A new planet is discovered orbiting the star Vega in a circular orbit. The planet takes 55 earth years to complete one orbit around the star. Vega's mass is 2.1 times the sun's mass. What is the radius of the planet's orbit? Give your answer as a multiple of the radius of the earth's orbit.

Video transcript

Hey, everyone. So this problem is working with Kepler's Third Law. Let's see what it's asking us. A circular orbiting planet has been discovered around the star Zoria. The planet's orbital period around Zion is 48 earth years. The mass of Zor is 1.8 times the mass of the sun, determine the radius of the planet's orbit are sub P arounds Orion expressed as a multiple of the radius of earth's orbit are sub E around the sun. And then they give us a hint here to utilize Kepler's third law to derive the radius of the planet's orbit. So our multiple choice answers here are given uh in terms of our sub P. So a 28 R sub E B 24 R, sub E C, 32 R, sub E or D 16 R seven. So the first thing we're going to do here is take their hint using Kepler's third law. So we can recall that Kepler's third law is given by T squared is equal to four high squared divided by G multiplied by M and then all of that multiplied by R Q. So we can solve Kepler's third law from both the perspective of the planet around Zora. So that will be the first thing. The first equation we solve or the first situation that we solve Kepler's third law for. And then the second is the second scenario is going to be the earth around the sun. So the planet arounds Orion, we're going to isolate the radius. So R is equal to the cube group of T P squared. Sorry, this is R A P is equal to the cubed root of T P squared multiplied by G and, and that's mass Zoria divided by four by square. We are told that T P is equal to 48 T E and M Z is equal to 1.8 M sorry MS mass the sun. So this is the planet that the or the celestial body that these planets are orbiting around. So in this case, uh earth is orbiting around the sun. And so we can plug that in for our sub P. So we have R and P is equal to the, the cube root of TB squared multiplied by G multiplied by 1.8 and S sorry mass of the sun all divided by four pi squared. Now for the second scenario where we have the earth orbiting around the sun, our arts of eve is going to be T E squared multiplied by G multiplied by MS mass, the sun multiplied by four pi squared. So we can substitute R sub E into this R sub P equation where we have AC P is equal to through uh T squared multiply by G multiplied by MS divided by four pi squared multiplied by, we're gonna pull out these um constant. So we'll take the cubed root of 48 squared multiplied by 1.8. So we can recognize that these are equal to each other. So this arse of P, this arse of E term here, we will substitute in or R sub B and then, and then plug the second half of the equation into our calculators. And that is 16. So part of P equals 16 are seven. And when we look at our multiple choice answers that aligns with answer choice D, so that is the answer to this problem. And that's all we have for this one. We'll see you in the next video.