 Greetings and welcome to the Introduction to Astronomy. In this lecture we are going to look at determining masses and sizes of stars, and look at general at the overall properties of stars. So, let's take a census of the stars and see what kind of stars are the typical ones. Well, the typical stars are not the stars that we see in the night sky. The stars that we see in the night sky are large and bright stars that can be seen over very large distances. A typical star is smaller and cooler than our sun. So, let's take a look here at stars within 21 light years of the sun. These are stars in our solar neighborhood. Our sun is a G-type star. There are seven of those. And there are three stars then that are hotter than our sun of higher classes, class A and F. And there are 111 stars of class K and M. So, we see that there are far more stars on the cooler side of the sun than on the hotter side. Now, we also see a few white dwarf stars and brown dwarf stars. To be honest, the brown dwarf stars may be underestimated still. We are still detecting more. These are hard to see. In fact, most of these stars that we see are not visible. Most of these M-type stars are very cool, and we really can't see them with the naked eye. Now, we have some here things with selection effects here. Since the stars that we see in the night sky are not typical stars, this is an example of what we call a selection effect. The bright stars can be seen for hundreds or even thousands of light years. Faint stars are hard to see even if they are close to us. So, what we are seeing in the sky are the more unusual stars. We see O and B stars. Remember, there were none of those in close to us. Red giants and super giants, which are also far away. Our nearest red giant is more than 100 light years away. The nearest O-type star is 500 light years away or so. So, we are looking at very large distances. However, we can see these brightly even over those distances because they are so tremendously bright. So, the stars you see when you go out at night are not the typical types of stars that exist. So, how do we determine masses of stars? Well, mass is a hard thing to determine, and that's because we need something orbiting a star so that we can figure out its mass. In order to get this, we look at things like binary stars. If we have a binary star, then we can determine the mass from the orbit. And we have three types of binaries we want to discuss. There is the visual binary, which is shown here where you can actually see two stars in an image. And if you watch over long periods of time, you could see that they are slowly orbiting around each other. We also have spectroscopic binaries where we cannot see two individual stars, but we can see them in the spectrum. So, when we look at the spectrum here, we can see that there are lines from star A and B when one star is moving toward us and one star is moving away, then we will get shifts in the opposite direction and those lines will separate. When they are going tangentially as we see in the image here, then we will see no shift, so that would be similar to the top and the bottom sections here. So, this was an example of what we call a spectroscopic binary. We cannot see the two stars individually, but we can see evidence from the spectrum that there are two stars orbiting. We also have eclipsing binary stars, such as shown here. In this case, a star will dim when an unseen companion passes in front of it and we can see that in the light curve. And we can use those properties to help us determine the mass of a star. Now, how do we determine that mass? Well, we go back and remember Kepler's third law from an earlier lecture. Kepler's third law stated that the cube of the semi-major axis was equal to the square of the period of revolution. If we use d for the semi-major axis, which is what the textbook is using here, and p for the period of revolution, Newton tells us that d cubed equals the sum of the masses times p squared. Now, actually, there's a constant out here that we are not going to worry about. In our case, we can simplify this and look at just this portion of it, because if we measure these in specific units, then we don't need a constant. We can make that constant equal to one. So we measure d in astronomical units. We measure the period in years, and in that case, the mass will be in solar masses. So if we determine the semi-major axis and period of a star, we can determine the sum of the two masses of those stars. We can determine the mass of that system. Now, what kind of range do we get in these? Well, the largest stars are limited to a little over 100 solar masses. Why? Because radiation pressure becomes so intense that if star is producing so much energy that the radiation pressure will overwhelm gravity and will keep the star from adding more material there. So radiation pressure exceeds the gravitational force. Now, if we want to look at this theoretically, there is actually an upper limit of about 300 solar masses. These are for stars with no metals. And remember, a metal is anything other than hydrogen or helium. So stars like this would have existed only very early after the Big Bang. Any current star forming will have at least some amount of metals. So those are the most massive stars that we see. The lowest mass stars are about one-twelfth of a solar mass. This is where they never get hot enough to fuse hydrogen into helium, and that becomes, then, a brown dwarf star. Now, we also find what we call the mass-luminosity relationship as shown here, that there is a relationship between the mass as you get larger and larger masses, we get larger and larger luminosities, and it actually goes up as a relatively high power. So a star that is twice as massive will not be twice as luminous, but will be many times more luminous. For example, we look at a star like our Sun, which would have one solar luminosity and one solar mass. If we went to something that was 10 times more massive, we might originally expect that, oh, that would be about 10 times as luminous. But if we move up the diagram here, we find that it is closer to 100,000 times more luminous. So the mass and luminosity increases very rapidly with the mass. That's one of the reasons these stars can be seen over such tremendous distances. So that's a little bit about masses. How do we determine the diameter of a star? Well, a vast majority of stars look only like points of light, even through very large telescopes. However, we do have some methods to determine the diameter. If the moon passes in front of a star, we can use that to measure by the amount of time it takes for it to dim as the moon passes in front of that. Eclipsing binary stars can also be used to do that when one star passes in front of another. If the star were infinitely small, it would just immediately drop down and go like that. The amount of time it takes to dim tells us something about the size of that star. And we can use the radiation law. The radiation law has a bunch of constants here. 4 pi and sigma are just constants. So it relates the luminosity, the radius, and the temperature. Now, if we can determine two of those, then we can calculate the third. So if we can figure out the temperature and the luminosity, we could then solve the equation and find what the radius would be. So that's another way that we can do it if we are able to determine the other two numbers. Now, what do we see when we look at the diameters of stars? The smallest stars are about the size of Jupiter. These are the most common stars that we see. Now, they are the size of Jupiter. Remember, that is diameter. That is not mass. They are many times more massive than Jupiter. The largest stars are much larger than our Sun. We have giant stars that are 10 to 100 times larger than the Sun. Supergiant stars 100 times larger. The largest known star, Vy Canis Majoris, 1,400 times larger than our Sun, which would extend out beyond Jupiter if we placed it where the Sun was in our solar system. Another star known as Stevenson 2-18, possibly over 2,000 times the diameter of our Sun, and would extend out beyond Saturn in our solar system. So, while there's a big range in mass, there is an even bigger range in sizes, and, as we'll see, luminosities within the types, different types of stars. So, let's go ahead and finish up with our summary, and what we've looked at today is that the stars have a wide variety of both masses and diameters. We've talked about various methods to use to determine how big a star is and the mass of the star. Large stars are more commonly seen in the sky, but the small stars are the more common ones in the universe. So, that concludes this lecture on determining masses and sizes 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.