In 2005 astronomers announced the discovery of a large black hole in the galaxy Markarian 766 having clumps of matter orbiting around once every 27 hours and moving at 30,000 km/s. (b) What is the mass of this black hole, assuming circular orbits? Express your answer in kilograms and as a multiple of our sun's mass.

Video transcript

Welcome back everybody. We are looking at a chunk of ionized matter that is orbiting a black hole. So here's our black hole and here's the matter right there. Now. We are told that it makes one full revolution Every 12 days. Now, I'm actually gonna convert that into seconds here because we're gonna need to use it in terms of seconds there 24 hours in a day and 3600 seconds In an hour, giving us one million 36, 800 seconds. Now we are told that its orbital speed but you could I guess make synonymous with a tangential velocity is 190 kilometers a second or 100 90,000 m per second. And we are asked to find what let's see, well, a what the mass of the black hole is and then be the mass of the black hole. In terms of some constant times the mass of the sun. Which will be really easy. And it sounds a lot more complicated than it actually is. So what is the mass of a black hole? Well, according to kepler's third law here, we have that. The mass of the black hole is given by this equation four pi squared times the radius of orbit cubed all over Newton's gravitational constant times the time it takes or the for one orbit. Now we have everything here except this radius. So let's go ahead and find that radius first. We have that the time it takes for one full orbit of this chunk of ionized matter is given by this formula two pi times r desired radius all over. The orbital speed ranging. Some things around. We get the radius is equal to the orbital speed times time. All over too high. So let's go and plug in our values here and then we will find our radius. So our orbital speed is 1.9 times. Oops. Sorry. Yeah, 1.9 times 10 to the 5th Times are time of 1,036, seconds. All divided by two I. Which when you plug this into your calculator, we have a radius orbit of 3.13 times 10 to the 10 m. Great. Now that we found the time, let's go ahead and find the mass of our black hole. As for a black hole. Well, we'll just use the formula and plug in our values that we know. So we have four pi squared times 3.13 Times 10 to the 10th, which is the radius we just found all divided by while we have Newton's gravitational constant of 6.67 times 10 to the negative. 11 Times our time of 1,036,800 seconds. Which when you plug this into your calculator, you get 1.69 times 10- Villa rams. Great. So we found that. But now we need to find it in terms of some constant times the mass of the sun. Well, the mass of the sun is just equal to two times 10 to the 30th. So we'll just divide the mass we just found by that number and we will get some multiple times the mass of the sun. So when you divide 1.69 times 10 to the 31st, divided by two times 10 to the 30th, you get that. The mass of this black hole is 8.5 times the mass of the sun. So now we have found the answer to part A and part B. Giving us a final answer choice of C. Thank you all so much for watching. Hope this video helped. We will see you all in the next one.