Madison 2003 AAPT National Tim Erickson Epistemological Engineering 5269 Miles Avenue, Oakland, CA 94618 866.341.3377 voice, 866.879.7797 fax (toll-free)
 Introducing Fathom We get some data from the SDSS Web Site In this section, we looked at the H-R diagram activity from their site. On their site, click on Advanced under Science projects, and follow you nose to the Hertzsprung-Russell diagram. Then follow though the pages until you get to the page where they give you colors and magnitudes of the 28 or so nearest stars. When you drag the URL-goody into Fathom, Fathom gets the data (though the attribute names may need some help) and puts it into a Fathom collection. [If you are using the free evaluation copy of Fathom, this will not work. If you're desperate to try this anyway, contact me!] Drag a case table off the shelf to see the data in spreadsheet form. Drag a graph off the shelf to get a new (empty) graph. To make a graph, drag the name of an attribute to an axis of the graph. We looked at a number of graphs. The main point is that we can see relationships and patterns that correspond to some pattern or relationship among the stars. So we can ask questions such as, what does it MEAN that the Sun is an outlier in this graph: The Pleiades The Hipparchos catalog search page; we should look at RA =56.7, Dec = 24.2, 3 degrees But it will be easier if we use the one I have prepared in the Data Zoo. (You can use data from our project's data zoo freely.) We skipped looking at the Pleiades in the actual talk. The point there was severalfold: To figure out how to decide which stars are members of the cluster. (We used parallax—all cluster members should be at about the same distance; we could also use proper motion. How else could we decide?) To see how clear the resulting pattern is in the H-R diagram. That is, in this cluster, bright stars seem to be blue, and dim ones seem to be red. This does not tell us why this should be so, but picking out the cluster members makes this empirical relationship clear. We noted that magnitude is backwards; if we want bright stars at the top of the diagram, we have to reverse "V". (In the left-hand diagram, appMag = –V. The selected (red) stars are candidates for cluster membershup based on being selected in the right-hand graph.) What's The Point? Why use this particular example? Because it's an example of doing genuine data analysis with astronomical data. Importantly, it's data analysis that requires no calculation, but still requires students to understand graphs and use that understanding. This is not a trivial matter. In moving from interacting with real objects, to looking at images, to making graphs, to manipulating symbolic representations, students are having to become more and more abstract. I believe that first coming to underestand graphs is one of the hardest, most mysterious steps. The H-R diagram for a cluster tells an important story about what is going on, but it doesn't look a bit like the cluster itself. What can we do to help? Whenever possible, make graphs that correspond somehow to the physical reality. In this case, we can make plots of RA versus DEC and see how the plots correspond to the pictures (even if they're backwards!). In ordinary graphs, take the time to ask really simple questions: which point is the Sun? How do you know? If this point is down and to the right of that point, what does it mean? (This problem is not limited to astronomy. We work in regular physics classes; students time balls rolling down a ramp. They plot results—by hand at first, of course!—and we ask them, what does this point represent? Even if they have made the point a few minutes before, many students lose track of its provenance. It's as if the abstraction and the emerging pattern on the paper obscure the fact that the dot represents one roll of the ball and the time it took.) The Hyades and Aldebaran What we were going to do with the Pleiades we did with the Hyades (below). Here we added the Really Cool Question, is Aldebaran (the big red one) a member of the cluster. We saw how we could make a convincing argument that Aldebaran does not belong (it's too close). Back to the Hipparchos catalog search page; this time we should look at RA = 66.5, Dec = 15.5, r=6 degrees But really we'll just go straight to the Zoo. What else can we ask or do? So—Aldebaran is really close (it has a large parallax), so it's not a member of the cluster. In the picture, it LOOKS really bright. If it were in the cluster, would it still look bright, or would it just be another humdrum red star? Also, we can use the distance (1000/parallax, in parsecs) and the angles RA and DEC to make a 3-D model of the cluster. The 3D trig may be too tough, but you can easily make a top view using just the distance and the RA. Jupiter's Moons We'll go to JPL to their Jupiter Ephemeris site Tabbed text in case it doesn't work We didn't do this, but it's really cool. It does, however, require more mathematical baggage. On the other hand, it may provide a context in which students can use their math to do something mroe riveting than math problems. Use the JPL site to get predicted positions for one or more moons of Jupiter. Use mjd for a time attribute, and get the positions of the moons relative to Jupiter. These are called dRA and ddec. For time span, choose what you like. Once every twelve hours for a month or two gives you plenty of information for the Galilean moons. Then plot the dRA, say, against mjd to see the almost-but-not-quite-perfect sine wave. Use Fathom to plot a sinusoid. Use a slider as a variable parameter that will become your period. Adjust the period (and any other parameters) to match the data. It might look like this: Then we match the period better (note my sleight-of-hand using the constant 52854.2; this is a convenience, setting the zero-point close to a spot where the points cross the t-axis). Then we make a Residual Plot to see how our model matches the data. Clearly we need more refinement, but we can now use this plot to fine-tune our fit: We could do a whole workshop on parameters and residual plots, but this gives you an idea. Can we do this without trig? You bet. Data in Depth (Erickson, 2000 -- it comes with Fathom) has an activity where you use Jupiter's Moons data and essentially chop the graph up and overlay it, using a varialble period. When it lines up best, the period is right. This method requires simple (but little-taught) math of integer parts and modular arithmetic. In Fathom, you could make a new attribute, phase, and give it a formula such as modulo(mjd, period). (use similar functions in Excel or other programs) Then plot the RA against phase and adjust the latter. Where to go from here? Have different groups get periods—and amplitudes—for different moons, then pool the data. How do the periods and amplitudes correspond? Students will find a power-law pattern (the same one Kepler found) to the data.