Now let's proceed to cosmology, the science to study our universe. And we begin with a question why the night sky is dark. Okay, you may think this question is stupid. Why the night sky is dark? Because we are on the other side of the earth, we're not seeing the sun, the sun is bright and we're not seeing the sun, so the night sky is dark. All right? That's right, that is the answer. But if you think about this answer in a little bit more details, you will get confused. Why you will get confused because if you think about why the sun is bright. Why the sun is bright, there are two possible reasons. The one reason is the sun spends a larger solid angle to you. So if you look at, imagine you look at too bright. If you look at the sun, one edge of the sun and the other edge of the sun. There is a solid angle of the sun in front of you. And this solid angle is a finite angle, is pretty large and if you look at the star, a star is very distant, so the solid angle of the star is extremely small, all right? This is one possible reason that the sun is bright, and another possible reason that you might imagine is per solid angle. Per solid angle, the energy emitted by the sun maybe greater compared to the energy emitted by the star per unit angle because the sun is closer, which is the reason. Is the sun brighter because of the solid angles larger, the sun looks larger, or it is per unit angle? The energy is a larger, which is the reason. To answer this question, we will calculate what is the power from the sun received by us, by our observers per unit solid angle. How to calculate that, first of all the power emitted by the sun is the brightness ,the luminosity of the sun, el san. And then if the radius of the sun here is and radius between us and the sun is capital R. In this case how to calculate this power received per solid angle, first of all, what is the power that we received from the sun? So first of all the luminosity of the sun, the total power. And then we need to divide it by part of the power because our telescope is with finite size, okay? If the area of our telescope is A, and that A should be divided by the 4 pi r squared. The total spherical surface from the sun to us. And this is the power received by us. And now what is the power received by us per unit solid angle? Then we have to divide this result by what is the solid angle of the sun. And what is the solid angle of the sun? That is the ratio between, so the whole sky, how many solid angles are there and projected to this distance. And that will be 4pi the solid angle and projected to the distance between us and the sun, which is R squared, 4pi R squared is the sphere, okay? Between us and the sun, and the solid angle occupied by the sun. So here there will be pi R sun squared. This is the solid angle occupied by the sun and heavily interestingly is 4 pi R squared cancelled. So the power received by us per solid angle is first of all the A, which is the size of our telescope, okay? This is what we can determine, we build a larger or smaller telescope. And afterwards there will be the luminosity of the sun divided by pi r squared, which is the radius of the sun, all right? So this result has two parts. The one part is defined by us, which is the size of the telescope and the other part is defined by the nature of the sun, which is known as the surface brightness of the sun. So the power received by us per unit solid angle is our area multiplying the surface brightness of the sun. And we denote the surface brightness by sigma. And what is surface brightness of the sun? That is five times 10 to the minus three was per meter squared per other second squared, okay? This is the surface brightness of the sun, and here we notice that the surface brightness of the sun is independent of the distance between us and the sun. It only depends on the brightness of the sun and the radius of the sun. What that means? That means when we were asking the question, the sun looks brighter. Is it because it takes a larger solid angle or is because per solid angle it looks brighter or both of them, okay? Now we know that the sun is very bright because it takes very large solid angle compared to others does but per solid angle. The sun is as bright as any other star, no difference. Assuming the average luminosity of the sun and the star of the same older size of the same order, which is not absolutely right, but it's not wrong by so many orders of magnitude, okay? Then the sun looks brighter because it has a larger solid angle to us and per solid angle the energy that we receive from the sun and from the star, they are the same. Which is the surface brightness of the star of the sun, okay? Now we are on a better ground to understand the problem why our night sky is dark, okay? This is the surface brightness of the sun. And what is the surface brightness of the night sky? The surface brightness of the night sky is approximately five times 10 to the minus 17 of the same unit, all right? Extreme difference between here and here. And that means how our night sky is darker compared to the second and how darker it is. So, there is 10 to the 14 times difference, 14 orders of magnitude difference. There is a factor of 10 to the minus 14 darker in the night sky compared to the surface of the sun. All right, this is how we describe the dark night sky, then we will encounter a problem. The problem that we encounter is, let me ask you, is our universe finite or infinite, okay? Finite or infinite in some sense that I will specify later but a seal our universe just infinite. And everywhere the same, it will appear the same to us, then there will be a problem. What is the problem? The problem is if the universe is infinite and everywhere appears the same to us. Then we can look infinitely distant from the universe and which our direction which our small, extremely small solid angle you are looking into. Eventually you will encounter a star and once you encounter a star, the brightness of the star per unit solid angle will be as bright as the sun. And then the whole universe everywhere, every solid angle you will encounter star, and that star for unit solid angle is as bright as the sun everywhere. Our night sky is as bright as the sun, okay? This is the problem. And this problem is known as the Olbers paradox although it was discussed much earlier but people forgot that. And remembered Olbers. All right, this is the problem. And this problem can either be understood in a slightly different way which is the star light which travels to you which diverges as the radius between you and the star distance to the minus 2 power. And the number of star increases by R to the 2 power. If we study a shell by a shell, this shell and the larger shell, the number in each shell increases R to the two power. So comparing these two quantities that are cancelled so each distance we are contribute to you the same brightness and then adding all this brightness together. You will have the night sky to be bright, all right? So there is a paradox. But here, if I take this approach, it looks like we are unable to do a real calculation. But if you take this approach, we can actually calculate something. What is the thing that we are going to calculate?