Welcome back. Today we're going to talk about Stellar Evolution. Starting with Stellar Structure, then turning to the energy source of stars, Nuclear Fusion, and then using our understanding of this to look how at how star evolve, age and eventually die. So let's begin, with the equation of Hydrostatic Equilibrium. You've seen this slide before, when we talked about the atmosphere of planets. And remember, Hydrostatic Equilibrium means we need to have a balance between gravity pushing down and pressure pushing up. On an atmosphere or a planet like the Earth, we have dense gas near the surface, lower density up here and, we have a pressure gradient balancing the gravity. So the heavier material on the bottom. Supports it against the force of gravity. The same thing happens in stars, except for when we think about stars, we have to think about things being kind of spherically distributed. So in a star, we have gravity pushing towards the center. Gravity trying to compress the star. And we have pressure because we have hot dense gas in the center, pushing up. And, the basic idea for a star is the star is in constant balance, a balance between gravity pushing in, pressure pushing out. In order to have pressure push out. We need the material in the center of a star to be hot and dense. Recall the Ideal Gas Law that says pressure depends on the number density of particles and the temperature, since the number density of particles goes as their density divided by the mass of the typical nucleon. We can say the pressure goes as density times temperature. So in the center of the star, we'll have higher density and higher temperature. In fact, the temperature gradients in stars are quite large. For a star like the sun, the central temperature is about 14 million degrees. While the surface temperature, is only about 5800 degrees. Now, once you have this hot region in the center, that hot energy is going to flow from the hot region to the cold region. Thermal dynamics basically implies that energy is always going to flow from hot material. To cold material. And there are three dominant ways that energy will flow in a star. Convection, as hot material sometimes rises, the same way in boiling water we have hot, the hot steam rises and boils, the heat moves convection. We have heat moving by conduction. As it does in a metal. As electrons scatter and carry energy outward from the hot region to the cold region. And by radiation, as the photons in the center of the sun move outwards, carriage, carrying energy with them. Now in the case with the sun, it takes a long time for the photons to get out. That's because the sun is so dense, that the typical photon can't travel very far, before it collides off of an electron and scatters off in a different direction. So you can think of a photon wandering slowly outwards as it heads out of the sun and eventually heads out. It actually takes a photon about 100,000 years. To travel from the sun's very hot center to the exterior. So as it travels, takes such a long time to travel. That slows the rate of energy transfer. And that's one of the reasons why, you can have such a large temperature gradient between the sun's very hot interior. And the outside. But left to itself, the sun if it had no energy source, would eventually cool. The energy would diffuse out, and the sun would reach a nearly constant temperature. You would no longer have a variation in pressure, that would cause the sun to collapse under it's own gravity. This was understood very well by 19th century physicists. And famously, Kelvin was, worked out the age of the sun because he assumed that this is all that was going on, that it was supported by pressure against gravity. He knew the mass of the sun, so he could infer what the pressure gradient was. Kelvin knew that the center would have to be very hot and that energy would diffuse out. And Kelvin knew, the luminosity of the sun and, how much energy was available by gravity and worked out, it wouldn't live very long. And Kelvin's result. Contradicted the estimates that geologists made and is very clear contradiction with the ages that we have today by radioactive dating. Today, we know that the earliest rocks on the earth are nearly four billion years old. And on moon and Mars, nearly four and a half billion years old. And on some of the meteorites we find rocks we can age date by looking at radio-active dating that their ages are at least 4.6 billion years old and just as an aside, you see a very interesting trend, that the smaller an object is, the older the rocks are. And that's because material cooled first here. In small rocks, sooner on Mars and the Moon. And it took the Earth, which is larger, longer to cool. So we can see a clear kind of gradient in age. But, nevertheless, this all suggests an age for our sun, of about 4.6 billion years, far longer than Kelvin would estimate. So now I'd like you to go off and do Kelvin's calculation. Given the gravitational binding energy of the sun, which he assumed to be the energy source, look up the sun's mass, the radius, and the strength of gravity, the sun's luminosity. I'd like you to go estimate its age.