 Greetings and welcome to the introduction to astronomy. In this lesson we are going to look at some of the properties of the stars and in particular how we determine the masses and the sizes of the stars. So how can we determine these in detail based on measurements that we can make? So let's start off looking at kind of a census of the stars. What kind of stars are typical? And what we find that these are not the stars that we see in the night sky. When you go out at night and look up at the stars, those are not the typical stars that exist in the universe. These are very large and bright stars that can be seen over tremendous distances. And many of those stars are hundreds or even thousands of light years away. The better idea of typical stars is the stars near the sun. So the average star is actually smaller and cooler than the sun. Now if we look at stars within 21 light years of the sun, these are the nearest stars within our solar neighborhood. We find that there are two class A stars, 1F, 7Gs, 17Ks, 94Ms. So if we look at the typical types of stars at the normal spectral classifications, remember that there's also O and B stars? They do not exist within 21 light years of the sun. They are very, very rare. Even A and F stars, I mean among the over 150 stars that are listed in this table, only three are A and F. When we get down to stars like the sun, we see a few more, but far, far more of the much cooler M-type stars. We also see white dwarf stars and brown dwarf stars. These could be, well, the white dwarf stars, especially the brown dwarf stars, could be underestimated. These are very hard to find. Even 21 light years away, while it's easier to find a star, an object that is much cooler is very hard to detect. So these brown dwarfs are very faint, and we are still detecting more. So likely this is even larger, and I would suspect that the number of brown dwarfs actually exceeds the number of M-type stars. That there would actually be more brown dwarfs within 21 light years than there are cool red dwarf stars. Now, why do we see this? Why do we see when we look out at the sky all of these bright stars, but they are not typical? This is an example of what we call a selection effect. A selection effect happens quite often. Typically what it means in astronomy here for stars, bright stars can only be seen, or can be seen, even if they are hundreds or thousands of light years away. So we can see incredibly distant bright stars, but faint stars at the same distance would be almost invisible. As an example for this, our own sun, which we know is decently bright as a G-class. It is not one of the coolest stars that is around, but it would not be visible if we took it just 30 light years away. It would be one of the fainter stars seen in the sky. It would not be a very bright star, even if seen from 30 light years. And if we went out to a hundred light years or a thousand light years, it would be invisible to the naked eye. So even our sun would not be that easy to see, so you can imagine these even fainter stars are really hard to see. Some of them, within even 20 light years, have to be seen through a telescope. They cannot be seen at all with the naked eye. So what do we see in the sky? Again, are the more unusual stars because of this selection effect. We see the O and B stars. We see red giants and super giants. But the nearest red giant is a hundred light years away. The nearest O-type star is 500 light years away. They're not common because we don't find any nearby. But they are visible across very large regions of space. So we can see these stars even though they are very far away. Whereas a typical star, or even a star like our sun, would be invisible at these distances. Now let's look at how we can go about determining masses of stars. And that can be something difficult because it's important to know, but it can be hard to determine. In order to determine the mass of an object, we need an orbit. We need something orbiting it. And we can do that for binary stars. Now we find that probably half the stars in the universe are part of binary systems, which is great for determining orbits, but it still is not an easy thing to find. So we need to use that, and we not only need to be able to determine that it is a binary, but we need to be able to measure the properties of the orbit. Now there are three specific types of binaries that we want to look at. First is the visual binary, which we see here where you can see two very distinct stars. So when we take an image of this part of the sky, we could see two stars, and we could note that over time they may be revolving around each other. We could track out that orbit and be able to determine the orbital properties so that we could figure out the masses. So a visual binary is one example. You actually see two stars over tens or hundreds of years you could follow their orbit. Now another type of binary is a spectroscopic binary. In this case, we cannot see two stars, so we cannot separate them visually, but we can see two different sets of spectra, and we can watch their motions. And what we would see is if we're looking here at various times, we would see this star coming around, and here it would be coming towards us, and here it would be going away from us. So in this case, we would see a blue shift as it's coming towards us, a red shift later as it's going away, and that would shift all of its spectral lines. So when things are going across our line of sight like this, there is no Doppler effect, so we would see just one set of lines. Other times we would see a sets of lines are split apart, and we can use that to be able to help determine the orbital parameters. So that's an example of a spectroscopic binary, and that is the most common type. The most common type that we see is a spectroscopic binary. So another and the third type that we want to look at is the eclipsing binary. Now the eclipsing binary, again, we cannot see the two separate stars, but we can see their effects. In this case, we have it in a very edge on view, meaning that one star passes directly in front of the other. So as this orbits at position three to four, then this star passes in front of this other star and dims its light. Here on the back, at position one to two, this larger star actually blocks out the entire star and causes it to dim. So we can actually look at what we call the light curve, which measures the brightness over time. So at position one, we are seeing the light from both stars, and we get the full brightness. At position two, the very small but bright star is blocked out, so the brightness drops significantly at that time. As it comes back out, then we get the normal light curve again. And at position three, we're seeing the light of both stars. At position four, now the brighter star is blocking out part of the fainter star, so you will get some drop in brightness, but not as much as you had previously. So these are the three different types. One, the visual binary. Two, the spectroscopic binary. Three, the eclipsing binary. And the spectroscopic binary is by far the most common, because all you have to be able to do is get a spectrum. In order to separate the stars in a visual binary, it has to be relatively close to you. And in order to