[SOUND] So first let's start with some basic ideas about how stars work. And there are four important ideas or issues to consider. First, since most stars seem to last very long time, that means they're stable, and that, among other things, means they have to be in hydrostatic equilibrium. And that means that gas pressure inside stars has to balance the gravity that tries to keep the star together. That is providence of classical physics, a little bit of, maybe, modern atomic physics, but even back in the early 20th century people like Addington and son figured out how stars must work even though they didn't know exactly what the source of energy was. The second things is the starts have to be in thermal equilibrium. Meaning the amount of energy they produce in the middle has to match the luminosity, otherwise either stars gets extinguished or it explodes. The origin of the energy or the thermonuclear reaction fusion of lighter elements inside of the stellar core which hot and dense enough to happen. And then finally that energy has to get out somehow that's know as the energy transport. So these things are encapsulated in what's called the equations of the stellar structure. And in a somewhat simplified version, it goes like this. First of all, there is continuity of mass. As you look at the radial shell and what the radius are, the element of the mass is the area of a shell times the element of the radius. That gives you the gradient of mass as a function of radius which is related to the local density. The same thing with luminosity. In principle some layer inside the sun, usually near the core, energy is being generated. The amount of energy that's being generated is similar to the amount mass, but here we use quantity called the energy generation rate per unit mass, which is something that's computed from nuclear physics. And once you know that from nuclear reactions, then you can establish the gradient of luminosity inside a star. Now hydrostatic equilibrium. If you look at a shell of material inside the sun or any star, there is a gravitational force pulling you down. There is pressure on the inside and the outside from thermal motions, and the difference in those two pressures has to match the gravitational force that pulls the shell inwards. So since we know what element of mass versus radius is, you can then simply differentiate this and you find the gradient of pressure with radius is proportional to the enclosed mass, inversely proportional to the square of the radius, simple Newtonian gravity, and the density, at that point. Now pressure of what? Gas, obviously, but also radiation. Radiation has its own pressure, photons would hit some surface, there's momentum transfer so there is a force acting on a surface, and that radiation pressure can be computed in a moment. But for the gas, you probably know from your statistical mechanics or thermodynamics, it is simple relation that pressure is proportional to the temperature and proportionality is the density of particles. In volume. Another way to do it in terms of the gas density is that you replace the number density of particles with the mass density. And then, because it's the number of particles, you divide the mass density with mean mass of particles. And we like to express this in the units of masses of hydrogen. So, this quantity new is called the new molecular weight and it is the average mass of particles. If you had pure neutral hydrogen that would be one. If you had fully ionized hydrogen, now you have two particles and same amount of mass so new would be one half. If you start adding heavier elements, then that's going to go up. The composition of solar plasma is roughly speaking 70% or so of hydrogen, little shy of 30% of helium, and a couple percent of heavier elements that we're inherited from some previous stellar generations. So it works out on average to be roughly 0.8. This is a reasonable number to remember. So I remember what the ideal gas constant is, it's Boltzmann constant divided by the atomic mass unit or mass of hydrogen. And then you can express this as, what's really a somewhat familiar relationship from thermodynamics, the pressure is gas constant times density times the temperature. Second component of this is the radiation pressure. And to a very good approximation, radiation field inside stars is like a blackbody radiation. And the formula for this comes directly from blank distribution and it's proportional to the fourth power of temperature and constant of proportionality is Stefan–Boltzmann constant. And the question is, why is this important? You'd think that radiation pressure would be too slight to matter. But in fact, in really hot stars, it dominates over the mass density. The weight of the photons, if you will, is bigger. So there is so much radiation that radiation pressure dominates all the dynamics. And that can be computed nicely. So altogether, those four equations are called the equations of stellar structure. They're differential equations so they're usually solved numerically. Or you can make a toy model of what you think in theory your distribution of density might be, people do that too, and then plug in some constants from atomic physics, and so on, and you can compute what the star might look like. And if you know what the composition is, you can compute opacity, though it will not go that. The energy generation rate and voila, you can figure out how a the star is. So this is, more or less, complete treatment. Even so, we have neglected several important things. We assumed the star is spherical. A lot of them pretty much are, but not exactly. We didn't take into account rotation. That turns out to be important because, in some sense, rotation works as antigravity centrifugal force that's to pull stuff out. And it depends where in the star you are. And we neglected magnetic fields, which also can play a significant role in moving the stuff around. So in more modern models, people take that into account as best they can. But we can do them just back of the envelope simple estimates. So let's see if we can figure out what is the pressure and temperature inside the sun. You need force acting upon an area and so simple force would be g times square of the solar mass divided by square of solar radius. You know the area divide it too, and you estimate pressure of 10 to the 15 dynes per square centimeter. And that's, for sure, going to be underestimate, because we took outer surface area. So in reality, when you do a proper computation, it's 100 times more, it's two times ten to the 17th in CGS units. We do better with temperature. We can say, let's equate thermal kinetic energy three halves K T because this was essentially protons. So that's why three halves, not five halves. But that's equal to segregational binding energy of a proton, so radius and multiply those numbers and we've come up with something like 16 million degrees kelvin. Turns out that's, more or less, exactly right. So the density's also inside sun, in middle it's of the order of 150 grams per cubic centimeter. Surface it's much less. So high densities plus high temperatures is what's needed for stars to do nuclear reactions. So here is a simplified version of what's now called a standard solar model. People do all the computation properly. And this is how mass and luminosity generation grows the radius. So in the very center, you don't have anything, then as you go further out you encompass more and more volume in which nuclear reactions happen. At some point density and temperature drop sufficiently that you can no longer do nuclear fusion, and then it remains constant. So solar luminosity is established somewhere deep inside the sun. The mass is a function of radius, depends on the density distribution, denser in the middle and the outside. And so here you see roughly how that works. So by about half the solar radius, you have 90% of solar mass. So that's the basic ideas behind stellar structure. How about another plot? This is density and temperature as a function of radius and not temperature. It goes down from 16 million kelvin to essentially zero at stratosphere and the density is about 160 grams per cubic centimeter, again, goes to something very light, no surface.
