Hey, everyone. So in this video, we're going to talk about some fascinating objects in our universe which are called black holes. Alright? So there's one important equation that you need to know to solve problems. So let's go and take a look and we'll do a quick example, alright? So, a black hole is an object that has an enormous amount of mass in a relatively tiny space. And what I mean by that is that something like 500 kilometers is pretty big to us, but relative to the solar system, it's an incredibly small distance, alright? So it's relatively tiny. Now, this object is so massive that not even light can escape. So basically anything that falls into a black hole, whether it's a planet or a star or something like that or a spaceship or even light itself, once it falls in, it can never come back out. And the reason for that is if you've seen escape velocity, it's okay if you haven't. Basically, the escape speed would have to be faster than the speed of light, which is 3×108 m/s, but nothing can go faster than light. So essentially, anything that falls in is just doomed and will never be able to come back out. And that's why it looks black to us because no light escapes it and actually reaches our eyes or our telescopes and stuff like that. Alright? So there's an important equation called the Schwarzschild radius, and it's the equation that relates the mass of the black hole and the physical size of the black hole. The way that we define the size of the black hole is basically the radius from the center out to this boundary here where you see all the lights, and that's called the Schwarzschild radius. The equation is actually pretty straightforward. It's 2GM_BH/c2. And so basically, the surface of this boundary here, the boundary where everything gets sort of dark, is called the event horizon. So this boundary here is called the event horizon. It's more of a mathematical boundary. It's not like if you were to sort of pass through it, you would feel like a barrier or something like that. But basically, what happens is that once you pass across this boundary, you would have to go faster than the speed of light to come back out again, but that's impossible. So this is the boundary where nothing can possibly escape, and you're doomed if you fall in. Alright? So everything else that we've learned in this chapter, all the stuff about forces, satellite motion, all of our equations are still valid. We just now have an extra one, the Schwarzschild radius equation. Alright? So let's go ahead and take a look at our example here. So we have a team of astronomers who are imaging a black hole at the center of our galaxy or the galaxy MEDIT statement. By the way, this is a real thing that happened. And what they determined is anything closer than this distance here, a 120 AU, falls in and never escapes. Now, you should recognize that as basically the distance in which something comes into the black hole and then never comes back out. So this is going to be the Schwarzschild radius. We want to calculate the mass of this black hole, but we want it in terms of solar masses, in terms of basically as a multiple of how massive our sun is. Okay, so we want the M_BH, but we want it in terms of M_sun, right? So if we want the mass of the black hole, we're going to have to use our new equation, the Schwarzschild radius. So this is R_S = 2GM_BHc2. If we want this M_BH, we're just going to have to rearrange. The c2 goes up to the top. The 2Gs come down, and you're going to have R_Sc2 × c2 / 2 G, and this is going to equal the mass of your black hole. Alright, so we know what the Schwarzschild radius is, it's the 120 AU, but because we're calculating the mass, we need everything to be in SI units. So this R_S here, which is 120 AU, we're going to have to convert it to meters. And I have this conversion factor right here. So if you want it in terms of meters, you're going to have to cancel out AU on the bottom. So this is going to be 1.5×1011 meters, and that's going to get rid of your AUs. Alright, so this just becomes 1.8×1013 meters. Okay? So now this is going to be 1.8×1013 and then you're going to do 3×108, that's the speed of light but you have to square it and now you divide by 2 × 6.67 × 10 minus 11 that's your big G, and then when you work this out, what you're going to get here is you're going to get a number that's 1.21×1040 kilograms. Alright? Now, this might not seem so, you know, it might seem like any ordinary big number. But remember, we want to express this in terms of solar masses. Okay. So there's one last conversion I need to do, which is I want to convert these kilograms here to M_sun. Alright, so what I have to do is I'm going to have to multiply this so that the kilograms cancel on the bottom. So this is going to be 2×1030. This is going to be 1. This is going to be M_sun. So when you work this out, your kilograms should cancel and the mass of your black hole is going to be 6×109 M_sun. So if you, so for those of you who, realize what this number is, this is going to be about 6,000,000,000. So 6,000,000,000 times the mass of our sun. So, in other words, this black hole is 6,000,000,000 times heavier than our Sun.