Lecture #5 -- Stellar Spectra Jan 31, 2003

I. Line broadening
Other factors in the environment where the lines form may also shift lines a little bit at random, and by looking at the sum of all these random shifts, we get a total degree of line "broadening". By studying this broadening, we learn a lot about the region where the lines form. In particular, these collisions with nearby atoms and electrons will "warp" the atom that is emitting or absorbing light, slightly changing the frequency at which it radiates. These frequency shifts can be upward or downward, and happen at random as the collisions are occurring, so the net effect is simply to broaden the line. The collision rate is related to the pressure, so the width of the line in an observed spectrum tells us the pressure in the line-forming region.
II. Stellar size on the Main Sequence
Understanding the pressure near the surface of a star is crucial for determining the size of the star. Why? Because the size of the star will affect its gravity at the surface, and the gravity determines the pressure. The catch is you have to be a bit careful to get the dependence correct. If you look at terrestrial planets, they have a fairly fixed density because they are mostly rock, so the mass is proportional to the radius cubed. The acceleration of gravity at the surface (called simply the "surface gravity") is proportional to the mass divided by the square of the radius (right?). Putting these together, the surface gravity of a rock is proportional to its radius (check this).
Do stars work that way? Nope. Stars are made of gas, so the density is not fixed. So what is fixed? Well, main-sequence stars are undergoing hydrogen fusion in their cores, and this occurs whenever the temperature reaches a certain value, so for main-sequence stars, it is the interior temperature that stays fairly fixed from star to star. So what we have to do is figure out how the surface gravity depends on size when the core temperature stays fixed. The way to do that is to use the virial theorem, noting that the total kinetic energy in the star is proportional to the core temperature times the mass (right?), while the gravitational potential energy is proportional to the the square of the mass divided by the radius (right?). The virial theorem says that these are proportional to each other, and if we keep the temperature fixed, this says that the radius is proportional to the mass! This is only approximately true because the core temperature does not stay completely fixed, but it's a good starting point. OK, so if the radius is proportional to the mass, how does surface gravity depend on radius? It should be easy to see that the surface gravity is inversely proportional to radius for a main-sequence star, not proportional like for a rock. A very key point to understand!
For stars off the main sequence the situation is more complicated, because you cannot assume the interior structure is similar if you change the elements being fused. But nevertheless it still holds that big stars have weak gravity at the surface, and so have small surface pressure, so show very narrow absorption lines.
III. The Distance Scale
So why is the size of the star so key? Because we know the temperature and the surface flux density from the thermal spectrum, and now we also know the radius, so we can get the total surface area and thus the total luminosity. This allows us to use the spectrum of stars to determine their luminosity as well as their spectral class! The spectral class is essentially given by the surface temperature, and the luminosity also involves the radius. Once we know the luminosity, we can get the distance to the star by looking at how bright it appears. Now we really know something!
IV. Luminosity class
When you do this for actual stars, you find that stars at a given temperature appear to break down into several different groups of luminosities, called luminosity classes. There are three main luminosity classes, but there are five in total, denoted by roman numerals. So class V is the smallest, called dwarfs, and includes main sequence stars and old stars called white dwarfs. Class III is larger, called giants, and is made of stars that have evolved quite a bit. Class I is the largest (and highest luminosity), and are called supergiants. Again these are evolved stars, but they have higher masses so get really bright.
V. The HR Diagram
Plotting the luminosity class as a function of spectral type yields a very important diagram called the Hertzsprung-Russell diagram. You will want to understand this diagram quite well!
VI. Key Aspects of a Star
Kind of like for racehorse jockeys, the key attributes of a star are its mass, its age, and its composition (what it's "made of"). If you know these three things, you pretty much know the star, although whether or not the star is rotating can at times be a relevant factor as well. Since most stars have solar-like composition, we will focus here on the mass and age of the star.
VII. Stellar Clusters
Knowing the distance to one star can help us know the distance to a whole cluster, since stars form in clusters and spend much of their lives there, especially if they form in a globular cluster (which is tightly bound). Making HR diagrams for entire clusters is especially easy, because the distance may be assumed to stay fixed, so we easily determine the luminosity even for stars not on the main sequence that are harder to model. If you have a bunch of stars at the same distance and of the same age and with the same composition (i.e., a cluster), then the only thing that is different about the stars is their mass. Simulations of what such clusters would look like can be linked to below, as a function of the age of the cluster: 1 million years old
10 million years old
100 million years old
1 billion years old
10 billion years old
Note the obvious "main sequence", and the different "turnoff" points from that main sequence line. (Always remember that the main sequence is not a "sequence" in the sense of time or age, but rather a sequence in mass. The only way it is a sequence in age is if you think of how the turnoff point moves like a zipper along the main sequence with the age of the cluster.)