power is equal to R. C over R. B. Which then multiplying both sides by R. B. We get that are desired. R C. Is equal to T. C over TB. All raised to start here two thirds times R. B. We know all of the values of those variables. So let's just go ahead and plug them in. We have the R. C. Is equal to T. C. Which is 16.2 days divided by T B. Which is 13.9 days are of two thirds times our distance between 36 B. And 36 which is 0.1153 astronomical units. Giving us a final answer of 0.12 astronomical units corresponding to answer choice. A Thank you all so much for watching. Hope this video helped. We will see you all in the next one