Stellar Properties from Spectral Lines
Introduction
Note: This exercise is for advanced students.

The spectral lines of a star contain a wealth of information because their formation is sensitive to the conditions present in the star's atmoshere. For the astronomer this is simply fabulous, because it means that the light from a star can tell us a lot about what's going on!

Composition:
The spectral lines are formed by the absorption of photons as electrons jump from a lower energy level to a higher one. The specific wavelength (color) of light which is absorbed depends on the size of the jump (bigger jump = more energy = bluer/shorter wavelength light absorbed). Because every element and molecule has a unique set of allowed jumps, this means that the pattern of light absorbed by each element is unique - sort of a fingerprint left behind in the light.

What gets tricky is taking the amount of light absorbed (the strength of a spectral line) and using this to figure out the actual abundance of an element. The absorption you find in a line depends on both the number of absorbers present (the abundance of an element) and what fraction of that element's electrons are in the proper energy level (and able to absorb that specific color of light to jump to an allowed higher level). This will depend on conditions such as temperature, pressure (which vary with depth), and microturbulent velocity.

But if you know these quantities, you know where the electrons are. Then, by observing how much absorption takes place, you can use the strength of a spectral line to determine how many atoms of the element must be present to match the observed strength of the line (the abundance of that element in the star's atmosphere).

Temperature:
Consider the formation of spectral lines by a single element, say iron. Each of the allowed electron transitions is the result of an electron jumping upward from a specific energy level. Of course the number of electrons in each level is to first order a function of temperature (c.f., the Boltzmann equation). So, if you look at a number of different Fe lines, from a variety of different energy levels, the strengths of those lines depends on both the number of absorbers (Fe atoms) and the temperature.

Think about working the problem backwards. You know you have the same number of absorbers (Fe atoms), no mater which Fe line you look at. Now assume a certain effective temperature for the star and calculate how much Fe must be present to match the observed absorption in each of the Fe lines (i.e., looking at lines from electrons starting from many different energy levels). If you have the temperature right, you should need the same number of absorbers (same Fe abundance) for all of them.

But if you have assumed a wrong temperature, you'll find that different lines (from different initial energy levels) require different abundances to match the observed absorption. For example, if your assumed temperature is too high, you will have too many electrons in the higher energy levels and too few in the lower levels. The lines caused by electrons in the lower energy levels will bit fit by larger abundances than those from the higher excitation potentials (becuase there are few electrons in the low energy states, more atoms are needed to macth the observed absorption from such lines). So, the test as to whether you've got the temperature right is that all the Fe observed Fe lines will require the same Fe abundance, regardless of excitation potential.

Pressure/Density/Surface Gravity
First note that these three properties are all tied together. Surface gravity is a function of mass and radius - but mass and radius are very difficult to determine, so we often just stick to surface gravity as that's what will affect the formation of spectral lines anyway. The greater the surface gravity, the greater the pressure and the higher the density of the gas at any given level in the atmosphere. Note of course that pressure and density are also related to temperature through the perfect gas law, so to really do this, you'll also need to know how temperature changes with depth too (and as long as we're wallowing in details, you'll also need to worry about hydrostatic equilibrium to really figure out the pressues and densities as a function of depth!).

But how does pressure/density affect the formation of spectral lines? The greater the density, the larger the number of absorbers (atoms) per cubic cm, and the stronger the lines will be. But there's even more to this! As discussed above, the strength of a line also depends on the fraction of the absorbers (atoms) with electrons that can do the absorbing. Again temperature comes into play because we not only have to worry about what energy level the electons are starting off at, but also how much ionization has taken place.

That's where pressure/surface gravity come into play. Sure, the higher the temperature, greater the rate of ionization (think about collisions). But the reverse process, recombination, depends both on temperature and on density (or pressure - the electron pressure to be specific) - see the Saha Equation. The greater the density of electrons, more frequently recombination will take place, and the fewer ions you'll end up with (at a given temperature).

So, when considering spectral lines, the strength of a line will also depend on the electron pressure, which is also related to the density and the surface gravity. As an added bonus, in most stellar spectra you can often find lines due to atoms in two different ionization states (metals in moderate temperature stars are an excellent example). Again, as with temperature, if you look at lines due to atoms in two different ionization states, you can calculate the number of absorbers which must be present (given the temperature and surface gravity which derermine how many of the atoms have electrons in the proper energy level to cause the spectral line). You'll know you have the right surface gravity (pressure) when you derive the same number of atoms for the two ionization states.