see an eclipsing binary, the stars have to be lined up properly, so it has to be tilted in exactly the right direction. So how can we use this to determine the mass of a star? Well, we have to go back to Kepler's third law. And Kepler's third law told us that the cube of the semi-major axis is equal to the period of revolution squared. So if we use d to represent the semi-major axis, p for the period of revolution, Newton modified this to give us that d cubed is equal to the sum of the masses times the square of the period. So if we can determine the semi-major axis and we can determine the period, then we can calculate the sum of the masses of the two objects, or m1 plus m2, equals d cubed divided by p squared. So if we can measure these two, we then can get the total of the masses. If we measure d in astronomical units in p in years, then the mass will be in solar masses. If you want to use other sets of units, there's actually a constant that you have to put out here that I'm not going to go into at this point, but there is a way to be able to determine if you wanted the mass in grams, you could put a specific constant out there that would convert the solar masses into grams. But the easiest way to do it, and to get a number that we typically use, would be just to use the period in years and the semi-major axis in astronomical units, and that gives us the answer in solar masses. But it only works if we can determine the orbital parameters. We need something orbiting. If we have a star that's sitting there all by itself and we cannot detect a planet around it, we can only compare it to similar stars to get an approximation of its mass. We cannot actually directly measure it. Now, what kind of masses do we find when we do this? Well, typically, largest stars are limited to about 120 solar masses. Why? Why can we not make a star that is 1,000 or 2,000 solar masses? Well, as that star forms, the radiation pressure will exceed the gravitational force. So if the gravitational force pulling material in is exceeded by the radiation pressure pushing outward, you have gravity pulling down, you have radiation pressure pushing out, then the radiation pressure will be able to overcome gravity and will be able to expel out the outer layers and prevent further material from adding to it. So there's a limit at about 120 solar masses, although theoretically very early stars could have been about 300 solar masses because those very early stars would not have had any metals. Recall that metals means anything other than hydrogen or helium. So you technically could have made in the very early universe even larger stars, but now they're limited to about 120 times the mass of our sun. Now the lowest mass stars are about one-twelfth of a solar mass. If we get below that, there's not enough mass and does not heat up enough at the center to fuse hydrogen to helium, and it becomes a brown dwarf star instead. So this is the ranges that we see. The smallest star is about one-twelfth of a solar mass. The largest star is now about 120 solar masses or so. Now how about determining the diameter of a star? Well we can do that by a couple of different methods, and in fact when we just look at stars, the vast majority of them only look like points of light even through a gigantic telescope. So no matter how much we image them, we still only see points of light. There are some very large stars that are relatively close to us that we can actually see sizes of and measure a diameter directly, but those are very, very rare. So the methods that we use to determine the diameter include lunar occultations. If the moon passes in front of the star, so as the moon is moving across the sky, it may pass directly in front of a star blocking out its light. We can then time how long it takes that star's light to disappear. If the star is so far away that it's just a point, it will just blink out. It will disappear immediately. However, if it has some size, even if we can't measure it with a telescope, it might take some tiny fraction of a second for the star's light to disappear, and that, depending on the distance, would then be able to give us a diameter for that star. We can do a similar thing with the eclipsing binary stars. If you note we looked at this light curve earlier, but as it eclipses, it does not drop straight down. It does not just disappear immediately, and then does not come straight back up. There's actually a slight angle to this. That tells us how long it takes that star to actually be completely eclipsed by the other and gives us a measure of the size. So we can determine the size of one star here, in this case the smaller star. Over here, we can determine the size of the other star by how long it takes, so we can determine the sizes from eclipsing binary stars as well. And a third method we can use is the radiation law. If we can measure the luminosity of a star. Remember, that's difficult because we need to know a distance to get the luminosity. We can use this equation, which relates the luminosity, the radius, and the temperature. Temperature is not too difficult to get. Luminosity, if we know a distance and a parent brightness we can determine, and that means that we could then calculate the radius and the other numbers in here, the 4 and the pi and the sigma, are just constants. So if we can determine the luminosity and the temperature of a star, we can then use those numbers to estimate the radius of the star. What do we find when we calculate radii of stars? Well, there's a big range. So the smallest stars are actually the size of Jupiter. They're not much bigger than Jupiter in size. Remember, they are still more massive than Jupiter, but not much bigger in size. These are the most common stars that we see. Largest stars, much, much larger than the Sun. We have giant stars, which are defined to be between 10 and 100 times larger than the Sun. Supergiant stars are more than 100 times larger than the Sun. And the very largest known star, known as Vy Canis Majoris, is 1,400 times larger than our Sun. That means it would extend out beyond the planet Jupiter if we placed it at the center of our solar system. So Mercury, Venus, Earth, Mars, and even Jupiter would be inside this star if it were placed at the center of our solar system. So that gives us an idea of how tremendously large these stars can get. So let's finish up here with our summary. And what we find is that stars do have a wide variety of masses and sizes. So they vary very widely from very small mass and small diameter stars to very large mass and large diameter stars. We've talked about some of the various methods that we can use to determine the sizes and the masses of stars. And typically what we see, the large stars are more commonly seen in the sky, large size, large mass. But it is the small size and mass stars that are really more common in the universe. The difference is we can see these stars from very large distances. So we can see them even if they are not close to us. And the small stars can be hard to find even if they are close to us. So that concludes our lecture on determining sizes and masses of stars. We'll be back again next time for another topic in astronomy. So until then, have a great day, everyone, and I will see you in class.