In today's lab we used my EJS Gnomon simulation to track the motion of the Sun using shadows. The simulation can display the shadow cast by a vertical stick (or a sundial gnomon that is aligned with Earth's axis, but we didn't use that feature) at any time of day, on any day of the year, from any location on Earth. My students explore how the shadow can be used to track the direction (azimuth, using the direction of the shadow) and height (altitude, using the length of the shadow and some trigonometry) of the Sun in the sky over the course of a day, and how these directions vary throughout the year. They make observations from different locations on Earth to see how this affects the patterns that they find.
They pretty quickly discover some interesting regularities. Some are fairly obvious. The Sun rises due East and sets due West only on the equinoxes, while in summer it rises northeast and sets northwest and in winter it rises southeast and sets southwest. They find that in the northern hemisphere the shadow at local noon points North. Then they find that in the Southern hemisphere the noon shadow points South. On the equator they find that it sometimes points north and sometimes points south. Once they have narrowed down the range of latitudes for which the direction can changes (23.5 degrees S to 23.5 degrees N) I try to help them make the connection to our notion of the "tropics" and specifically to the Tropic of Cancer and the Tropic of Capricorn.
Moving on they discover that the median noon altitude for the Sun over a year is just 90 degrees minus the observer's latitude. The variation in altitude is plus or minus 23.5 degrees (there it is again). Once they have the pattern down they can predict the noon altitude as a function of time of year for any latitude. They notice that the curve gets flipped in the southern hemisphere (so that highest altitudes are near the winters solstice - the southern summer). They also discover that for some latitudes on some days the noon altitude is negative, so the sun never rises on those days in those places. By exploring the latitudes for which this occurs (above 66.5 degrees N and below 66.5 degrees S) they can define the Arctic and Antarctic circles.
It's pretty cool to see students finally understand these features on the globe that they have heard about for years. They know the tropics are near the Equator where it is hot, and the Arctic/Antarctic circles are near the poles where its cold. But they have never known why those lines should be drawn at particular latitudes rather than other latitudes. Now they know. And hopefully they have 23.5 degrees burned into their brains by this point.
Everything mentioned above can be explained fully with a simple Celestial Globe model. But the best part of the lab is when they take a closer look at the noon shadows and find that even in the northern hemisphere (above the tropics) the noon shadow doesn't always point exactly north. This is because the simulation keeps time by the mean sun, so local noon occurs when the sun is on the meridian ON AVERAGE. The true sun does not move uniformly across the Celestial Sphere, so sometimes it runs ahead of the mean sun (meaning it is east of the mean sun, and thus produces a noon shadow that point just west of north) and sometimes it runs behind (it is west of the mean sun and produces a noon shadow slightly east of north). By examining the variations in the shadow direction over the course of a year, my students find that the Sun moves faster relative to the stars in winter and slower in summer. This variation in the Sun's speed across the Celestial Sphere will be important when we discuss why Hipparchus displaced the center of the Sun's orbit away from Earth (and why Copernicus displaced the center of Earth's orbit away from the Sun).
Overall I really like this lab. There is so much more that could be done with it. In the future I want to add a project in which students must set up their own gnomon and make their own shadow observations in the real world. They can't do as much as they can do with my simulation, but a handful of observations in the real world may be enough to make the virtual world seem more realistic.
I'm not the only one who teaches this stuff to college students. Joe Heafner at Catawba Valley CC has built a substantial curriculum around observing shadows, and some of his materials are available at SticksAndShadows.com. He may even be using my Gnomon simulation. Anyway, his stuff is worth checking out. He and I share a vision of teaching astronomy in a radically different way, a way that emphasizes scientific inquiry and critical thinking rather than surveying vast amounts of factual information. We go about things somewhat differently, but I think the spirit of our efforts is quite similar.
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