Microturbulence
If you stop and think about it, you expect the gas in the atmosphere of a star to be moving. With this motion comes Doppler shifts, which will slightly change the wavelengths which are absorbed (and emitted) by the atoms - some a little too red, some a little too blue, essentially making the line wider than it normally would be. One might first think of the thermal motions of the atoms as the only source of line broadening, but there are actually many of them. Others include the natural line width due to the uncertainty principle and pressure broadening due to nearby atoms. But one of the most significant broadening sources in an average star is microturbulence, a non-thermal "bulk" motion in the atmosphere of a few km/sec.

The microturbulent motion also has an effect on spectral lines, particularly the strong ones. This is because at the wavelength center of a very strong line, most of the light is absorbed (that's why it's so strong/dark). So, if you add more absorbers to the mix, there's just not that much light at that wavelength for the new atoms to absorb, so the line really doesn't get any stronger. Such a line is said to be "saturated." But if there's also microturbulent motion, some of the absorbers will be redshifted/blueshifted away from the central wavelength of the line. And there is light out there away from the central wavelength - in the "wings" of the line - which they can absorb, increasing the total amount of light that is absorbed in the line.

So, think about two lines, one strong, one weak. In the weak line, if the microturbulence is low, most of the absorbers will absorb near the central wavelength, and there are plenty of photons around to be absorbed, so the strength of the line is sensitive to the number of absorbers. Likewise if the microturbulence is high - all this does is shift some absorbers to the red/blue, but the total number of photons absorbed (and the strength of the line) doesn't change. In the strong line, its a different story. With a low microturbulence, there are lots of absorbers capable of absorbing near the line center. But with most of the photons removed, adding more absorbers doesn't change much, so the line is relatively insenitive to the abundance of absorbers. But if you increase the microturbulence, some of the atoms can absorb photons away from the line center, so you'll get more total flux removed (a stronger line) than in the case of a low microturbulence.

What all this means is that (just like above), you can predict how many atoms need to be in the atomsphere to get the absorption you see at a given temperature, surface gravity, and microturbulence. When you look at lines of different strengths due to one kind of atom, you expect to get the same required abundance regardless of strength. This will only be true if you have the correct microturbulence (assuming your temperature and gravity are correct too). If your microturbulence is too low, you'll need too many atoms for the strong lines (compared to the weak ones). If it's too high, you'll need too few.

Putting It All Together
What all the words above bascially mean is that it's possible to look at the spectral lines of a star and get a pretty good idea of its temperature (effective temperature), surface gravity, and microturbulence. As an added bonus, as you figure these out, you can also figure out the abundaces of different atoms.

In practice, it's a little more complicated and all these quantities are very deeply related. In fact, you have to essentially figure them out all at the same time. But if you keep in mind the following basic relations you should be able to do it:

Required abudances of a givien element from each line must be the same for all excitation potentials - temperature

Required abundances of a given element must be the same for all ionization stages - surface gravity

Required abudances of a givien element from each line must be the same for all equivalent widths (amount of absorption) - microturbulence

You Tyr It
The link below will take you to a web interface to some of our analysis codes. What you will be given is a set of measurements of line strengths in the star III-96 of the Galactic globular cluster M5. The spectra were taken on the Lick 3-m telescope and the lines were analyized by Sneden et al. in 1992 (see their paper in the Astronomical Journal for details).

All of the measurments have been input for you. You just need to pick an effective temperature (hint: somewhere between 3750 and 8000K), surface gravity (hint: somewhere between 0.0 < log g < 5.0), and a microturbulence (hint: somewhere between 1.75 and 3.0 km/s).

Once you've picked your input parameters (and the above hints are the limits of acceptable input), the computer will calculate an appropriate model atmosphere using the MARCS program (see Gustafsson et al. 1975). This atmosphere (a temperature/pressure relation for an atmosphere) will be fed to the MOOG spectrum synthesis program (see Snden 1973) and the abundances required to fit each of the observed lines (given your inputs) will be calculated and returned.

Look at your output. How did you do with the abundances from each line as a function of excitation potential? Pay particular attention to the elements with many lines and mulitple ionization states present (Fe is a good one). Were the resulting abundances the same for Fe I as for Fe II? Did the abundance from a line depend on its equivalent width? If there are trends in these results that you don't like, try a different set of parameters.

If you have a temperature/surface gravity/microturbulence that you think works, you can also take a look at what abundances fit lines due to other elements too. The composition of a star provides valuable insight into where a star came from and what's going on in its core!

So, what is the temperature, surface gravity, and microturbulence for M5 III-96?