The point of this exercise is to give you a feel for how astronomers try to map the spiral arms of our Galaxy by measuring the Doppler shifts of 21cm radiation from neutral hydrogen clouds. You will have to make the same assumptions as astronomers make, in order to carry this out. Some of these assumptions are
Zeilik Figure 15-7, and the associated text, show how you can express the radial velocity v_r of a hydrogen cloud (or other galactic object), relative to the observer, in terms of the rotation curve and galactic longitude l. Using angular velocities w and w_0 for the cloud and the observer, we find (Zeilik Eq.15-9)
Here, R_0 is the distance of the observer from the center of the Galaxy. Since you either know or measure everything in this equation except for w, you can solve for it and therefore the rotational velocity of the cloud. The rotation curve tells you how far away the cloud is from the center of the Galaxy, and combining that with the galactic longitude tells you where in the Galaxy that cloud is located. By mapping out the positions of the clouds, you can trace out the spiral arms. At least in principle.
You will use the CUPS Astronomy program "Galaxies". The program is located on the X-disk, in the folder GALAXIES inside CUPSAS inside CUPS. Choose the subprogram "The spiral structure of a galaxy" when you startup GALAXIES.EXE. (The other subprograms allow you to do things like design your own galaxy, with and without dark matter, but that's not necessarily part of this exercise.)
Before actually working through the exercise, it is a good idea to try out the program using the default spiral arm structure. Go straight to the "profiles" menu, and you can see the result of the 21cm Doppler shift profiles for this structure. The upper left plot shows the arms and the location of the observer, as well as the direction (i.e. galactic longitude) in which you are looking. The upper right plot shows the radial velocity as a function of distance from the observer, along the line of sight. This is the curve that is calculated once the rotation curve is known. (The program does this for you. All you need to do is read off the results.) The actual 21cm profiles are shown on the bottom plot.
Use the F2 function key to step through galactic longitude. The default is to use 15 degree steps, but you can set the value to something better if you prefer. You can read values off the profile and radial velocity plots just by clicking on the plot itself. Run through these procedures using the default setup and convince yourself that the plots look right and that you can read the appropriate numbers.
Now move on to the actual exercise. We'll do this twice, once for a very simple "spiral arm", and second for an arm or arms you can design yourself. Keep your results on the worksheet.
1) A very simple arm. Use the GALAXY menu to "plot the galaxy using current data". This draws a pattern of stars which represent the default galaxy, but without any spiral arms. Then, use the ARMS menu to draw a straight line from the center out to the edge of the galaxy. (This is about the simplest arm I can think of.) Next, use the VIEWPOINT menu to pick a new position of the observer. Put it relatively near the arm, but not directly on it. (What would happen if you did put the observer directly on this radial straight-line arm? Maybe you'd like to try it.)
Now step through the 21cm profiles. Determine the radial velocity of each profile peak, and turn that into a distance D with the radial velocity plot. Record the galactic longitude l, the radial velocity, and the distance D for each peak. Plot the positions of your measurements and see if they indeed reproduce a straight line. (If they don't, you did something wrong.) You can make this plot by turning l and D into x and y points using x=Dcosl and y=Dsinl, or by making a polar plot.
2) Design your own arms. Repeat this procedure, but give yourself some more complicated spiral structure. Don't go overboard, but your grade will depend to some extent on how ambitious you get. Realize that with more than one arm, or with an arm that curves around, you will get more than one profile for each galactic longitude. Plot all the points, and see if you can tell there are spirals. You and your lab partner might trade plots with your neighbors to see if you can figure out what kind of arm structure you each started with.